THE TAMM—DANCOFF METHOD

V. P. Silin and V. Ya. Fainberg

INTRODUCTION

The development of quantum field theory over the past several years has led to the unquestionable success of quantum electrodynamics. At the present time all conclusions of quantum electrodynamics accessible to experimental verification are fully confirmed by experiment, although a consistent treatment of the theory encounters a number of difficulties. The success of the theory is connected with the smallness of the fine-structure constant \(e^2/\hbar c\) and with the possibility of applying perturbation theory. In contrast to electrodynamics, perturbation theory is certainly unsuitable in the case of quantum mesodynamics. It is quite possible, however, that the modern meson theory will prove applicable in a certain range of energies. To answer this question it is necessary to use methods not connected with expansion in powers of the interaction constant. One such method, to the investigation of which much work has been devoted in the last 3–4 years, is the Tamm—Dancoff method*).

The present article is devoted to an exposition of the foundations of this method and its application to various concrete problems of the quantum theory of mesons. The main attention is given to general questions concerning the formulation of the method and to an analysis of the difficulties arising in attempts to renormalize the equations obtained according to the T. D. method.

The Tamm—Dancoff method was first formulated (in 1945) in a noncovariant form by I. E. Tamm¹, and then, in 1950, by Dancoff² in application to problems of quantum mesodynamics**). It should be noted that an analogous method had already been applied by Fock in 1934 to certain questions of quantum electrodynamics³. The essence of the T. D. method consists in truncating, according to the number of particles, the system

) Hereafter abbreviated: T. D. method.
*) According to generally accepted terminology, the original formulation of the method is called the old T. D. method.

strict equations of quantum mesodynamics and subsequently exactly solving such an approximate system of equations. Questions connected with the noncovariant formulation of the old T. D. method are discussed in § 1. Attempts to solve the equations directly in noncovariant notation encounter difficulties of two kinds: first, it is not clear how to eliminate uniquely the infinities of the proper-energy type, and, second, additional infinities connected with closed vacuum loops appear in the equations. The covariant formulation of the old T. D. method, proposed by Chew\(^{4}\), is set forth in § 2. Here the question of boundary conditions and the transition to the momentum representation are also discussed.

In § 3, using as examples the equations for two nucleons and for a nucleon and a meson, the difficulties arising in attempts to renormalize the equations of the old T. D. method written in covariant form are analyzed. It is shown that the renormalization method proposed by Chew\(^{4}\) is erroneous. In the case of the nucleon proper energy (mass operator), Chew’s method leads to incorrect finite additions, and in the case of the meson proper energy (polarization operator) it leads to additional divergences upon transition to the momentum representation\(^{5}\).

The covariant form of writing the equations of the old T. D. method also does not eliminate the difficulties connected with vacuum loops.

In § 4 various attempts connected with the application of the old T. D. method to the interaction of nucleons are discussed.

§ 5 is devoted to solving the problem of scattering of \(\pi\)-mesons by nucleons in a state with \(I = {}^{3}/_{2}\)* within the framework of the old T. D. method, without taking into account the contribution from vacuum loops and proper-energy terms. This problem was solved in work\(^{6}\) for the case of a symmetric pseudoscalar theory with pseudoscalar coupling. It then turned out that, in contrast to perturbation theory, the T. D. method gives good agreement with experiment.

In §§ 6—8 a new formulation of the T. D. method** proposed by Dyson\(^{7-9}\) is considered, with the aim of avoiding the above-mentioned difficulties with renormalization in the old T. D. method.

In § 6 the connection between the amplitudes of the old and the new T. D. method is set forth. It is shown how, proceeding from this connection, one can formulate the correct boundary conditions in the equations of the new method\(^{10}\), and a covariant formulation of this method is given. An essential advantage of the equations of the new T. D. method is—

*) \(I\) is the isotopic spin of the system \(\pi\)-meson + nucleon. When this problem is solved in a state with \(I = {}^{1}/_{2}\), additional difficulties arise, due to an inadmissible singularity of the kernel in this state (see the end of § 3).

**) In what follows: the new T. D. method.

is the absence in them of infinities associated with vacuum loops.

In § 7 the renormalization of the nucleon self-energy in the equation for two nucleons, obtained in the lowest approximation of the new T.–D. method, is considered. The investigation shows that all infinities arising in such an equation can be eliminated by a covariant renormalization of the masses of both nucleons and of the coupling constant \(g\). The problem of renormalizing the polarization operator arising in the equation for the meson + nucleon system in the new T.–D. method turns out to be more complicated. § 8 is devoted to the investigation of this question. The analysis shows that, although the polarization operator is transformed to a covariant form, the subtraction from it of infinite quantities leads to the appearance of two renormalized charges \({}^{11}\). In addition, taking into account the finite additions remaining after renormalization leads to the appearance in the equation of additional poles that have no direct physical meaning \({}^{10,12}\). The presence of such poles in renormalized equations is a general property of the approximate equations of modern quantum field theory and indicates the limited domain of their applicability \({}^{13-16}\).

Finally, in § 9 the question of the connection between equations of the T.–D. type and covariant equations \({}^{17,18}\) is briefly considered.

§ 1. THREE-DIMENSIONAL FORMULATION OF THE OLD TAMM–DANCOFF METHOD

In this section we shall briefly discuss the original \({}^{1,2}\) noncovariant form of the old T.–D. method. This will facilitate, on the one hand, an understanding of the subsequent exposition and, on the other hand, will make it possible to clarify better the essence of this method (truncation of the equations according to the number of particles, the meaning of amplitudes, questions of normalization and orthogonality). Here it is necessary to emphasize that all concrete calculations within the method are simpler to carry out precisely in the three-dimensional form, whereas for understanding the nature of the difficulties connected with the elimination from the equations of divergent quantities (renormalization), the covariant form of notation proposed by Chini \({}^{4}\) is more convenient.

We shall formulate the T.–D. method using as an example the interaction of meson and nucleon fields. We shall start from the Schrödinger equation for stationary states

\[ (H_0+H')\Psi=W\Psi, \tag{1.1} \]

where \(H_0\) and \(H'\) are, respectively, the Hamiltonians of the free fields and of the interaction, \(\Psi\) is the state vector (functional) in the second-quantized representation, and \(W\) is the eigenvalue of the total-energy operator in this state.

We shall seek the solution of (1.1) in the form of an expansion in the eigenfunctions \(\Phi_\lambda(N)\) of the free Hamiltonian:

\[ \Psi=\sum_{\lambda,N} a_\lambda(N)\Phi_\lambda(N), \tag{1.2} \]

\[ \Psi_0=\sum_{\lambda,N} \beta_\lambda(N)\Phi_\lambda(N). \]

Here \(\Psi_0\) is the vector of the physical vacuum, the index \(N\) characterizes the number of free mesons, nucleons, and antinucleons in the state \(\Phi_\lambda(N)\); \(\lambda\) denotes the remaining variables of these particles (for example, momenta and spins), and the expansion coefficients \(a_\lambda(N)\) have the meaning of probability amplitudes for finding \(N\) free particles in the state \(\Psi\).

The functions \(\Phi_\lambda(N)\) form a complete set of normalized and mutually orthogonal eigenfunctions of the operator \(H_0\). In explicit form one may write

\[ \Phi_\lambda(N)=[\Pi(N)]^{-1/2} C_\lambda(N)\Phi_0. \tag{1.3} \]

Here \(C_\lambda(N)\) is a product of \(N\) creation operators; for example,

\[ C_\lambda(N)=Q^*(\mathbf{k}_1)Q^*(\mathbf{k}_2)\ldots Q^*(\mathbf{k}_r) B^{n_1^*}(\mathbf{p}_1)B^{n_2^*}(\mathbf{p}_2)\ldots B^{n_m^*}(\mathbf{p}_m), \]

where \(Q^*(\mathbf{k})\) is the creation operator of a meson with momentum \(\mathbf{k}\), \(B^{n^*}(\mathbf{p})\) is the creation operator of a nucleon with momentum \(\mathbf{p}\) (\(n=1,2\) and characterizes the direction of the nucleon spin), and \(B^{n^*}(\mathbf{p})\) is likewise for an antinucleon with momentum \(\mathbf{p}\), when \(n=3,4\); the operators \(Q\), \(Q^*\) and \(B^n\), \(B^{n^*}\) obey the usual commutation rules:

\[ [Q(\mathbf{k}_1),Q^*(\mathbf{k}_2)]_-=\delta_{\mathbf{k}_1\mathbf{k}_2}; \qquad [B^n(\mathbf{p}_1),B^{n_2^*}(\mathbf{p}_2)]_+=\delta_{n_1 n_2}\delta_{\mathbf{p}_1\mathbf{p}_2}, \tag{1.3'} \]

where \([\ ,\ ]_{\mp}\) denotes, respectively, the commutator and the anticommutator; \(\Phi_0\) is the vacuum-state vector of the noninteracting fields. By definition of the vacuum,

\[ Q(\mathbf{k})\Phi_0=B^n(\mathbf{p})\Phi_0=0. \tag{1.4} \]

\(\Pi(N)=(N_1!)(N_2!)\ldots\) is the normalization constant, where \(N_1\), \(N_2\), etc. are occupation numbers. The orthogonality condition for \(\Phi_\lambda(N)\) reads:

\[ (\Phi_{\lambda'}^*(N'),\Phi_\lambda(N))=\delta_{\lambda\lambda'}\delta_{NN'}. \tag{1.5} \]

Using (1.3) and (1.5), one may write an explicit expression for the amplitudes \(a_\lambda(N)\) in terms of the annihilation operators \(Q\) and \(B^n\):

\[ a_\lambda(N)=(\Phi_0^* Q(k_1)\ldots Q(k_r)B^{n_1}(p_1)\ldots B^{n_m}(p_m)\Psi). \tag{1.6} \]

If one assumes that the state vector \(\Psi\) is normalized, for example, to unity,

\[ (\Psi^*,\Psi)=1, \tag{1.7} \]

then from this, taking into account (1.2) and (1.5), the normalization condi-

THE TAMM–DANCOV METHOD

normalization for the amplitudes \(\alpha_\lambda(N)\):

\[ \sum_{\lambda,N} |\alpha_\lambda(N)|^2 = 1. \tag{1.8} \]

Let us now substitute the expansion (1.2) into (1.1). Then, taking into account formulas (1.3)—(1.5), we obtain the following infinite system of integral equations for the amplitudes (the Schrödinger equation in the occupation-number representation):

\[ (W - E_{\lambda,N})\alpha_\lambda(N) = \sum_{\lambda',N'} (\lambda,\, N|H'|\lambda',\, N')\alpha_{\lambda'}(N'), \tag{1.9} \]

where \((\lambda,\, N|H'|\lambda',\, N')=(\Phi_\lambda^*(N),\, H'\Phi_{\lambda'}(N'))\) is the matrix element of \(H'\), and \(E_{\lambda,N}\) is the eigenvalue of \(H_0\) in the state \(\Phi_\lambda(N)\),

\[ H_0\Phi_\lambda(N)=E_{\lambda,N}\Phi_\lambda(N) = \left( \sum_{i=1}^{n}\omega_{k_i} + \sum_{i=1}^{m}E_{p_i} \right)\Phi_\lambda(N), \]

where \(\omega_k=+\sqrt{\mu^2+\mathbf{k}^2}\) and \(E_p=+\sqrt{M^2+\mathbf{p}^2}\) are, respectively, the energy of a free meson with momentum \(\mathbf{k}\) and of a free nucleon (or antinucleon) with momentum \(\mathbf{p}\)*.

The system of equations (1.9) is the exact system of equations of the quantum theory of the meson field.

Up to now no methods have been found for an exact solution of this system of equations. In the case of electrodynamics it proves quite sufficient to solve the system (1.9) by means of perturbation theory in an approximation appropriate to each particular problem. However, in the case of meson theory, because of the strong interaction of the fields, solution by the method of perturbation theory leads to completely incorrect results. There arises the need to construct another method for solving the system (1.9). Such a method, essentially different from perturbation theory, is the Tamm–Dancov method. The essence of the T. D. method is as follows.

To construct an approximate solution of equations (1.9), or, what is the same thing, of equation (1.1), it is assumed that the state vector is not an infinite superposition of states of free particles, as apparently occurs in the case of an exact solution, but a finite superposition. In other words, in the expansion (1.2) only those amplitudes \(\alpha(N)\) are taken to be different from zero which correspond to a number of virtual particles \(N\) smaller than some \(N_0\). For the system of equations (1.9) this corresponds to discarding all equations whose left-hand sides contain amplitudes for numbers of particles exceeding \(N_0\), and to neglecting such amplitudes in the remaining equations.

\[ \text{* ) Everywhere in what follows } \hbar=c=1. \]

From the physical point of view, such a cutoff is based on the assumption that states with a number of virtual particles \(>N_0\) make a negligibly small contribution to the process under consideration. From the mathematical point of view, the Tamm–Dancoff method represents the use, in quantum mesodynamics, of the Ritz–Galerkin method in function space.

Indeed, here, as in the Ritz–Galerkin method, the exact functional of the system (1.2), depending on an infinite number of functions \(a_\lambda(N)\), is approximated by a finite number of such functions.

The study of questions concerning the convergence of successive approximations of this method (in mesodynamics) is a very complex and as yet unsolved problem. The physical hopes in this respect are connected mainly with the successful application of the method to a number of concrete physical problems, in which in a number of cases it has been possible to obtain qualitative, and sometimes (meson scattering on nucleons in the state with \(I=3/2\)) fairly good quantitative agreement with experiment\(^6\).

Mathematically, the conditions of applicability of the Tamm–Dancoff method may be formulated as follows:

\[ \sum_{\lambda,\;N>N_0} |a_\lambda(N)|^2 \ll 1 . \tag{1.10} \]

Equations (1.9) are written in an explicitly relativistically noninvariant form. In connection with this, when one attempts to solve these equations by the T. D. method, substantial difficulties arise already in the lower approximations. First, it is not clear how to carry out unambiguously the renormalization of divergent terms of the self-energy type that appear in these equations. For this reason, in most investigations of one or another concrete problem carried out up to now by the old T. D. method, the proper-energy terms were simply crossed out. Second, in the truncated equations (1.9) there appear additional infinite terms due to vacuum loops. In contrast to the theory of the \(S\)-matrix, in the old T. D. method it is not possible to eliminate infinities of the vacuum type by means of a renormalization of the functional of the system\(^*\). To investigate these questions, Chin\(^4\) proposed a covariant formulation of the Tamm–Dancoff method.

§ 2. COVARIANT FORM OF THE OLD TAMM–DANCOFF METHOD

For a covariant formulation of the T. D. method it is convenient to start from the interaction representation. In this representation the state vector \(\Psi(\sigma)\), depending on the spacelike surface \(\sigma\), satisfies the equation

\[ i\,\frac{\delta\Psi(\sigma)}{\delta\sigma(x)}=H'(x)\Psi(\sigma), \tag{2.1} \]

\(^*\) See also § 3.

or, in integral form,

\[ \Psi(\sigma)=\Psi(-\infty)-i\int_{-\infty}^{\sigma} H'(x')\Psi(\sigma')\,dx', \tag{2.2} \]

where \(H'(x)\) is the interaction Hamiltonian.

In the case of the pseudoscalar symmetric meson theory with pseudoscalar coupling\(^*\)

\[ \begin{aligned} H'(x) &=(ig/2)\bigl(\bar\psi_\alpha(x)\psi_\beta(x)-\psi_\beta(x)\bar\psi_\alpha(x)\bigr) (\gamma_5\tau_i)_{\alpha\beta}\varphi_i(x) \\ &= ig\,\bar\psi_\alpha(x)(\gamma_5\tau_i)_{\alpha\beta}\psi_\beta(x)\varphi_i(x)^{**}. \end{aligned} \tag{2.3} \]

Here \(\psi_\beta(x)\) and \(\varphi_i(x)\) \((i=1,2,3)\) are the operators of the free nucleon and meson fields; \(\tau_i\) are the operators of the isotopic spin of the nucleon; the indices \(\alpha\) and \(\beta\) each run through eight values and characterize both the spinor and the isotopic components of the nucleon operator.

The Fourier expansion of the operators has the form:

\[ \left. \begin{aligned} \psi_\alpha^{(+)}(x) &=\sum_{n=1}^{2}\int u_\alpha^{n}(p)b^{n}(p)\hat\delta(p^2+M^2)\eta^{+}e^{ipx}\,d^4p \\ &=\frac{1}{(2\pi)^3}\sum_{n=1}^{2}\int u_\alpha^{n}(p)B^{n}(p) \exp\bigl[i(\mathbf{p}\mathbf{r}-E_p t)\bigr]\,d^3p, \\[6pt] \psi_\alpha^{(-)}(x) &=\sum_{n=3}^{4}\int v_\alpha^{n}(p)b^{n}(p)\hat\delta(p^2+M^2)\eta^{-}(p)e^{ipx}\,d^4p \\ &=\frac{1}{(2\pi)^3}\sum_{n=3}^{4}\int v_\alpha^{n}(p)B^{n*}(p) \exp\bigl[-i(\mathbf{p}\mathbf{r}-E_p t)\bigr]\,d^3p, \\[6pt] \varphi_i^{(+)}(x) &=\int q_i(k)\hat\delta(k^2+\mu^2)\eta^{+}(k)e^{ikx}\,d^4k \\ &=\frac{1}{(2\pi)^3}\int Q_i(\mathbf{k})(2\omega_k)^{-1/2} \exp\bigl[i(\mathbf{k}\mathbf{r}-\omega_k t)\bigr]\,d^3k, \\[6pt] \varphi_i^{(-)}(x) &=\int q_i^{*}(k)\hat\delta(k^2+\mu^2)\eta^{-}(k)e^{ikx}\,d^4k \\ &=\frac{1}{(2\pi)^3}\int Q_i^{*}(\mathbf{k})(2\omega_k)^{-1/2} \exp\bigl[-i(\mathbf{k}\mathbf{r}-\omega_k t)\bigr]\,d^3k, \end{aligned} \right\} \tag{2.4} \]

\(^*\) All subsequent calculations will be carried out with the Hamiltonian (2.3).

\(^ {**}\) This follows from the fact that \(\operatorname{Sp}(\gamma_5 S_{\alpha\beta}(0))=0\), where \(S_{\alpha\beta}(x)\) is the nucleon permutation function.

where

\[ \eta^{+}(k)=1,\quad \eta^{-}(k)=0 \quad \biggr\}\; k_0>0 \qquad \eta^{+}(k)=0,\quad \eta^{-}(k)=1 \quad \biggr\}\; k_0<0. \]

\(\psi^{(+)}\) is the nucleon annihilation operator, \(\psi^{(-)}\) is the antinucleon creation operator, \(\varphi^{(+)}\) is the meson annihilation operator, \(\varphi^{(-)}\) is the meson creation operator. The index \(n\) in formulas (2.4) corresponds to the number of a solution of the Dirac equation with prescribed spin, isotopic spin, and energy. In this case

\[ u^{n*}(\mathbf p)u^{n'}(\mathbf p)=\delta_{nn'},\qquad v^{n*}(\mathbf p)v^{n'}(\mathbf p)=\delta_{nn'}, \]

\(u^n\) is a solution of the Dirac equation in a state with positive energy, and \(v^n\) is the same for antinucleons in a state with positive energy. Further,

\[ \psi(x)=\psi^{(+)}(x)+\psi^{(-)}(x);\qquad \bar\psi(x)=\overline{\psi^{(+)}}+\overline{\psi^{(-)}}, \]

\[ \varphi(x)=\varphi^{(+)}(x)+\varphi^{(-)}(x). \]

Between the operators there hold the usual commutation relations\(^{1*}\)

\[ \left. \begin{aligned} [\psi_\alpha(x),\bar\psi_\beta(x')]_+ &=-iS_{\alpha\beta}(x-x'),\\ [\psi_\alpha^{(+)}(x),\bar\psi_\beta^{(+)}(x')]_+ &=[\psi_\alpha^{(+)}(x),\bar\psi_\beta^{(-)}(x')]_+ =-iS_{\alpha\beta}^{(+)}(x-x'),\\ [\varphi_i(x),\varphi_j(x')]_- &=i\delta_{ij}\Delta(x-x'),\\ [\varphi_i^{(+)}(x),\varphi_j^{(-)}(x')]_- &=i\delta_{ij}\Delta^{(+)}(x-x') =-i\delta_{ij}\Delta^{(-)}(x'-x). \end{aligned} \right\} \tag{2.5} \]

The remaining brackets are equal to zero. The invariant permutation functions entering into (2.5) are equal to

\[ \left. \begin{aligned} \Delta(x) &=-\frac{2i}{(2\pi)^3}\int e^{ikx}\delta(k^2+\mu^2)\,\varepsilon(k)\,d^4k\\ &=-\frac{1}{(2\pi)^3}\int e^{i\mathbf{k}\mathbf{r}}(\omega_k)^{-1}\sin\omega_k t\,d^3k;\\[4pt] \Delta^{(\pm)}(x) &=-\frac{i}{(2\pi)^3}\int e^{ikx}\delta(k^2+\mu^2)\frac{2\varepsilon(k)\pm1}{2}\,d^4k\\ &=\frac{\pm i}{2(2\pi)^3}\int(\omega_k)^{-1}\exp[i(\mathbf{k}\mathbf{r}\pm\omega t)]\,d^3k;\\[4pt] \Delta(x)&=\Delta^{(+)}(x)+\Delta^{(-)}(x),\\[4pt] S(x)&=\left(\gamma_\mu\frac{\partial}{\partial x_\mu}-M\right)\Delta(x);\\[4pt] S^{(\pm)}&=\left(\gamma_\mu\frac{\partial}{\partial x_\mu}-M\right)\Delta^{(\pm)}(x);\\[4pt] \varepsilon(k)&=\pm \tfrac12 \quad \text{for } k_0\gtrless 0. \end{aligned} \right\} \tag{2.6} \]

THE TAMM—DANKOV METHOD

Let us also write down the invariant functions \(\Delta^{(1)}\), \(\bar{\Delta}\), \(\Delta_F\), \(S^{(1)}\), \(\bar{S}\), and \(S_F\), which we shall need in what follows:

\[ \left. \begin{aligned} \Delta^{(1)}(x) &= \frac{1}{(2\pi)^3}\int e^{ikx}\delta(k^2+\mu^2)\,d^4k,\\[4pt] \bar{\Delta}(x) &= -\varepsilon(x)\Delta(x) = \frac{1}{(2\pi)^4}\,\mathcal{P}\int \frac{\exp(ikx)}{k^2+\mu^2}\,d^4k,\\[4pt] \Delta_F(x) &= \Delta^{(1)}(x)-2i\bar{\Delta}(x);\\[4pt] S^{(1)}(x) &= \left(\gamma_\mu\frac{\partial}{\partial x_\mu}-M\right)\Delta^{(1)}(x) \quad \text{and so on.} \end{aligned} \right\} \tag{2.7} \]

The covariant amplitudes in the old Tamm—Dankov method are defined in the following way*):

\[ \left. \begin{aligned} (\Phi_0,\varphi^{(+)}(x)\Psi(\sigma)) &\equiv {}_{\sigma}\!\left<\varphi^{(+)}(x)\right>_{\sigma},\\[4pt] (\Phi_0,\psi^{(+)}(x)\Psi(\sigma)) &\equiv {}_{\sigma}\!\left<\psi^{(+)}(x)\right>_{\sigma},\\[4pt] &\ldots\\[4pt] (\Phi_0,\overline{\psi^{(-)}}(x_1)\ldots \overline{\psi^{(-)}}(x_n) \psi^{(+)}(y_1)\ldots \psi^{(+)}(y_m) \times\\ \qquad \times \varphi^{(+)}(z_1)\ldots \varphi^{(+)}(z_r)\Psi(\sigma)) &\equiv {}_{\sigma}\!\left< \overline{\psi^{(-)}}(x_1)\ldots \overline{\psi^{(-)}}(x_n)\times \right.\\ \qquad \left. \times \psi^{(+)}(y_1)\ldots \psi^{(+)}(y_m) \varphi^{(+)}(z_1)\ldots \varphi^{(+)}(z_r) \right>_{\sigma}. \end{aligned} \right\} \tag{2.8} \]

Here \({}_{\sigma}\!\left<\varphi^{(+)}(x)\right>_{\sigma}\) is a one-meson amplitude,
\({}_{\sigma}\!\left<\psi^{(+)}(x)\right>_{\sigma}\) is a one-nucleon amplitude;
\({}_{\sigma}\!\left< \overline{\psi^{(-)}}(x_1)\ldots \overline{\psi^{(-)}}(x_n) \psi^{(+)}(y_1)\ldots \psi^{(+)}(y_m) \varphi^{(+)}(z_1)\ldots \varphi^{(+)}(z_r) \right>_{\sigma}\) is an \(n+m+r\)-particle amplitude \((m\) nucleons, \(n\) antinucleons, and \(r\) mesons).

\(\Phi_0\) is the state vector of the mathematical vacuum. By definition,

\[ \psi^{(+)}\Phi_0=\overline{\psi^{(-)}}\Phi_0=\varphi^{(+)}\Phi_0=0; \tag{2.9} \]

\(\Psi(\sigma)\) is a state vector satisfying equation (2.1).

We note that in (2.8) the four-dimensional points \(x_1\ldots x_n\), \(y_1\ldots y_m\), \(z_1\ldots z_r\), generally speaking, need not lie on \(\sigma\).

Owing to (2.9), in the old Tamm—Dankov method all amplitudes containing on the right the creation operators \(\varphi^{(-)}\), \(\psi^{(-)}\), and \(\bar{\psi}^{(+)}\) are identically equal to zero.

*) For the relation of the covariant amplitudes (2.8) to the three-dimensional ones (1.6), see below (2.17) and (2.18).

V. P. Silin and V. Ya. Fainberg

From (2.1), (2.2), and (2.8) one readily obtains an infinite system of equations for amplitudes of the type (2.8):

\[ i\,\frac{\delta}{\delta\sigma(x)} \left\langle \overline{\psi^{(-)}}(x_1)\ldots \overline{\psi^{(-)}}(x_n) \psi^{(+)}(y_1)\ldots \psi^{(+)}(y_m) \varphi^{(+)}(z_1)\ldots \varphi^{(+)}(z_l) \right\rangle_{\sigma} \]
\[ = \left( \Phi_0,\, \overline{\psi^{(-)}}(x_1)\ldots \overline{\psi^{(-)}}(x_n) \psi^{(+)}(y_1)\ldots \psi^{(+)}(y_m) \varphi^{(+)}(z_1)\ldots \varphi^{(+)}(z_l) H'(x)\Psi(\sigma) \right), \tag{2.10} \]

or, in integral form,

\[ \left\langle \overline{\psi^{(-)}}(x_1)\ldots \varphi^{(+)}(z_l) \right\rangle_{\sigma} = \left\langle \overline{\psi^{(-)}}(x_1)\ldots \varphi^{(+)}(z_l) \right\rangle_{-\infty} - \]
\[ -\,i\int_{-\infty}^{\sigma} dx'\, \left( \Phi_0,\, \overline{\psi^{(-)}}(x_1)\ldots \varphi^{(+)}(z_l) H'(x')\Psi(\sigma') \right). \tag{2.11} \]

In order to express the right-hand side of equations (2.10) and (2.11) explicitly in terms of amplitudes of the type (2.8), it is necessary, using the permutation relations (2.5), to place all annihilation operators to the right of the creation operators (\(N\)-ordering \(^{20,21}\)). In the simplest cases we have:

\[ i\,\frac{\delta}{\delta\sigma(x)} \left\langle \varphi_k^{(+)}(x_1)\right\rangle_{\sigma} = \]
\[ = -\,g(\tau_i\gamma_5)_{\alpha\beta} \left\{ \partial_{ik}\Delta^{(+)}(x_1-x) \left\langle \overline{\psi_{\alpha}^{(-)}}(x)\psi_{\beta}^{(+)}(x) \right\rangle_{\sigma} \right. \]
\[ \left. -\,i \left\langle \overline{\psi_{\alpha}^{(-)}}(x)\psi_{\beta}^{(+)}(x) \varphi_k^{(+)}(x_1)\varphi_i^{(+)}(x) \right\rangle_{\sigma} \right\} \tag{2.12} \]

for the one-meson amplitude, and

\[ i\,\frac{\delta}{\delta\sigma(x)} \left\langle \psi_k^{(+)}(x_1)\right\rangle_{\sigma} = \]
\[ = g(\tau_i\gamma_5)_{\alpha\beta} \left\{ S_{\lambda\alpha}^{(+)}(x_1-x) \left\langle \psi_{\beta}^{(+)}(x)\varphi_i^{(+)}(x) \right\rangle_{\sigma} \right. \]
\[ \left. -\,i \left\langle \overline{\psi_{\alpha}^{(-)}}(x)\psi_{\lambda}^{(+)}(x_1) \psi_{\beta}^{(+)}(x)\varphi_i^{(+)}(x) \right\rangle_{\sigma} \right\} \tag{2.13} \]

for the one-nucleon amplitude.

In (2.12) and (2.13), for definiteness, we have introduced spinor \((\alpha,\beta,\lambda)\) and isotopic \((i,k)\) indices.

The system of equations (2.10) (or (2.11)) decomposes into independent subsystems according to the magnitude of the nuclear charge: the difference between the number of operators \(\psi^{(+)}\) and the number of conjugate operators \(\overline{\psi^{(-)}}\) in each subsystem is constant.

THE TAMM—DANCOFF METHOD

To obtain an approximate truncated system of equations (see § 1), one must, in the exact system of equations (2.10), (2.11), discard all amplitudes with a number of particles greater than some \(N_0\). As a result one obtains a complete and, as can be shown, consistent system of equations of the old T. D. method for determining amplitudes of the type (2.8) with a number of particles not exceeding \(N_0\).

Here it is important to emphasize that many authors \(^{22-25}\), working with the equations of the old T. D. method, understand by this term one equation for one of the amplitudes of the type (2.8), corresponding to the system of particles actually present in the given problem (for example, nucleon + meson in scattering theory, or two nucleons in the theory of the deuteron), and this equation is obtained from the system of equations (2.10) or (2.11) by eliminating from it all amplitudes except the one sought*). This elimination usually reduces to the fact that the kernel of the exact integral equation for the selected amplitude is expanded in a series in powers of the interaction constant and, in practice, is cut off at some power of this constant. Since, however, such an expansion is, generally speaking, divergent, such a simplification of the system of truncated equations may lead to incorrect results. Therefore the system of equations of the type (2.10) that arises after truncation must be solved exactly. This requirement is an essential component of the Tamm—Dancoff method. Let us note that in the first approximation the equations obtained by the two different approaches indicated coincide.

Let us consider the transition to the momentum representation in equations (2.10) and (2.11). For this purpose we use the explicit dependence of \(\Psi(\sigma)\) on time for stationary states. We choose as \(\sigma\) the plane surface \(t=\mathrm{const}\). The state vectors in the interaction representation, \(\Psi(t)\), and in the Schrödinger representation, \(\widetilde{\Psi}(t)\), are related by a relation of the form

\[ \Psi(t)=\exp[iH_0t]\,\widetilde{\Psi}(t), \tag{2.14} \]

where \(H_0\) is the Hamiltonian of the free fields.

For a stationary state with total energy \(W\) (in the center-of-inertia system) we have:

\[ \widetilde{\Psi}(t)=\exp[-iWt]\,\Psi . \tag{2.15} \]

From (2.14) and (2.15) it follows that

\[ \Psi(t)=\exp[-i(W-H_0)t]\,\Psi . \tag{2.16} \]

*) The so-called Lévy—Klein method \(^{22,23}\).

\(\Psi\) coincides with the state vector in the Heisenberg representation, by means of which the amplitudes of the old Tamm—Dancoff method were originally defined (see (1.2)).

With the aid of formulas (2.4) and (2.16) one can express the time-dependent covariant amplitudes (2.8) in terms of time-independent amplitudes of the type (1.6). For the simplest cases we obtain:

\[ \left<\varphi_i^{(+)}(x)\right>_t = \]

\[ = \int \frac{d\mathbf{k}}{(2\pi)^3}\,(2\omega_{\mathbf{k}})^{-1/2} \exp\,[i\mathbf{k}\mathbf{x}-i(W-\omega_{\mathbf{k}})t]\, (\Phi_0,Q_i(\mathbf{k})\Psi), \tag{2.17} \]

\[ \left<\psi^{+}(x)\right>_t = \sum_{n=1}^{2}\int \frac{d\mathbf{p}}{(2\pi)^3}\, u^n(\mathbf{p})\exp\,[i\mathbf{p}\mathbf{x}-i(W-E_{\mathbf{p}})t]\, (\Phi_0,B^n(\mathbf{p})\Psi). \tag{2.18} \]

With the aid of relations analogous to (2.17) and (2.18), and also of the expressions for the permutation functions (2.6), the system of covariant equations (2.10) or (2.11) can be transformed into the system of equations (1.9) for stationary amplitudes of the type (1.6) in three-dimensional momentum space. Thus, for example, equations (2.12) and (2.13) in the momentum representation take the form

\[ (W-\omega_{\mathbf{k}})\left<Q_i(\mathbf{k})\right>_0 = \]

\[ = \frac{ig}{(2\pi)^3(2\omega_{\mathbf{k}})^{1/2}} \sum_{n=1}^{2}\sum_{n'=3}^{4} \int d\mathbf{p}\,\overline{v}^{\,n'}(\mathbf{p}+\mathbf{k})(\tau_i\gamma_5)\times \]

\[ \times u^n(\mathbf{p})\left<B^{n'}(\mathbf{p}+\mathbf{k})B^n(\mathbf{p})\right>_0 + ig(2\pi)^{-6}\sum_{n=1}^{2}\sum_{n'=3}^{4} \int \overline{v}^{\,n'}(\mathbf{p}+\mathbf{k}')\times \]

\[ \times(\tau_j\gamma_5)u^n(\mathbf{p})(2\omega_{\mathbf{k}})^{-1/2} \left<B^{n'}(\mathbf{p}+\mathbf{k}')B^n(\mathbf{p})Q_i(\mathbf{k})Q_j(\mathbf{k}')\right>_0 \,d\mathbf{p}\,d\mathbf{k}', \tag{2.19} \]

\[ (W-E_{\mathbf{p}})\left<B^n(\mathbf{p})\right>_0 = ig(2\pi)^{-3}\sum_{n'=1}^{2} \int \overline{u}^{\,n}(\mathbf{p})(\gamma_5\tau_i)u^{n'}(\mathbf{p}-\mathbf{k})\times \]

\[ \times(2\omega_{\mathbf{k}})^{-1/2} \left<B^{n'}(\mathbf{p}-\mathbf{k})Q_i(\mathbf{k})\right>_0 \,d\mathbf{k}+ \]

\[ + ig\int \sum_{n_1=1}^{2}\sum_{n'=3}^{4} \overline{v}^{\,n'}(\mathbf{p}'+\mathbf{k}')(\gamma_5\tau_i)u^{n_1}(-\mathbf{p}') (2\omega_{\mathbf{k}'})^{-1/2}\times \]

\[ \times\left<B^{n'}(\mathbf{p}'+\mathbf{k}')B^{n_1}(-\mathbf{p}')B^n(\mathbf{p})Q_i(\mathbf{k}')\right>_0 \,d\mathbf{k}'\,d\mathbf{p}'. \tag{2.20} \]

Let us dwell briefly on the question of the choice of boundary conditions for

equations (1.9), (2.10)*) and (2.11). We shall start from the impulse representation. In the general case, the solution of an equation of type (1.9) can be written as the sum of the solutions of the homogeneous (free) and inhomogeneous equations. In the case of bound states the solution of the homogeneous equation is absent, since the factor \((W-E_{\lambda,N})\) cannot vanish for any values of the particle momenta, while the solution of the inhomogeneous equation can be written in the form

\[ \frac{ \sum_{\lambda' N'}(\lambda,N|H'|\lambda',N')\,a_{\lambda'}(N') }{ (W-E_{\lambda,N}) }. \tag{2.21} \]

In the case of a scattering problem, for those equations for which \((W-E_{\lambda,N})\) can vanish, to the solution, in addition to the term (2.21), one may add a term of the form \(\delta(W-E_{\lambda,N})\), multiplied by an arbitrary factor\({}^{26}\). If the division in (2.21) in this case is understood in the sense of the principal value, then this factor is chosen, for example, so that the asymptotic solution of the equation describes (in \(\mathbf r\)-space) incoming and outgoing waves.

It follows from this that, in covariant form, the boundary conditions in passing from (2.10) to (2.11) are formulated as follows. For bound states it is necessary to set

\[ \left\langle \overline{\psi}^{(-)}(x_1)\ldots \psi^{(+)}(z_r) \right\rangle_{-\infty}=0. \]

Formally this can be achieved if one adds to \(W\) a positive infinitesimal increment. In the case of scattering processes,

\[ \left\langle \ldots \right\rangle_{-\infty} \]

must describe incoming and outgoing waves.

§ 3. RENORMALIZATION IN THE EQUATIONS OF THE OLD TAMM—DANCOFF METHOD

The elimination of divergent expressions arising in the equations of the old Tamm—Dancoff method encounters serious difficulties. The covariant formulation of this method, proposed by Chini, made it possible to take a considerable step forward in understanding the nature of these difficulties, but it could not completely overcome them.

We shall consider the questions arising here by taking as examples the equation for two nucleons and the equation for the nucleon–meson system, taken in the first nonvanishing approximation of the method.

The choice of these two equations is not accidental. First, in these equations the characteristic difficul-

*) This question is discussed in more detail in § 6.

...associated with renormalization, and, secondly, the nucleon–nucleon and nucleon–meson systems are the simplest physical systems whose investigation opens the most direct path for elucidating both the value of the method itself and the limits of applicability of the modern meson theory to reality.

We begin with the two-nucleon problem.

The equation for the amplitude of two nucleons
\(\left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2)\right\rangle_{\sigma}\), according to (2.10), has the following form:

\[ i\frac{\delta}{\delta\sigma(x)} \left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2)\right\rangle_{\sigma} = ig(\gamma_5\tau_k)_{\alpha\beta}\times \]
\[ \times \left\{ \left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2) \psi_\alpha^{(-)}(x)\psi_\beta^{+}(x)\varphi_k^{+}(x) \right\rangle_{\sigma} - \right. \]
\[ \left. -\, iS_{\delta\alpha}^{(+)}(x_2-x) \left\langle \psi_\gamma^{+}(x_1)\psi_\beta^{+}(x)\varphi_k^{+}(x) \right\rangle_{\sigma} + \right. \]
\[ \left. +\, iS_{\gamma\alpha}^{(+)}(x_1-x) \left\langle \psi_\delta^{+}(x_2)\psi_\beta^{+}(x)\varphi_k^{+}(x) \right\rangle_{\sigma} \right\}. \tag{3.1} \]

Next let us write the equations for the amplitudes standing on the right-hand side of (3.1), and let us discard, in the right-hand sides of these equations, all amplitudes containing more than two nucleons and zero mesons. As a result we obtain:

\[ i\frac{\delta}{\delta\sigma(x')} \left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2) \overline{\psi_\alpha^{-}}(x)\overline{\psi_\beta^{-}}(x) \varphi_k^{+}(x) \right\rangle_{\sigma} = \]
\[ = -\,g(\gamma_5\tau_i)_{\mu\nu}\delta_{ki}\Delta^{(+)}(x-x') \left\{ -\,S_{\beta\mu}^{(+)}(x-x')S_{\nu\alpha}^{(-)}(x'-x)\times \right. \]
\[ \left. \times \left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2)\right\rangle_{\sigma} + S_{\delta\mu}^{(+)}(x_2-x')S_{\nu\alpha}^{(-)}(x'-x)\times \right. \]
\[ \left. \times \left\langle \psi_\gamma^{(+)}(x_1)\psi_\beta^{+}(x)\right\rangle_{\sigma} - S_{\gamma\mu}^{(+)}(x_1-x)S_{\nu\alpha}^{(-)}(x'-x)\times \right. \]
\[ \left. \times \left\langle \psi_\delta^{+}(x_2)\psi_\beta^{+}(x)\right\rangle_{\sigma} \right\}; \tag{3.2} \]

\[ i\frac{\delta}{\delta\sigma(x')} \left\langle \psi_\gamma^{+}(x_1)\psi_\beta^{+}(x)\varphi_k^{+}(x) \right\rangle_{\sigma} = -\,ig(\gamma_5\tau_i)_{\mu\nu}\delta_{ki}\Delta^{(+)}(x-x')\times \]
\[ \times \left\{ -\,S_{\beta\mu}^{(+)}(x-x') \left\langle \psi_\gamma^{+}(x_1)\psi_\nu^{+}(x')\right\rangle_{\sigma} + \right. \]
\[ \left. + S_{\gamma\mu}^{(+)}(x_1-x') \left\langle \psi_\beta^{+}(x)\psi_\nu^{+}(x')\right\rangle_{\sigma} \right\}; \tag{3.3} \]

\[ i\frac{\delta}{\delta\sigma(x')} \left\langle \psi_\delta^{+}(x_2)\psi_\beta^{+}(x)\varphi_k^{+}(x) \right\rangle_{\sigma} = -\,ig(\gamma_5\tau_i)_{\mu\nu}\delta_{ki}\Delta^{(+)}(x-x')\times \]
\[ \times \left\{ -\,S_{\beta\mu}^{(+)}(x-x') \left\langle \psi_\delta^{+}(x_2)\psi_\nu^{+}(x')\right\rangle_{\sigma} + \right. \]
\[ \left. + S_{\delta\mu}^{(+)}(x_2-x') \left\langle \psi_\beta^{+}(x)\psi_\nu^{+}(x')\right\rangle_{\sigma} \right\}. \tag{3.4} \]

Integrating these equations, taking into account the boundary conditions*)

\[ \underset{0}{\left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2)\bar{\psi}_\alpha^{-}(x)\psi_\beta^{+}(x)\varphi_k^{+} \right\rangle}_{-\infty} = \underset{0}{\left\langle \psi_\gamma^{+}(x)\psi_\beta^{+}(x)\varphi_k^{+}(x) \right\rangle}_{\infty-} = \underset{0}{\left\langle \psi_\delta^{+}(x_2)\psi_\beta^{+}(x)\varphi_k^{+}(x) \right\rangle}_{-\infty} =0 \]

and substituting the result obtained into (3.1), we find the desired equation for the amplitude \(\underset{0}{\left\langle\psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2)\right\rangle}_{\sigma}\):

\[ \begin{aligned} i\frac{\delta}{\delta\sigma(x)} \underset{0}{\left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2)\right\rangle}_{\sigma} &= -g^2\int_{-\infty}^{\sigma} dx'\,\Delta^{(+)}(x-x')\times \\ &\times\Bigl\{ -6Sp\bigl(\gamma_5 S^{(+)}(x-x')\gamma_5 S^{(-)}(x'-x)\bigr) \underset{0}{\left\langle \psi_\gamma^{+}(x_1)\psi_\delta^{+}(x_2)\right\rangle}_{\sigma'} \\ &\quad -3\bigl(S^{(+)}(x_1-x)\gamma_5 S^{(+)}(x-x')\bigr)_{\gamma\beta} \underset{0}{\left\langle \psi_\beta^{+}(x')\psi_\delta^{+}(x_2)\right\rangle}_{\sigma'} \\ &\quad +3\bigl(S^{(+)}(x_1-x')\gamma_5 S^{(-)}(x'-x)\bigr)_{\gamma\beta} \underset{0}{\left\langle \psi_\beta^{+}(x)\psi_\delta^{+}(x_2)\right\rangle}_{\sigma'} \\ &\quad -3\bigl(S^{(+)}(x_2-x)\gamma_5 S^{(+)}(x-x')\bigr)_{\delta\beta} \underset{0}{\left\langle \psi_\gamma^{+}(x_1)\psi_\beta^{+}(x')\right\rangle}_{\sigma'} \\ &\quad +3\bigl(S^{(+)}(x_2-x')\gamma_5 S^{(-)}(x'-x)\bigr)_{\delta\beta} \underset{0}{\left\langle \psi_\gamma^{+}(x_1)\psi_\beta^{+}(x)\right\rangle}_{\sigma'} \\ &\quad +\bigl(S^{(+)}(x_2-x)\gamma_5\tau_k\bigr)_{\delta\beta} \bigl(S^{(+)}(x_1-x')\gamma_5\tau_k\bigr)_{\gamma\nu} \underset{0}{\left\langle \psi_\beta^{+}(x)\psi_\nu^{+}(x')\right\rangle}_{\sigma'} \\ &\quad -\bigl(S^{(+)}(x_1-x)\gamma_5\tau_k\bigr)_{\gamma\beta} \bigl(S^{(+)}(x_2-x')\gamma_5\tau_k\bigr)_{\delta\nu} \\ &\qquad\qquad\times \underset{0}{\left\langle \psi_\beta^{+}(x)\psi_\nu^{+}(x')\right\rangle}_{\sigma'} \Bigr\}. \end{aligned} \tag{3.5} \]

Figure 1 shows the diagrams corresponding to each term of the kernel in this equation.

The first term in the kernel of equation (3.5) (see Fig. 1 (1)) corresponds to the vacuum loop; the 2nd and 3rd to the self-energy of the first nucleon; the 4th and 5th to the self-energy of the second nucleon. All these terms contain infinities. The 6th and 7th terms represent the contribution of ordinary scattering chains, which contain no divergences.

Let us first consider the self-energy terms themselves (restricting ourselves, for simplicity, to the first nucleon). In integral form the contribution of these terms to the kernel of equation (3.5) is written as

\[ \begin{aligned} 3g^2\int_{-\infty}^{\sigma} dx'\int_{-\infty}^{\sigma'} dx''\, \Delta^{(+)}(x'-x'')\times \Bigl\{ &\bigl(S^{(+)}(x_1-x')\gamma_5 S^{(+)}(x'-x'')\gamma_5\bigr)_{\gamma\beta} \underset{0}{\left\langle \psi_\beta^{+}(x'')\psi_\delta^{+}(x_2)\right\rangle}_{\sigma''} \\ &-\bigl(S^{(+)}(x_1-x'')\gamma_5 S^{(-)}(x''-x')\gamma_5\bigr)_{\gamma\beta} \underset{0}{\left\langle \psi_\beta^{+}(x')\psi_\delta^{+}(x_2)\right\rangle}_{\sigma''} \Bigr\}. \end{aligned} \tag{3.6} \]

*) We assume that the total energy \(W\) of the system is insufficient for particle production.

Changing, in the second term, the order of integration and replacing the variables \(x' \rightleftarrows x''\) gives:

\[ \begin{aligned} 3g^2\Bigg\{& \int_{-\infty}^{\sigma} dx' \int_{-\infty}^{\sigma'} dx''\, \Delta^{(+)}(x'-x'') \bigl(S^{(+)}(x_1-x')\gamma_5 S^{(+)}(x'-x'')\gamma_5\bigr)_{\gamma\beta} \\ &\qquad\qquad\qquad\qquad\times \left\langle 0\left|\psi_\beta^+(x'')\psi_\delta^+(x_2)\right|\right\rangle_{\sigma'} \\ &+ \int_{-\infty}^{\sigma} dx' \int_{\sigma'}^{\sigma} dx''\, \Delta^{(-)}(x'-x'') \\ &\qquad\qquad\qquad\qquad\times \bigl(S^{(+)}(x_1-x')\gamma_5 S^{(-)}\gamma_5\bigr)_{\gamma\beta} \left\langle 0\left|\psi_\beta^+(x')\psi_\delta^+(x_2)\right|\right\rangle_{\sigma'} \Bigg\}. \end{aligned} \tag{3.7} \]

In Chini’s work\(^4\) an attempt was made to renormalize expression (3.7) directly in the covariant form of notation.

Fig. 1.

For this purpose Chini, using the relation (see 2.7) between the causal \(\bigl(\Delta_F(x)\) and \(S_F(x)\bigr)\) and the commutation functions,

\[ \Delta_F(x_1-x_2)= \begin{cases} 2i\Delta^{(+)}(x_1-x_2), & \text{for } x_1>x_2,\\ -\,2i\Delta^{(-)}(x_1-x_2), & \text{for } x_1<x_2, \end{cases} \]

\[ iS_F(x_1-x_2)= \begin{cases} 2iS^{(+)}(x_1-x_2), & \text{for } x_1>x_2,\\ -\,2iS^{(-)}(x_1-x_2), & \text{for } x_1<x_2, \end{cases} \tag{3.8} \]

made in (3.7) the following replacement:

in the first term \((x' > x'')\)

\[ \Delta^{(+)}(x' - x'') S^{(+)}(x' - x'') \quad \text{by} \quad -\frac{1}{4}\Delta_F(x' - x'')S_F(x' - x'') \tag{3.9} \]

and in the second term \((x' \leq x'')\)

\[ \Delta^{(-)}(x' - x'') S^{(-)}(x' - x'') \quad \text{by} \quad -\frac{1}{4}\Delta_F(x' - x'')S_F(x' - x''). \tag{3.10} \]

In the expression obtained instead of (3.7),

\[ -\frac{3}{4}g^2 \left\{ \int_{-\infty}^{\sigma} dx' \int_{-\infty}^{\sigma'} dx'' \left(S^{(+)}(x_1-x')\dot{M}_F(x'-x'')\right) \times \left\langle \psi_\beta^+(x'')\psi_\delta^+(x_2)\right\rangle_{\sigma''} \right. \]

\[ \left. + \int_{-\infty}^{\sigma} dx' \int_{\sigma'}^{\sigma} dx'' \left(S^{(+)}(x_1-x')M_F(x'-x'')\right)_{\gamma\beta} \times \left\langle \psi_\beta^+(x')\psi_\delta^+(x_2)\right\rangle_{\sigma''} \right\}, \tag{3.11} \]

where

\[ M_{F_c}(x'-x'')=\Delta_F(x'-x'')\gamma_5 S_F(x'-x'')\gamma_5, \tag{3.12} \]

Chini carried out the renormalization in the usual covariant manner,

\[ M_{F_c}(x)=A\delta(x)+B\left(\gamma_\mu \frac{\partial}{\partial x_\mu}+M\right)\delta(x)+M_{F_c}(x), \tag{3.13} \]

where \(M_{F_c}(x)\) is a quantity which no longer has inadmissible singularities and is finite. We shall now show that the renormalization method proposed by Chini is erroneous. The reason for the error lies in the fact that the replacement (3.9) and (3.10) in (3.7) is illegitimate, since the functions \(\Delta_F(x)\) and \(\Delta^{(\pm)}(x)\) (respectively \(S_F(x)\) and \(S^{(\pm)}(x)\)) possess completely different singularities precisely at the point \((x=0)\), which is responsible for the divergence of expression (3.7). To prove our assertion, we shall pass in equation (3.5) to the momentum representation, writing explicitly only the second and third terms that interest us. First we write the Fourier expansion for the amplitude of two nucleons

\[ \left\langle \psi_\alpha^+(x_1)\psi_\beta^+(x_2)\right\rangle_{\sigma} = (2\pi)^{-6} \sum_{n_1,n_2=-1}^{1} \int dp_1\,dp_2\,u_\alpha^{n_1}(p_1)u_\beta^{n_2}(p_2) \times \]

\[ \times \exp\left[ip_1x_1+ip_2x_2-i(W-E_{p_1}-E_{p_2})t\right] \times \left(\Phi_0,B^{n_1}(p_1)B^{n_2}(p_2)\Psi\right). \tag{3.14} \]

Taking further into account formulas (2.6), after simple transformations we obtain from (3.5):

\[ (W-E_{p_1}-E_{p_2})\langle B^{n'_1}(p_1)B^{n_2}(p_2)\rangle = \]

\[ =3g^2\sum_{n_1=1}^{2}\overline{u}^{\,n'_1}(p_1)\, M^{(1)}(p_1,W-E_{p_2})\,u^{n_1}(p_1)\times \]

\[ \times \langle B^{n_1}(p_1)B^{n_2}(p_2)\rangle_0+R(p_1,p_2). \tag{3.15} \]

Here \(M^{(1)}(p_1,W-E_{p_2})\) is the mass operator of the first nucleon

\[ M^{(1)}(p_1,W-E_{p_2})= \frac{1}{(2\pi)^3}\int \frac{dk}{2\omega_k}\,\gamma_5\times \]

\[ \times\left\{ \frac{\Lambda^{(1)}_{+}(p_1+k)} {E_{p_1+k}+\omega_k-(W-E_{p_2})} - \frac{\Lambda^{(1)}_{-}(p_1+k)} {E_{p_1+k}+\omega_k+2E_{p_1}-(W-E_{p_2})} \right\}\gamma_4\gamma_5, \tag{3.16} \]

where

\[ \Lambda_{\pm}(p)=[E_p\pm(\alpha p+\beta M)](2E_p)^{-1} \]

is the projection operator, and \(R(p_1,p_2)\) is the contribution of the remaining terms.

The covariant mass operator of the nucleon

\[ M_F(p_1)=-\frac{i}{(2\pi)^4}\int dk\,\gamma_5 S_F(p_1+k)\gamma_5\Delta_F(k)^*), \tag{3.17} \]

which arises, for example, in the equation for two nucleons in the new T. D. method (see § 7), after integration over \(dk_0\) can be written in the form (putting \(p_{10}=W-E_{p_2}\))

\[ M_F(p_1,W-E_{p_2})= \frac{1}{(2\pi)^3}\int \frac{dk}{(2\omega_k)}\,\gamma_5\times \]

\[ \times\left\{ \frac{\Lambda^{(1)}_{+}(p_1+k)} {E_{p_1+k}+\omega_k-(W-E_{p_2})} - \frac{\Lambda^{(1)}_{-}(p_1+k)} {E_{p_1+k}+\omega_k+(W-E_{p_2})} \right\}\gamma_4\gamma_5 . \tag{3.18} \]

\[ *)\quad S_F(p)=\left(i\hat p+M\right)^{-1},\qquad \Delta_F(k)=(k^2+\kappa^2)^{-1}; \]

\[ \Delta_F(x)=\frac{2}{i(2\pi)^4}\int e^{ikx}\Delta_F(k)\,dk . \]

Comparing formulas (3.16) and (3.18), we see that in \(M^{(1)}(\mathbf p_1, W-E_{\mathbf p_2})\), \(\mathbf p_1\) and \((W-E_{\mathbf p_2})\) do not form a 4-vector and, consequently, the mass operator in the old Tamm—Dancoff method, unlike \(M_F(\mathbf p_1, W-E_{\mathbf p_2})\), cannot be transformed to covariant form.

It is not difficult to verify that the replacement (3.9) and (3.10), which Chini made in his work in order to transform the mass operator to covariant form, leads in the momentum representation to the following mass operator in equation (3.15):

\[ \widetilde M(\mathbf p_1, W-E_{\mathbf p_2}) = M_F(\mathbf p_1, W-E_{\mathbf p_2}) + M_F(\mathbf p_1, -W+E_{\mathbf p_2}), \tag{3.19} \]

where \(M\) is the covariant mass operator (3.18). This mass operator differs essentially from the corresponding one (see (3.16)) obtained by direct passage in equation (3.5) to the momentum representation.

Let us emphasize the difference between expressions (3.16) and (3.19). First, (3.16), unlike (3.19), cannot be transformed to covariant form; second, the charge renormalization in expression (3.19) is equal to zero, whereas expression (3.16) leads to a finite charge renormalization; finally, third, these two expressions lead to different finite additions. Let us explain the second point. The expansion of the mass operator in covariant form (see (3.18)) is equivalent to the expansion of the quantity \(M\) in three-dimensional notation in powers of \(W-E_{\mathbf p_1}-E_{\mathbf p_2}\). Since (3.19) is a symmetric function of \(W-E_{\mathbf p_2}\), the first derivative

\[ \left. \frac{\partial \widetilde M}{\partial (W-E_{\mathbf p_2})} \right|_{W-E_{\mathbf p_2}=E_{\mathbf p_1}} \]

vanishes, and charge renormalization is absent in this case. If the renormalization of expression (3.16) is carried out in an analogous way, then, as is easy to see,

\[ \left. \frac{\partial M^{(1)}}{\partial (W-E_{\mathbf p_2})} \right|_{W-E_{\mathbf p_2}=E_{\mathbf p_1}} \]

will be a finite quantity.

Thus, comparison of expressions (3.16) and (3.19) shows that the replacement (3.8) and (3.9) made by Chini is illegitimate, and the mass operator arising after such a replacement turns out to be incorrect.

Let us note that here there is also revealed an essential difference between the nucleon mass operators in the old and the new

methods of T. D., since in the latter case (see § 6) the charge renormalization coincides with the covariant one and is finite.

Let us now consider the contribution to the kernel of equation (3.5) from the first (vacuum) term. The decisive difference from perturbation theory (the \(S\)-matrix) is manifested here in the fact that in the momentum representation this term depends on \(W\). Therefore the infinite contribution corresponding to this term cannot, by analogy with the \(S\)-matrix, be eliminated by means of some unitary transformation (renormalization) of the amplitude

\[ \left\langle \psi^{+}(x_1)\psi^{+}(x_2)\right\rangle_{0} \;\to\; \left\langle \psi^{+}(x_1)\psi^{+}(x_2)\right\rangle_{0}\exp[ia], \]

where \(\alpha\) is a (generally speaking, infinite) real constant. The simple deletion of vacuum terms in the old T. D. method, which is done in all papers, is an arbitrary operation and in essence is not justified by anything.

We note that precisely the difficulties connected with the appearance of vacuum infinities in the old T. D. method constitute the most appreciable shortcoming of this method and served as one of the main reasons for the formulation of the new T. D. method, free from these difficulties.

We next write equation (3.5) in the momentum representation. Denote by

\[ a(\mathbf{p})=\left(\Phi_0,\;B^{n_1}(\mathbf{p})B^{n_2}(-\mathbf{p})\Psi\right) \tag{3.20} \]

(\(\mathbf{p}\) is the momentum in the center-of-inertia system) the amplitude of two nucleons in the center-of-inertia system. Then, after simple transformations, we obtain from (3.5):

\[ \begin{aligned} (W-2E_p)a(\mathbf{p})={}& 3g^2\left[\overline{u^{(1)}(\mathbf{p})}\,M^{(1)}(\mathbf{p},W-E_p)u^{(1)}(\mathbf{p})\right]+{}\\ &+3g^2\left[\overline{u^{(2)}(-\mathbf{p})}\,M^{(2)}(-\mathbf{p},W-E_p)u^{(2)}(-\mathbf{p})\right]a(\mathbf{p})+{}\\ &+\frac{g^2}{(2\pi)^3}\int\frac{d\mathbf{k}}{\omega_k}\times\\ &\times\left\{ \frac{\left[\overline{u^{(1)}(\mathbf{p})}\gamma_5\tau_k u^{(1)}(\mathbf{p}+\mathbf{k})\right] \left[\overline{u^{(2)}(\mathbf{p})}\gamma_5\tau_k u^{(2)}(-\mathbf{p}-\mathbf{k})\right]} {E_p+E_{\mathbf{p}+\mathbf{k}}+\omega_k-W} \right\}a(\mathbf{p}+\mathbf{k}). \end{aligned} \tag{3.21} \]

Here we have omitted the vacuum term.

Let us pass to the investigation of the questions of renormalization in the meson–nucleon equation.

By analogy with the equation for two nucleons, in this case it is necessary to write equations of type (2.10) for the amplitudes

\[ \left\langle \psi^{+}(x_1)\varphi_i^{+}(x_2)\right\rangle_{0},\quad \left\langle \psi^{+}(x)\right\rangle_{0},\quad \left\langle \psi^{+}(x_1)\overline{\psi}^{-}(x)\psi^{+}(x)\right\rangle_{0}, \]

\[ \left\langle \psi^{+}(x)\varphi_i^{+}(x_2)\varphi_k^{+}(x)\right\rangle_{0} \quad\text{and}\quad \left\langle \psi^{+}(x_1)\overline{\psi}^{-}(x)\psi^{+}(x)\varphi_i^{+}(x_2)\varphi_k^{+}(x)\right\rangle_{0}. \]

and discard, in the right-hand sides of the equations for the last four amplitudes, all amplitudes except the first. Then, after eliminating from the equation for the amplitude \(\langle \psi^{+}(x_1)\psi_i^{+}(x_2)\rangle\) all the remaining amplitudes, we obtain the following equation\(^*\) for the meson + nucleon system (in the first approximation of the method):

\[ \begin{aligned} i\,\frac{\delta}{\delta\sigma(x)} \left\langle \psi^{+}(x_1)\varphi_i^{+}(x_2)\right\rangle_{0,\sigma} ={}&-g^2\int_{-\infty}^{\sigma} dx'\,\Bigl\{ -6\Delta^{(+)}(x-x')\times \\ &\times P(x-x')\left\langle \psi^{+}(x_1)\varphi_i^{+}(x_2)\right\rangle_{0,\sigma'}- \\ &-2\Delta^{(+)}(x_2-x)P(x-x') \left\langle \psi^{+}(x_1)\varphi_i^{+}(x')\right\rangle_{0,\sigma'}- \\ &-2\Delta^{(+)}(x_2-x')P(x-x') \left\langle \psi^{+}(x_2)\varphi_i^{+}(x)\right\rangle_{0,\sigma'}+ \\ &+3\Delta^{(+)}(x-x')S^{(+)}(x_1-x')\gamma_5 S^{(-)}(x'-x)\gamma_5 \left\langle \psi^{+}(x)\varphi_i^{+}(x_2)\right\rangle_{0,\sigma'}- \\ &-3\Delta^{(+)}(x-x')S^{(+)}(x_1-x)\gamma_5 S^{(+)}(x-x')\gamma_5\times \\ &\times \left\langle \psi^{+}(x')\varphi_i^{+}(x_2)\right\rangle_{0,\sigma'}+ \\ &+\Delta^{(+)}(x_2-x')S^{(+)}(x_1-x')\gamma_5 S^{(-)}(x'-x)\gamma_5(\tau_i\tau_k)\times \\ &\times \left\langle \psi^{+}(x)\varphi_k^{+}(x)\right\rangle_{0,\sigma'} -\Delta^{(+)}(x_2-x)S^{(+)}(x_1-x)\times \\ &\times \gamma_5 S^{(+)}(x-x')\gamma_5(\tau_i\tau_k) \left\langle \psi^{+}(x')\varphi_k^{+}(x')\right\rangle_{0,\sigma'}+ \\ &+\Delta^{(+)}(x_2-x) \times \\ &\times S^{(+)}(x_1-x')\gamma_5S^{(-)}(x'-x)\gamma_5(\tau_k\tau_i) \left\langle \psi^{+}(x)\varphi_k^{+}(x')\right\rangle_{0,\sigma'}- \\ &-\Delta^{(+)}(x_2-x')S^{(+)}(x_1-x)\gamma_5 S^{(+)}(x-x')\gamma_5\times \\ &\times(\tau_k\tau_i) \left\langle \psi^{+}(x')\varphi_k^{+}(x)\right\rangle_{0,\sigma'} \Bigr\}, \end{aligned} \tag{3.21'} \]

where

\[ P(x-x')=\operatorname{Sp}\{S^{(+)}(x-x')\gamma_5S^{(-)}(x'-x)\gamma_5\}. \]

The chains corresponding to each term of the kernel of this equation are shown in Fig. 2.

The first term in the kernel of equation (3.21) corresponds to the vacuum loop (Fig. 2 (1)) and coincides exactly with the corresponding term

\(^*\) As in the case of two nucleons, we assume that the total energy \(W\) of the system is insufficient for the formation of new particles.

in the nucleon–nucleon equation. The renormalization of this term is difficult for the reasons indicated above; therefore, in what follows we shall not return to its consideration. The 4th and 5th terms correspond to the proper energy of the nucleon (see Fig. 2 (4) and (5)). These terms coincide with the proper-energy terms in the nucleon–nucleon equation. Therefore everything said there is also valid in this case. The 6th and 7th terms correspond to chains with initial absorption of the meson (chains with absorption), while the 8th

Fig. 2.

and 9th correspond to chains with initial emission of the meson (chains with emission). From the standpoint of renormalizations in equation (3.21), the 2nd and 3rd terms are of greatest interest; they are responsible for the proper energy of the meson (or vacuum polarization; see Fig. 2 (2) and (3)). An attempt to renormalize analogous terms by the method proposed by Chini was made in Lehmann’s paper\(^5\). The inadmissibility of replacing the permutation functions \(S^{(\pm)}(x)\) \(\left(\Delta^{(\pm)}(x)\right)\) at the point \(x=0\) by causal functions \(S_F(x)\) \(\left(\Delta_F(x)\right)\) is manifested particularly clearly in this case. Because of lack of space, we shall briefly formulate only the main result. Lehmann showed that the finite additions, calculated in the \(x\)-representation, which arise after renormalization of the mass (polarization) operator of the meson by Chini’s method, lead to new infinities upon transition to the momentum representation. This paradoxical result is essentially due to the fact that the finite expression \(\int P(x)e^{ipx}dx\)

is replaced in the Chew method by an infinite quantity equal to

\[ \frac{1}{4}\operatorname{Sp}\int S_F(x)\gamma_5 S_F(-x)\gamma_5 e^{ipx}\,dx . \]

Below we shall show that the terms corresponding to the meson self-energy in the old T. D. method cannot be renormalized in an unambiguous way.

In the momentum representation the contribution of the meson self-energy terms to equation (3.21) is written in the form (in the center-of-inertia system)

\[ (W-E_{\mathbf p}-\omega_{\mathbf p})\,\langle B(\mathbf p)Q(-\mathbf p)\rangle_0 =(\omega_{\mathbf p})^{-1}g^2 P(\mathbf p,W-E_{\mathbf p}) \tag{3.22} \]

\[ \langle B(\mathbf p)Q(-\mathbf p)\rangle_0+\text{remaining terms}, \]

where

\[ P(\mathbf p,W-E_{\mathbf p}) = \frac{1}{(2\pi)^3}\operatorname{Sp}\int d\mathbf p'\left\{ \frac{\Lambda_+(\mathbf p+\mathbf p')\gamma_4\gamma_5\Lambda_-(\mathbf p')} {E_{\mathbf p+\mathbf p'}+E_{\mathbf p'}-(W-E_{\mathbf p})} + \right. \]

\[ \left. + \frac{\Lambda_-(\mathbf p+\mathbf p')\gamma_4\gamma_5\Lambda_+(\mathbf p')} {E_{\mathbf p+\mathbf p'}+E_{\mathbf p'}+2\omega_{\mathbf p}-(W-E_{\mathbf p})} \right\}\gamma_4\gamma_5 . \tag{3.23} \]

— the polarization (or mass) operator of the meson.

The covariant expression for the polarization operator (cf. also § 7)

\[ P_F(p)=-\frac{i}{(2\pi)^4}\operatorname{Sp}\int dp'\,S_F(p+p')\gamma_5 S_F(p') \tag{3.24} \]

after integration over \(dp_0'\) (in the three-dimensional form) is equal to

\[ P_F(\mathbf p,p_0=W-E_{\mathbf p}) = \frac{1}{(2\pi)^3}\operatorname{Sp}\int d\mathbf p'\left\{ \frac{\Lambda_+(\mathbf p+\mathbf p')\gamma_4\gamma_5\Lambda_-(\mathbf p')} {E_{\mathbf p+\mathbf p'}+E_{\mathbf p'}-(W-E_{\mathbf p})} + \right. \]

\[ \left. + \frac{\Lambda_-(\mathbf p+\mathbf p')\gamma_4\gamma_5\Lambda_+(\mathbf p')} {E_{\mathbf p+\mathbf p'}+E_{\mathbf p'}+(W-E_{\mathbf p})} \right\}\gamma_4\gamma_5 . \tag{3.25} \]

Let us emphasize the difference between (3.23) and (3.25). First, in the first expression, unlike the second, \(\mathbf p\) and \(W-E_{\mathbf p}\) do not form a 4-vector, and, secondly, (3.25) is a symmetric function of \(W-E_{\mathbf p}\), whereas (3.23) has no evident symmetry with respect to this quantity. Owing to the first difference, the expression for (3.23) cannot be transformed to covariant form and, consequently, the renormalization cannot be carried out covariantly. The second difference leads to very serious difficulties in attempting to carry out the renormalization of the polarization operator \(P(\mathbf p,W-E_{\mathbf p})\) in the old T. D. method. The essence of this difficulty is as follows. The covariant renormalization \(P_F\) reduces to the раз-

to the expansion of this quantity in powers of \(p^{2}=\mathbf p^{2}-p_{0}^{2}=\mathbf p^{2}-(W-E_{p})^{2}\) at the point \(p^{2}=-\mu^{2}\). In the three-dimensional form (3.25) this expansion is equivalent to an expansion in powers of \((W-E_{p})^{2}\) at the point \((W-E_{p})^{2}=\omega_{p}^{2}\). If one attempts to carry out an analogous expansion of the quantity \(P(\mathbf p,W-E_{p})\), then the first terms of the expansions (3.23) and (3.25) turn out to be equal:

\[ P_{F}(\mathbf p,\omega_{p})=P(\mathbf p,\omega_{p}). \]

However, the first derivative

\[ \left. \frac{\partial P(\mathbf p,W-E_{p})} {\partial(W-E_{p})} \right|_{W-E_{p}=\omega_{p}} \]

does not vanish (because of the absence of symmetry) and has a linear divergence. The second derivative diverges logarithmically (charge renormalization). The sum of the remaining terms gives a convergent remainder. The linearly divergent first derivative cannot be interpreted as a charge renormalization. A simple deletion of this term is not justified. Therefore the extraction of a finite remainder from the polarization operator in the old Tamm–Dancoff method is an ambiguous operation.

This is one of the principal differences between the renormalization of the mass and polarization operators in the old Tamm–Dancoff method.

Let us emphasize that there is yet another essential difference between the renormalization of the equation for two nucleons and the equation meson—nucleon. In the case of the equation for two nucleons, renormalization of the proper-energy kernels is sufficient for the solution of the equation to be finite. This is not so in the case of the meson—nucleon equation. Here one can no longer restrict oneself to the renormalization of divergent kernels. Owing to the field nature of the meson, the solution of the meson—nucleon equation with a finite kernel turns out to be divergent and requires additional renormalization. For details on this see \(^{27-35}\).

We shall now note only that this difference can be understood very clearly if the solution is represented in the form of a series calculated by perturbation theory. The solution of the nucleon—nucleon equation after renormalization of the divergent kernels corresponds only to finite chains which, in higher approximations in \(g^{2}\), are obtained by successive iteration of chains (6) and (7), Fig. 1. The solution of the meson—nucleon equation corresponds to the sum of all possible chains which arise under successive iteration of the chains ( )—(9), Fig. 2. In particular, this sum contains divergent chains of the complicated vertex and proper-energy type, which give an infinite contribution to the solution. For scattering with total isotopic moment \(I=3/2\), when in equation (3.21) only the chain with emission remains, the solution will be finite

and from the point of view of perturbation theory to correspond to taking into account only the so-called selected chains. The problem of renormalizing the solution for the case \(I = {}^{1}/_{2}\) in the old T. D. method has not yet been solved.

The shortcomings of the old T. D. method served as the main impetus for the formulation of the new T. D. method, which was given by Dyson\(^{7–9}\) and is discussed in detail in §§ 6–8.

§ 4. INTERACTION OF NUCLEONS IN THE OLD
TAMM—DANCOFF METHOD

The Tamm—Dancoff method was originally formulated as applied to the problem of two nucleons, in other words, as applied to the problem of nuclear forces. Indeed, the extremely large magnitude of nuclear forces makes the application of perturbation theory clearly unacceptable, which forces one to seek other methods for solving the equations of quantum field theory. The use for the two-nucleon problem of the first approximation of the T. D. method, analogous to that adopted in § 3, i.e. taking into account, in addition to the amplitude of two nucleons, also the amplitude of two nucleons and one meson, makes it possible to obtain a comparatively simple integral equation for the amplitude of two nucleons. Such an equation was studied in a number of works for various meson theories: scalar\(^{1,2}\), pseudoscalar with pseudoscalar\(^{36,22}\) and with pseudovector couplings\(^{36,37}\). Let us note that if, in obtaining the operator of the interaction energy of particles, one restricts oneself to the first nonvanishing term of the expansion in powers of \(v/c\), where \(v\) is the velocity of the nucleon, then the pseudoscalar and pseudovector couplings give the same operator of the interaction energy\(*\)

\[ V(\mathbf r)=\left(\frac{g}{2M}\right)^2(\tau_1\tau_2)(\sigma_1\nabla)(\sigma_2\nabla)\frac{e^{-r}}{r}; \tag{4.1} \]

\(V(\mathbf r)\) is the operator arising when the pseudoscalar coupling is used. The corresponding pseudovector operator differs by the factor \((2M)^2\).

However, the above expansion, as a result of which the adiabatic potential (4.1) arises, is valid only at large distances. In the region of small distances \((\ll \hbar/\mu c)\), the pseudoscalar and pseudovector nuclear forces differ sharply. The pseudovector forces, in this case, possess an inadmissible singularity, which, in particular, does not permit the pseudovector coupling to be used consistently.

Attempts to compare with experiment the theoretical results obtained for nuclear forces in the first approximation of the T. D. method in the case of pseudoscalar coupling showed a substantial discrep—

\(*\) Below a system of units is used in which \(\mu = 1\).

of the theory and experiment. This indicated the necessity of taking higher approximations into account. Up to now, within the framework of the T. D. method, such an account has not been carried out in any sufficiently consistent form. Therefore, below we shall briefly discuss only some, in fact unfinished, attempts to take higher approximations into account.

One of the first such attempts is represented by the work of Lévy^22, in which the interaction-energy operator was considered in the form of a series in powers of \(g^2\). It should be pointed out that interest in the T. D. method in the foreign literature especially increased after the work of Lévy, who obtained a potential of nuclear forces consistent with experiment and leading to strong repulsion at small distances.

However, the Lévy potential in the approximation used by him was obtained inaccurately. Subsequent works by Klein^23 and others^38–40, 42, 43, in which the necessary refinements were made, led to an adiabatic potential of nuclear forces that differs substantially from the Lévy potential. However, such a difference by no means signifies that the pseudoscalar coupling contradicts experiment. First of all, the potential in the works indicated above was obtained in the adiabatic approximation, while analysis of nonadiabatic corrections indicates their essential role^41, 44–46, leading to a significant change in the interaction-energy operator. In addition, constructing the potential in the form of a series in powers of \(g^2\) is a substantially approximate procedure. Such an approximation may be rather inaccurate because of the poor convergence of the resulting series^47–48*). It seems to us advisable, for the construction of a correct theory of nuclear forces, to consider a system of integral equations for several amplitudes and, in any case, not to confine oneself to the adiabatic approximation.

It is necessary to note one more shortcoming, common to almost all works devoted to the interaction of two nucleons, and especially important for the problem of the bound state—the deuteron.

The point is that, when normalizing the wave function, and also when calculating mean quantities or values of matrix elements, in most works only the amplitude of two nucleons is taken into account, while, for example, the contribution of the amplitude of two nucleons and a meson is not taken into account at all. This contribution, for example, is infinite in the norm. Therefore, in order to carry out the normalization of functions correctly, one must formulate rules for handling the divergences that arise. Recently works have appeared in which attempts are made to solve this problem^54–55. However, no completeness in the solution of this problem has yet been achieved.

*) In this connection, approximate methods for constructing the interaction operator of two nucleons, close to the Tamm–Dancoff method, are of interest; that is, methods that do not use an expansion in powers of the coupling constant^41, 49–53.

§ 5. INTERACTION OF THE π-MESON AND THE NUCLEON IN THE OLD TAMM–DANCOFF METHOD

As was already said above, the success of meson theory in explaining the experimental data on the scattering of π-mesons by nucleons, achieved with the aid of the T. D. method, is one of the grounds for hopes for the successful application of the T. D. method also to other problems. In the present section we shall consider in detail the application of the T. D. method to the scattering of π-mesons by nucleons. Of greatest interest in this respect are the results of work ⁶, which we shall set forth in comparatively great detail. In this work the scattering of π-mesons was considered according to the symmetric pseudoscalar theory with pseudoscalar coupling. An analogous consideration, though considerably cruder, was carried out in work ⁵⁶. Finally, in works ⁵⁷–⁶⁵ the scattering of π-mesons, as well as certain other questions, were considered according to the symmetric pseudoscalar theory with pseudovector coupling.

Before proceeding to the presentation of the results of the theory, let us recall the principal results of experiments on the scattering of π-mesons by nucleons in the energy region up to 200 MeV (see in more detail ⁶⁶–⁶⁸). First, it turns out that the experimental data do not contradict the hypothesis of isotopic invariance, which makes it possible to characterize the states of the π-meson—nucleon system by the eigenvalues of the isotopic-spin operator, equal to one half or three halves. Second, the angular distributions and the energy dependence of meson scattering can be well interpreted by taking into account only states with orbital angular momentum zero and one, i.e. the \(S_{1/2}\)-, \(P_{1/2}\)-, and \(P_{3/2}\)-states. Owing to the fact that the system can possess two different values of the isotopic spin, the total number of states taken into account in interpreting the experimental data is six. Third, in the energy region 150–200 MeV the phase shift of the state with isotopic and mechanical angular momenta equal to \(3/2\) turns out to be considerably larger than the phase shifts of all the other states and, at an energy \(\sim 195\) MeV, passes through a value equal to \(\pi/2\), thus reaching a resonance at this point ⁶⁹, ⁷⁰. This fact is the most striking and, one may say, the principal result of the experiments on meson scattering, which led to the appearance of various phenomenological and semi-phenomenological theories of π-meson scattering (see, for example, ⁷¹–⁷³).

It is necessary to emphasize the following. The symmetric pseudoscalar theory, both with pseudovector and with pseudoscalar coupling, in the nonrelativistic approximation gives equivalent results for the interaction in \(P\)-states. At the same time, for the \(^{3/2}P_{3/2}\) state, in contrast to other states taken into account in the interpretation of scattering, different approximate methods lead to an attractive potential ⁷⁴–⁷⁷. Such a law of interact—

action makes it possible to hope to obtain, in the \({}^{3}/_{2}P_{3/2}\) state, a scattering resonance.

However, perturbation theory, as is well known, does not give such a resonance \(^{78-81}\). It is precisely this fact that, first of all, compels one to turn to methods that differ from perturbation theory.

The treatment of \(\pi\)-meson scattering carried out in Ref. \(^{6}\) (see also \(^{82-85}\)) corresponds to the approximation used in § 3 to describe the \(\pi\)-meson and nucleon system. Therefore, in our exposition we shall use the results of that section. First of all, taking into account the difficulty of renormalizations in the old T. D. method, we shall, following Ref. \(^{6}\), completely omit the expressions corresponding to the diagrams of the nucleon self-energy, vacuum polarization, and vacuum self-energy. Then the equation for the meson and nucleon amplitude assumes the following form:

\[ i \frac{\partial}{\partial \sigma(x)} \left\langle \psi^{(+)}(x_1)\varphi_i^{(+)}(x_2)\right\rangle_\sigma = g^2 \int_{-\infty}^{\sigma} dx' \left\{ \Delta^{(+)}(x_2-x)\times \right. \]

\[ \times S^{(+)}(x_1-x)\gamma_5 S^{(+)}(x-x')\gamma_5 N_2 \left\langle \psi^{(+)}(x')\varphi_k^{(+)}(x')\right\rangle_{\sigma'} - \]

\[ -\Delta^{(+)}(x_2-x')S^{(+)}(x_1-x')\gamma_5 S^{(-)}(x'-x)\gamma_5 N_2 \times \]

\[ \times \left\langle \psi^{(+)}(x)\varphi_k^{(+)}(x)\right\rangle_{\sigma'} + \]

\[ +\Delta^{(+)}(x_2-x')S^{(+)}(x_1-x)\gamma_5 S^{(+)}(x-x')\gamma_5 N_1 \times \]

\[ \times \left\langle \psi^{(+)}(x')\varphi_k^{(+)}(x)\right\rangle_{\sigma'} - \]

\[ \left. -\Delta^{(+)}(x_2-x)S^{(+)}(x_1-x')\gamma_5 S^{(-)}(x'-x)\gamma_5 N_1 \times \right. \]

\[ \left. \times \left\langle \psi^{(+)}(x)\varphi_k^{(+)}(x')\right\rangle_{\sigma'} \right\}, \tag{5.1} \]

where \(N_1\) and \(N_2\) are operators in the space of isotopic spin:

\[ N_1=\tau_{\lambda\nu}^{k}\tau_{\nu\mu}^{i},\qquad N_2=\tau_{\lambda\nu}^{i}\tau_{\nu\mu}^{k} \tag{5.2} \]

with the following eigenvalues:

\[ \begin{aligned} N_1&=-1,\qquad &N_2&=3 \quad &&\text{for } I=1/2,\\ N_1&=2,\qquad &N_2&=0 \quad &&\text{for } I=3/2, \end{aligned} \tag{5.3} \]

where \(I\) denotes the total isotopic spin of the nucleon \(+\ \pi\)-meson system. Below we shall consider states with definite values of the isotopic angular momentum, in which the operators \(N_1\) and \(N_2\) are diagonal, and therefore we shall not write the isotopic indices \(\lambda\) and \(i\).

Equation (5.1) is conveniently investigated in the momentum representation in the center-of-inertia system. For this it is necessary, following the formu-

according to (2.18), introduce the time-independent amplitude of two particles, the meson and the nucleon,

\[ \left\langle \stackrel{\circ}{B}^{\,n}(\mathbf p)Q^{(+)}(-\mathbf p)\right\rangle = a_n(\mathbf p). \tag{5.4} \]

Here the index \(n\) corresponds to the two solutions of the Dirac equation with different spins and positive energy; below we shall use the Pauli matrices \(\sigma_{mn}\), which will act on \(a_n\) as on an ordinary spinor. Then in the momentum representation equation (5.1) may be written in the following form:

\[ (W-E-\omega)a(\mathbf p)=\frac{g^2}{32\pi^3}\int d\mathbf p'\,R(\mathbf p,\mathbf p')a(\mathbf p'), \tag{5.5} \]

\[ R(\mathbf p,\mathbf p')=\varphi(\mathbf p,\mathbf p')\{N_1S(\mathbf p,\mathbf p')+N_2T(\mathbf p,\mathbf p')\}, \tag{5.6} \]

where

\[ \varphi(\mathbf p,\mathbf p')=\frac12 \sqrt{\frac{(E+M)(E'+M)}{EE'\omega\omega'}}, \]

\[ S(\mathbf p,\mathbf p')=[(W-M)m+(E+E'+\omega+\omega'-M-W)n]+ \]

\[ +\frac{\boldsymbol\sigma\mathbf p}{E+M}\frac{\boldsymbol\sigma\mathbf p'}{E'+M}\times \]

\[ \times[(W+M)m+(E+E'+\omega+\omega'+M-W)n], \tag{5.7} \]

\[ T(\mathbf p,\mathbf p')=\frac{2}{W+M}+ \frac{2}{W-M}\frac{\boldsymbol\sigma\mathbf p}{E+M} \frac{\boldsymbol\sigma\mathbf p'}{E'+M}, \tag{5.8} \]

\[ m=\{E_q(E_q+E+E'-W)\}^{-1},\quad n=\{E_q(E_q+\omega+\omega'-W)\}^{-1}, \]

\[ E_q=\sqrt{(\mathbf p+\mathbf p')^2+M^2}. \]

For the scattering problem of interest to us, the asymptotic behavior of \(a(\mathbf p)\) has the following form:

\[ a(\mathbf p)=\hat{\delta}(\mathbf p-\mathbf p_0)+f(\mathbf p)\hat{\delta}_+(E+\omega-W), \tag{5.9} \]

where \(\hat{\delta}_+(x)=i\pi\hat{\delta}(x)-x^{-1}\), and \(\mathbf p_0\) is the momentum of the incident wave, corresponding to \(E_0+\omega_0=W\). Then for the amplitude of the outgoing wave we obtain the following equation:

\[ f(\mathbf p)=\frac{g^2}{32\pi^3}R(\mathbf p,\mathbf p_0)+ \frac{ig^2}{32\pi^2}\int d\mathbf p'\,R(\mathbf p,\mathbf p')f(\mathbf p')\hat{\delta}(E'+\omega'-W)+ \]

\[ +\frac{g^2}{32\pi^3}\int d\mathbf p'\, \frac{R(\mathbf p,\mathbf p')f(\mathbf p')}{W-E'-\omega'}. \tag{5.10} \]

To solve equation (5.10) it is necessary to separate the angular variables. For this purpose we expand the amplitude of the outgoing wave in a series in orthogonal polynomials

\[ f(\mathbf p)=\sum L_l^{\pm}\left(\frac{\mathbf p}{p},\frac{\mathbf p_0}{p_0}\right)f_{Jl}(p). \tag{5.11} \]

An analogous expansion must also be carried out for the kernel \(R(\mathbf p,\mathbf p')\). In relation (5.11) the \(L^\pm\) have the form \({}^{73}\):

\[ \left. \begin{aligned} L_l^{+}(\mathbf n,\mathbf n')&=(l+1)P_l(\cos\theta)-i\sigma[\mathbf n\mathbf n']P_l^{1}(\cos\theta) \\ &\hspace{4.3cm}\text{for } j=l+\frac12, \\[0.6em] L_l^{-}(\mathbf n,\mathbf n')&=lP_l(\cos\theta)+i\sigma[\mathbf n,\mathbf n']P_l^{1}(\cos\theta) \\ &\hspace{4.3cm}\text{for } j=l-\frac12. \end{aligned} \right\} \tag{5.12} \]

As a result of separating the angular variables we obtain the following equation for the amplitude of the outgoing wave in a state with prescribed values of the total and orbital angular momenta:

\[ f_{jl}(p)=\frac{g^{2}jl}{32\pi^{3}}R(p,p_0) \left\{1+i\frac{g^{3}}{2}\frac{p_0E_0\omega_0}{E_0+\omega_0}f_{jl}(p_0)\right\} + \frac{g^{2}}{8\pi^{2}}\int \frac{p'^2\,dp'\,{}^{jl}R(p,p')f_{jl}(p')}{W-E'-\omega'} . \tag{5.13} \]

In equation (5.13) the kernel \({}^{jl}R\) is related to \({}^{jl}S\) and \({}^{jl}T\) by formula (5.6). Here

\[ {}^{jl}S=[(W-M)J_{k_1}(E+E'-W)+ \]

\[ +(E+E'+\omega+\omega'-M-W)J_{k_1}(\omega+\omega'-W)]+ \]

\[ +\frac{p}{E+M}\frac{p}{E'+M}[(W+M)J_{k_2}(E+E'-W)+ \]

\[ +(E+E'+\omega+\omega'+M-W)J_{k_2}(\omega+\omega'-W)], \tag{5.14} \]

\[ {}^{jl}T=\delta_{j,l'_2}\delta_{l,0}\frac{1}{W+M} +\delta_{j,l'_2}\delta_{l,1}\frac{1}{W-M} \frac{p}{E+M}\frac{p'}{E'+M}, \tag{5.15} \]

where

\[ J_k(z)=\frac12\int_{-1}^{+1}P_k(x)\frac{dx}{E_q(E_q+z)},\qquad E_q=\sqrt{M^2+p^2+p'^2+2pp'x}, \]

\(P_k(x)\) are Legendre polynomials. For the state \(j=\frac12\) and \(l=0\) \((S_{1/2})\), \(k_1=0,\ k_2=1\); for \(j=\frac12,\ l=1\) \((P_{1/2})\), \(k_1=1,\ k_2=0\); for \(j=\frac12,\ l=1\) \((P_{3/2})\), \(k_1=1,\ k_2=2\), etc. The kernel \({}^{jl}T\) corresponds to the chain with absorption. At large \(p\) this kernel possesses an inadmissible singularity which, as was said in § 3, leads to the absence of finite solutions. Therefore, following Ref. \({}^{6}\), we shall restrict ourselves to considering states for which the kernel \(T\) gives no contribution. We note that the kernel \(T\) arises only in states with isotopic and mechanical spins equal to one half [see (5.2) and (5.15)].

The function $f_{jl}$ satisfies equation (5.13), which contains complex expressions. One can pass to an equation with real coefficients and for a real function. To do this, we carry out the following transformation*):

\[ u_{jl}=f_{jl}\left[1+\frac{ig^2}{2}\frac{p_0\omega_0E_0}{E_0+\omega_0}f_{jl}(p_0)\right]^{-1}. \tag{5.16} \]

Then for the function $u_{jl}$ we obtain the following equation:

\[ u_{jl}(p)=\frac{g^2}{32\pi^3}R(p,p_0)+\frac{g^2}{8\pi^2}\int\frac{p'^2dp'}{W-E'-\omega'}\,jlR(p,p')u_{jl}(p'). \tag{5.17} \]

The function $u_{jl}(p_0)$ determines the value of the phase shift for scattering of a $\pi$-meson by a nucleon by means of the following formula:

\[ \delta_{jl}=-\operatorname{arc\,tg}\frac{4\pi^2p_0\omega_0E_0}{E_0+\omega_0}u_{jl}(p_0). \tag{5.18} \]

In paper $^{6}$ equation (5.17) was solved numerically for states with isotopic spin $I=3/2$ and $j=1/2,\ l=0$ (${}^{3/2}S_{1/2}$ state) and $j=3/2,\ l=1$ (${}^{3/2}P_{3/2}$ state). However, before presenting the results of the numerical solution, we shall make several remarks concerning equation (5.17). A nonrelativistic consideration of equation (5.17) leads to the conclusion that for $I=3/2$ in the ${}^{3/2}S_{1/2}$ and ${}^{3/2}P_{1/2}$ states the effective potential of the interaction between the nucleon and the meson corresponds to repulsion of the particles. Conversely, in the ${}^{3/2}P_{3/2}$ state the effective potential corresponds to attractive forces, which agrees with the results mentioned at the beginning of the present paragraph. We also note that the first term on the right-hand side of equation (5.17) is the Born approximation to the function $u_{jl}(p)$. Therefore we denote

\[ u^{\mathrm{B}}_{jl}(p)=\frac{g^2}{32\pi^3}\,jlR(p,p_0). \tag{5.19} \]

In states with an effective attractive potential $R$ is negative; conversely, in the case of repulsion $R$ is positive. Further, the denominator of the integrand in equation (5.17) is negative for most values of $p'$. All this leads to the fact that for states with repulsion the solution of equation (5.17) proves to be smaller than the Born approximation, and, conversely, for states with attraction it is larger than the Born approximation.

*) The function $u$ introduced in this way is connected with the function $f$ of paper $^{6}$ by the following relation:

\[ u=4\pi\frac{p_0\omega_0E_0}{E_0+\omega_0}\sqrt{\frac{E+M}{E_0+M}}\,f. \]

We indicate, finally, how one can determine the asymptotic behavior of the solutions of equation (5.17). For this purpose, as an example, let us consider, following\({}^{84}\), the asymptotic behavior of the solution

Fig. 3. The wave function\({}^{6}\) and its Born approximation for \(S_{1/2}\), \(I=3/2\), at
\[ \frac{g^2}{4\pi}=10. \]

Fig. 4. The wave function\({}^{6}\) and its Born approximation for \(P_{3/2}\), \(I=3/2\), at
\[ \frac{g^2}{4\pi}=4.6\pi \]
and \(P_0=0.22\,M\).

of equation (5.17) for the state \({}^{3/2}P_{3/2}\). In the region \(p\gg M\), equation (5.17) can be approximately represented in the following form:

\[ u_{ass}(p)=\frac{C}{p^{5/2}}+\frac{g^2}{32\pi^2}\int_q^p dp'\,\frac{p'^{3/2}}{p^{5/2}}\,u_{ass}(p')+\frac{g^2}{32\pi^2}\int_p^\infty \frac{dp'}{p'^{3/2}}\,p'^{1/2}u_{ass}(p'), \tag{5.20} \]

where \(p\gg q\gg M\), \(C\) is a constant arising from the Born term and from the integral over \(p'\) from zero to \(q\). Equation (5.20) can easily be reduced to the differential equation

\[ \left(\frac{1}{p^2}(p^{5/2}u_{aSS})'\right)'= -\frac{3g^2}{32\pi^3}\frac{u_{aSS}}{p^{3/2}}. \tag{5.21} \]

It is easy to see that

\[ u_{aSS}\sim p^n, \]

where

\[ n=-1\pm\sqrt{\frac{9}{4}-\frac{3g^2}{32\pi^2}}. \tag{5.22} \]

The solution corresponding to the \(+\) sign in (5.22) must be discarded, since it corresponds to a singular solution. Thus,

\[ u_{aSS}\sim p^{-1-\frac{3}{2}\sqrt{1-\frac{g^2}{24\pi^2}}}. \tag{5.23} \]

Fig. 5. Dependence of the phase shift for the \(S_{1/2}\)-wave on the momentum \(P_0\) in the center-of-mass system; \(\dfrac{g^2}{4\pi}=15\).

A similar investigation of the asymptotic behavior of the wave function can also be carried out for other states.

As a result of numerical calculations in Ref. 6, wave functions were obtained in the \({}^{3/2}S_{1/2}\)- and \({}^{3/2}P_{3/2}\)-states, as well as the corresponding phase shifts as functions of the energy (see Figs. 3–6). In the remaining states, estimates give small phase shifts, not exceeding a few degrees. From Fig. 3 it is seen that the exact wave function of the \({}^{1/2}S_{1/2}\)-state is smaller than the wave function of the Born approximation. This, as was said above, is connected with the presence of an effective repulsive potential in the \({}^{3/2}S_{1/2}\)-state. Conversely (Fig. 4), the exact wave function of the \({}^{1/2}P_{3/2}\)-state turns out to be larger than that obtained in the Born approximation, which corresponds to attractive forces.

Fig. 6. Dependence of the phase shift for the \(P_{3/2}\)-wave on the kinetic energy of the meson in the laboratory system.

Comparison with the experimental data for the \({}^{3}/_{2}P_{3/2}\)-state, as is seen from Fig. 6, shows that the theory gives a quite satisfactory energy behavior of the phase shift. The best agreement is obtained for \(g^{2}/4\pi=16\). The situation is more complicated with the \({}^{3}/_{2}S_{1/2}\)-state. The experimental dependence of the corresponding phase \((\delta_3)\) on the energy can be described by the following formula:

\[ \delta_3 = 11^\circ - 130^\circ \left(\frac{p_0}{M}\right). \tag{5.24} \]

The theory for \(g^{2}/4\pi=13\) gives [cf. also Fig. 5]

\[ \delta_3 = -160^\circ \left(\frac{p_0}{M}\right). \tag{5.25} \]

Thus, the term in formula (5.24) depending linearly on the momentum is close to that obtained from the theory. However, the theory gives no energy-independent term. This difference is due to the fact that the effective range of the forces corresponding to equation (5.25) turns out to be considerably smaller than \(\hbar/\mu c\) and corresponds to a repulsive potential. On the contrary, (5.24) corresponds to attractive forces at large distances \((\sim \hbar/\mu c)\), which gives an energy-independent term, and at small distances to repulsive forces, which lead to an energy dependence. It should be noted that the experimental value of the radius of the repulsive forces is equal to \(4.8\cdot 10^{-14}\,\text{cm}\), while the theoretical value is \(5.9\cdot 10^{-14}\,\text{cm}\). Thus, one may say that the repulsive forces in the \({}^{3}/_{2}S_{1/2}\)-state are correctly described by the theory set forth above; on the other hand, in order to obtain the attractive forces in this state, a more detailed consideration is apparently necessary \(^{86-87}\).

A study of the question of the behavior of the \(S\)-phases of \(\pi\)-meson scattering using the T. D. method is the subject of work \(^{88}\), in which the nonrelativistic approximation was considered. For two variants of truncation (for \(p=M\) and for \(p=2M\)) the \(S\)-phases were calculated. The phase shift for the \({}^{1}/_{2}S_{1/2}\)-state at small energies changes sign.

However, in this work no renormalizations were carried out, and therefore the results for the state with isotopic spin \(1/2\) may be substantially changed by renormalization. Similar calculations were carried out in works \(^{89-90}\).

The T. D. method as applied to meson scattering in the symmetric pseudoscalar theory with pseudovector coupling was used in works \(^{51-65}\). As was said at the beginning of the paragraph, the nonrelativistic approximation of the pseudoscalar and pseudovector coupling gives coinciding expressions for the interaction in \(P\)-states; in this connection we shall not dwell on these works in detail. Let us note,

that a pseudovector coupling, just as was shown above, gives a resonance behavior of the scattering of \(\pi\)-mesons*).

On the whole, the application of the T. D. method to meson scattering allows one to say that the contemporary meson theory does not contradict experiment and, in any case, is still far from exhausted.

§ 6. THE NEW TAMM—DANCОFF METHOD

In order to avoid the difficulties characteristic of the original formulation of the T. D. method (see § 3), Dyson \(^{7-9}\) proposed a certain modification of the T. D. method, called in the literature the “new Tamm—Dancoff method” (N. T. D.). The distinctive feature of the N. T. D. consists in the use of a representation of the vacuum state of the interacting fields. Instead of considering the amplitudes (1.6), as was done in the old T. D. method (O. T. D.), Dyson proposed considering matrix elements of the following form:

\[ a(N,N')=[\Pi(N)\Pi(N')]^{-1/2}\left(\Psi_0^* C(N)A(N')\Psi\right), \tag{6.1} \]

where, in contrast to relation (1.6), instead of \(\Phi_0\) one uses \(\Psi_0\)—the vector of the vacuum state of the interacting fields. Thanks to the use of \(\Psi_0\), formula (6.1) corresponds to matrix elements not only of particle-absorption operators \(A(N')\), as was the case in O. T. D., but also to matrix elements of particle-creation operators. It should be borne in mind that, in calculating the matrix elements (6.1), the particle-absorption operators must be placed to the right of the creation operators. Thus expression (6.1) is a matrix element of an \(N\)-ordered product of field operators \(^{20,21}\). The matrix elements (6.1), following the terminology adopted in the literature, we shall call N. T. D. amplitudes. The meaning of the N. T. D. amplitudes is less evident than that of the amplitudes of the old T. D. method. Therefore, first of all we shall consider the relation between the two kinds of amplitudes.

Substituting the expressions (1.2) for the state vectors into formula (6.1), it is easy to obtain the following relation:

\[ \begin{aligned} a(N,N') &={} \\ &=\sum_M \beta^*(N+M)a(N'+M) \left(\frac{N+M}{M}\right)^{1/2} \left(\frac{N'+M}{M}\right)^{1/2}, \end{aligned} \tag{6.2} \]

where

\[ \binom{N_1}{N}=\Pi(N_1)[\Pi(N)\Pi(N_1-N)]^{-1}. \tag{6.3} \]

*) It is essential that all calculations in the case of a pseudovector coupling are carried out by introducing a cutoff at large momenta.

The relation (6.2) makes it possible to express the N. T. D.-amplitude in terms of the S. T. D.-amplitudes. The inverse relation, as is not difficult to verify by direct substitution*), has the following form:

\[ \beta^*(N_1)\alpha(N_2)= \sum_M (-1)^M a(N_1+M,\; N_2+M)\times \]

\[ \times \binom{N_1+M}{M}^{1/2} \binom{N_2+M}{M}^{1/2}, \tag{6.4} \]

where \((-1)^M\) denotes \((-1)^{\sum M}\), and \(\sum M\) is the sum of the occupation numbers \(M\). With the aid of relation (6.4), knowing all \(a(N,\;N')\) corresponding to a given state, one can determine all the S. T. D.-amplitudes of this state, and also all the S. T. D.-amplitudes of the vacuum of the interacting fields. Thus, knowledge of the N. T. D.-amplitudes makes it possible to describe completely both the state \(\Psi\) and the state \(\Psi_0\).

Dyson obtained the normalization condition for the N. T. D.-amplitudes:

\[ \sum_N \sum_{N'} |a(N,\;N')|^2=\mathrm{const}. \tag{6.5} \]

It should be noted, however, that in deriving formula (6.5) the assumption was used that the sums

\[ \sum_N |\alpha(N)|^2 \quad \text{and} \quad \sum_N |\beta(N)|^2 \]

are bounded.

In modern field theory this is in fact not so. To obtain convergent expressions it is necessary to carry out a renormalization of the amplitudes \(\alpha(N)\) and \(\beta(N)\). The problem of such renormalization has not yet been solved. The normalization condition (6.5) was also obtained under the assumption that the S. T. D.-amplitudes for detecting a number of particles greater than some bounded number (say \(N_0\)) are equal to zero both for the state \(\Psi\) and for \(\Psi_0\). Such an assumption, as was discussed in detail above (see § 3), always holds in S. T. D. Moreover, owing to relation (6.2), the assumption that the amplitudes of the old method vanish,

\[ \alpha(N>N_0)=0,\qquad \beta(N'>N'_0)=0 \]

corresponds to the condition

\[ a(N'>N'_0,\; N>N_0)=0. \]

*) In doing so one should take into account the fact that, for fixed \(M+M'\), the sum

\[ \sum_M (-1)^M \binom{M+M'}{M} \]

is always equal to zero, except in the case \(M+M'=0\).

In complete analogy with the way this was done in the S-matrix, in the new Tamm–Dancoff method it is proposed, for each particular approximation, to calculate the Tamm–Dancoff amplitudes for occupation numbers greater than certain ones as equal to zero. The convergence of the solutions obtained as a result of such successive approximations, just as in the old method, has not yet been investigated (cf. \(^{91}\)).

Let us next consider the general question of the boundary (or, respectively, initial) conditions for the Tamm–Dancoff amplitudes \(^{9,10}\). In doing so, we shall first consider this question for noncovariant equations of motion. Suppose there is a stationary state with energy \(W\). Then the equation of motion for the amplitude \(a(N,N')\) has the following form:

\[ (W+E_N-E_{N'})a(N,N')=[\Pi(N)\Pi(N')]^{-1/2}\times \]
\[ \times(\Psi_0^*[C(N)A(N'),H]\Psi). \tag{6.6} \]

The general form of the solution of such an equation may be represented as follows (see, for example, \(^{26}\)):

\[ a(N,N')=P\,\frac{f(N,N')}{W+E_N-E_{N'}}+c\delta(W+E_N-E_{N'}), \tag{6.7} \]

where \(P\) means that the singularity is to be understood in the sense of the principal value, and \(c\) is an undetermined constant which must be determined from the boundary conditions of the problem. Such a question arises, generally speaking, also for the amplitudes of the old method. However, for S-matrix amplitudes this question is resolved simply. Namely, the presence of \(\delta\)-functions corresponds, for example in the scattering problem, to the presence at infinity of an incident plane wave, and also (in the corresponding combination with the first term on the right-hand side of formula (6.7)) of an outgoing spherical wave. In this case the appearance or nonappearance of \(\delta\)-functions is determined first of all by energy considerations. It is precisely the energy that must be sufficiently large for the question of the appearance of \(\delta\)-functions to arise at all.

In the case of Tamm–Dancoff amplitudes, however, there is a complication connected with the appearance of “minus-particle” amplitudes, i.e. amplitudes for which in formula (6.2) \(N\) is different from zero. In this case energy considerations cannot always forbid the appearance of \(\delta\)-functions. An ambiguity of solutions arises. To remove such an ambiguity, let us turn to formula (6.2), from which it follows that if boundary conditions are specified for \(\alpha\) and \(\beta\), then the boundary conditions for \(a(N,N')\) will thereby be specified. Boundary conditions for \(\alpha\) were discussed above (§ 3). Let us now consider the question of boundary conditions for the vacuum amplitudes. It may be asserted that the appearance of \(\delta\)-functions is impossible for vacuum amplitudes. In support of this one may give the following argument. First of all, from considerations of relativistic invariance, the energy of the vacuum state...

the sum \(W_0\) must be equal to zero. Owing to this, no ambiguity arises in solving the system of equations for the amplitudes \(\beta(N)\).

It follows from what has been said that, according to formula (6.2), \(\delta\)-functions of the form \(\delta(W+E_N-E_{N'})\) cannot arise unless \(N\) is equal to zero. If, however, \(N=0\), then \(\delta\)-functions may arise, and their appearance or nonappearance in this case is determined by the usual boundary conditions.

Let us now turn to the covariant formulation of the new method of T. D. In order to avoid ambiguity in isolating divergent quantities, we shall consider the equation in the coordinate representation, while the matrix elements will be constructed from operators in the interaction representation by means of wave functionals of the same representation (cf. § 2).

Thus, below we shall consider matrix elements of \(N\)-ordered products of operators of the following form (cf. also \(^{92}\)):

\[ \begin{aligned} &\bigl(\Psi_0^*(\sigma)N\{\bar\psi_{\nu_1}(x_1)\ldots \bar\psi_{\nu_n}(x_n) \psi_{\mu_1}(y_1)\ldots \psi_{\mu_m}(y_m) \\ &\qquad\qquad\qquad \times \varphi_{a_1}(z_1)\ldots \varphi_{a_r}(z_r)\}\Psi(\sigma)\bigr)\equiv \\ &\equiv \left\langle \bar\psi_{\nu_1}(x_1)\ldots \bar\psi_{\nu_n}(x_n) \psi_{\mu_1}(y_1)\ldots \psi_{\mu_m}(y_m)\times \varphi_{a_1}(z_1)\ldots \varphi_{a_r}(z_r)\right\rangle_a . \end{aligned} \tag{6.8} \]

Since the wave functionals \(\Psi(\sigma)\) and \(\Psi_0(\sigma)\) satisfy equation (2.1), for the matrix element (6.8) the following equation of motion arises:

\[ i\frac{\delta}{\delta\sigma(\xi)} \bigl(\Psi_0^*(\sigma)N\{\ \}\Psi(\sigma)\bigr) = \bigl(\Psi_0^*(\sigma)[N\{\ \},H(\xi)]\Psi(\sigma)\bigr). \tag{6.9} \]

As a result of the \(N\)-ordering of the commutator appearing on the right-hand side of equation (6.9), one obtains, generally speaking, matrix elements different from the H. T. D.-amplitude standing on the left-hand side of equation (6.9). The complete system of equations for the amplitudes thus connected with one another turns out to be infinite. Below we shall consider such a system of equations in complete analogy with how this was done in the preceding paragraph when considering the system of equations for the S. T. D.-amplitudes, i.e. in each particular approximation only a few amplitudes will be regarded as different from zero, the equations for which must be solved exactly.

Let us finally point out the difference between equation (6.9) and the corresponding equation (2.10), obtained in the old T. D. method. Owing to the presence of the commutator in equation (6.9), the number of operators in the \(N\)-products on the right-hand side may exceed the number of operators in the \(N\)-product on the left-hand side by only one (for \(H\sim \bar\psi O\psi\varphi\)).

This leads to the fact that diagrams corresponding to vacuum closed loops of the type shown in Fig. 7, and similar to them, do not arise[^7]. Indeed, in order to obtain such loops it is necessary for three particles to arise: a field quantum and a pair. In the case of equation (2.10) of the old T.–D. method, however, the simultaneous appearance of three particles was possible, which there led to the appearance of vacuum divergences. The absence of such divergences in the new T.–D. is an essential merit of this method. In accordance with what was stated earlier for noncovariant equations of motion, one can formulate boundary conditions for equation (6.9). It is convenient to do this by passing from (6.9) to an integral equation taking the boundary conditions into account. Such an equation has the following form:

Fig. 7.

\[ i\left(\Psi_0^*(\sigma_x)N\{\,\}\Psi(\sigma_x)\right) = \int_{-\infty}^{+\infty} d\xi\,\varepsilon(x-\xi)\times \]
\[ \times\left(\Psi_0^*(\sigma_\xi)[N\{\,\},\,H(\xi)]\Psi(\sigma_\xi)\right) + a(x_1,\ldots,z_r), \tag{6.10} \]

where the integration is carried out over the whole space-time, \(\varepsilon(x)=\pm \frac12\) \((x_0 \gtreqless 0)\). Here the integral should be understood in such a way that, for \(t=\pm\infty\), the integrand tends sufficiently rapidly to zero.

In the momentum representation, the integral from \(-\infty\) to \(+\infty\) and \(\varepsilon(x-\xi)\) lead to the appearance of a singular denominator, more precisely, to the appearance of the principal value of such a singularity. The function \(a(x_1,\ldots,z_r)\) corresponds to \(\delta\)-functions of the momentum representation. According to what was stated earlier, for the case of minus-particles \(a(x_1,\ldots,z_r)\) must be omitted. For the case of plus-particles, however, the appearance of \(a(x_1,\ldots,z_r)\) is determined by energy considerations and by the concrete conditions of each problem1.

In conclusion of the present paragraph let us dwell on the connection between the amplitudes (6.8) of the coordinate representation and the time-independent amplitudes of the momentum representation. For simplicity let us consider the case \(t=\mathrm{const}\). Owing to the fact that the time dependence of the state vectors is determined by the formulas (cf. formula (2.16)):

\[ \Psi(s)=e^{i(H_0-\mathcal E)t}\Psi \quad\text{and}\quad \Psi_0(s)=e^{i(H_0-\mathcal E_0)t}\Psi_0, \tag{6.11} \]

where \(\mathcal E\) and \(\mathcal E_0\) are, respectively, the energies of the state under consideration

and the energy of the physical vacuum, one can represent, for example, the amplitude of one nucleon in the following form:

\[ \langle \psi(x_1)\rangle_t = \frac{1}{L^{3/2}}\sum_n\sum_{\mathbf p} u_\alpha^n(\mathbf p)e^{i\mathbf p x_1+i(p_0-W)t} \langle B_n(\mathbf p)\rangle . \tag{6.12} \]

Analogously, for the amplitude of one meson one may write the following expansion:

\[ \langle \varphi_s(x_1)\rangle_t = \frac{1}{L^{3/2}}\sum_{\mathbf k} \frac{1}{\sqrt{2\omega_k}} e^{i\mathbf k x_1+i(\omega_k-W)t} \langle Q_s(\mathbf k)\rangle . \tag{6.13} \]

In the formulas given, \(W=\mathcal E-\mathcal E_0\) and is equal to the observed value of the energy of the system.

The amplitudes \(\langle B\rangle\) and \(\langle Q\rangle\) introduced in formulas (6.12), (6.13) do not depend on time and are determined by the formulas:

\[ \langle B_n(\mathbf p)\rangle \equiv (\Psi_0^{*}B_n(\mathbf p)\Psi), \qquad \langle Q(\mathbf k)\rangle \equiv (\Psi_0^{*}Q(\mathbf k)\Psi). \tag{6.14} \]

For more complicated amplitudes one may likewise introduce time-independent amplitudes by formulas analogous to (6.12)—(6.14).

Finally, for time-independent amplitudes, equation (6.7) leads, as is readily seen from the form of the functions \(\Psi(\tau)\) and \(\Psi_0(\sigma)\), determined by formula (6.11), to the equation of motion (6.6).

In the following two sections we shall consider concrete problems of the interaction of two nucleons and of a nucleon and a meson in the symmetric pseudoscalar theory. These problems will demonstrate the merits of the new Tamm—Dancoff method, as well as the difficulties standing in the way of applying the method, the solution of which has not yet been found.

§ 7. INTERACTION OF TWO NUCLEONS ACCORDING TO THE NEW TAMM—DANCOFF METHOD

In the present section we shall consider the problem of the interaction of two nucleons according to the new Tamm—Dancoff method. In the approximation adopted below, the equations obtained can be completely renormalized, which is an essential advantage of the new method. In addition, taking account of “minus-particles” makes it possible simply to obtain, in fact, a higher approximation with respect to the number of particles than was the case in the old method.

For the problem of the interaction of two nucleons we shall take the following amplitudes as different from zero:

\[ \langle \psi(x_1)\psi(x_2)\rangle_s, \qquad \langle \psi(x_1)\psi(x_2)\varphi_a(x_3)\rangle_\sigma . \tag{7.1} \]

All other amplitudes in this approximation we shall neglect.

Then, according to equation (6.9), we obtain:

\[ i\frac{\delta}{\delta j(x)} \left\langle \psi_\rho(x_1)\psi_\lambda(x_2)\right\rangle_{\sigma} = g\bigl(S(x_2-x)\gamma_5\tau_a\bigr)_{\lambda\mu} \left\langle \varphi_a(x)\psi_\rho(x_1)\psi_\mu(x)\right\rangle_{\sigma} + g\bigl(S(x_1-x)\gamma_5\tau_a\bigr)_{\rho\mu} \left\langle \varphi_a(x)\psi_\mu(x)\psi_\lambda(x_2)\right\rangle_{\sigma}, \tag{7.2} \]

\[ i\frac{\delta}{\delta j(x')} \left\langle \varphi_a(x)\psi_\rho(x_1)\psi_\mu(x)\right\rangle_{\sigma} = -ig\bigl(ZS(x-x')\Delta(x'-x)\gamma_5\tau_a\bigr)_{\mu\chi} \left\langle \psi_\rho(x_1)\psi_\chi(x')\right\rangle_{\sigma} + ig\bigl(ZS(x_1-x')\Delta(x'-x)\gamma_5\tau_a\bigr)_{\rho\chi} \left\langle \psi_\mu(x)\psi_\chi(x')\right\rangle_{\sigma}. \tag{7.3} \]

The sign \(Z\) has the following meaning:

\[ \bigl(ZS(x')\Delta(x'')\bigr) = S^{(+)}(x')\Delta^{(-)}(x'') - S^{(-)}(x')\Delta^{(+)}(x''). \tag{7.4} \]

In obtaining equation (7.2) no approximations were made, whereas in equation (7.3) matrix elements of four nucleon-field operators of the form

\[ \left\langle \psi_\rho(x_1)\psi_\mu(x)\bar{\psi}_\nu(x')\psi_\chi(x')\right\rangle_{\sigma} \]

were neglected.

The resulting system of equations should, according to the Tamm—Dancoff method, be solved exactly. Above, in discussing the C.T.D., it was said that there exists an approximate way of eliminating all amplitudes except one\(^{22-25}\) (the so-called Lévy—Klein method). The comparative simplicity of such an approach led to the appearance of a number of works in which nuclear-force potentials are constructed by means of the new Tamm—Dancoff method\(^{93-95}\). However, because of the poor convergence of the resulting series, the validity of such a method is doubtful\(^{23}\).

We shall restrict ourselves to energies at which meson bremsstrahlung by the colliding nucleons is impossible. In this case, when integrating equation (7.3), according to relation (6.10), one may, as was stated above, omit the term corresponding to \(a(x_1,\ldots)\). Having integrated equation (7.3) and substituted the amplitude of two nucleons and a meson, determined in this way, into equation (7.2), we obtain the following equation for the amplitude of two nucleons:

\[ i\frac{\delta}{\delta\sigma(x)} \left\langle \psi_\rho(x_1)\psi_\lambda(x_2)\right\rangle_{\sigma} = -3g^2\int dx'\,\varepsilon(x-x')\times \]

\[ \times \left\{ \bigl(S(x_2-x)\gamma_5[ZS(x-x')\Delta(x'-x)]\gamma_5\bigr)_{\lambda\chi} \left\langle \psi_\rho(x_1)\psi_\chi(x')\right\rangle_{\sigma'} + \right. \]

\[ \left. + \bigl(S(x_1-x)\gamma_5[ZS(x-x')\Delta(x'-x)]\gamma_5\bigr)_{\rho\chi} \left\langle \psi_\chi(x')\psi_\lambda(x_2)\right\rangle_{\sigma'} \right\} + \]

\[ + g^2\int dx'\,\varepsilon(x-x') \left\{ -\bigl(S(x_1-x)\gamma_5\tau_a\bigr)_{\rho\mu} \bigl([ZS(x_2-x')\Delta(x'-x)]\gamma_5\tau_a\bigr)_{\lambda\chi} \right. \]

\[ \left. + \bigl(S(x_2-x)\gamma_5\tau_a\bigr)_{\lambda\mu} \bigl([ZS(x_1-x')\Delta(x'-x)]\gamma_5\tau_a\bigr)_{\rho\chi} \right\} \left\langle \psi_\mu(x)\psi_\chi(x')\right\rangle_{\sigma'} . \tag{7.5} \]

Two terms in the first integral on the right-hand side of (7.5) correspond to the diagrams of the nucleon self-energy (Fig. 8, \(a, b\)). The second integral corresponds to the scattering diagrams (Fig. 8, \(c, d\)).

The expressions corresponding to diagrams \(a\) and \(b\) are infinite and subject to renormalization. Therefore, first of all, we shall consider these expressions.

Fig. 8.

First of all, with the aid of formula (2.7) one can obtain the following relation:

\[ \begin{aligned} M(x-x') &\equiv -\,\varepsilon(x-x')\gamma_{5} \left[ZS(x-x')\Delta(x'-x)\right]\gamma_{5} \\ &= -\frac{1}{8}\left\{ -\gamma_{5}S_{F}(x-x')\gamma_{5}\Delta_{F}(x-x') +\left[\gamma_{4}\gamma_{5}S_{F}(x'-x)\gamma_{5}\gamma_{4} \times \Delta_{F}(x'-x)\right]^{*} \right\}, \end{aligned} \tag{7.6} \]

where the sign \(*\) denotes Hermitian conjugation. Thus, relation (7.6) makes it possible to express \(M(x)\) in terms of singular functions that usually occur in perturbation theory. To separate out the divergent expressions from \(M(x)\), one may therefore use the methods developed in perturbation theory. Namely,

\[ M(x)=A_{N}\delta(x)+B_{N}\left(\gamma_{\mu}\frac{\partial}{\partial x_{\mu}}+M\right)\delta(x)+M_{c}(x). \tag{7.7} \]

Here \(A_{N}\) and \(B_{N}\) are undetermined (infinite) constants, while \(M_{c}(x)\) no longer contains divergent quantities and has the following form:

\[ M_{c}(x)=\frac{1}{(2\pi)^{4}}\int e^{ipx}M_{c}(p)\,dp, \tag{7.8} \]

where

\[ M_{c}(p)=(i\hat{p}+M)A(p^{2})+B(p^{2}), \tag{7.9} \]

\[ A(p^{2})=-\frac{i}{16\pi^{2}}\int_{0}^{1}du\,(1-u) \left\{ \frac{2M^{2}u^{2}}{\mu^{2}(1-u)+M^{2}u^{2}} + \ln\left| \frac{\mu^{2}(1-u)+M^{2}u^{2}} {\mu^{2}(1-u)+p^{2}u(1-u)+uM^{2}} \right| \right\}, \tag{7.10} \]

\[ B(p^{2})=-\frac{iM}{16\pi^{2}}\int_{0}^{1}du\,u\, \ln\left| \frac{\mu^{2}(1-u)+M^{2}u^{2}} {\mu^{2}(1-u)+p^{2}(u-u^{2})+uM^{2}} \right|. \tag{7.11} \]

Relations (7.6)—(7.7) make it possible to write the first integral of the right-hand side of equation (7.5) in the following form:

\[ \begin{gathered} \left\{3g^2 A_N S_{\lambda x}(x_2-x)+3g^2 B_N \left(S(x_2-x)\left(\gamma_\mu\frac{\partial}{\partial x_\mu}+M\right)\right)_{\lambda x}\right\}\times\\ \times \left\langle \psi_\rho(x_1)\psi_x(x)\right\rangle_\sigma +\left\{3g^2 A_N S_{\rho x}(x_1-x)+\right.\\ \left. +3g^2 B_N\left(S(x_1-x)\left(\gamma_\mu\frac{\partial}{\partial x_\mu}+M\right)\right)_{\rho x}\right\} \left\langle \psi_x(x)\psi_\lambda(x_2)\right\rangle_\sigma+\\ +3g^2\int dx'\left\{(S(x_2-x)M_c(x-x'))_{\lambda x} \left\langle \psi_\rho(x_1)\psi_x(x')\right\rangle_{\sigma'}+\right.\\ \left. +(S(x_1-x)M_c(x-x'))_{\rho x} \left\langle \psi_x(x')\psi_\lambda(x_2)\right\rangle_{\sigma'}\right\}. \end{gathered} \tag{7.12} \]

The operator \(\gamma_\mu\dfrac{\partial}{\partial x_\mu}\) acts not only on \(\psi_x(x)\), but must also be applied to \(\sigma\). In the particular case of \(\sigma\), corresponding to \(t=\mathrm{const}\), the application of \(\gamma_\mu\dfrac{\partial}{\partial x_\mu}\) to \(\sigma\) corresponds to \(\gamma_4\dfrac{\partial}{\partial x_4}\).

The divergent terms in (7.12), containing \(A_N\), can be eliminated by means of a renormalization of the nucleon mass. Indeed, if the expression \(\delta M\bar{\psi}\psi\) is added to the Hamiltonian (2.3), then, according to formula (6.9), in equation (7.2) and, consequently, in equation (7.5), expressions arise that are similar to the terms in (7.12) and proportional to \(A_N\). In this case, putting \(\delta M=i3g^2 A_N\), one can eliminate the indefinite constant \(A_N\). Next, the elimination of \(B_N\) is carried out by means of charge renormalization. First of all, let us note that, since the interaction representation is used for the operator \(\psi_x(x)\), then (for simplicity we take \(\sigma\) corresponding to \(t=\mathrm{const}\))

\[ \left(\gamma_\mu\frac{\partial}{\partial x_\mu}+M\right) \left\langle \psi_\rho(x_1)\psi_x(x)\right\rangle_t =-i\gamma_4\frac{\partial}{\partial t} \left\langle \psi_\rho(x_1)\psi_x(x)\right\rangle_t . \tag{7.13} \]

On the right-hand side the differentiation operator is applied only to the argument of the state vectors and is not applied to the argument \(\psi(x)\). If we now integrate both sides of equation (7.5) over the three-dimensional space \(\mathbf{x}\), then on the left-hand side there arises a derivative with respect to time, while on the right-hand side the terms containing \(B_N\) lead to the following:

\[ 3g^2B_N i\frac{\partial}{\partial t}\int d\mathbf{x}\, \left(\Psi_0^*(t)N\{\psi_\rho(x_1)(S(x_2-x)\gamma_4\psi(x))_\lambda+\right. \]

\[ \left. +(S(x_1-x)\gamma_4\psi(x))_\rho\psi_\lambda(x_2)\}\Psi(t)\right). \tag{7.14} \]

In expression (7.14) the operator \(\dfrac{\partial}{\partial t}\) is applied in the same way as in for-

module (7.13), only to the arguments of the state vectors. Taking into account the relation

\[ \psi(x)=\int S(x-x')\gamma_4\psi(x')\,dx', \tag{7.15} \]

we obtain, instead of (7.14), the following expression:

\[ 6g^2B_N i\,\frac{\partial}{\partial t}\left<\psi_p(x_1)\psi_\lambda(x_2)\right>_t . \tag{7.16} \]

Therefore one can eliminate the indefinite constant \(B_N\) by introducing the renormalized charge:

\[ g'^2=\frac{g^2}{1+6g^2B_N}. \tag{7.17} \]

Thus, after renormalization of the nucleon mass and after renormalization of the coupling constant, the equation for the amplitude of two nucleons takes the form:

\[ \begin{aligned} i\,\frac{\delta}{\delta\sigma(x)} \left<\psi_p(x_1)\psi_\lambda(x_2)\right>_\sigma ={}& 3g'^2 \int dx'\, \bigl(S(x_2-x)M_c(x-x')\bigr)_{\lambda\chi} \left<\psi_p(x_1)\psi_\chi(x')\right>_{\sigma'} \\ &+3g'^2 \int dx'\, \bigl(S(x_1-x)M_c(x-x')\bigr)_{p\chi} \left<\psi_\chi(x')\psi_\lambda(x_2)\right>_{\sigma'} \\ &+g'^2 \int dx'\,\varepsilon(x-x')\Bigl\{ -\bigl(S(x_1-x)\gamma_5\tau_\alpha\bigr)_{p\mu} \bigl([ZS(x_2-x')\Delta(x'-x)]\gamma_5\tau_\alpha\bigr)_{\lambda\chi} \\ &\qquad\qquad\qquad +\bigl(S(x_2-x)\gamma_5\tau_\alpha\bigr)_{\lambda\mu} \bigl([ZS(x_1-x')\Delta(x'-x)]\gamma_5\tau_\alpha\bigr)_{p\chi} \Bigr\} \left<\psi_\mu(x)\psi_\chi(x')\right>_\sigma . \end{aligned} \tag{7.18} \]

This equation no longer contains divergent quantities and can be used for the analysis of the interaction of two nucleons. We note that, in fact, (7.18) is a system of four equations for four amplitudes

\[ \left<\psi_p^{\varepsilon_1}(x_1)\psi_\lambda^{\varepsilon_2}(x_2)\right>_\sigma, \]

where \(\varepsilon_1\) and \(\varepsilon_2\) are \(+\) or \(-\). In practice it is not possible to eliminate three amplitudes and obtain a single equation. This elimination can be carried out using perturbation theory. It is precisely perturbation theory, i.e. successive iterations leading to the interaction operator in the form of a series in powers of the coupling constant, that is used in the Lévy–Klein method. In the zero approximation one may then neglect the amplitudes for minus-particles and the equations for

them. Then instead of (7.18) we obtain:

\[ i\frac{\delta}{\delta\sigma(x)} \left\langle \psi_\rho^{(+)}(x_1)\psi_\lambda^{(+)}(x_2)\right\rangle_\sigma = \]

\[ =3g^{\prime 2}\int dx'\,(S^{(+)}(x_2-x)M_c(x-x'))_{\lambda\chi} \left\langle \psi_\rho^{(+)}(x_1)\psi_\chi^{(+)}(x')\right\rangle_{\sigma'} + \]

\[ +3g^{\prime 2}\int dx'\,(S^{(+)}(x_1-x)M_c(x-x'))_{\rho\chi} \left\langle \psi_\chi^{(+)}(x')\psi_\lambda^{(+)}(x_2)\right\rangle_{\sigma'} + \]

\[ +g^{\prime 2}\int dx'\,\varepsilon(x-x') \{(S^{(+)}(x_1-x)\gamma_5\tau_a)_{\rho\mu} \times \]

\[ \times(S^{(+)}(x_2-x')\Delta^{(+)}(x-x')\gamma_5\tau_a)_{\lambda\chi} - (S^{(+)}(x_2-x)\gamma_5\tau_a)_{\lambda\mu} \times \]

\[ \times(S^{(+)}(x_1-x')\Delta^{(+)}(x-x')\gamma_5\tau_a)_{\rho\chi} \left\langle \psi_\mu^{(+)}(x)\psi_\chi^{(+)}(x')\right\rangle_{\sigma'}\}. \tag{7.19} \]

The last integral in equation (7.19) coincides exactly with the interaction operator of two nucleons obtained in the equation for two particles according to the old T. D. method. The only difference is that in equation (7.19) the corresponding term is preceded by the renormalized constant, whereas in equation (3.5) it was the unrenormalized one. Another difference between equation (7.19) and (3.5) consists in the different self-energy terms. Namely, in the old method the self-energy terms were infinite and were not renormalized. In the equation of the new method the renormalization has been carried out, and in (7.13) there are only finite expressions.

§ 8. INTERACTION OF THE π-MESON AND THE NUCLEON ACCORDING TO THE NEW TAMM—DANKOFF METHOD

The new T. D. method makes it possible, in principle, to advance somewhat further in the study of the scattering of π-mesons by nucleons than was achieved in the old method. It should be noted that, as will be shown below, despite a certain success in the treatment of self-energy terms, the new T. D. method still does not give a complete solution of the arising renormalization problems. What is valuable, however, is that the new method does make it possible to consider a higher approximation and thus to assess the correctness of the results obtained by the old method as applied to meson scattering.

In the first approximation of the new T. D. method for the problem of π-meson scattering by a nucleon we shall set equal to zero all amplitudes except

\[ \left\langle \psi(x)\right\rangle_\sigma,\quad \left\langle \psi(x_1)\varphi_a(x_2)\right\rangle_\sigma,\quad \left\langle \psi(x_1)\varphi_a(x_2)\varphi_\beta(x_3)\right\rangle_\sigma, \]

\[ \left\langle \bar{\psi}(x_1)\psi(x_2)\psi(x_3)\right\rangle_\sigma. \tag{8.1} \]

Thanks to the inclusion of minus-particles, this approximation covers considerably larger amplitudes than the first approximation of T. D. D.

Then, with the aid of equation (6.9), it is easy to obtain the following system of equations:

\[ i\,\frac{\delta}{\delta\sigma(x')}\left<\psi(x)\right>_{\sigma} = gS(x-x')\gamma_{5}\tau_{\alpha}\left<\psi(x')\varphi_{\alpha}(x')\right>_{\sigma}, \tag{8.2} \]

\[ \begin{aligned} i\,\frac{\delta}{\delta\sigma(x)} \left<\psi_{\rho}(x_{1})\varphi_{\alpha}(x_{2})\right>_{\sigma} &= -ig\bigl(ZS(x_{1}-x)\Delta(x-x_{2})\bigr) \\ &\quad\times (\gamma_{5}\tau_{\alpha})_{\rho\delta} \left<\psi_{\delta}(x)\right>_{\sigma} + g\bigl(S(x_{1}-x)\gamma_{5}\tau_{\beta}\bigr)_{\rho\delta} \left<\psi_{\delta}(x)\varphi_{\beta}(x)\varphi_{\alpha}(x_{2})\right>_{\sigma} \\ &\quad - g\Delta(x_{2}-x)(\gamma_{5}\tau_{\alpha})_{\lambda\delta} \left<\bar{\psi}_{\varepsilon}(x)\psi_{\delta}(x)\psi_{\rho}(x_{1})\right>_{\sigma}, \end{aligned} \tag{8.3} \]

\[ \begin{aligned} i\,\frac{\delta}{\delta\sigma(x')} \left<\psi(x)\varphi_{\beta}(x)\varphi_{\alpha}(x_{2})\right>_{\sigma} &= -ig\,[ZS(x-x')\Delta(x'-x_{2})] \\ &\quad\times \gamma_{5}\tau_{\alpha} \left<\psi(x')\varphi_{\beta}(x)\right>_{\sigma} - ig\,[ZS(x-x')\Delta(x'-x)] \\ &\quad\times \gamma_{5}\tau_{\beta} \left<\psi(x')\varphi_{\alpha}(x_{2})\right>_{\sigma}, \end{aligned} \tag{8.4} \]

\[ \begin{aligned} i\,\frac{\delta}{\delta\sigma(x')} \left<\bar{\psi}_{\varepsilon}(x)\psi_{\delta}(x)\psi_{\rho}(x_{1})\right>_{\sigma} &= \\ &= -ig\,[ZS(x-x')\gamma_{5}\tau_{\alpha}S(x'-x)]_{\delta\varepsilon} \left<\psi_{\rho}(x_{1})\varphi_{\alpha}(x')\right>_{\sigma} \\ &\quad+ ig\,[ZS(x_{1}-x')\gamma_{5}\tau_{\alpha}S(x'-x)]_{\rho\varepsilon} \left<\psi_{\delta}(x)\varphi_{\alpha}(x')\right>_{\sigma}. \end{aligned} \tag{8.5} \]

In the system of equations (8.2)—(8.5), equations (8.2) and (8.3) are exact; on the contrary, equations (8.4) and (8.5) are approximate—they were obtained by neglecting four-particle amplitudes, in other words, matrix elements containing the \(N\)-product of four operators. According to the Tamm—Dancoff method, the approximate system of equations (8.2)—(8.5) should be solved exactly.

To solve the system of equations (8.2)—(8.5) it is necessary to use boundary conditions. Considering the energy region in which the formation of nucleon–antinucleon pairs is impossible and the formation of mesons is impossible (or negligibly small), one can, in equations (8.2), (8.4), and (8.5), get rid of the variational derivative with respect to \(\sigma\), passing, according to formula (6.10), to an integral equation. In doing so, according to what was said above, for all these equations one should omit in formula (6.10) the term corresponding to \(a(x_{1},\ldots,z')\). This makes it possible to express the amplitudes \(\left<\psi(x)\right>_{\sigma}\), \(\left<\psi(x_{1})\varphi_{\alpha}(x_{2})\varphi_{\beta}(x_{3})\right>_{\sigma_i}\) and \(\left<\bar{\psi}(x_{1})\psi(x_{2})\psi(x_{3})\right>_{\sigma}\) through \(\left<\psi(x_{1})\varphi_{\alpha}(x_{2})\right>_{\sigma}\).

Therefore one can obtain one equation for the amplitude
\(\langle \psi(x_1)\varphi_a(x_2)\rangle_\sigma\). This equation has the following form:

\[ i\frac{\delta}{\delta \sigma(x)}\langle \psi(x_1)\varphi_a(x_2)\rangle_\sigma = g^2\int dx'\,\varepsilon(x-x')\times \]

\[ \times\{N_1Y_1+N_2Y_2+Y_H+Y_M\}, \tag{8.6} \]

where

\[ Y_1=S(x_1-x)\gamma_5\bigl(ZS(x-x')\Delta(x_2-x')\bigr)\gamma_5 \langle \psi(x')\varphi_\beta(x)\rangle_{\sigma'} - \]

\[ -\Delta(x_2-x)\bigl(ZS(x_1-x')\gamma_5S(x'-x)\gamma_5\bigr) \langle \psi(x)\varphi_\beta(x')\rangle_{\sigma'}, \tag{8.7} \]

\[ Y_2=\bigl(ZS(x_1-x)\Delta(x_2-x)\bigr)\gamma_5S(x-x')\gamma_5 \langle \psi(x')\varphi_\beta(x')\rangle_{\sigma'}, \tag{8.8} \]

\[ Y_H=3S(x_1-x)\gamma_5\bigl(ZS(x-x')\Delta(x'-x)\bigr)\gamma_5 \langle \psi(x')\varphi_a(x_2)\rangle_{\sigma'}, \tag{8.9} \]

\[ Y_M=-2\Delta(x_2-x)\operatorname{Tr}\,[ZS(x'-x)\gamma_5S(x-x')\gamma_5]\times \]

\[ \times \langle \psi(x_1)\varphi_a(x')\rangle_{\sigma'}. \tag{8.10} \]

The symbol “Tr” denotes the trace of the matrix over spinor indices. \(N_1\) and \(N_2\) are operators in the isotopic-spin space (5.2) with eigenvalues (5.3). \(I\) is the value of the total isotopic spin of the meson + nucleon system. Below we shall consider states of such a system for a definite value of the total isotopic spin. Since in such states the operators \(N_1\) and \(N_2\) are diagonal, we shall nowhere write out the isotopic indices and shall understand by the operators \(N_1\) and \(N_2\) their eigenvalues. The following diagrams (Fig. 9) may be put in correspondence with expressions (8.7)—(8.10).

Fig. 9.

The first two expressions correspond to the scattering of a meson by a nucleon, with \(Y_1\) corresponding to scattering with initial emission of the meson (“chain with emission”), and \(Y_2\) to initial absorption (“chain with absorption”). \(Y_H\) and \(Y_M\) correspond to the self-energy of the nucleon and of the meson. The last expressions are infinite and subject to renormalization. Divergences are also caused by the appearance of

and \(Y_2\) (see below). Any consistent elimination of the divergent expressions in equation (8.6) encounters a number of difficulties. First, the renormalization of the self-energy kernels leads to the appearance of two renormalized charges\(^{11}\); second, the finite additions arising after the separation of infinities from the self-energy terms lead to the appearance of additional poles in the equations\(^{10,12}\); and, finally, third, for states with isotopic spin one half, the kernel of equation (8.6) turns out to possess a singularity leading to the absence of finite solutions, which necessitates an additional renormalization\(^{27–34, 68}\).

Let us begin with the consideration of the kernels \(Y_H\) and \(Y_M\). The kernel \(Y_H\), corresponding to the self-energy operator of the nucleon, coincides with the same operator in the equation for two nucleons. Therefore the renormalization of the divergent expressions arising in the kernel \(Y_H\) is carried out in complete analogy with how this was done in the two-nucleon problem. As a result of such renormalization, in equation (8.6) the following substitution should be made:

\[ g^2 \to g_1^2=\frac{g^2}{1+3g^2B_N}, \tag{8.11} \]

\[ \int dx'\,\varepsilon(x-x')Y_H \to 3\int dx'\,S(x_1-x)M_c(x-x')\times \]

\[ \times \left<\psi(x')\varphi_a(x_2)\right>_{a'}. \tag{8.12} \]

The renormalization of divergences from the polarization operator corresponding to \(Y_M\) was not considered by us above. As it turns out, in the approximation chosen by us such a renormalization encounters certain difficulties. In analogy with how this was done above for the self-energy operator of the nucleon, using formulas (2.7), one can obtain the following relation:

\[ P(x-x')\equiv \varepsilon(x-x')\operatorname{Tr}\{Z\gamma_5S(x'-x)\gamma_5S(x-x')\}= \]

\[ =-\frac{i}{4}\operatorname{Im}\operatorname{Tr}\{S_F(x'-x)\gamma_5S_F(x-x')\gamma_5\}, \tag{8.13} \]

which permits one to express \(P(x)\) through the polarization operator ordinarily arising in the first nonvanishing approximation of perturbation theory. Therefore, to isolate the divergences one may use the results of perturbation theory. Then \(P(x)\) may be represented in the form

\[ P(x)=A_M\delta(x)+B_M(\Box-\mu^2)\delta(x)+P_c(x), \tag{8.14} \]

where \(A_M\) and \(B_M\) are undetermined (infinite) constants. \(P_c(x)\)

already contains no divergent quantities and has the following form:

\[ P_c(x)=-\frac{1}{(2\pi)^4}\int e^{ipx}P_c(p^2)\,dp, \tag{8.15} \]

where

\[ P_c(p^2)=\frac{i}{4\pi^2}\int_0^1 du\left\{[3p^2(u-u^2)+M^2]\ln\left|\frac{M^2+p^2(u-u^2)}{M^2-\mu^2(u-u^2)}\right|-\right. \]

\[ \left. -\frac{(p^2+M^2)(u-u^2)[M^2-3\mu^2(u-u^2)]}{M^2-\mu^2(u-u^2)} \right\}. \tag{8.16} \]

Analogously to the way in which, in the renormalization of the infinities arising from the self-energy integral, \(A_N\) corresponded to the renormalization of the nucleon mass, so for (8.14) \(A_M\) corresponds to the renormalization of the meson mass. Therefore, by adding to the Hamiltonian the expression \(\xi\mu^2\varphi_\alpha\varphi_\alpha\), one can eliminate from equation (8.6) the divergent constant \(A_M\). For this it is necessary to assume that \(\xi\mu^2=ig^2A_M\). The role of the undetermined constant \(B_M\) is more conveniently considered in the momentum representation. According to the general formulas (2.18), using the rest frame of the center of inertia, we introduce the following time-independent amplitudes:

\[ \langle B^n(\mathbf p)Q^\xi(-\mathbf p)\rangle=b^\xi_{\nu\alpha}(\mathbf p) =\left(\frac{\sigma\mathbf p}{p}\right)^{\frac{1-\varepsilon}{2}}_{\nu\psi}a^\xi_{\mu\alpha}(\mathbf p). \tag{8.17} \]

The index \(\varepsilon\) takes the values \(+1\) for \(n=1,2\) (plus-nucleons) and \(-1\) for \(n=3,4\) (minus-antinucleons); \(\xi\) is equal to \(\pm1\), respectively, for plus- or minus-mesons; \(\nu\) characterizes the direction of the mechanical spin of the nucleon (or antinucleon), \(\alpha\) is the isotopic spin index. \(\sigma_{\nu\mu}\) are the Pauli spin matrices. As stated above, we shall consider states with definite isotopic spin, and therefore the index \(\alpha\) will be omitted below.

Equation (8.6) can then be written in the following form:

\[ (W-\varepsilon E-\xi\omega)b^{\varepsilon\xi} =\xi g_1^2B_M\frac{(W-\varepsilon E)^2-\omega^2}{\omega} \,[b^{\varepsilon\xi}+b^{\varepsilon,-\xi}]+g_1^2V_c^{\varepsilon\xi}, \tag{8.18} \]

where \(V_c^{\varepsilon\xi}\) contains finite terms arising from \(Y_H\) and \(Y_M\), as well as the corresponding expressions from \(Y_1\) and \(Y_2\). With the aid of simple transformations (8.18) can be written in the following form:

\[ (W-\varepsilon E-\xi\omega)b^{\varepsilon\xi} =g_{11}^2V_c^{\varepsilon\xi} +\frac{W-\varepsilon E-\xi\omega}{\xi\omega}(g_{11}^2-g_1^2)[V_c^{\varepsilon\xi}+V_c^{\varepsilon,-\xi}], \tag{8.19} \]

where

\[ g_{11}^{2}=\frac{g_{1}^{2}}{1-2g_{1}^{2}B_{M}} =\frac{g^{2}}{1-3g^{2}B_{N}-2g^{2}B_{M}} . \tag{8.20} \]

Thus, one may say that an attempt to eliminate \(B_M\) leads to the appearance of two renormalized coupling constants \(g_1\) and \(g_{11}\). This makes the quantity \(B_M\) observable. Thus, one may say that the approximate equation for the meson and nucleon considered by us is not renormalizable. We note that below, when considering the effect of self-energy additions, we shall neglect the difference between these two coupling constants. Such a neglect can find some justification only insofar as, in the equations considered by us, all terms leading to kernels of order higher than \(g^2\) have been discarded from the very beginning. However, as is apparently already clear to the reader, carrying out a consistent program of renormalizations for the meson–nucleon problem requires special investigation.

We note that in a number of works\(^{12,31,32}\) devoted to the consideration of equations of interacting particles, divergent constants of the type \(A_N, B_N, A_M\), and \(B_M\) are treated much more freely than in the exposition proposed by us. Often, renormalization is understood simply as the separation of such constants and their discarding. Such an approach does not seem to us completely justified. However, the following argument may be adduced in favor of discarding divergent constants. In a renormalizable theory all divergent constants can ultimately be eliminated by renormalizing the coupling constant and the particle masses; therefore all observable quantities of the theory can be obtained as a result of discarding the divergent constants and replacing the coupling constant and the particle masses by their observable values.

The system of equations for the four amplitudes of two particles, a meson and a nucleon, obtained as a result of discarding the divergent constants or as a result of the renormalization carried out above and neglecting the difference between \(g_1^2\) and \(g_{11}^2\), has, in the momentum representation, the following form:

\[ \sum_{\varepsilon'\xi'} r_{\varepsilon'\xi'}^{\varepsilon\xi}\, a^{\varepsilon'\xi'}(\mathbf p) = \frac{g^{2}}{32\pi^{3}} \sum_{\varepsilon'\xi'} \int d\mathbf p'\, R_{\varepsilon'\xi'}^{\varepsilon\xi}(\mathbf p,\mathbf p')\, a^{\varepsilon'\xi'}(\mathbf p'), \tag{8.21} \]

\[ \begin{aligned} r_{\varepsilon'\xi'}^{\varepsilon\xi} &= \delta_{\varepsilon\varepsilon'}\delta_{\xi\xi'} \left\{(W-\varepsilon E-\xi\omega)[1+A'(\xi)] -\varepsilon\frac{M}{E}B'(\xi)+\xi C'(\varepsilon)\right\} \\ &\quad +\delta_{\varepsilon\varepsilon'}\delta_{\xi,-\xi'}\,\xi C'(\varepsilon) +\delta_{\varepsilon,-\varepsilon'}\delta_{\xi,\xi'}\,B'(\xi), \end{aligned} \tag{8.22} \]

\[ R_{\varepsilon'\xi'}^{\varepsilon\xi}(\mathbf p,\mathbf p') = \varphi(\mathbf p,\mathbf p') \left\{ N_1 S_{\varepsilon'\xi'}^{\varepsilon\xi}(\mathbf p,\mathbf p') + N_2 T_{\varepsilon'\xi'}^{\varepsilon\xi}(\mathbf p,\mathbf p') \right\}. \tag{8.23} \]

where

\[ \begin{gathered} A'(\xi)=3ig^2 A\,[p^2-(W-\xi\omega)^2],\\ B'(\xi)=-\frac{3ig^2}{M}\,B(p^2-[W-\xi\omega]^2),\\ C'(\varepsilon)=2ig^2 P_c[p^2-(W-\varepsilon E)^2], \end{gathered} \tag{8.24} \]

\[ \varphi(p,p')=\frac12\sqrt{\frac{(E+M)(E'+M)}{EE'\omega\omega'}}, \tag{8.25} \]

\[ \begin{aligned} S_{\varepsilon'\xi'}^{\varepsilon\xi}(p,p')={}& \varepsilon\varepsilon'\xi \left(\frac{p}{E+M}\right)^{\frac{1-\varepsilon}{2}} \left(\frac{p'}{E'+M}\right)^{\frac{1-\varepsilon'}{2}} \{(W-M)m_{\varepsilon\varepsilon'} \\ &+(\varepsilon E+\varepsilon'E'+\xi\omega+\xi'\omega'-M-W)n_{\xi\xi'}\}\\ &+\left(\frac{E+M}{p}\right)^{\frac{1-\varepsilon}{2}} \left(\frac{E'+M}{p'}\right)^{\frac{1-\varepsilon'}{2}} \frac{\boldsymbol{\sigma}\mathbf p}{E+M} \frac{\boldsymbol{\sigma}\mathbf p'}{E'+M}\times\\ &\times[(W+M)m_{\varepsilon\varepsilon'} +(\varepsilon E+\varepsilon'E'+\xi\omega+\xi'\omega'+M-W)n_{\xi\xi'}], \end{aligned} \tag{8.26} \]

\[ \begin{aligned} T_{\varepsilon'\xi'}^{\varepsilon\xi}(p,p')=(\varepsilon+\xi)\Bigg\{& \frac{1}{W+M} \left(\frac{M-E}{p}\right)^{\frac{1-\varepsilon}{2}} \left(\frac{M-E'}{p'}\right)^{\frac{1-\varepsilon'}{2}} \\ &+\frac{1}{W-M} \left(\frac{E+M}{p}\right)^{\frac{1-\varepsilon}{2}} \left(\frac{E'+M}{p'}\right)^{\frac{1-\varepsilon'}{2}} \frac{\boldsymbol{\sigma}\mathbf p}{E+M} \frac{\boldsymbol{\sigma}\mathbf p'}{E'+M} \Bigg\}. \end{aligned} \tag{8.27} \]

\[ \begin{gathered} m_{\varepsilon\varepsilon'}=\{E_q(\varepsilon E_q+\varepsilon E+\varepsilon'E'-W)\}^{-1},\\ n_{\xi\xi'}=\{E_q(\xi E_q+\xi\omega+\xi'\omega'-W)\}^{-1},\\ E_q=\sqrt{(\mathbf p+\mathbf p')^2+M^2}. \end{gathered} \tag{8.28} \]

The matrix \(S_{\varepsilon'\xi'}^{\varepsilon\xi}\) arose in equation (8.21) owing to the inclusion of the finite terms of the proper-energy integrals. Neglecting such terms makes the matrix \(r\) diagonal and equal to

\[ \delta_{\varepsilon\varepsilon'}\delta_{\xi\xi'}(W-\varepsilon E-\xi\omega). \tag{8.29} \]

It should be noted that neglecting the finite proper-energy additions, and also taking account of the amplitude and, correspondingly, of the equation only for plus-particles, leads to the equation of the old T. D. method, solved in work \(^{6}\).

In accordance with what was said in § 6 about the choice of boundary conditions, in the problem of \(\pi\)-meson scattering on a nucleon the asymptotic behavior of the function \(a(+,+)(\mathbf p)\) must correspond to the incident and outgoing waves. Therefore we set

\[ a(+,+)(\mathbf p)=\delta(\mathbf p-\mathbf p_0)+f(+,+)(\mathbf p)\delta_+(E+\omega-W), \tag{8.30} \]

where \(\delta_{+}(x)= i\pi\delta(x)-\dfrac{1}{x}\), and for the remaining functions

\[ a^{\varepsilon\xi}(\mathbf p)= \frac{f^{\varepsilon\xi}(\mathbf p)}{W-\varepsilon E-\xi\omega}. \tag{8.31} \]

For the functions \(f^{\xi}\) we obtain from (8.21) the following system of equations:

\[ \sum_{\varepsilon'\xi'}\Delta_{\varepsilon'\xi'}^{\varepsilon\xi} f^{\varepsilon'\xi'}(\mathbf p) = \frac{g^2}{32\pi^3}R_{++}^{\varepsilon\xi}(\mathbf p,\mathbf p_0) + \frac{i g^2}{32\pi^2} \int d\mathbf p'\, R_{++}^{\varepsilon\xi}(\mathbf p,\mathbf p')\times \]

\[ {}\times f^{(+,+)}(\mathbf p')\delta(E'+\omega'-W) + \]

\[ {}+ \frac{g^2}{32\pi^3} \sum_{\varepsilon'\xi'}\int d\mathbf p'\, \frac{R_{\varepsilon'\xi'}^{\varepsilon\xi}(\mathbf p,\mathbf p')\,f^{\varepsilon'\xi'}(\mathbf p')} {W-\varepsilon'E'-\xi'\omega'}, \tag{8.32} \]

where

\[ \Delta_{\varepsilon'\xi'}^{\varepsilon\xi} = r_{\varepsilon'\xi'}^{\varepsilon\xi}(W-\varepsilon'E-\xi'\omega)^{-1}. \tag{8.33} \]

The angular variables can be separated by using the orthogonal system of polynomials \(L_l^{\pm}\), considered in Ref. \(^{73}\) (see above, § 5). In this case, owing to the introduction, according to formula (8.17), of the functions \(a^{\varepsilon\xi}(\mathbf p)\), for separating the angular variables it is sufficient to expand the functions \(f^{\varepsilon\xi}\) and the kernels \(R_{\varepsilon'\xi'}^{\varepsilon\xi}\) in the polynomials \(L_l^{\pm}\).

Substituting into equation (8.32) the expansion

\[ f^{\varepsilon\xi}(\mathbf p) = \sum_j L_l^{\pm}\left(\frac{\mathbf p}{p},\frac{\mathbf p_j}{p_0}\right) f_{jl}^{\varepsilon\xi}(p) \tag{8.34} \]

and eliminating the angular variables, we obtain a system of equations for the amplitudes \(f_{jl}^{\varepsilon\xi}\), corresponding to the specified value of the total \((j)\) and orbital \((l)\) angular momenta of the meson–nucleon system. The resulting integral equations are already one-dimensional, since the functions \(f_{jl}^{\varepsilon\xi}\) depend only on the modulus \(p\):

\[ \sum_{\varepsilon'\xi'}\Delta_{\varepsilon'\xi'}^{\varepsilon\xi}(p) f_{jl}^{\varepsilon'\xi'}(p) = \frac{g^2}{32\pi^3}\,{}^{jl}R_{++}^{\varepsilon\xi}(p,p_0)\times \]

\[ {}\times \left\{ 1+i\frac{g^2}{2}\, \frac{p_0E_0\omega_0}{E_0+\omega_0}\, f_{jl}^{(+,+)}(p_0) \right\} + \]

\[ {}+ \frac{g^2}{8\pi^2} \sum_{\varepsilon'\xi'} \int \frac{p'^2\,dp'\,{}^{jl}R_{\varepsilon'\xi'}^{\varepsilon\xi}(p,p')} {W-\varepsilon'E'-\xi'\omega'} \,f_{jl}^{\varepsilon'\xi'}(p'). \tag{8.35} \]

The kernels ${}^{\mu}R$ are related to the functions ${}^{\mu}S$ and ${}^{\mu}T$ by formula (8.23), where

\[ \begin{aligned} {}^{\mu}S_{\varepsilon'\xi'}^{\varepsilon\xi} &=\varepsilon\varepsilon'\xi \left(\frac{p}{E+M}\right)^{\frac{1-\varepsilon}{2}} \left(\frac{p'}{E'+M}\right)^{\frac{1-\varepsilon'}{2}} \times \\ &\quad\times \left[\varepsilon(W-M)J_{k_1}(E+\varepsilon\varepsilon' E' - \varepsilon W)+\right. \\ &\quad\left. +\xi(\varepsilon E+\varepsilon' E' + \xi\omega+\xi'\omega' - M - W) J_{k_1}(\omega+\xi\xi'\omega' - \xi W)\right]+ \\ &\quad+ \xi \left(\frac{p}{E+M}\right)^{\frac{1+\varepsilon}{2}} \left(\frac{p'}{E'+M}\right)^{\frac{1+\varepsilon'}{2}} \times \\ &\quad\times \left[\varepsilon(W+M)J_{k_2}(E+\varepsilon\varepsilon' E' - \varepsilon W)+\right. \\ &\quad\left. +\xi(\varepsilon E+\varepsilon' E' + \xi\omega+\xi'\omega' + M - W) J_{k_2}(\omega+\xi\xi'\omega' - \xi W)\right], \end{aligned} \tag{8.36} \]

\[ {}^{\mu}T_{\varepsilon'}^{\varepsilon\xi} = \delta_{j,1/2}\hat{\varepsilon}_{l0}(\varepsilon+\xi) \frac{1}{W+M} \left(\frac{M-E}{p}\right)^{\frac{1-\varepsilon}{2}} \left(\frac{M-E'}{p'}\right)^{\frac{1-\varepsilon'}{2}} + \]

\[ +\delta_{j,1/2}\delta_{l,1}(\varepsilon+\xi) \frac{1}{W-M} \left(\frac{p}{E+M}\right)^{\frac{1+\varepsilon}{2}} \left(\frac{p'}{E'+M}\right)^{\frac{1+\varepsilon'}{2}} . \tag{8.37} \]

In (8.36) the notation has been introduced (cf. § 5)

\[ J_k(z)=\frac{1}{2}\int_{-1}^{+1} P_k(x)\, \frac{dx}{E_q(E_q+z)}, \]

\[ E_q=\sqrt{M^2+p^2+p'^2+2pp'x}, \tag{8.38} \]

where $P_k(x)$ are Legendre polynomials. For the state $S_{1/2}$ ($j=1/2$, $l=0$) one must put $k_1=0$, $k_2=1$; for the state $P_{1/2}$, $k_1=1$, $k_2=0$; for the state $P_{3/2}$, $k_1=1$, $k_2=2$. The kernels ${}^{\mu}T$ differ from zero only for states with total angular momentum one-half, i.e., only for the states $S_{1/2}$ and $P_{1/2}$. This corresponds to the fact that the kernel $T$ arises owing to the “chain with absorption” (see Fig. 8 above). In other words, in the corresponding intermediate state there is only one nucleon, which, naturally, is at rest in the center-of-inertia system and, consequently, has total angular momentum $j=1/2$. This also leads to the fact that the kernel $N_2T$ differs from zero only for states with isotopic spin one-half.

Let us now dwell on the physical meaning of the amplitudes $f(\mathbf p)$. The boundary conditions used by us correspond to the fact that only the amplitude of the nucleon with positive energy

and to a meson with positive frequency there corresponds, at infinity, an incident and an outgoing wave. Owing to this, as is evident from relation (6.4), the incident and outgoing waves turn out to coincide for the amplitudes of the old and the new T.D. method. But since the square of the corresponding amplitude of the old method gives the scattering cross section, the same is given by the amplitude \(f(+, +)(p)\).

Equations (8.35) for the functions \(f_{jl}\) contain imaginary coefficients. However, by means of the transformation

\[ u_{jl}^{\xi}=f_{jl}^{\xi}\left[1+i\,\frac{g^{2}}{2}\,\frac{p_{0}E_{0}\omega_{0}}{E_{0}+\omega_{0}}\,f_{jl}^{(+,+)}(p_{0})\right]^{-1} \tag{8.39} \]

one can pass to equations with real coefficients for the real functions \(u_{jl}^{\xi}\). These equations differ from (8.35) only by the replacement of \(f\) by \(u\) and by the absence of the imaginary term in brackets.

The phase of scattering of \(\pi\)-mesons by nucleons in a state with given \(j, l, I\), as is not difficult to verify, is determined by the formula

\[ \eta_{0jl}=-\operatorname{arctg}\frac{4\pi^{2}p_{0}\omega_{0}E_{0}}{E_{0}+\omega_{0}}\,u_{jl}^{++}(p_{0}). \tag{8.40} \]

The system of integral equations to which the functions \(u_{jl}^{\xi}\) are subject cannot be solved analytically, just as was the case for the equation of the meson and the nucleon in the old T.D. method. However, besides such a purely technical difficulty, the investigation of the equations for the functions \(u\), or, what is the same, of equations (8.35), encounters more serious difficulties.

First of all, let us dwell on the shortcoming, common to both the old and the new T.D. method, of the meson and nucleon equations under consideration. As was said in § 3, the “chain with absorption” leads in the old T.D. method to divergent expressions. The same also takes place in the case of the new method. This is clear if only from the fact that \(j_{i}T_{++}^{++}\) exactly coincides with the corresponding kernel of the equation of the old method. In practice, the renormalization of the divergent expressions which arise here has not yet been carried out. However, such a renormalization can apparently be carried out in the spirit of the ideas set forth in works \(^{27-34}\) devoted to the renormalization of covariant equations.

Another shortcoming of equations (8.35) is connected with the peculiar behavior of the matrix \(\Delta_{\varepsilon'\varepsilon}^{\xi}\). The determinant of this matrix at large values of \(p\), for sufficiently small \(g^{2}\), has the following form:

\[ \Delta \cong \left(1-\frac{3g^{2}}{32\pi^{2}}\ln p\right)^{2}\left(1-\frac{11g^{2}}{32\pi^{2}}\ln p\right)^{2}. \tag{8.41} \]

Thus, \(\Delta\) can vanish. It must be emphasized that the possibility of \(\Delta\) vanishing is retained also for large values of \(g^2\). At the same time, as \(g^2\) increases, the value of \(p\) at which the determinant becomes equal to zero decreases. The same picture also occurs in the approximation that neglects all amplitudes except \(a^{(l+1)12}\).

The vanishing of \(\Delta\) corresponds to the appearance, in the Green’s functions of the particles, of additional poles \(^{13-14}\) which have no direct physical meaning and which indicate at least a restricted domain of applicability of the approximate equations. One may think that the inclusion of higher approximations could substantially change \(\Delta\). However, to clarify this a special investigation is necessary.

Because of the presence of the indicated difficulties, it is expedient—just as was done in the case of the old T.D. method—to restrict oneself initially to considering states in which the operator \(T\) is equal to zero and, moreover, to neglect the finite additions from the self-energy terms. The interest of such an approximation is as follows. As was already said above (see § 1), the T.D. method can be truly assessed only when it becomes clear that, in solving physical problems, it is sufficient to restrict oneself to a small number of virtual particles. Therefore the indicated approximation can show whether, in the problem of scattering of \(\pi\)-mesons by nucleons, it is indeed possible to restrict oneself to the approximation considered in the old method, or whether the result of work \(^{6}\) is to some extent accidental. Numerical calculations in the approximation that neglects the self-energy terms are currently being carried out.

§ 9. CONNECTION OF THE TAMM–DANCOFF METHOD WITH COVARIANT METHODS

Alongside the Tamm–Dancoff method, in recent years much attention has been devoted to another approximate method for investigating the equations of modern field theory, one that does not use an expansion in powers of the coupling constant. This is the method of obtaining approximate covariant equations of the Bethe–Salpeter type \(^{96-98}\). Initially, a covariant equation was obtained for two nucleons \(^{96}\); then the method was extended to a meson–nucleon system \(^{18}\). In addition, various authors obtained an infinite system of covariant equations both for the Green’s functions of particles \(^{99,100,103}\) and for covariant wave functions \(^{101}\), generalizing the Bethe–Salpeter equations \(^{96-98}\). Truncation of such a system gives an approximate system of covariant equations \(^{99,100-103,108}\). The relation between such an approximate system of covariant equations and equations of the Bethe–Salpeter type is analogous to the relation between the Tamm–Dancoff method and the Lévy–Klein method (see § 2).

In this section we shall briefly consider the question of the relation of the Tamm—Dancoff method to covariant methods. The study of this question is of great interest, since both of these methods possess, as it were, mutually complementary advantages. In the first method, unlike the second, the meaning of the wave function is completely clear and solving the system of equations is a simpler problem; however, as we saw in §§ 3, 7, 8, serious difficulties arise here with the renormalization of divergent quantities, whereas in the covariant equations renormalization can be carried out in a general form \(^{104,105}\).

Many authors have dealt with questions concerning the relation of the two methods indicated \(^{22,24,101,106,107}\). However, most of them considered the question of deriving the three-dimensional equations in the Levy—Klein approximation from covariant equations. The problem of deriving the Tamm—Dancoff system of equations from the system of covariant equations in an arbitrary approximation has not yet been solved. Therefore, we shall analyze the question of the relation of these two methods in the lowest approximation, using as examples the equation for two nucleons and for the system meson + nucleon.

The equation for the Green’s function \(G(x,y;\,x',y')\)*) of two nucleons in the lowest approximation, without taking into account self-energy terms, has the form \(^{96}\):

\[ G(x,y;\,x',y')= \frac{1}{4}S_F^{(1)}(x-x')S_F^{(2)}(y-y')+ \]

\[ +\frac{g^2}{8}\int \bigl(S_F(x-x_1)\tau_k\gamma_5\bigr)^{(1)} \bigl(S_F(y-y_1)\gamma_5\tau_k\bigr)^{(2)} \Delta_F(x_1-y_1)\times \]

\[ \times G(x_1,y_1;\,x',y')\,dx_1dy_1 . \tag{9.1} \]

Here \(S_F\) and \(\Delta_F\) are the distribution functions of a free nucleon and meson (see § 2; formulas (2.7)); the indices \((1)\) and \((2)\) indicate the number of the nucleon.

From the point of view of perturbation theory, equation (9.1) takes into account all Feynman diagrams of the “ladder type” (Fig. 10).

The wave function \(\psi(x,y)\) of two interacting nucleons is defined with the aid of the Green’s function as follows:

\[ \psi(x,y)= \]

\[ =- \int_{x'_0=y'_0\to-\infty} G(x,y;\,x',y')(\gamma_4)^{(1)}(\gamma_4)^{(2)} =\psi_0(x',y')\,dx'dy', \tag{9.2} \]

where \(\psi_0(x,y)\) is the wave function of two free nucleons.

*) The Green’s function of two nucleons is defined as follows \(^{98}\):
\(G(x,y;\,x'y')=(\Psi_0,T[\psi_1(x)\bar\psi_1(x')\psi_1(y)\bar\psi_1(y')]\Psi_0)\), where \(\psi_1\) are Heisenberg field operators, and \(\Psi_0\) is the vacuum in the Heisenberg representation.

From (9.2) and (9.1) follows⁹⁶ the equation for \(\psi(x,y)\):

\[ \psi(x,\ y)=\psi_0(x,\ y)+\frac{g^2}{8}\int \bigl(S_F(x-x_1)\gamma_5\tau_k\bigr)^{(1)} \bigl(S_F(y-y_1)\gamma_5\tau_k\bigr)^{(2)} \times \]

\[ \times \Delta_F(x_1-y_1)\psi(x_1,y_1)\,dx_1dy_1, \tag{9.3} \]

or, in differential form,

\[ \left(\gamma_\mu\frac{\partial}{\partial x_\mu}+M\right)^{(1)} \left(\gamma_\mu\frac{\partial}{\partial y_\mu}+M\right)^{(2)} \psi(x,y)= \]

\[ =-\frac{g^2}{2}\,(\gamma_5\tau_k)^{(1)}(\gamma_5\tau_k)^{(2)} \Delta_F(x-y)\psi(x,y). \tag{9.4} \]

The solution of equation (9.3) may be sought in the particular form

\[ \psi(x,\ y)=e^{iPX}\psi(\zeta), \tag{9.5} \]

where \(X=\dfrac{x+y}{2}\) and \(\zeta=x-y\) are, respectively, the coordinates of the “generalized center of inertia” and the relative coordinates of the two particles.

Fig. 10.

In the center-of-inertia system \(P=(0,0,0,W)\), where \(W\) is the eigenvalue of the problem.

For bound states \(W<2M\). For scattering \(W>2M\). In the first case, in equation (9.3) \(\psi_0(x,y)=0\), since this term represents the free motion of the particles⁹⁶.

We shall restrict ourselves to the case of bound states. Expand \(\psi(\zeta)\) in free solutions:

\[ \psi(\zeta)=\frac{1}{(2\pi)^4}\int u_\sigma^{(1)}(p)u_\sigma^{\prime(2)}(p)\psi^{\varepsilon\sigma}(p)\exp(ip\zeta)\,dp, \tag{9.6} \]

where the indices \(\varepsilon\) and \(\sigma\) number the solutions of the Dirac equation with positive \((\varepsilon,\sigma=+1)\) and negative \((\varepsilon,\sigma=-1)\) energies. Transforming equation (9.3) with the aid of (9.5) and (9.6) into momentum pro-

space, we find:

\[ \psi^{\varepsilon\sigma}(p)= \sum_{\varepsilon',\sigma'} \frac{i g^2}{(2\pi)^4}\int \frac{ \Gamma^{(1)}_{\varepsilon\varepsilon'}\Gamma^{(2)}_{\sigma\sigma'}\Delta_F(p-p') }{ \left(\varepsilon E_p-p_0-\frac{W}{2}\right) \left(\sigma E_p+p_0-\frac{W}{2}\right) } \psi^{\varepsilon'\sigma'}(p')\,dp'; \tag{9.7} \]

here

\[ \Gamma^{(1)}_{\varepsilon\varepsilon'}= \bar u^{(1)}_{\varepsilon}(p)(\gamma_5\tau_k)u^{(1)}_{\varepsilon'}(p'), \qquad \Gamma^{(2)}_{\sigma\sigma'}= \bar u^{(2)}_{\sigma}(-p)\gamma_5\tau_k u^{(2)}_{\sigma'}(-p'). \tag{9.8} \]

Moreover, we have used the convenient representation \(S_F(p)\)

\[ S_F(p)=\sum_{\varepsilon} \frac{\Lambda_{\varepsilon}(p)\gamma_4} {\varepsilon E_p-p_0-\frac{W}{2}}. \tag{9.9} \]

Define the three-dimensional function \(a(\xi)\) in the center-of-mass system as follows\(^{22,96}\):

\[ a(\xi)=\psi(\xi,0). \tag{9.10} \]

In the momentum representation

\[ a^{\varepsilon\sigma}(p)=\int_{-\infty}^{\infty}\psi^{\varepsilon\sigma}(p)\,dp_0. \tag{9.11} \]

It is easy to see that equation (9.7) does not reduce directly to an equation for \(a^{\varepsilon\sigma}(p)\), since the kernel of equation (9.2) depends on \(p'_0\).

Such a reduction can be made only approximately*). In the zeroth adiabatic approximation, when retardation is completely neglected, one must replace \(\Delta_F\) in the kernel of equations (9.7) by \(\frac{1}{2\pi i}\int \Delta_F(p)\,dp_0\). In this case the dependence of \(\psi^{\varepsilon\sigma}(p)\) on \(p_0\) will be proportional to the factor

\[ \left(\varepsilon E_p-p_0-\frac{W}{2}\right)^{-1} \left(\sigma E_p+p_0-\frac{W}{2}\right)^{-1}, \]

i.e.

\[ \psi^{\varepsilon\sigma}(p)= \frac{A^{\varepsilon\sigma}(p,W)} { \left(\varepsilon E_p-p_0-\frac{W}{2}\right) \left(\sigma E_p+p_0-\frac{W}{2}\right) } \,a^{\varepsilon\sigma}(p). \tag{9.12} \]

The proportionality factor \(A^{\varepsilon\sigma}\) is found from condition (9.11) and is equal (when the poles are bypassed, an imaginary negative addition is added to the particle masses)

\[ A^{\varepsilon\sigma}(p,W)= \delta_{\varepsilon\sigma}(2E_p-\varepsilon W)(2\pi i)^{-1}. \tag{9.13} \]

* If in the kernel of the equation for \(a(p)\) one retains only terms \(\sim g^2\).

Substituting (9.12) into (9.7) and integrating with respect to \(dp_0\) and to \(dp'_0\), we obtain an equation for \(a^{\varepsilon \varepsilon'}(\mathbf p)\) in the first approximation (allowing for recoil):

\[ (2E_{\mathbf p}-\varepsilon W)a^{\varepsilon\varepsilon}(\mathbf p)= \sum_{\varepsilon'} \frac{g^2}{(2\pi)^3}\int \frac{d\mathbf p'}{\omega_{\mathbf p-\mathbf p'}} \Gamma^{(1)}_{\varepsilon\varepsilon'}\Gamma^{(2)}_{\varepsilon\varepsilon'} K^{\varepsilon}_{\varepsilon'}(\mathbf p,\mathbf p')a^{\varepsilon'\varepsilon'}(\mathbf p'), \tag{9.14} \]

where the kernel is

\[ K^{\varepsilon}_{\varepsilon'}(\mathbf p,\mathbf p')= -\left(\omega_{\mathbf p-\mathbf p'}+E_{\mathbf p}+E_{\mathbf p'}-\frac{\varepsilon+\varepsilon'}{2}W\right)^{-1}. \tag{9.15} \]

We see that in this approximation\(^*\)

\[ a^{+-}(\mathbf p)=a^{-+}(\mathbf p)=0. \tag{9.16} \]

If in (9.14) we neglect the states of nucleons with negative energy, i.e., set \(a^{--}(\mathbf p)=0\), then we obtain:

\[ (-2E_{\mathbf p}+W)a^{++}(\mathbf p)= -\frac{g^2}{(2\pi)^3}\int \frac{d\mathbf p'}{\omega_{\mathbf p-\mathbf p'}} \Gamma^{(1)}_{++}\Gamma^{(2)}_{++}\times \]

\[ \times(\omega_{\mathbf p-\mathbf p'}+E_{\mathbf p}+E_{\mathbf p'}-W)^{-1} a^{++}(\mathbf p'), \tag{9.17} \]

This equation coincides exactly with equation (3.21) for two nucleons, obtained in the first approximation of the old Tamm—Dancoff method.

From the point of view of perturbation theory, the meaning of the approximation which reduces equation (9.7) to equations (9.14) and (9.17) is as follows. Equation (9.7) takes into account all diagrams shown in Fig. 10. Each such diagram is equivalent to \(n!\) (where \(n\) is the order of the diagram) time-ordered diagrams (of the type shown in Figs. 1 and 2). In particular, an \(n\)-th order diagram may contain an ordered diagram corresponding to the simultaneous presence of \(n/2\) virtual mesons. If in equation (9.7) we neglect the contribution of all ordered diagrams containing more than one virtual meson simultaneously, then we obtain equation (9.14). If, in addition to this, we neglect the possibility of formation of virtual pairs, then we arrive at the equation of the old Tamm—Dancoff method (9.17).

One can obtain (see \(^{24,101}\)) an equation of the Levi—Klein type for \(a^{++}(\mathbf p)\), strictly equivalent to the covariant equation (9.7). However, the kernel of such an equation will be an infinite, generally divergent series in powers of the coupling constant \(g^2\).

Let us now turn to the equation for the meson + nucleon system. In this case the situation is somewhat more complicated, since in the lowest approximation one can obtain several different equations for the Green function meson + nucleon \(^{18,99,102}\). We shall proceed

\(^*\) The influence of the negative components was considered in work \(^{109}\).

from the equation proposed by Martin and Dyson^18. The advantage of this equation is that, in the language of perturbation theory, it takes into account finite chains of the form*) (Fig. 11), which make the main contribution to scattering with total isotopic spin \(I=3/2\).

Fig. 11.

In the lowest approximation this equation has the form (without taking self-energy terms into account)

\[ G_{ik}(x\xi; x'\xi')= \frac{1}{4}\delta_{ik}S_F(x-x')\Delta_F(\xi-\xi') + \sum_j \frac{g^2}{8}\int S_F(x-x_1)(\gamma_5\tau_i)S_F(x_1-x_2)(\gamma_5\tau_j)\times \]

\[ \times G_{jk}(x_2\xi; x'\xi')\Delta_F(x_1-\xi)\,dx_1dx_2 + \]

\[ +\sum_j \frac{g^2}{8}\int S_F(x-x_1)(\gamma_5\tau_j)S_F(x_1-x_2)(\gamma_5\tau_i)\times \]

\[ \times G_{jk}(x_2,x_1; x'\xi')\Delta_F(x_2-\xi)\,dx_1dx_2. \tag{9.18} \]

Here \(G_{jk}(x\xi; x'\xi')\) is the Green’s function of the meson–nucleon equation**). The first term in the kernel of this equation corresponds to a chain with absorption; the second, to a chain with emission.

Unlike (9.2), the wave function \(\varphi_i(x\xi)\) of the interacting meson and nucleon is related to \(G_{ik}(x\xi; x'\xi')\) in the following way***):
\[ \varphi_i(x,\xi)= \]

\[ = -\sum_j \int_{x'_0=\xi'_0\to -\infty} \left[ G_{ij}(x\xi; x'\xi')\gamma_4,\frac{\partial}{\partial \xi'_0} \right]_{-} \varphi_j^0(x',\xi')\,dx'\,d\xi', \tag{9.19} \]

where \(\varphi_j^0(x,y)\) is the wave function of the free nucleon and meson. The equation for \(\varphi_j(x,y)\), which follows from (9.18) with the help of (9.19),

*) The so-called “picket-fence” chains.

**) By definition
\[ G_{ik}(x\xi; x'\xi')=(\Psi_0,T\{\psi_r(x)\bar{\psi}_r(x')\varphi_i(\xi)\varphi_k(\xi')\}\Psi_0). \]

***) The difference is due to the fact that the meson wave function obeys a second-order equation.

has the form

\[ \varphi_i(x,\xi)=\varphi_i^0(x,\xi)-\sum_j {g^2\over 8}\int S_F(x-x_1)(\gamma_5\tau_i)\times \]

\[ \times S_F(x_1-x_2)(\gamma_5\tau_j)\Delta_F(x_1-\xi)\varphi_j(x_2,x_2)\,dx_1dx_2+ \]

\[ +\sum_j {g^2\over 8}\int S_F(x-x_1)(\gamma_5\tau_j)S_F(x_1-x_2)(\gamma_5\tau_i)\Delta_F(x_2-\xi)\times \]

\[ \times\varphi_j(x_2,x_1)\,dx_1dx_2 . \tag{9.20} \]

The meson—nucleon equation is, as a rule, used for investigating the scattering problem. Therefore we shall make the transition to the three-dimensional meson—nucleon equation directly in equation (9.18) for the meson—nucleon Green’s function*).

Passing in (9.18) to the momentum representation, in the center-of-inertia system we obtain:

\[ G_{ik}(p,p';W)=\delta_{ik}\Omega(p,W)\delta(p-p')+ \]

\[ +\lambda\Omega(p,W)\sum_j\int \left\{(\gamma_5\tau_i)S_F(P)(\gamma_5\tau_j)+ \right. \]

\[ \left. +(\gamma_5\tau_j)S_F(p+p'')(\gamma_5\tau_i)\right\}G_{jk}(p'',p';W)\,dp''\ldots , \tag{9.21} \]

where \(p\) and \(p'\) are, respectively, the initial and final relative 4-momenta of the meson—nucleon system; \(W\) is the total energy of the system,

\[ \Omega(p,W)=S_F\left(p+{P\over 2}\right)\Delta_F\left(p-{P\over 2}\right);\quad \lambda={ig^2\over (2\pi)^4}. \]

By analogy with the three-dimensional wave function (see 9.14), the three-dimensional Green’s function is defined by the relation

\[ G_{ik}(p,p';W)=\int G_{ik}(p,p';W)\,dp_0dp'_0= \]

\[ =\int G_{ik}(p,p';W)\,dp_0, \tag{9.22} \]

where

\[ G_{ik}(p,p;W)=\int G(p,p';W)\,dp'_0 . \]

Arguing in the same way as in the case of two nucleons, we find that in the first approximation

\[ G_{ik}(p,p';W)={i\over (2\pi)}(2\omega_p)\Omega(p,W)(\Lambda_+(p)(W-E_p-\omega_p)+ \]

\[ +\Lambda_-(p)(W+E_p+\omega_p))\gamma_4G_{ik}(p,p';W). \tag{9.23} \]

*) The transition to the three-dimensional equation for the meson—nucleon wave function is carried out similarly to the way this was done above for two nucleons.

We next introduce the Green’s function corresponding to a given energy of the nucleon and meson in the final state,

\[ G_{ik}^{\varepsilon\sigma}(p,\mathbf p';W) = \frac{\sigma\Lambda_{\varepsilon}(\mathbf p)}{2\omega_{\mathbf p}} \left(\sigma\omega_{\mathbf p}-p_0+\frac{W}{2}\right) G_{ik}(p,\mathbf p';W), \tag{9.24} \]

where \(\varepsilon\) and \(\sigma\) take the values \(\pm 1\). Then, if we define

\[ G^{\varepsilon\sigma}(\mathbf p,\mathbf p';W) = \int G^{\varepsilon\sigma}(p,\mathbf p';W)\,dp_0, \tag{9.25} \]

then, proceeding in the same way as in the case of two nucleons, we obtain for \(G^{\varepsilon\sigma}\) the following equation:

\[ \begin{aligned} \bigl[W-\varepsilon(E_{\mathbf p}+\omega_{\mathbf p})\bigr] G_{ik}^{\varepsilon\sigma}(\mathbf p,\mathbf p';W) &= \frac{2\pi}{i}\,\delta_{ik}\Lambda_{\varepsilon}(\mathbf p)\gamma_4(2\omega_{\mathbf p})^{-1} \delta(\mathbf p-\mathbf p') \\ &\quad +\frac{g^2}{(2\pi)^3} \sum_{j,\varepsilon'} \frac{\Lambda_{\varepsilon}(\mathbf p)}{2\omega_{\mathbf p}} (\tau_i\tau_j) \left(\frac{W-\gamma_4M}{W^2-M^2}\right) \int G_{ik}^{\varepsilon'\varepsilon'}(\mathbf p'',\mathbf p';W)\,d\mathbf p'' \\ &\quad -\sum_j \frac{g^2}{(2\pi)^3} \frac{\Lambda_{\varepsilon}(\mathbf p)}{2\omega_{\mathbf p}} (\tau_j\tau_i) \\ &\quad\times \int \Biggl\{ \left( \frac{\Lambda_{\varepsilon}(\mathbf p+\mathbf p'')} {E_{\mathbf p+\mathbf p''}+E_{\mathbf p}+E_{\mathbf p''}-W} - \frac{\Lambda_{-\varepsilon}(\mathbf p+\mathbf p'')} {E_{\mathbf p+\mathbf p''}+\omega_{\mathbf p}+\omega_{\mathbf p''}-W} \right) G_{jk}^{\varepsilon\varepsilon}(\mathbf p'',\mathbf p';W) \\ &\qquad + \left( \frac{\Lambda_{\varepsilon}(\mathbf p+\mathbf p'')} {E_{\mathbf p}+\omega_{\mathbf p}+E_{\mathbf p''}} - \frac{\Lambda_{-\varepsilon}(\mathbf p+\mathbf p'')} {E_{\mathbf p+\mathbf p''}+\omega_{\mathbf p''}+E_{\mathbf p}} \right) G_{jk}^{-\varepsilon-\varepsilon}(\mathbf p'',\mathbf p';W) \Biggr\} \,d\mathbf p''. \end{aligned} \tag{9.26} \]

In the approximation considered here (see (9.16)),

\[ G^{+-}(\mathbf p,\mathbf p'';W)=G^{-+}(\mathbf p,\mathbf p';W)=0. \tag{9.27} \]

If in (9.26) one neglects states with negative energy and introduces, instead of \(G_{ik}^{++}\), the scattering amplitude

\[ (W-E_{\mathbf p}-\omega_{\mathbf p})^{-1}u_{ik}^{++}(\mathbf p,W) = \]

\[ = \frac{i}{2\pi} G_{ik}^{++}(\mathbf p,W)(W-E_{\mathbf p}-\omega_{\mathbf p}) - \delta_{ik}\delta(\mathbf p-\mathbf p')(2\omega_{\mathbf p})^{-1}\Lambda_{+}(\mathbf p)\gamma_4, \]

then the equation for \(u^{++}\) will coincide exactly with equation (5.5) for meson + nucleon in the old Tamm–Dancoff method.

We note that equation (9.26) was studied in the nonrelativistic approximation in work \(^{110}\). The most interesting result

THE TAMM—DANCOV METHOD

of this investigation is that, in a state with \(l=3/2\) and \(l=5\) (where \(l\) is the orbital angular momentum), the system pion + nucleon, for \(g^2 \simeq 13\) and \(W-M-\mu \simeq 35\,M_{\text{ev}}\), forms a virtual state with a lifetime \(\sim 10^{-10}\) sec., which agrees in order of magnitude with the lifetime of the \(\Lambda\)-particle*). However, the calculation was carried out too crudely and needs refinement.

In conclusion to this paragraph we shall dwell on the relation between equations (9.3) and (9.21) and the corresponding equations of the new Tamm—Dancov method.

The rules for going around the poles in integration over \(dp_0\) in formulas (9.11) and (9.22) are determined according to Feynman, i.e. an imaginary negative addition is added to the particle masses. Let us define new three-dimensional functions for two nucleons and for a meson and a nucleon by relations (9.11) and (9.22), assuming that, in integration over \(dp_0\), an imaginary positive addition is added to \(W\)**) \(^{25}\).

Then, in the first approximation, the three-dimensional equations for these functions, which follow from equations (9.3) and (9.21), will have the following form:

in the case of two nucleons

\[ \begin{aligned} \left[ W-(\varepsilon+\sigma)E_{\mathbf p}\right] a^{\varepsilon\sigma}(\mathbf p) &= \frac{g^2}{(2\pi)^3} \sum_{\varepsilon',\,\sigma'} \int \frac{d\mathbf p'}{2\omega_{\mathbf p-\mathbf p'}} \, \Gamma_{\varepsilon\varepsilon'}^{(1)} \Gamma_{\sigma\sigma'}^{(2)} \\ &\quad \times \left( \frac{\sigma}{ E_{\mathbf p}+\omega_{\mathbf p-\mathbf p'}+\sigma\varepsilon' E_{\mathbf p'}-\sigma W} + \frac{\varepsilon}{ E_{\mathbf p}+\omega_{\mathbf p-\mathbf p'}+\varepsilon\sigma' E_{\mathbf p'}-\varepsilon W} \right) a^{\varepsilon'\sigma'}(\mathbf p') ; \end{aligned} \tag{9.28} \]

in the case of a meson and a nucleon

\[ \begin{aligned} (W-\varepsilon E_{\mathbf p}-\sigma\omega_{\mathbf p}) G_{ik}^{\varepsilon\sigma}(\mathbf p,\mathbf p';W) &= \left(\frac{2\pi}{i}\right) \hat{\delta}_{ik}\Lambda_{\varepsilon}(\mathbf p)\gamma_4 (2\omega_{\mathbf p})^{-1} \\ &\quad \times \delta(\mathbf p-\mathbf p')\delta_{\varepsilon\sigma} + \frac{g^2}{(2\pi)^3} \frac{\Lambda_{\varepsilon}(\mathbf p)}{2\omega_{\mathbf p}} \sum_{j,\,\varepsilon',\,\sigma'} \Bigg\{ \delta_{\varepsilon\sigma}(\tau_i\tau_j) \left(\frac{W-\gamma_4 M}{W^2-M^2}\right) \\ &\quad \times \int G_{jk}^{\varepsilon'\sigma'}(\mathbf p'',\mathbf p';W)\,d\mathbf p'' \\ &\quad - (\tau_j\tau_i) \int \left( \frac{ \sigma\Lambda_{\varepsilon}(\mathbf p+\mathbf p'') }{ E_{\mathbf p+\mathbf p''}+E_{\mathbf p} +\varepsilon\varepsilon' E_{\mathbf p''}-\varepsilon W } \right. \\ &\qquad\qquad\left. - \frac{ \sigma\Lambda_{-\varepsilon}(\mathbf p+\mathbf p'') }{ E_{\mathbf p+\mathbf p''}+\omega_{\mathbf p} +\sigma\sigma'\omega_{\mathbf p'}-\sigma W } \right) G_{jk}^{\varepsilon'\sigma'}(\mathbf p'',\mathbf p';W) \Bigg\}\,d\mathbf p'' . \end{aligned} \tag{9.29} \]

) See also \(^{111}\).
*) The so-called Dyson analytic continuation \(^{112}\).

Equations (9.28) and (9.29) coincide with the corresponding equations in the new Tamm–Dancoff method.

Let us emphasize the important distinction between equations (9.3) and (9.21), on the one hand, and equations (9.28) and (9.29), on the other. First, in equations (9.3) and (9.21) only the \((++ )\)- and \((--)\)-components of the wave functions are different from zero, whereas in equations (9.28) and (9.29) all components are different from zero. Second, in equations (9.28) and (9.29) there appear the so-called “spurious poles”*); the presence of “spurious poles” in the equations of the new Tamm–Dancoff method leads to additional difficulties in the renormalization of the equations. In equations (9.3) and (9.21) “spurious poles” do not arise.

Above, using the simplest examples, we considered the connection between equations of the Tamm–Dancoff type and covariant equations of the Bethe–Salpeter type. In doing so, two essential restrictions were made: first, in the original covariant equations (9.1) and (9.18) the self-energy terms were discarded, and, second, we restricted ourselves to the first nonvanishing approximation for these equations. Taking account of the self-energy terms in equations (9.1) and (9.18) makes it possible, after carrying out renormalization, to obtain from the covariant equations three-dimensional renormalized equations of the T.D. type. However, the appearance of additional poles in the renormalized propagator functions (see 10, 12) substantially narrows the domain of applicability of such equations (both the covariant ones and the three-dimensional equations following from them). The investigation of the questions that arise here is at present of undoubted interest.

In conclusion, it should be said that although meson theory is still far from a complete quantitative description of nuclear phenomena, the absence of qualitative contradictions between the predictions of meson theory and the experimental data of nuclear physics compels one to think that the construction of correct mathematical methods for solving the equations of the meson field will make it possible to obtain (perhaps in a restricted energy region, for example \(\ll Mc^2\)) a quantitative theory of nuclear phenomena \({}^{113}\).

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1. Let us note the following fact. In the T.–D. method, when considering problems of collision (or, in general, interaction) of particles, one may formulate the problem in such a way that, for example, two colliding particles, being sufficiently far from each other, do not interact with one another, but do interact with their own field. In other words, one may consider the problem of the interaction of real, and not “bare,” particles. ↩