THE FUNCTIONAL METHOD IN QUANTUM FIELD THEORY

Yu. V. Novozhilov, A. V. Tulub*)

CONTENTS

I. The functional method in quantum field theory . . . . . . . . . . . . . . . . . . . 53
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
§ 1. Quantum field theory and functionals . . . . . . . . . . . . . . . . . . . . . . 54
§ 2. Fock’s functional method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
    1. The idea of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
    2. Generating functional for probability amplitudes . . . . . . . . . . . . . . . . 60
    3. The functional method and Fermi statistics . . . . . . . . . . . . . . . . . . . 62
    4. Equations for the state functional . . . . . . . . . . . . . . . . . . . . . . . . 65

§ 3. Generating functional for amplitudes of the new Tamm—Dancoff method . . . . 72
II. Generating functionals for relativistic functions and functional integration . . . 75
§ 4. Generating functionals for relativistic functions . . . . . . . . . . . . . . . . 76
    1. \(T\)-functions and the generating functional . . . . . . . . . . . . . . . . . . 76
    2. Feynman amplitudes and the generating functional . . . . . . . . . . . . . . . 82
    3. \(\rho\)-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

§ 5. Space-time treatment of quantum field theory and functionals . . . . . . . . . 86
    1. Basic equations for the four-dimensional state vector . . . . . . . . . . . . . 86
    2. Generalized Fock functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
    3. Functional Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . 92

§ 6. Variation of the operator and functional integration in the case of a Fermi field . . . 94

I. THE FUNCTIONAL METHOD IN QUANTUM FIELD THEORY

Introduction

In recent years functional methods have been widely applied in quantum field theory. The attention devoted to these methods is due to the hope of obtaining, with their aid, exact solutions of certain problems, or at least of finding some exact relations in field theory. Until recently almost all work in quantum field theory was carried out on the basis of perturbation theory.

At present most physicists share the conviction that, even in quantum electrodynamics, where the interaction constant is small, perturbation theory cannot serve as a basis for the investigation of fundamental questions, since a number of approximations may diverge. Methods for solving the equations of quantum field theory in meson theory are urgently needed, since the interaction constant of nucleons with the meson field cannot be considered small.

In contrast to other methods, functional methods make it possible to formulate rigorously equations for field functions and provide the possibility of obtaining

*) §§ 3 and 4.2 were written by A. V. Tulub; the remaining sections by Yu. V. Novozhilov.

formal solution of the problem of interacting fields. This feature of the functional method is important both for investigations of a fundamental nature and for the development of approximate methods for solving field equations that differ from perturbation theory.

At the present time, works on the functional method may be divided (from the point of view of the use of the functional apparatus) into two groups: works on the study of generating functionals and works connected with the use of functional integration.

The idea of the method of the generating functional was advanced by Academician V. A. Fock in 1928[^1] and was developed in detail in his work of 1934[^2]. This method was used in subsequent years for solving a number of problems[^5–^8], but the functional method received wide application and development only in recent years, in connection with the general development of quantum field theory, which brought to the fore the requirements of relativistic covariance of the field equations and the possibility of solving them without perturbation theory. The most important development of the functional method is associated with the functionals of external sources introduced by Schwinger[^9], which are generating functionals for relativistic field functions, and with Feynman’s equivalent Lagrangian formulation[^10].

Some works on functional methods were discussed in the reviews of Berestetskii and Galanin[^3], and of Silin and Feinberg[^4]. We therefore devote comparatively little space to the questions touched upon in these articles.

In Part I of the review we consider in detail the foundations of the functional method in quantum field theory and the generating functionals for nonrelativistic functions.

Questions concerning generating functionals for relativistic functions, functionals in the space-time treatment, and functional integration over the Fermi field are set apart in Part II of the review.

§ 1. Quantum Field Theory and Functionals

As is known, from the classical point of view a field is characterized by one or several functions of the coordinates \(\varphi(\mathbf{x})\)—the field potentials, satisfying the wave equation. The field may be considered as a mechanical system with an infinitely large number of degrees of freedom. Indeed, in order to determine in the classical way the state of a system with a finite number of degrees of freedom \(n\), it is necessary to specify \(n\) independent coordinates \(q_l\) and \(n\) conjugate momenta \(p_l\) \((l = 1, 2 \ldots n)\). The state of the field will be known if, for each of the points \(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3 \ldots\) of space, we know the potential \(\varphi\) and the time derivative \(\partial \varphi/\partial t\), i.e. if the infinite set of quantities is known:
\[ \varphi(\mathbf{x}_1),\quad \frac{\partial \varphi(\mathbf{x}_1)}{\partial t};\quad \varphi(\mathbf{x}_2),\quad \frac{\partial \varphi(\mathbf{x}_2)}{\partial t};\ldots \]

In the case of a system with a finite number of degrees of freedom one has to deal with functions \(f(q_1 \ldots q_n\, p_1 \ldots p_n)\) of the coordinates \(q_l\) and canonical momenta \(p_l\) (energy, momentum, and so forth). In the case of a field, however, the energy and other quantities of the field will depend on an infinite number of “coordinates” \(\varphi(\mathbf{x}_i)\), where \(\mathbf{x}_i\) runs through all points of space, and of conjugate “momenta.” Thus, in field theory one must consider functions of an infinite number of variables \(\varphi(\mathbf{x}_1),\ldots,\varphi(\mathbf{x}_i)\), or functions of the function \(\varphi(\mathbf{x})\)—functionals of the function \(\varphi(\mathbf{x})\).

Functional dependence will be denoted by braces: \(F\{\varphi(\mathbf{x})\}\) or \(F\{\varphi\}\) is a functional of \(\varphi(\mathbf{x})\). The simplest example of a functional is the integral \(F\{\varphi\}=\int f(\mathbf{x})\varphi(\mathbf{x})\,d\mathbf{x}\).

Let us consider the variation \(\delta F\{\varphi\}\) of an arbitrary functional \(F\{\varphi\}\), caused by a variation of the function \(\varphi(\mathbf{x})\) at the point \(\mathbf{x}'\): \(\delta F\{\varphi\}=F\{\varphi(\mathbf{x})+\lambda\delta(\mathbf{x}-\mathbf{x}')\}-\)

— \(F\{\varphi\}\). The functional derivative of \(F\{\varphi\}\) with respect to \(\varphi(x')\) is the quantity

\[ \frac{\delta F\{\varphi\}}{\delta \varphi(x')}=\lim_{\lambda\to 0}\frac{1}{\lambda}\,\delta F\{\varphi\}. \tag{1.1} \]

Instead of (1.1) one may also write:

\[ \delta F\{\varphi\}=\int \frac{\delta F\{\varphi\}}{\delta \varphi(x)}\,\delta\varphi(x)\,dx . \tag{1.2} \]

From this we find, putting

\[ F\{\varphi\}=\varphi(x)=\int \delta(x-x')\varphi(x')\,dx', \]

that

\[ \frac{\delta\varphi(x)}{\delta\varphi(x')}=\delta(x-x'). \tag{1.3} \]

If \(F\{\varphi\}\) is a functional of the function \(\varphi(x)\), which in turn is a functional of the function \(\gamma(x)\), then the functional derivative of \(F\) with respect to \(\gamma(x')\) is equal to

\[ \frac{\delta F\{\varphi\{\gamma\}\}}{\delta \gamma(x')}= \int \frac{\delta\varphi(x)}{\delta\gamma(x')} \frac{\delta F\{\varphi\{\gamma\}\}}{\delta\varphi(x)} \,dx . \tag{1.4} \]

We shall consider the definition of the functional integral later.*)

As is well known, in quantum mechanics of systems with a finite number of degrees of freedom \(n\), the fundamental commutation relation between the coordinate operators \(\hat q_l\) and momenta \(\hat p_{l'}\) has the form

\[ [\hat p_{l'},\hat q_l]=\hat p_{l'}\hat q_l-\hat q_l\hat p_{l'}=-i\delta_{ll'}. \tag{1.5} \]

Since all the coordinate operators \(\hat q_l\) commute with one another, we can describe the system by means of the wave function \(\Psi(q_1,\ldots,q_n)\), with respect to which \(\hat q_l\) is a number (the operator \(\hat q_l\) is the operator of multiplication by \(q_l\)). As is well known, it follows from (1.5) that then \(\hat p_l\) is a differential operator: \(\hat p_l=-i\frac{\partial}{\partial q_l}\).

If we now pass to an infinite number of degrees of freedom, letting \(n\) tend to infinity, then the index \(l\) will run through a continuous range of values. The infinite sets of coordinates \(\hat q_l\) and momenta \(\hat p_l\) obtained in this way may be regarded as functions \(q(l)\) and \(p(l)\), depending on the variable \(l\) as on a parameter. In this limiting case the commutation relation (1.5) should be written in the form

\[ [\hat p(l),\hat q(l')]=-i\delta(l-l'), \tag{1.6} \]

replacing, on the right-hand side, \(\delta_{ll'}\) by \(\delta(l-l')\), since \(l\) varies continuously. The wave function \(\Psi\) will depend on an infinite number of coordinates \(q_l\), or on the function \(q(l)\), i.e. \(\Psi\) will be a functional of \(q(l)\): \(\Psi=\Psi\{q(l)\}\).

In quantum field theory the fundamental commutation relation for fields with integer spin has precisely the form (1.6). The operator of the meson field \(\varphi(x)\) (the meson “potential”) depends on the space-time coordi-

*) Some mathematical questions connected with the functional formulation of quantum field theory are considered in Friedrichs’ book \(^{11}\) and in the appendix to Simanchik’s article \(^{12}\).

depend on \(x\) as on a parameter. If the times of the operators \(\varphi(x')\) and \(\partial\varphi(x)/\partial x_0\) are equal, then between them there is the commutation relation

\[ \left[\frac{\partial\varphi(x)}{\partial x_0},\, \varphi(x')\right] = - i\delta^3(x-x'), \qquad x_0=x_0'. \tag{1.7} \]

Since the operators \(\varphi(x)\) and \(\varphi(x')\) commute with each other for different points \(x\) and \(x'\) at equal times \(x_0=x_0'\), we may choose \(\varphi(x)\) (at a fixed time) as the coordinate function \(\hat q(l)\), i.e. regard \(\varphi(x)\) as the operator of multiplication by the function \(\varphi'(x)\), and describe the field by means of a functional \(\Omega\{\varphi'\}\) of the spatial function \(\varphi'(x)\). Then from the commutation relation (1.7) it follows that the operator \(\partial\varphi/\partial x_0\) plays the role of the momentum function \(\hat p(l)\). Relation (1.7) will, according to (1.3), be satisfied if we put

\[ \frac{\partial\varphi(x)}{\partial x_0}\Omega\{\varphi'\} = - i\,\frac{\partial}{\partial\varphi'(x)}\Omega\{\varphi'\}. \tag{1.8} \]

Thus the functional formulation in quantum field theory is a natural consequence of the fact that the field possesses an infinite number of degrees of freedom.

An important place in quantum field theory belongs to the concept of a generating functional. Suppose that the functional \(F\{\varphi\}\) of the function \(\varphi(x)\) can be expanded in a power functional series

\[ F\{\varphi\}=\sum_n F_n\{\varphi\}; \tag{1.9} \]

\[ F_n\{\varphi\}=(n!)^{-1/2}\int f_n(x_1\ldots x_n)\,\varphi(x_1)\ldots\varphi(x_n)\,dx_1\ldots dx_n. \tag{1.10} \]

Then \(F\{\varphi\}\) is a generating functional for the functions \(f_n(x_1\ldots x_n)\) symmetric with respect to the variables \(x_1\ldots x_n\). The concept of a generating functional is a generalization of the concept of a generating function. In the special case when the variable \(x\) can take only one value \(x=a\), we obtain from (1.10), putting \(\varphi(x)=\xi\delta(x-a)\):

\[ F\{\varphi(a)\}\equiv F(\xi)=\sum_{n=0}^{\infty}\frac{1}{\sqrt{n!}}\,f_n\xi^n \]

—the generating function for the coefficients \(f_n\).

Between the functional \(F_n\{\varphi\}\) and the function \(f_n(x_1\ldots x_n)\) there is a one-to-one correspondence, which makes it possible to use the functional \(F_n\{\varphi\}\) instead of the function \(f_n\). This means that, instead of the infinite collection of functions \(f_n(x_1\ldots x_n)\), \(n=0,1,2,\ldots\), contained in (1.10), one can deal with a single quantity—the functional \(F\{\varphi\}\).

The method of the generating functional can be generalized to the case when the functions \(f_n(x_1\ldots x_n)\) are antisymmetric with respect to the variables \(x_1\ldots x_n\). In this case the quantities \(\varphi(x)\) contained in the integral \(F_n\{\varphi\}\) (formula (1.10)) cannot be functions. Indeed, if \(\varphi(x)\) is a function, and \(f_n(x_1\ldots x_n)\) is antisymmetric with respect to the variables \(x_1\ldots x_n\), then the integral \(F_n\) is equal to zero. In order that the integral \(F_n\) not change under interchange of two variables \(x_i\) and \(x_k\), the quantities \(\varphi(x)\) must anticommute:

\[ \varphi(x_i)\varphi(x_k)+\varphi(x_k)\varphi(x_i)=0. \]

The meaning of the anticommuting quantities \(\varphi\) will be clarified in § 2.

The method of the generating functional is closely connected with the corpuscular aspect in quantum field theory. The interpretation of transitions and states of the field is based on the idea of corpuscular manifestations of the field. So long as this idea remains in force, the basic quantities describing the field should be considered the wave functions \(f_n(x_1\ldots x_n)\), referring to a definite number

particles \(n\) and possessing the required symmetry properties with respect to the variables of the particles \(x_1 \ldots x_n\). If the number of particles can vary, then in the general case the state or transitions of the field must be described by means of an infinite set of functions \(f_n(x_1 \ldots x_n)^{13}\), \(n=0,1,2,\ldots\):

\[ \begin{gathered} f_0,\\ f_1(x_1),\\ f_2(x_1,x_2),\\ \ldots\ldots\ldots\ldots\\ f_n(x_1,x_2,\ldots,x_n),\\ \ldots\ldots\ldots\ldots \end{gathered} \]

Instead of the set of functions \(f_n(x_1\ldots x_n)\), one may consider a generating functional of type (1.10).

Thus, a functional can describe a field state or transition processes only when it is a generating functional with respect to some functions \(f_n(x_1\ldots x_n)\). The simplest functions depending on the variables of \(n\) particles are the probability amplitudes \(\Psi_n(x_1\ldots x_n)\), the squared absolute values of which \(|\Psi_n|^2\) give the probability density that the particles are in the states \(x_1\ldots x_n\). The generating functional for probability amplitudes is the “Fock functional.”

In addition to the set of probability amplitudes \(\Psi_n(x_1\ldots x_n)\), other functions are now known (four-dimensional wave functions, Green’s functions), by means of which one can describe field states and transitions between them; the interpretation in terms of probability amplitudes is close to nonrelativistic quantum mechanics, whereas in the definitions of four-dimensional wave functions and Green’s functions the requirement of relativistic invariance of the theory is explicitly taken into account. These relativistic functions are connected with Schwinger’s functional of external sources.

Four-dimensional wave functions, Green’s functions, and other relativistic functions do not have so simple a meaning as probability amplitudes.

The functional method will be presented for the example of a nucleon field interacting with a neutral pseudoscalar meson field. The operators of the free nucleon field are denoted by \(\psi_\alpha(x)\) and \(\bar\psi_\beta(x)\) \((\bar\psi=\psi^*\gamma_4)\), and \(\varphi(x)\) is the operator of the free meson field. Spin variables \(\alpha,\beta\) will usually be denoted together with the coordinates by a single letter, for example \(\psi(x)\), \(\bar\psi(y)\). In this case integration over the coordinates \(x,y\) will also include summation over the remaining variables.

For what follows it is necessary to introduce creation and annihilation operators. If \(f^{(k)}(x)\) is a system of positive-frequency solutions of the Klein–Fock equation \((\Box-\mu^2)f^{(k)}(x)=0\), orthogonal and normalized in the sense

\[ -i\int d\sigma_\mu f^{(j)}(x)\frac{\overleftrightarrow{\partial}}{\partial x_\mu}f^{(k)}(x)=\delta_{jk} \tag{1.11} \]

(\(d\sigma_\mu\) is the four-dimensional vector of the element of a spacelike hypersurface), then the annihilation operator \(c_k\) for the meson field \(\varphi(x)\) is defined by the formula

\[ c_k=\frac{1}{i}\int d\sigma_\mu\,\varphi(x)\frac{\overleftrightarrow{\partial}}{\partial x_\mu}f^{(k)}(x). \tag{1.12} \]

In (1.11) and (1.12) the notation used was

\[ A\frac{\overleftrightarrow{\partial}}{\partial x_\mu}B = A\frac{\partial B}{\partial x_\mu} - \frac{\partial A}{\partial x_\mu}B. \tag{1.13} \]

Let \(u^{(k)}\) and \(\bar v^{(k)}\) be systems of positive-frequency solutions of the Dirac equation \((u^{(k)})\) and of the adjoint equation \((\bar v^{(k)})\), which are orthogonal and normalized: \((\bar v = v^* \gamma_4)\)

\[ \left. \begin{gathered} \int \bar u^{(k)} \gamma_\mu u^{(j)}\, d\sigma_\mu=\delta_{kj},\\ \int \bar v^{(k)} \gamma_\mu v^{(j)}\, d\sigma_\mu=\delta_{kj},\qquad \int \bar u^{(k)} \gamma_\mu v^{(j)}\, d\sigma_\mu=0. \end{gathered} \right\} \tag{1.14} \]

Then the absorption operators \(a_k\) and \(b_k\) for the nucleon field are defined by the equations

\[ \left. \begin{aligned} a_k&=\int \bar u^{(k)}(x)\gamma_\mu \psi(x)\,d\sigma_\mu,\\ b_k&=\int \bar\psi(x)\gamma_\mu v^{(k)}(x)\,d\sigma_\mu. \end{aligned} \right\} \tag{1.15} \]

The creation operators \(a_k^+\), \(b_k^+\), \(c_k^+\) are Hermitian conjugates of the absorption operators \(a_k\), \(b_k\), \(c_k\). If the field is considered at some time \(x_0\) and the states of the particles differ in the values of the momentum, then the absorption and creation operators defined by formulas (1.12) and (1.15) are the coefficients in the corresponding Fourier expansions:

\[ \varphi(x)=\frac{1}{(2\pi)^{3/2}}\int (2k_0)^{-1/2}\left[c(k)e^{ikx}+c^+(k)e^{-ikx}\right]\,d^3k, \tag{1.16} \]

\[ \psi(x)=\frac{1}{(2\pi)^{3/2}}\int \left[u(p)a(p)e^{ipx}+v(p)b^+(p)e^{-ipx}\right]\,d^3p, \tag{1.17} \]

\[ \bar\psi(x)=\frac{1}{2(\pi)^{3/2}}\int \left[\bar u(p)a^+(p)e^{-ipx}+\bar v(p)b(p)e^{ipx}\right]\,d^3p, \tag{1.18} \]

where \(kx=(\mathbf{k},\mathbf{x})-k_0x_0\); \(u\) and \(v\) are Dirac spinors; in (1.17) and (1.18) summation over polarizations is included in the integration over \(p\).

The commutation relations between the creation operators \(a^+\), \(b^+\), \(c^+\) and the absorption operators \(a\), \(b\), \(c\) have the form

\[ \{a^+(p),a(p')\}=\delta^3(p-p'), \tag{1.19} \]

\[ \{b^+(p),b(p')\}=\delta^3(p-p'), \tag{1.20} \]

\[ [c(p),c^+(p')]=\delta^3(p-p'). \tag{1.21} \]

§ 2. The Fock Functional Method

1. The idea of the method.

We shall first present the idea of the method in the simplest example, in order to avoid complications introduced by the introduction of functionals.

Thus, for methodological purposes, let us consider the case when mesons can be found only in one state (the same for all particles). We shall not consider the nucleon field for the time being. Let \(\Psi_n\) be the probability amplitude for the presence in the field of \(n\) mesons (which depends only on the number \(n\)). For an indefinite number of mesons, in order to describe the field completely it is necessary to know the entire set of quantities \(\Psi_n\), \(n=0,1,2,\ldots,\infty\).

In the Fock functional method, instead of the infinite set \(\Psi_n\), the state of the field is characterized by one quantity:

\[ \omega(\bar c)=\sum_{n=0}^{\infty}\frac{1}{\sqrt{n!}}\Psi_n \bar c^n=\sum_{n=0}^{\infty}\omega_n, \tag{2.1} \]

where \(\bar c\) is an auxiliary constant. It is obvious that specifying \(\omega_n\) uniquely determines \(\Psi_n\) and conversely. Thus, in the simplest case under consideration,

example; \(\omega\) is a function of \(\bar c\) and is the generating function for the probability amplitudes \(\Psi_n\).

In order that, by means of the single quantity \(\omega(\bar c)\), one might compute the mean values of field quantities, in order to write the equation for \(\omega(\bar c)\), it is necessary to determine how the field operators act on \(\omega(\bar c)\).

The creation and annihilation operators \(c^+\) and \(c\) in our example satisfy the commutation relation

\[ cc^+ - c^+c = 1, \tag{2.2} \]

and the particle-number operator \(\hat n\) is

\[ \hat n = c^+c . \tag{2.3} \]

Let us show that \(\omega(\bar c)\) is the wave function of the field in the representation in which \(c^+\) is the operator of multiplication by \(\bar c\):

\[ c^+\omega(\bar c) = \bar c\,\omega(\bar c). \tag{2.4} \]

If (2.4) holds, then from the commutation relations (2.2) it follows that \(c\) is the operator of differentiation with respect to \(\bar c\):

\[ c\omega(\bar c) = \frac{d}{d\bar c}\,\omega(\bar c). \tag{2.5} \]

The equation for the eigenfunctions \(\lambda_n\) of the particle-number operator \(\hat n\) can now be written in the form

\[ \bar c\,\frac{d}{d\bar c}\lambda_n(\bar c) = n\lambda_n(\bar c), \tag{2.6} \]

where \(n\) (a positive integer) is the number of particles.

The solution of (2.6) is a power function of \(\bar c\):

\[ \lambda_n = A_n \bar c^{\,n}, \tag{2.7} \]

where \(A_n\) is a normalization factor, determined from the condition that the scalar product of \(\lambda_n\) with itself be equal to unity: \((\lambda_n,\lambda_n)=\lambda_n^*\lambda_n=1\). Hence we immediately find that \(|A_0|=1\), and, consequently, in this representation the normalized state function with no particles is \(\lambda_0=1\). Bearing this in mind, we may also represent \(\lambda_n\) in the form

\[ \lambda_n = A_n(c^+)^n\lambda_0 . \]

The scalar product of the eigenfunctions \(\lambda_n\) and \(\lambda_m\) will be equal to

\[ (\lambda_n,\lambda_m)=A_n^*A_m\bigl((c^+)^n\lambda_0,(c^+)^m\lambda_0\bigr)= \]

\[ = A_n^*A_m\bigl(\lambda_0,c^n(c^+)^m\lambda_0\bigr) = A_n^*A_m\left.\frac{d^n}{dc^n}c^m\right|_{c=0} = \]

\[ = |A_n|^2 n!\,\delta_{nm},\quad \text{or}\quad A_n=(n!)^{-1/2}, \tag{2.8} \]

where (2.4), (2.5), and the Hermitian conjugacy of the operators \(c\) and \(c^+\) have been used.

Thus, the series (2.1) is an expansion of the function \(\omega(\bar c)\) in the eigenfunctions \(\lambda_n(\bar c)\) of the meson-number operator, with coefficients \(\Psi_n\). Since the meaning of \(\Psi_n\) is known to us in advance (the probability amplitude for the presence of \(n\) mesons in the field), \(\omega(\bar c)\) is indeed the wave function of the field.

The scalar product of the wave functions \(\omega\) and \(\omega'\), according to (2.1) and (2.8), is equal to

\[ (\omega',\omega)=\sum_n \Psi_n^{\prime *}\Psi_n, \tag{2.9} \]

and for the probability amplitude \(\Psi_n\) one may write the expression

\[ \Psi_n=(n!)^{-1/2}(\lambda_0,c^n\omega), \tag{2.10} \]

or

\[ \Psi_n=(n!)^{-1/2}\frac{d^n}{d\bar c^n}\omega(\bar c)\bigg|_{\bar c=0}. \tag{2.11} \]

By virtue of (2.10), for the generating functional \(\tilde\omega\) for the functions \(\tilde\Psi_n=\sqrt{n!}\Psi_n\) the equality

\[ \tilde\omega(\bar c')=(\lambda_0,e^{c\bar c'}\omega) \tag{2.12} \]

will hold.

To explain (2.12), let us note that in the representation where \(c^+\) is the multiplication operator, the eigenfunction \(\omega_{c'}\) of the absorption operator \(c\) is determined from the equation:

\[ \frac{d}{d\bar c}\,\omega_{c'}(\bar c)=c'\omega(\bar c), \]

whence

\[ \omega_{c'}\sim \exp[c'\bar c], \]

where \(c'\) is the eigenvalue of the operator \(c\).

1. The generating functional for probability amplitudes. Let us pass to the real case. We shall denote the state of a meson by the momentum \(\mathbf p\). The probability amplitude of the presence in the field of \(n\) mesons with momenta \(\mathbf p_1,\mathbf p_2,\ldots,\mathbf p_n\) will now be a function \(\Psi_n(\mathbf p_1\ldots \mathbf p_n)\) of the meson variables. The state of the field in the general case will still be completely characterized by the collection of an infinite number of amplitudes \(\Psi_n(\mathbf p_1\ldots \mathbf p_n)\), \(n=0,1,2,\ldots,\infty\), each of which describes a system of \(n\) mesons in momentum configuration space. The functions \(\Psi_n(\mathbf p_1\ldots \mathbf p_n)\) are symmetric with respect to their variables.

We introduce an auxiliary function of the vector argument \(\bar c(\mathbf p)\) and, instead of the wave function \(\Psi_n(\mathbf p_1\ldots \mathbf p_n)\), shall consider the functional of the function \(\bar c(\mathbf p)\):

\[ \Omega_n\{\bar c\}=\frac{1}{\sqrt{n!}}\int \Psi_n(\mathbf p_1\ldots \mathbf p_n)\bar c(\mathbf p_1)\ldots \bar c(\mathbf p_n)\,d^3\mathbf p_1\ldots d^3\mathbf p_n. \tag{2.13} \]

Since specifying \(\Omega_n\{\bar c\}\) uniquely determines \(\Psi_n\), and conversely, instead of the collection of probability amplitudes \(\Psi_n\) the state of the field may be described by the generating functional:

\[ \Omega\{\bar c\}=\sum_{n=0}^{\infty}\Omega_n\{\bar c\}. \tag{2.14} \]

Let us clarify the meaning of the functional \(\Omega\{\bar c\}\). We shall see that, as in the elementary example of § 2, point 1, \(\Omega\{\bar c\}\) is the state vector of the field in the representation where the creation operator \(c^+(\mathbf p)\) is the multiplication operator. To see this, let us consider the state vector \(\Phi\) in such a representation. Since now the operator \(c^+(\mathbf p)\) depends on the meson momentum as on a parameter, the result of its action on the state vector \(\Phi\) is multiplication of \(\Phi\) by a function of \(\mathbf p\). Put

\[ c^+(\mathbf p)\Phi=\bar c(\mathbf p)\Phi \tag{2.15} \]

where on the right-hand side stands the same auxiliary function \(\bar c(\mathbf p)\) as in formula (2.13). In § 1, in the discussion of formulas (1.5)—(1.7), it was clarified that from an equality of the form (2.15), in conjunction with the commutation relation

\[ [c(\mathbf p),c^+(\mathbf p')]=\delta^3(p-p') \tag{2.16} \]

it follows that \(\Phi\) must be a functional of the function \(\bar c(\mathbf p)\). Moreover, the operator \(c(\mathbf p)\) must be the operator of functional differentiation with respect to \(\bar c(\mathbf p)\). In other words, one must have

\[ c(\mathbf p)\Phi\{\bar c\}=-\frac{\delta}{\delta \bar c(\mathbf p)}\Phi\{\bar c\}. \tag{2.17} \]

Formulas (2.15) and (2.17) replace, in the general case, formulas (2.4) and (2.5) of the elementary case, when only one state is possible for the mesons.

In the representation under consideration, the meson-number operator has the form

\[ \hat n=\int c^+(\mathbf p)c(\mathbf p)d^3p =\int \bar c(\mathbf p)\frac{\delta}{\delta \bar c(\mathbf p)}d^3p. \tag{2.18} \]

It is easy to verify that the eigenfunctions of \(\hat n\) will be products of the functions \(\bar c(\mathbf p)\): the functional

\[ \Lambda_n\{\bar c\}=A_n\bar c(\mathbf p_1)\bar c(\mathbf p_2)\ldots \bar c(\mathbf p_n) \tag{2.19} \]

(\(A_n\) is a normalization factor) belongs to the eigenvalue \(n\). In the particular case when the momenta of all the mesons are equal, \(\mathbf p_1=\mathbf p_2\ldots=\mathbf p_n\), we obtain formula (2.7) of the elementary example. Of special importance is the vacuum functional \(\Lambda_0\) (more precisely, the free-vacuum functional, since we are using operators of the noninteracting meson field). According to (2.19), the normalized vacuum functional is \(\Lambda_0=1\) in the representation under consideration. Therefore (2.19) is equivalent to the expression

\[ \Lambda_n=A_n c^+(\mathbf p_1)c^+(\mathbf p_2)\ldots c^+(\mathbf p_n)\Lambda_0. \tag{2.20} \]

By definition, the operators \(c^+(\mathbf p)\) and \(c(\mathbf p)\) are Hermitian conjugates. Therefore the scalar product \((\Omega_n,\Omega')=\Omega_n^*\Omega'\) of the functionals \(\Omega_n\) (formula (2.13)) and an arbitrary functional \(\Omega'\) can be written in the form

\[ (\Omega_n,\Omega')= \frac{1}{\sqrt{n!}}\int \Psi_n(\mathbf p_1\ldots \mathbf p_n)c^+(\mathbf p_1)\ldots c^+(\mathbf p_n)\Lambda_0\, d^3p_1\ldots d^3p_n,\Omega' = \]

\[ =\frac{1}{\sqrt{n!}}\int \Psi_n^*(\mathbf p_1\ldots \mathbf p_n)(\Lambda_0,c(\mathbf p_1)\ldots c(\mathbf p_n)\Omega')\,d^3p_1\ldots d^3p_n. \tag{2.21} \]

If one takes into account that \(c(\mathbf p)\Lambda_0=0\) and \(\Lambda_0^*c^+(\mathbf p)=0\), then the matrix element on the right in (2.21) can also be represented in the form

\[ (\Lambda_0,c(\mathbf p_1)\ldots c(\mathbf p_n)\Omega') = \left. \frac{\delta^n\Omega'}{\delta\bar c(\mathbf p_1)\ldots \delta\bar c(\mathbf p_n)} \right|_{\bar c(\mathbf p)=0}, \tag{2.22} \]

where, after carrying out the functional differentiation, one must set \(\bar c(\mathbf p_i)=0\). In particular, for \(\Omega'=\Omega'_m\) we find \((\Omega_n,\Omega'_m)=\delta_{nm}(\Omega_n,\Omega'_n)\), or, in the general case,

\[ (\Omega,\Omega')=\sum_n\int \Psi_n^*(\mathbf p_1\ldots \mathbf p_n) \Psi'_n(\mathbf p_1\ldots \mathbf p_n) d^3p_1\ldots d^3p_n. \tag{2.23} \]

If, however, in formula (2.22) we set \(\Omega'=\Lambda_n\), then for the normalization constant \(A_n\) (formula (2.19)) we obtain the value \(A_n=(n!)^{-1/2}\). Thus

Thus, the series (2.13) is an expansion of the generating functional in the eigenfunctionals of the particle-number operator \(\Lambda_n\). Since we give the meaning of probability amplitudes to the expansion coefficients \(\Psi_n(\mathbf p_1\ldots \mathbf p_n)\), it follows that \(\Omega\{\bar c\}\) coincides with the state vector \(\Phi\) in a representation in which \(c^+(\mathbf p)\) is the operator of multiplication by \(\bar c(\mathbf p)\).

We can now define the amplitude \(\Psi(\mathbf p_1\ldots \mathbf p_n)\) as the quantity

\[ \Psi_n(\mathbf p_1\ldots \mathbf p_n) = \frac{1}{\sqrt{n!}}(\Lambda_0,\ c(\mathbf p_1)\ldots c(\mathbf p_n)\Omega) = \]

\[ = \frac{\delta^n \Omega\{\bar c\}}{\delta \bar c(\mathbf p_1)\ldots \delta \bar c(\mathbf p_n)} \frac{1}{\sqrt{n!}} \bigg|_{\bar c=0}. \tag{2.24} \]

From (2.24) one immediately obtains the relations between the functions \(\Psi_n\) and \(\Psi'_n\) in the functionals \(\Omega\) and \(\Omega'\), if \(\Omega'=c^+(\mathbf p)\Omega\). Substituting \(\Omega'\) into (2.24) and carrying the function \(\bar c(\mathbf p)\) to the left through the functional derivatives, we have:

\[ \Psi'_n(\mathbf p_1\mathbf p_2\ldots \mathbf p_n) = \frac{1}{\sqrt n} \left[ \Psi_{n-1}(\mathbf p_2\mathbf p_3\ldots \mathbf p_n)\, \delta^3(\mathbf p-\mathbf p_1) \right]_p, \tag{2.25} \]

where \([\ ]_p\) denotes symmetrization of the expression inside the brackets with respect to the variables \(\mathbf p\) and \(\mathbf p_j\):

\[ [F(\mathbf p_1\ldots \mathbf p_n\mathbf p)]_p = F(\mathbf p_1\ldots \mathbf p_n\mathbf p) + F(\mathbf p_1\ldots \mathbf p_{n-1}\mathbf p\mathbf p_n) +\cdots \]

If \(\Omega''=c(\mathbf p)\Omega\), then in (2.24) an extra derivative is added, and we have:

\[ \Psi''_n(\mathbf p_1\ldots \mathbf p_n) = \sqrt{n+1}\, \Psi_{n+1}(\mathbf p\mathbf p_1\ldots \mathbf p_n). \tag{2.26} \]

Let us find the eigenfunctionals of the absorption operator \(c(\mathbf p)\). According to (2.17), for the eigenvalue \(c'\),

\[ c(\mathbf p)F_{c'}\{\bar c\} = \frac{\delta}{\delta \bar c(\mathbf p)} F_{c'}\{\bar c\} = c'(\mathbf p)F_{c'}\{\bar c\}, \]

or

\[ F_{c'}\{\bar c\}\sim \exp\left[ \int c'(\mathbf p)\bar c(\mathbf p)\,d^3p \right]. \tag{2.27} \]

Consider now

\[ (\Lambda_0,\exp\{\int c(\mathbf p)\bar c'(\mathbf p)\,d^3p\}\Omega). \tag{2.28} \]

Expanding the exponential in a series and recalling the definition (2.24) of the amplitude \(\Psi_n\), one can verify that (2.28) is nothing other than the generating functional (of the function \(\bar c'(\mathbf p)\)) for the amplitudes \(\widetilde\Psi_n=\sqrt{n!}\Psi_n\).

3. The method of functionals and Fermi statistics. The generalization of the method of the generating functional to the case of fields obeying Fermi statistics encounters difficulties. The permutation relations (1.19)—(1.20) for the creation operators \(a^+\), \(b^+\) and absorption operators \(a\), \(b\) have a plus sign instead of the minus sign for Bose-field operators, and the operators \(a^+\) and \(b^+\) anticommute:

\[ \{a^+(\mathbf p),\,a^+(\mathbf p')\}=0, \qquad \{a^+(\mathbf p),\,b^+(\mathbf p')\}=0. \tag{2.29} \]

In this case one cannot introduce a representation in which the action of the operator \(a^+(\mathbf p)\) on the functional \(\Omega\) would reduce to multiplication by the function \(\bar a(\mathbf p)\): if one admits that the equality

\[ a^+(\mathbf p)\Omega=\bar a(\mathbf p)\Omega, \]

then the equality

\[ a^+(\mathbf p)a^+(\mathbf p')\Omega =\bar a(\mathbf p)\bar a(\mathbf p')\Omega =\bar a(\mathbf p')\bar a(\mathbf p)\Omega, \]

would also have to hold, contradicting the commutation relation (2.29), from which, moreover, it follows that \(\bar a(\mathbf p)^2=0\).

One way of generalization is connected with a formula analogous to formula (2.20). According to (2.20), the expansion of \(\Omega\) in the functional power series (1.15) is equivalent to an expansion in a series in products of creation operators acting on the normalized vacuum functional \(\Lambda_0\). The products

\[ \left. \begin{gathered} \Lambda_{nml}=(n!m!l!)^{-1/2} c^+(\mathbf k_1)\ldots c^+(\mathbf k_l)b^+(\mathbf q_m)\ldots b^+(\mathbf q_1) a^+(\mathbf p_n)\ldots a^+(\mathbf p_1)\Lambda_0,\\ \left[a(\mathbf p)\Lambda_0=0,\qquad b(\mathbf q)\Lambda_0=0,\qquad c(\mathbf k)\Lambda_0=0,\qquad (\Lambda_0,\Lambda_0)=1\right] \end{gathered} \right\} \tag{2.30} \]

are eigenvectors of the particle-number operators \(\int a^+(\mathbf p)a(\mathbf p)\,d^3p\) and \(\int b^+(\mathbf q)b(\mathbf q)\,d^3q\) in the case of Fermi statistics. Let \(\Psi_{nml}(\mathbf p_1\ldots \mathbf p_n \mid \mathbf q_1\ldots \mathbf q_m \mid \mathbf k_1\ldots \mathbf k_l)\) denote the probability amplitude, referred to the instant of time \(t\), for the presence in the field of \(n\) nucleons with momenta \(\mathbf p_1,\ldots,\mathbf p_n\), \(m\) antinucleons with momenta \(\mathbf q_1,\ldots,\mathbf q_m\), and \(l\) mesons with momenta \(\mathbf k_1,\ldots,\mathbf k_l\). The function \(\Psi_{nml}\) is antisymmetric with respect to the variables \(\mathbf p_1,\ldots,\mathbf p_n\); \(\mathbf q_1,\ldots,\mathbf q_m\), and symmetric with respect to the variables \(\mathbf k_1,\ldots,\mathbf k_l\). The state vector \(\Omega\) is then equal to

\[ \Omega=\sum_{nml}^{\infty}\Omega_{nml} =\sum_{nml}(n!m!l!)^{-1/2} \int \Psi_{nml}(\mathbf p_1\ldots \mid \mathbf q_1\ldots \mid \mathbf k_1\ldots)\times \]

\[ \times c^+(\mathbf k_1)\ldots c^+(\mathbf k_l)b^+(\mathbf q_m)\ldots b^+(\mathbf q_1) a^+(\mathbf p_n)\ldots a^+(\mathbf p_1)\Lambda_0. \tag{2.31} \]

For the quantities \(\Psi_{nml}\) we obtain:

\[ \Psi_{nml}(\mathbf p_1\ldots \mathbf p_n \mid \mathbf q_1\ldots \mathbf q_m \mid \mathbf k_1\ldots \mathbf k_l)= \]

\[ =(n!m!l!)^{-1/2} (\Lambda_0,\, a^+(\mathbf p_1)\ldots a^+(\mathbf p_n)b^+(\mathbf q_1)\ldots \]

\[ \ldots b^+(\mathbf q_m)c^+(\mathbf k_1)\ldots c^+(\mathbf k_l)\Omega). \tag{2.32} \]

From (1.4) it is easy to find the result of the action of the creation operators \(a^+\) and annihilation operators \(a\) on \(\Omega\). If \(\Omega'=a^+(\mathbf p)\Omega\), then the amplitudes \(\Psi'_{nml}\) in \(\Omega'\) are related to the amplitudes \(\Psi_{nml}\) in \(\Omega\) by a relation obtained by substituting \(\Omega'\) instead of \(\Omega\) in (2.32) and moving the operator \(a^+(\mathbf p)\) to the left toward \(\Lambda_0\). Bearing in mind that \((\Lambda_0,a^+(\mathbf p)\Omega)=0\) for any \(\Omega\), we find

\[ \Psi'_{nml}(\mathbf p_1\ldots \mathbf p_n\mid \ldots \mid \ldots) =\frac{1}{\sqrt n}\{\delta^3(p-p_1)\Psi_{n-1\,ml}(\mathbf p_2\ldots \mathbf p_n\mid \ldots \mid \ldots)- \]

\[ -\delta^3(p-p_2)\Psi_{n-1\,ml}(\mathbf p_1,\mathbf p_3\ldots \mathbf p_n\mid \ldots \mid \ldots)+\ldots \]

\[ \ldots-(-1)^n\delta^3(p-p_n)\Psi_{n-1\,ml}(\mathbf p_1\ldots \mathbf p_{n-1}\mid \ldots \mid \ldots)\}. \tag{2.33} \]

If \(\Omega''=a(\mathbf p)\Omega\), then the corresponding amplitude \(\Psi''_{nml}\) is determined directly from (2.32):

\[ \Psi''_{nml}(\mathbf p_1\ldots \mathbf p_n\mid \ldots \mid \ldots) =\sqrt{n+1}\,\Psi_{n+1\,ml}(\mathbf p\,\mathbf p_1\ldots \mathbf p_n\mid \ldots \mid \ldots). \tag{2.34} \]

However, the representation of the state vector in the form (2.31) is, in essence, a departure from the functional method and a return to the operator method of second quantization. In order, in accordance with the commutation relations (2.29), to develop the functional method for Fermi statistics, it is necessary to define a system of such quantities \(\bar a(\mathbf p)\) and \(\bar b(\mathbf q)\), depending on the momenta of the nucleon \(\mathbf p\) and antinucleon \(\mathbf q\), which anticommute with one another

\[ \left. \begin{aligned} \{\bar a(\mathbf p),\bar a(\mathbf p')\}&=0,\qquad \{\bar b(\mathbf q),\bar b(\mathbf q')\}=0,\\ \{\bar a(\mathbf p),\bar b(\mathbf q)\}&=0 \end{aligned} \right\} \tag{2.35} \]

and with the nucleon-field operators and, at the same time, are proportional to the unit matrix in the space of the nucleon, antinucleon, and meson occupation numbers. This can be done with the aid of the notion of external sources of the nucleon field \(\eta(x)\) and \(\bar\eta(x)\), introduced by Schwinger\({}^{9}\). The sources \(\eta(x)\) and \(\bar\eta(x)\) may be interpreted as quantities belonging to some other prescribed field. Since the operators of different spinor fields anticommute with one another, the sources \(\eta\) and \(\bar\eta\) anticommute with the operators \(\Psi\) and \(\bar\Psi\). Moreover, \(\{\eta,\bar\eta\}=0\), since the spinor field \(\eta,\bar\eta\) is assumed to be prescribed. The quantities \(\bar a(k)\) and \(\bar b(q)\) with properties (2.35) may be identified with the components of the expansion of the negative-frequency parts of the external sources in a three-dimensional Fourier integral:

\[ \left. \begin{aligned} \bar\eta^{(-)}(x)&=\frac{1}{(2\pi)^{3/2}}\int \bar a(\mathbf p)\,\bar U(\mathbf p)\,e^{-ipx}\,d^3p,\\ \eta^{(-)}(x)&=\frac{1}{(2\pi)^{3/2}}\int \bar b(\mathbf p)\,\bar V(\mathbf p)\,e^{-ipx}\,d^3p, \end{aligned} \right\} \tag{2.36} \]

where \(V(\mathbf p)\) and \(U(\mathbf p)\) are Dirac spinors for the prescribed external field, analogous to the spinors \(v(\mathbf p)\) and \(u(\mathbf p)\) in formulas (1.17)—(1.18).

We can now pass to a representation in which \(a^{+}(\mathbf p)\) and \(b^{+}(\mathbf q)\) are multiplication operators:

\[ a^{+}(\mathbf p)\Omega=\bar a(\mathbf p)\Omega;\qquad b^{+}(\mathbf p)\Omega=\bar b(\mathbf p)\Omega . \tag{2.37} \]

In connection with the anticommutativity of \(\bar a\) and \(\bar b\), when defining functional derivatives with respect to \(\bar a\) and \(\bar b\) the order of the factors is essential. By definition,

\[ \left. \begin{aligned} \delta\Omega[\bar a]&=\int \delta\bar a(\mathbf p)\,\frac{\delta\Omega}{\delta\bar a(\mathbf p)}\,d^3p,\\ \delta\Omega[\bar b]&=\int \frac{\delta\Omega}{\delta\bar b(\mathbf p)}\,\delta\bar b(\mathbf p)\,d^3p . \end{aligned} \right\} \tag{2.38} \]

The functional-derivative operators with respect to \(\bar a\) and \(\bar b\) anticommute:

\[ \left\{\frac{\delta}{\delta\bar a(\mathbf p)},\,\frac{\delta}{\delta\bar b(\mathbf q)}\right\}=0,\qquad \left\{\frac{\delta}{\delta\bar a(\mathbf p)},\,\frac{\delta}{\delta\bar a(\mathbf p')}\right\}=0 . \]

It follows from definition (2.38) that

\[ \left\{\frac{\delta}{\delta\bar a(\mathbf p)},\,\bar a(\mathbf p')\right\}=\delta^3(p-p'),\qquad \left\{\frac{\delta}{\delta\bar b(\mathbf p)},\,\bar b(\mathbf p')\right\}=\delta^3(p-p'). \]

Hence it follows that, in complete agreement with the method of functionals for Bose statistics, one may put

\[ a(\mathbf p)=\frac{\delta}{\delta\bar a(\mathbf p)}\,\Omega;\qquad b(\mathbf p)=\frac{\delta}{\delta\bar b(\mathbf p)}\,\Omega . \tag{2.39} \]

By analogy with the case of Bose statistics, one may conclude that in the general case of interacting nucleon and meson fields the expansion of the state vector as a functional of \(\bar a\), \(\bar b\), and \(\bar c\) differs from (2.31) by replacing the products

\[ c^{+}(\mathbf k_1)\ldots c^{+}(\mathbf k_e)b^{+}(\mathbf q_m)\ldots b^{+}(\mathbf q_1)a^{+}(\mathbf p_n)\ldots a^{+}(\mathbf p_1)\Lambda_0 \]

by

\[ \bar c(\mathbf k_1)\ldots \bar c(\mathbf k_e)\bar b(\mathbf q_m)\ldots \bar b(\mathbf q_1)\bar a(\mathbf p_n)\ldots \bar a(\mathbf p_1). \]

Consider the operator

\[ R[\bar a',\bar b']\equiv \exp\left\{\int\left[\bar a'(\mathbf p)a(\mathbf p)+b(\mathbf p)\bar b'(\mathbf p)\right]\,d^3p\right\}, \tag{2.40} \]

which is a functional of \(\bar a'(p)\) and \(\bar b'(p)\). We have:

\[ \frac{\delta}{\delta \bar a'(p)}R=Ra(p),\qquad \frac{\delta^n R}{\delta \bar a'(p_1)\ldots \delta \bar a'(p_n)}=Ra(p_n)\ldots a(p_1) \]

and analogous formulas for the functional derivatives with respect to \(\bar b(q)\). Comparing these formulas with expression (2.32) for the probability amplitude, one can arrive at the conclusion that the generating functional \(\widetilde\Omega\) for the amplitudes \(\widetilde\Psi_{nml}=\sqrt{n!m!l!}\Psi_{nml}\) can be represented in the form

\[ \widetilde\Omega[\bar a',\bar b',\bar c']=\left(\Lambda_0,\; Re^{\int \bar c'(k)c(k)\,d^3k}\Omega\right). \tag{2.41} \]

Instead of formula (2.32), for the probability amplitudes \(\Psi_{nml}\) one may then write:

\[ \Psi_{nml}(p_1\ldots p_n\mid q_1\ldots q_m\mid k_1\ldots k_l)= \]

\[ =\frac{1}{\sqrt{n!m!l!}}\, \frac{\delta^{n+m+l}\Omega} {\delta\bar a(p_1)\ldots \delta\bar a(p_n)\delta\bar b(q_1)\ldots \delta\bar b(q_m)\delta\bar c(k_1)\ldots \delta\bar c(k_l)} \bigg|_{\bar a=\bar b=\bar c=0}. \tag{2.42} \]

Just as in the derivation of formula (2.23), it can be shown that from the condition of Hermitian conjugacy of the operators \(a\) and \(a^+\), \(b\) and \(b^+\) there follows the definition of the scalar product of two functionals \(\Omega\) and \(\Omega'\):

\[ (\Omega',\Omega)= \]

\[ =\sum_{nml}\int \psi^{\prime *}_{nml}(p_1\ldots p_n\mid q_1\ldots q_m\mid k_1\ldots k_l) \psi_{nml}(p_1\ldots p_n\mid q_1\ldots q_m\mid k_1\ldots k_l)\times \]

\[ \times d^3p_1\ldots d^3p_n d^3q_1\ldots d^3q_m d^3k_1\ldots d^3k_l. \tag{2.43} \]

4. Equations for the state functional. To derive the equation satisfied by the state functional considered in § 2, it is necessary to express the energy operator \(H\) through the creation operators \(a^+(p)\), \(b^+(q)\), \(c^+(k)\) and the annihilation operators \(a(p)\), \(b(q)\), \(c(k)\) of the various particles:

\[ H=H(a,b,c,a^+,b^+,c^+). \tag{2.44} \]

In the method of functionals, as was explained in § 2, one can introduce a representation in which the creation operators are multiplication operators by the auxiliary quantities \(\bar a(p)\), \(\bar b(q)\), \(\bar c(k)\). Then in this representation the action of the annihilation operators reduces to taking the functional derivative of the state vector \(\Omega\) with respect to these auxiliary quantities. Therefore, in order to obtain the equation of motion for the functional \(\Omega\), it is necessary in expression (2.44) to make the substitution:

\[ \begin{gathered} a^+(p)\to \bar a(p),\qquad b^+(q)\to \bar b(q),\qquad c^+(k)\to \bar c(k),\\ a(p)\to \delta/\delta\bar a(p),\qquad b(q)\to \delta/\delta\bar b(q),\qquad c(k)\to \delta/\delta\bar c(k), \end{gathered} \tag{2.45} \]

after which the Schrödinger equation can be written in the form

\[ H(\bar a,\bar b,\bar c,\delta/\delta\bar a,\delta/\delta\bar b,\delta/\delta\bar c)=i\partial\Omega/\partial t. \tag{2.46} \]

From the functional equation (2.46) one can obtain the equation for the probability amplitudes \(\Psi_{nml}\) (2.32).

In the problem under consideration of the interaction of a neutral pseudoscalar meson field with a nucleon field, the energy operator \(H\) has, in the case

of pseudoscalar coupling has the following form:

\[ \left. \begin{aligned} H&=H_1+H_2+H_{12},\\ H_1&=\frac12\int d^3x\left(\pi^2+(\nabla\varphi)^2+\mu^2\varphi^2\right),\\ H_2&=\int d^3x\left(\bar\psi(\gamma\nabla)\psi+m\bar\psi\psi\right),\\ H_{12}&=ig\int d^3x\,\bar\psi\gamma_5\psi\varphi . \end{aligned} \right\} \tag{2.47} \]

Using the Fourier expansion of the field operators (1.16)—(1.18), and taking (2.45) into account, we obtain for \(H_1\) and \(H_2\) the following expressions:

\[ H_1=\int d^3k\,k_0\bar c(\mathbf{k})\,\partial/\partial\bar c(\mathbf{k}), \]

\[ H_2=\sum_{i=1}^{2}\int d^3p\,E(p)\left\{\bar a_i(\mathbf{p})\,\partial/\partial\bar a_i(\mathbf{p})+\bar b_i(\mathbf{p})\,\partial/\partial\bar b_i(\mathbf{p})\right\}, \]

\[ k_0=+\sqrt{\mu^2+\mathbf{k}^2},\qquad p_0=E(p)=\sqrt{m^2+\mathbf{p}^2}. \]

For \(H_{12}\) one obtains a rather cumbersome expression, consisting of eight terms corresponding to different combinations of creation and annihilation operators of the meson and nucleon fields. The equations for the probability amplitudes (2.32) can be obtained if one uses the expansion (2.31) for the state functional \(\Omega\) and equates in equation (2.46) the terms containing the same number of auxiliary quantities \(\bar a,\bar b,\bar c\). Let us explain this on a simpler example, when the interaction operator of the meson and nucleon fields has the form:

\[ H_{12}=ig\gamma_5\varphi(x). \]

In this case closed loops corresponding to the creation of virtual nucleon–antinucleon pairs are not taken into account.

The equation for the wave functional will take the simpler form:

\[ (\gamma_\mu\partial/\partial x_\mu+m)\Omega(x)=-ig\gamma_5\varphi(x)\Omega(x), \tag{2.48} \]

where the operator \(\varphi(x)\) can be written in the form

\[ \varphi(x)=\frac{1}{(2\pi)^{3/2}}\int\frac{d^3k}{\sqrt{2k_0}} \left(e^{ik_\mu x_\mu}\partial/\partial\bar c(\mathbf{k})+e^{-ik_\mu x_\mu}\bar c(\mathbf{k})\right). \tag{2.49} \]

From (2.48) and (2.49) it follows that

\[ (\gamma_\mu\partial/\partial x_\mu+m)\Omega =\int d^3k\,[G^+(k)\partial\Omega/\partial\bar c(\mathbf{k})-G(k)\bar c(\mathbf{k})\Omega], \tag{2.50} \]

where

\[ G(k)=\frac{i}{(2\pi)^{3/2}}\,g\,\frac{e^{-ik_\mu x_\mu}}{\sqrt{2k_0}}\gamma_5 . \tag{2.51} \]

In equation (2.50) the coefficients \(G(k)\) and \(G^+(k)\) explicitly depend on time. This dependence can be eliminated if one passes to the Schrödinger representation:

\[ L_0=\exp(-iH_1t)L\exp(iH_1t),\qquad \Omega=\exp(+iH_1t)\Omega_0. \tag{2.52} \]

In this case the operators \(\partial/\partial \bar c(\mathbf{k})\), \(\bar c(\mathbf{k})\) transform according to

\[ \left. \begin{aligned} \exp(-iH_1t)\,\partial/\partial \bar c(\mathbf{k})\,\exp(iH_1t) &=\exp(ik_0t)\,\partial/\partial \bar c(\mathbf{k}),\\ \exp(-iH_1t)\,\bar c(\mathbf{k})\,\exp(iH_1t) &=\exp(-ik_0t)\,\bar c(\mathbf{k}). \end{aligned} \right\} \tag{2.53} \]

As a result, for the functional \(\Omega\) one obtains the following equation*):

\[ \left(\gamma_\mu \partial/\partial x_\mu + m + \gamma_4 \int k_0 \bar c(\mathbf{k})\,\partial/\partial \bar c(\mathbf{k})\right)\Omega = \]

\[ = \int d^3k\,\{G_0^{+}(\mathbf{k})\,\mathfrak{S}\Omega/\partial \bar c(\mathbf{k}) - G_0(\mathbf{k})\,\bar c(\mathbf{k})\Omega\}. \tag{2.54} \]

From equation (2.54), on the basis of (2.13) and (2.14), the following equation for the probability amplitude is obtained:

\[ \left(\gamma_\mu \partial/\partial x_\mu + m + \gamma_4 \sum_k N_k k_0\right) \psi_n(x,k_1\ldots k_n)= \]

\[ = \sqrt{n+1}\int d^3k\,G_0^{+}(k)\psi_{n+1}(x,k,k_1\ldots k_n)- \]

\[ -\frac{1}{\sqrt n}\{G_0(k)\psi_{n-1}(x,k_2\ldots k_n)\}_{\mathrm{sym}}. \tag{2.55} \]

Equation (2.55) relates the amplitude \(\psi_n\) to the amplitudes \(\psi_{n+1}\) and \(\psi_{n-1}\), for which, in turn, analogous equations can be written. In the end an infinite system of “coupled” equations is obtained. In the one-meson approximation, for example, we obtain:

\[ \left. \begin{aligned} (\gamma_\mu \partial/\partial x_\mu + m)\psi_0(x) &=\int d^3k\,G_0^{+}(k)\psi_1(k,x),\\ (\gamma_\mu \partial/\partial x_\mu + m+\gamma_4 k_0)\psi_1 &=-G_0(k)\psi_0(x). \end{aligned} \right\} \tag{2.56} \]

Equations (2.55) can also be derived directly from expression (2.32) for the probability amplitudes, if one makes use of the relation**):

\[ i\partial\psi_n/\partial t=(\Lambda_0,c^{+}(k_1)\ldots c^{+}(k_n)H_{int}\Omega). \tag{2.57} \]

The right-hand side of (2.57) can easily be calculated on the basis of (2.45) and (2.47).

In an analogous way, equations can be obtained for probability amplitudes containing Fermi-field operators.

The system of coupled equations (2.55) was obtained by V. A. Fock in 1934.\(^{2}\) From these equations, in the approximation (2.56), the Breit formula, the Møller formula were obtained, and the question of the natural width of spectral lines was considered. Subsequently these equations were applied by A. A. Smirnov\(^{5}\) and A. G. Vlasov\(^{6}\) to the study of the interaction of the electron with the electromagnetic field. The theory of the natural width of spectral lines in the two-photon approximation was considered in work \(^{7}\). The method of coupled equations acquired especially great practical importance in meson theory, since in the latter perturbation theory becomes unsuitable. In meson theory, independently of the preceding works in the field of quantum electrodynamics, equations (2.55) were obtained and applied to the study of the problem of nuclear forces by I. E. Tamm\(^{14}\) and S. M. Dancoff\(^{15}\).

The system of equations (2.55) can be solved by the method of perturbation theory and by the method of truncated equations. The latter is known in the literature under the name of the Tamm–Dancoff method and consists in the following.

*) The state functional is still denoted by the letter \(\Omega\) instead of \(\Omega_0\).

**) The interaction representation is used.

It is assumed that, when solving a system of equations of type (2.55), one may neglect all amplitudes that describe such field states in which there are \(N+1, N+2\), etc., mesons. In other words, in the expansion \(\Omega=\sum_{n=0}^{\infty}\Omega_n\) one retains, in this approximation, \(N+1\) terms, and the rest are set equal to zero. As a result one obtains a “truncated” system of equations, which must then be solved exactly (for example, by numerical integration) with definite boundary conditions. In the one-meson approximation (2.56) one can eliminate the amplitude \(\psi_1\) from the second equation and substitute it into the first; as a result one obtains one integral equation. In higher approximations such a procedure can be carried out only under the assumption that the probability amplitude corresponding to the minimally possible number of particles in the given problem is the principal one, while the others play the role of corrections\(^*\). Then for the principal amplitude one can obtain one integral equation. However, such a method is an oversimplification of the problem, since in this case the kernel will be represented in the form of a series in the constant \(g^2\), with the individual terms of the series being comparable in magnitude, and therefore such an expansion is unsuitable for investigation even in the asymptotic sense.

When solving the system of equations (2.55) by the method of perturbation theory\({}^{25}\), it should be noted that each of these equations is an inhomogeneous Dirac-type equation

\[ D(x)\psi=i(\gamma_\mu\partial/\partial x_\mu+m)\psi(x)=L(x). \tag{2.58} \]

The general solution of this equation may be represented in the form

\[ \left. \begin{aligned} \psi(x)&=\varphi(x)+\int K(x,x')L(x')\,d^4x',\\ D(x)\varphi(x)&=0. \end{aligned} \right\} \tag{2.59} \]

The Green function \(K(x,x')\) must be chosen in accordance with positron theory. The interaction described by the term \(L(x)\) must be introduced adiabatically at \(t=-\infty\) and switched off at \(t=+\infty\), attaining its full magnitude at all finite times. By successively eliminating amplitudes one can arrive at one integral equation, which corresponds to the integral equation in Feynman’s theory. Renormalization in this method of solution is carried out in the same way as in perturbation theory.

Along with probability amplitudes that are defined in momentum space, one may also consider probability amplitudes in coordinate space\({}^{13,20,24}\). The latter may be used to clarify the possibility of renormalization within the method of truncated equations\({}^{4,16}\) and to compare probability amplitudes with the relativistic wave functions that will be considered in § 4. To introduce probability amplitudes in coordinate space, let us represent the operators of the noninteracting fields \(\varphi(x)\) and \(\psi(x)\) as sums of terms containing only positive and only negative frequencies:

\[ \varphi(x)=\varphi^{(+)}(x)+\varphi^{(-)}(x),\quad \psi(x)=\psi^{(+)}(x)+\psi^{(-)}(x), \tag{2.60} \]

where

\[ \varphi^{(\pm)}(x)=\int d\sigma_\mu(x')\left\{\Delta^{(\pm)}(x-x')\overleftrightarrow{\partial}/\partial x_\mu'\,\varphi(x')\right\}, \tag{2.61a} \]

\[ \psi^{(\pm)}(x)=\int d\sigma_\mu(x')S^{(\pm)}(x-x')\gamma_\mu\psi(x'). \tag{2.61b} \]

\[ \text{}^* \text{Levy--Klein method}^{17,18}; \text{ see also }^{7}. \]

The operators \(\varphi^{(\pm)}(x)\), \(\psi^{(\pm)}(x)\) satisfy the following commutation relations:

\[ [\varphi^{(+)}(x),\varphi^{(-)}(x')] = i\Delta^{+}(x-x')=-i\Delta^{(-)}(x-x'), \tag{2.62} \]

\[ \left\{\psi_{\lambda}^{(+)}(x),\overline{\psi_{\mu}^{(+)}}(x')\right\} =-iS_{\lambda\mu}^{(+)}(x-x'), \tag{2.63} \]

\[ \Delta^{\pm}(x)=\mp i/(2\pi)^3\int \frac{d^3k}{2k_0}\exp(\mp ik_\mu x_\mu), \tag{2.64} \]

\[ S^{(\pm)}(x)=(\gamma_\mu \partial/\partial x_\mu-m)\Delta^{(\pm)}(x). \tag{2.65} \]

The absorption operators \(a(\mathbf p)\), \(b(\mathbf q)\), and \(c(\mathbf k)\) are expressed in terms of the field operators \(\psi(x)\), \(\overline{\psi}(x)\), \(\varphi(x)\) according to formulas (1.12) and (1.15). In these expressions only the positive-frequency parts of the operators contribute to the corresponding integrals; therefore the creation operators needed by us \(a^{+}(\mathbf p)\), \(b^{+}(\mathbf q)\), \(c^{+}(\mathbf k)\) can be represented by the following formulas *):

\[ \left. \begin{aligned} a^{+}(\mathbf p)&=\int d\sigma_\mu\,\overline{\psi}^{(-)}(x)\gamma_\mu u^{(p)}(x),\\ b^{+}(\mathbf q)&=\int d\sigma_\mu\,\overline{v}^{(q)}(x)\gamma_\mu\psi^{(-)}(x), \end{aligned} \right\} \tag{2.66} \]

\[ c^{+}(\mathbf k)=-i\int d\sigma_\mu\,\varphi^{(-)}(x)\frac{\overleftrightarrow{\partial}}{\partial x_\mu}f^{(k)}(x). \tag{2.67} \]

Let us now transform, with the aid of (2.67), expression (2.31) for the functional \(\Omega\) to coordinate space. For brevity of notation we shall carry out this transformation only for the amplitude of a neutral Bose field **). The functional
\(\Omega_1=\int d^3k\,\psi(k)c^{+}(k)\Lambda_0\), with the aid of (2.67), is transformed into the following form:

\[ \Omega_1=-i\iint d^3k\,d\sigma_\mu\,\Psi_1(k)\varphi^{(-)}(x) \frac{\overleftrightarrow{\partial}}{\partial x_\mu}f^{(k)}(x) = \]

\[ =-i\int d\sigma_\mu\,(\varphi)^{-}(x) \frac{\overleftrightarrow{\partial}}{\partial x_\mu}\Psi_1(x)\Lambda_0, \tag{2.68} \]

where \(\Psi_1(x)=\int \psi_1(k)f^{(k)}(x)\,d^3k\).

Similarly,

\[ \left. \begin{aligned} \Omega_n&=\frac{(-1)^n}{\sqrt{n!}}\int d\sigma_{\mu_1}(x_1)\cdots d\sigma_{\mu_n}(x_n) \prod_j \varphi^{(-)}(x_j)\frac{\overleftrightarrow{\partial}}{\partial x_{\mu_j}} \times\\ &\qquad\qquad\times \psi_n(x_1\ldots x_n,\sigma)\Lambda_0,\\ \psi_n(x_1\ldots x_n,\sigma)&=\int \psi(k_1\ldots k_n) f^{(k_1)}(x_1)\ldots f^{(k_n)}(x_n) \,d^3x_1\ldots d^3x_n . \end{aligned} \right\} \tag{2.69} \]

The amplitude \(\psi_n(x_1\ldots x_n)\) is symmetric with respect to its arguments and satisfies the following relation:

\[ \psi_n(x_1\ldots x_n\sigma)= \int d\sigma_\mu(x'_j) \left( \Delta^{(+)}(x_j-x'_j) \frac{\overleftrightarrow{\partial}}{\partial x'_j} \right) \times \]

\[ \times\psi_n(x_1\ldots x'_j\ldots x_n,\sigma), \tag{2.70} \]

whose validity follows from the independence of expression (2.70) from

*) See formulas (1.16)—(1.18).

**) Analogous transformations for a charged Bose field and for Fermi fields can be found in work \(^{24}\).

of the choice of the surface \(\sigma(x)\) and from the fact that the function \(\Delta^{(+)}(x-x')\) is a positive-frequency solution of the Klein—Fock equation.

The operators \(\varphi^{(+)}(x)\) and \(\varphi^{(-)}(x)\) have the following representation. If, as in § 2, we denote \(\Omega''=\varphi^{(+)}(x)\Omega\), then the functions

\[ \psi_n''(x_1\ldots x_n,\sigma)\quad \text{and}\quad \psi_n(x,x_1\ldots x_n,\sigma) \]

will be related in the following way:

\[ \psi_n''(x_1\ldots x_n,\sigma)=(n+1)^{1/2}\int d\sigma_\mu(x')\Delta^{(+)}(x-x')\frac{\overleftrightarrow{\partial}}{\partial x'_\mu}\Psi_n\times \]
\[ \times (x',x_1\ldots x_n,\sigma)=(n+1)^{1/2}\psi_{n+1}(x,x_1\ldots x_n,\sigma). \tag{2.71} \]

Similarly, starting from the relation \(\varphi^{(-)}(x)\Omega=\Omega'\), we obtain:

\[ \Psi_n'(x_1\ldots x_n)=\frac{i}{\sqrt{n!}}\left\{\Delta^{(+)}(x-x_1)\psi_{n-1}(x_2\ldots x_n)\right\}_{sym}. \tag{2.72} \]

In coordinate space the basis vectors will be the following expressions:

\[ \Lambda_n=\frac{1}{\sqrt{n!}}\varphi^{(-)}(x_1)\ldots \varphi^{(-)}(x_n)\Lambda_0. \tag{2.73} \]

The particle-number operator itself can be written in the form

\[ N=i^{-1}\int d\sigma_\mu\left(\varphi^{(-)}(x)\frac{\overleftrightarrow{\partial}}{\partial x_\mu}\varphi^{(+)}(x)\right). \tag{2.74} \]

It is easy to see that \(N\Lambda_n=n\Lambda_n\). Thus the expansion of the functional \(\Omega\) in the functionals \(\Omega_n\) is precisely the expansion in the eigenfunctionals of the particle-number operator. The “coefficients” of this expansion are the probability amplitudes for detecting \(n\) particles in coordinate space. The functionals \(\Omega_n\) and \(\Omega_{n'}\) with different subscripts are mutually orthogonal; moreover, as is not difficult to verify, the following relation holds:

\[ (\Omega,\Omega)=(\Lambda_0,\Lambda_0)+\sum_{n=1}^{\infty}(-1)^n\int d\sigma_{\mu_1}(x_1)\ldots d\sigma_{\mu_n}(x_n)\times \]
\[ \times \psi_n^*(x_1\ldots x_n,\sigma)\prod_j\frac{\overleftrightarrow{\partial}}{\partial (x_j)_{\mu_j}}\psi_n(x_1\ldots x_n,\sigma). \tag{2.75} \]

With the aid of formulas (2.71) and (2.72) one can show that

\[ (\Omega_n,\varphi^{(+)}(x)\Omega_{n+1})=(\varphi^{(-)}(x)\Omega_n,\Omega_{n+1}), \]

i.e. that \(\varphi^{(+)}(x)\) and \(\varphi^{(-)}(x)\) are Hermitian conjugates.

From expression (2.72) it follows that

\[ \psi_n(x_1\ldots x_n)=\frac{1}{\sqrt{n!}}\left(\Lambda_0,\varphi^{(+)}(x_1)\ldots \varphi^{(+)}(x_n)\Omega\{\sigma\}\right). \tag{2.76} \]

In the general case, the expression for the probability amplitude in coordinate space, describing a system of \(l\) nucleons, \(m\) antinucleons, and \(n\) mesons, can be written (omitting a numerical coefficient) in the form

\[ \psi_{lmn}=(x_1\ldots x_l\mid y_1\ldots y_m\mid z_1\ldots z_n)= \]

\[ =(\Lambda_0,\bar{\psi}^{(+)}(x_1)\ldots \bar{\psi}^{(+)}(x_l);\ \psi^{(+)}(y_1)\ldots \psi^{(+)}(y_m);\ \varphi^{(+)}(z_1)\ldots \varphi^{(+)}(z_n)\Omega\{\sigma\}). \tag{2.77} \]

In what follows we shall assume that in expression (2.77) the state vector \(\Omega\) is taken in the interaction representation:

\[ \frac{i\delta \Omega\{\sigma\}}{\delta\sigma(x)}=H_{int}(x)\Omega\{\sigma\}. \tag{2.78} \]

Then the equation for the amplitudes in the coordinate representation can be obtained in the same way as the equation for the amplitude (2.32):

\[ \frac{i\delta}{\delta\sigma(x)} \psi(x_1\ldots x_l\mid y_1\ldots y_m\mid z_1\ldots z_n)= \]

\[ =\left(\Lambda_0,\ \bar{\psi}^{(+)}(x_1)\ldots \psi^{(+)}(y_1)\ldots \varphi^{(+)}(z_1)\ldots H_{int}\Omega\right). \tag{2.79} \]

Here the right-hand side of the equation must be represented, with the aid of relations (2.63), in the form of \(N\) products; thereby the amplitude \(\psi_{lmn}\) under consideration will be connected with other amplitudes, for which analogous equations can be written. For example, for the one-meson amplitude, taking into account \(H_{int}=ig\bar{\psi}\gamma_\lambda\varphi\):

\[ \frac{i\delta}{\delta\sigma(x)} \left(\Lambda_0,\ \varphi^{(+)}(x_1)\Omega\right) = -g(\gamma_5)_{\alpha\beta}\left\{\Delta^{+}(x_1-x)\times\right. \]

\[ \left.\times\left(\Lambda_0,\ \bar{\psi}^{(+)}_\alpha\psi^{(+)}_\beta(x)\Omega\right) -i\left(\Lambda_0,\ \bar{\psi}^{(+)}_\alpha(x)\psi^{(+)}_\beta(x)\varphi^{(+)}(x_1)\varphi^{(+)}(x)\Omega\right)\right\}. \tag{2.80} \]

For the one-nucleon amplitude

\[ \frac{i\delta}{\delta\sigma(x)} \left(\Lambda_0,\ \psi^{(+)}_\lambda(x_1)\Omega\right) = g(\gamma_5)_{\alpha\beta}\left\{S^{(+)}_{\lambda\alpha}(x_1-x)\times\right. \]

\[ \left.\times\left(\Lambda_0,\ \psi^{(+)}_\beta(x)\varphi^{(+)}(x)\Omega\right) -i\left(\Lambda_0,\ \bar{\psi}^{(+)}_\alpha(x)\psi^{(+)}_\lambda(x_1)\psi^{(+)}_\beta(x)\varphi^{(+)}(x)\Omega\right)\right\}. \tag{2.81} \]

The amplitudes (2.77) were considered by Chini \(^{16}\) for the study of the possibility of renormalization in the method of truncated equations. A criticism of Chini’s method and a detailed investigation of equations (2.55) can be found in work \(^{4}\).

Let us pass to the question of the generating functional for the amplitudes

\[ \tilde{\psi}_{lmn}=\sqrt{l!m!n!}\,\psi_{lmn}. \]

The latter can be written in the form (see also (2.41))

\[ F=\left(\Lambda_0,\ \exp\left\{\int d^3x\left(\eta(x)\psi^{(+)}(x)+\psi^{(-)}(x)\eta^*(x)+c(x)\varphi^{(+)}(x)\right)\right\}\Omega\right)= \]

\[ =\sum_{l,m,n}\frac{1}{m!n!l!}\,F_{mnl}\Lambda_0 =\left(\Lambda_0,\ M\Omega\right), \tag{2.82} \]

where

\[ F_{lmn}\{\eta,\eta^*,c\}=\int \psi(x_1\ldots x_l\mid y_1\ldots y_m\mid z_1\ldots z_n)\times \]

\[ \times \eta^*(x_1)\ldots \eta(y)\ldots c(z)\ldots d^3x\ldots d^3y\ldots d^3z\ldots \tag{2.83} \]

\(F_{lmn}\) is a functional with respect to the functions \(c(z)\) and the quantities \(\eta^*(x)\) and \(\eta(y)\), which anticommute with one another. The equation of motion for the generating functional can be written in the form

\[ \frac{i\delta F}{\delta\sigma(x)} = \left(\Lambda_0,\ MH_{int}\Omega\{\sigma\}\right). \tag{2.84} \]

The method of truncated equations under consideration in the three-dimensional formulation has a number of advantages in comparison with the more consistent four-dimensional formalism, which will be considered in §§ 3, 4, 5. Thus, for example, the boundary conditions have a simple physical meaning, the calculations are less complicated, and in some problems the angular variables can be separated.

However, along with the advantages mentioned above, there are a number of serious difficulties connected with the nonrenormalizability of terms of the type of the self-energy of the meson or nucleon, owing to the noncovariance of these expressions. In addition, in the Tamm—Dancoff method difficulties arise in considering terms describing vacuum polarization. The latter arise because the interaction-energy operator (2.47) contains products of three field operators; therefore processes are possible in which the simultaneous creation or annihilation of three particles takes place. In practical calculations of the cross section for scattering of \(\pi\) mesons, the self-energy terms were simply omitted\(^{21,4}\), since methods for a consistent treatment of these expressions are unknown. Although the values of the phase shifts thereby obtained for states with isotopic spin \(T = 3/2\) are in good agreement with experiment, nevertheless such a simple deletion of a number of terms is unfounded from the theoretical point of view. It should also be noted that even if it were possible to renormalize the kernel of the integral equation for scattering, difficulties connected with the self-energy of the nucleon would still arise if one takes into account that the emitted “final” meson may in fact be absorbed again by the nucleon, if states with \(T = 1/2,\ j = 1/2\) are considered, and in solving the integral equation it will be necessary to integrate over the momentum of this meson.

In the method of truncated equations, moreover, additional divergences arise in passing to the momentum representation\(^{19}\), and the resulting infinite expressions cannot be removed by renormalization. The occurrence of these infinities is connected with the fact that, in passing from the differential equations (2.55) to the integral equations, one of the limits of integration remains finite, in contrast to the scattering matrix, in which the integration is performed over infinite limits. In general, the calculation of the mathematical expectation of field operators leads to infinite values when one attempts to fix the time boundaries rigorously. To remove these new infinities, Stückelberg proposed introducing a “diffuse” boundary of integration by replacing discontinuous functions with smoothed ones. However, this device is not a logical consequence of the theory.

The method of truncated equations was subsequently improved to a considerable extent by Dyson. This method, to the consideration of which we now turn, bears the name of the new Tamm—Dancoff method.

§ 3. Generating functional for the amplitude of the new Tamm—Dancoff method

As we saw in § 1, the sequence of probability amplitudes (2.32) gives a complete description of the quantum properties of the field. These amplitudes are defined through the operators of noninteracting fields, on the one hand, and the vacuum of noninteracting fields*) on the other. The concept of the mathematical vacuum as a state without particles does not correspond to the really existing vacuum, since in the latter, owing to the interaction

*) Mathematical vacuum.

between fields there occurs the continuous creation and absorption of virtual quanta, which are in “dynamical equilibrium” with the background. Dyson proposed such a modification of the theory, in which the state of the system is determined with respect to the real physical vacuum, and not with respect to the mathematical one1. The creation and absorption operators, however, are still connected with noninteracting fields. Thus the vacuum state is physical, while the operators continue to remain “mathematical.”

A consistent generalization of the theory would consist in introducing field operators obeying inhomogeneous equations of motion, but in this case it is no longer possible to introduce covariantly the creation and absorption operators through which the probability amplitude must be determined.

Since the application of the absorption operator to the state of the physical vacuum gives a result different from zero, amplitudes inevitably appear in the theory that contain, alongside absorption operators, also creation operators. These amplitudes are called the “amplitudes” of the new Tamm—Dancoff method and are introduced as follows:

\[ a(N,N')=\frac{1}{\sqrt{\Pi(N)\Pi(N')}}\left(\Omega_0^\Phi,\ C(N)A(N')\Omega\right). \tag{3.1} \]

In expression (3.1), following Dyson1, the following notation has been introduced: the product of all absorption operators is denoted by the single letter \(A(N')\), the product of all creation operators by \(C(N)\), where \(N\) and \(N'\) replace the triples of numbers \(m,n,l\) and \(m',n',l'\). \(\Omega_0^\Phi\) is the state of the physical vacuum; \(\Omega\) is the vector of the given state; \(\Pi(N)=N!\). In this notation formulas (2.30) and (2.32) are symbolically written in the following form:

\[ \psi(N)=\frac{1}{\sqrt{\Pi(N)}}\left(\Lambda_0,\ A(N)\Omega\right),\qquad \Lambda(N)=\frac{1}{\sqrt{\Pi(N)}}A(N)\Lambda_0. \tag{3.2} \]

Let us consider a system consisting of one particle and the physical vacuum. As was mentioned above, this system has an indefinite number of particles (an indefinite number of virtual quanta) and therefore cannot be described by any single probability amplitude; it must be described by a collection of different amplitudes. This circumstance is reflected in the fact that the amplitudes of the new method can be represented in the form of a linear combination of the amplitudes of the old method.

Let us consider the specific form of the individual Dyson amplitudes (3.1). For this purpose we rewrite the Fourier expansion of the field operators (1.17) and (1.18) in a somewhat different form:

\[ \left. \begin{aligned} \psi_\alpha(x)&=\frac{1}{(2\pi)^{3/2}}\sum_u\int d^3k\,u_\alpha b_{ku}\exp(ik_\mu x_\mu),\\ \overline{\psi}_\alpha(x)&=\frac{1}{(2\pi)^{3/2}}\sum_u\int d^3k\,\overline{u}_\alpha\,\overline{b}_{ku}\exp(-ik_\mu x_\mu). \end{aligned} \right\} \tag{3.3} \]

In the new Tamm—Dancoff method the one-nucleon amplitude is defined according to

\[ \left(\Omega_0^\Phi,\ b_{ku}\Omega\right)=a_1(ku). \tag{3.4} \]

Depending on which sign of the energy the spinor index \(u\) corresponds to, positive or negative, the quantity \(b_{ku}\) will be either an absorption operator or a creation operator; thereby expression (3.4) for \(a_1(ku)\) will correspond either to a one-proton amplitude, or to such

called the “minus one antiproton amplitude.” By “minus particles” one means particles that are absent in the state under consideration, but are present in the state of the physical vacuum. A minus particle is a kind of “hole” in the vacuum of interacting fields. The two-nucleon amplitude in the new method will have the form

\[ (\Omega_0^\Phi,\, N b_{p\mu} d_{q\nu}\Omega), \tag{3.5} \]

where the sign \(N\) denotes the normal product of operators. The other amplitudes are constructed in an analogous way.

The equation for the amplitudes (3.1) can be obtained in the following way. Denote by \(E\) and \(E_0\) the energy of the given state and of the physical vacuum:

\[ (H_0+H_1)\Omega=E\Omega;\qquad (H_0+H_1)\Omega_0^\Phi=E_0\Omega_0^\Phi, \tag{3.6} \]

where \(H_1\) is the interaction operator. We have:

\[ (\Omega_0^\Phi, C(N)A(N')(H_0+H_1)\Omega)=E(\Omega_0^\Phi, C(N)A(N')\Omega)= \]

\[ =(\Omega_0^\Phi,(H_0+H_1)C(N)A(N')\Omega)+(\Omega_0^\Phi,[C(N)A(N'),H_0+H_1]\Omega). \]

The commutator \([C(N)A(N')H_0]\) can be easily computed if one uses the relation

\[ [c(\mathbf{k}),H_0]=E(k)c(\mathbf{k}), \]

\[ [c^+(\mathbf{k}),H_0]=-E(k)c^+(\mathbf{k}). \]

Denoting \(\displaystyle \sum^N E(k)=E_N,\quad \sum^{N'} E(k)=E_{N'}\), we have:

\[ (\varepsilon+E_N-E_{N'})a(N,N')= \frac{1}{\sqrt{\Pi(N)\Pi(N')}}(\Omega_0^\Phi,[C(N)A(N'),H_1]\Omega). \tag{3.7} \]

The quantity \(\varepsilon\), equal to \(\varepsilon=E-E_0\), is already a finite quantity, although \(E\) and \(E_0\) are themselves infinite. This circumstance is a positive aspect of the theory. Owing to the fact that there is a commutator on the right-hand side of equation (3.7), the number of operators on the right differs from the number of operators on the left by one. Thus the amplitude \(a(N,N')\) is “coupled” with amplitudes in which the number of particles is greater or smaller by one. For these amplitudes similar equations can be written, as a result of which an infinite system of “coupled equations” arises.

Vacuum loops are absent in the new method, since the simultaneous creation or annihilation of three particles in this method is impossible\(^{4,23}\). In coordinate space the amplitudes of the new method can be written in the form

\[ (\Omega_0^\Phi,\, N\psi(x_1)\ldots \psi(x_l), \bar{\psi}(y_1)\ldots \bar{\psi}(y_m)\varphi(x_1)\ldots \varphi(x_n)\Omega\{\sigma\}). \tag{3.8} \]

The state of the system in (3.8) can be specified on the hypersurface \(\sigma(x)\). The equations for the amplitudes (3.8) can be easily obtained if one uses the Tomonaga–Schwinger equation for the state vector \(\Omega\).

In the new method, the Tamm–Dancoff renormalization in the problem of the interaction of two nucleons in the lowest approximation can be carried out quite consistently; in the scattering problem, however, difficulties arise, although the expression for the self-energy of a nucleon has a covariant form, in contrast to the old method. It should also be noted that difficulties arise in the theory connected with the appearance of “spurious poles” in the Green’s function, which apparently is a common shortcoming of quantum field theory.

The description of the quantum properties of the field in the new Tamm—Dancoff method can be obtained, just as in the old one, by introducing a generating functional for the amplitudes (3.1). The latter may be written in the form

\[ F\{c^{*}c\}= \left( \Omega_{0}^{\Phi}, R^{+}\exp\left(\int c^{*}(\mathbf{k})\,c^{+}(\mathbf{k})\,d^{3}k\right) R\exp\left(\int \bar c(\mathbf{k})\,c(\mathbf{k})\,d^{3}k\right)\Omega \right), \tag{3.9} \]

where \(R\) is defined according to (2.40).

Let us establish the relation between the generating functionals of the new (3.9) and the old (2.41) methods. For brevity of exposition we shall consider only the Bose field. Expression (3.9) may be represented in the form

\[ F\{c^{*}c\}= \sum_{n=0}^{\infty} \left( \Omega_{0}^{\Phi}, \exp\left(\int c^{*}(\mathbf{k})\,c^{+}(\mathbf{k})\,d^{3}k\right)\Lambda_{n} \right)\times \]

\[ \times \left( \Lambda_{n}, \exp\left(\int \bar c(\mathbf{k})\,c(\mathbf{k})\,d^{3}k\right)\Omega \right), \tag{3.10} \]

where \(\Lambda_n\) are quantities defined according to (2.20), and constitute a complete orthonormal system of functions. The second factor in (3.10) may be written in the following way:

\[ \left( \Lambda_{n}, \exp\left(\int c(\mathbf{k})\,c(\mathbf{k})\,d^{3}k\right)\Omega \right) = \frac{\partial^{n}}{\partial c^{n}(\mathbf{k})}\, \frac{1}{\sqrt{n!}}\times \]

\[ \times \left( \Lambda_{0}, \exp\left(\int \bar c(\mathbf{k})\,c(\mathbf{k})\,d^{3}k\right)\Omega \right). \tag{3.11} \]

From (3.11) it follows that

\[ F= \left( \Omega_{0}^{\Phi}, \exp\left(\int\left(c^{*}(\mathbf{k})+\partial/\partial\bar c(\mathbf{k})\right)c^{+}(\mathbf{k})\,d^{3}k\right)\Lambda_{0} \right)\times \]

\[ \times \left( \Lambda_{0}, \exp\left(\int \bar c(\mathbf{k})\,c(\mathbf{k})\,d^{3}k\right)\Omega \right) = \]

\[ = \left( \Omega_{0}^{\Phi}, \exp\left(\int\left(c^{*}(\mathbf{k})+\partial/\partial\bar c(\mathbf{k})\right)c^{+}(\mathbf{k})\,d^{3}k\right)\Lambda_{0} \right) \widetilde{\Omega}(\bar c). \tag{3.12} \]

Expression (3.12) also establishes the relation between the mentioned functionals\(^{12}\). Here \(\widetilde{\Omega}(\bar c)\) is given by formula (2.41).

II. GENERATING FUNCTIONALS FOR RELATIVISTIC FUNCTIONS AND FUNCTIONAL INTEGRATION

Fock’s method of functionals is one of the possible rigorous functional formulations of quantum field theory.

In his method Fock developed the basic idea of the generating functional by choosing, as the basic functions describing the field, an infinite sequence of probability amplitudes in configuration space. It is clear that we shall obtain another rigorous functional formulation of quantum field theory if the same idea of the generating functional is developed on the basis of a complete sequence of some other functions,

dependent on the variables of a definite number of particles. We could already convince ourselves of this by considering in § 3 the generating functional for Dyson amplitudes (the amplitudes of the new Tamm—Dancoff method). Both for the Fock functional and for the functional in the new Tamm—Dancoff method it is characteristic that they depend on functions of a vector argument, or, in the general case, on functions of a space-time point lying on a space-like surface.

In § 4 we shall consider the development of the idea of a generating functional in the case where the field is described by a set of relativistic functions: either \(T\)-functions, or Feynman amplitudes, or \(\rho\)-functions.

In relativistic functions the space-time variables of the particles may be arbitrary, and therefore the generating functionals in this case will be functionals of functions of a space-time point, which have the meaning of external sources of the fields. Functionals of external sources were introduced by Schwinger\(^{27}\). The equations for the Schwinger generating functional and for the functions listed above are derived simply from the equations for the field operators in the Heisenberg representation. The equations for the Feynman amplitudes may be regarded\(^{28}\) as a four-dimensional generalization of the Fock equations for probability amplitudes.

Feynman amplitudes and \(T\)-functions are directly connected with the transition amplitude between states at \(t=\pm\infty\). Describing the field with the help of these functions means a space-time approach to field theory. The space-time interpretation of field theory (first put forward by Feynman\(^{29}\)) will apparently be preserved also in a future theory free of the contradictions\(^{30}\) of the present theory. Indeed, since in the future theory the corpuscular aspect will be preserved, the principal quantities characterizing the state of the field will still be relativistic functions of the type of Feynman amplitudes\(^{31}\) or a generating functional for them. In addition, a number of considerations\(^{32,33}\) argue in favor of the fact that, in a space-time interpretation, where no development in time is considered and the canonical formalism is absent, it will be easier to formulate a theory that does not use the concept of “bare” particles. For these reasons, in some works\(^{34,35}\) a consistent construction of the “four-dimensional” formalism was undertaken from the very beginning, bypassing the “three-dimensional” interpretation. The space-time interpretation is the subject of § 5 of the present review. In the same section the functional Fourier transform is considered, and with its aid the general solution of the problem of interacting fields is found in the form of a continual integral over Bose and Fermi fields.

Integration over a Fermi field, i.e. over anticommuting functions, requires great caution. A method of integration over a Fermi field was indicated by Matthews and Salam\(^{36}\). In § 6 of the review, following the idea of these authors, we express the variation of the Fermi field operator in terms of variations of ordinary numbers. Thanks to this, the equations for the four-dimensional state vector can be represented in a form in which there are no anticommuting functional derivatives.

Approximate methods of functional integration and the problems of renormalization are not discussed.

§ 4. Generating Functionals for Relativistic Functions

1. \(T\)-function and generating functional.

The probability amplitudes \(\Psi_{nml}\) in coordinate space

\[ \Psi_{nml}(x\ldots|y\ldots|z\ldots)= (\Lambda_0,\psi_0^{(+)}x\ldots \bar{\psi}_0^{(+)}(y)\ldots \varphi_0^{(+)}(z)\ldots \Omega(\sigma)) \tag{4.1} \]

are not “four-dimensional” functions, since the coordinates \(x \ldots y \ldots z \ldots\) in (4.1) must lie on some space-like surface \(\sigma\). Moreover, as was shown in § 2.4, calculations with the probability amplitudes \(\Psi_{nml}\) encounter a number of difficulties, connected in the last analysis with the fact that the amplitudes \(\Psi_{nml}\) are the expansion coefficients of the state vector \(\Omega(\sigma)\) in the eigenfunctionals of the particle-number operators for noninteracting fields. Calculations with amplitudes of the Dyson type (§ 3) also have their own special difficulties, arising from their definition, which is nonsequential, since in it the operators of noninteracting fields act on the vacuum of interacting fields.

To define relativistic wave functions it is convenient to use the Heisenberg representation. In the Heisenberg representation the field operators \(\psi(x)\), \(\bar{\psi}(x)\), and \(\varphi(x)\) satisfy the equations:

\[ D(x)\psi(x)-g\gamma_{5}\varphi(x)\psi(x)=0, \tag{4.2} \]

\[ D(-x)\bar{\psi}(x)-g\gamma_{5}(x)\varphi(x)\bar{\psi}(x)=0, \tag{4.3} \]

\[ K(x)\varphi(x)-g\,\frac{1}{2}\,[\bar{\psi}(x),\gamma_{5}\psi(x)]=0, \tag{4.4} \]

where

\[ D(x)=i\left(\gamma_{\lambda}(x)\frac{\partial}{\partial x_{\lambda}}+m\right), \qquad K(x)=i(\Box_x-\mu^2), \]

and \(\gamma_{\lambda}(x)\) means that the matrix \(\gamma_{\lambda}\) is multiplied by the spinor depending on \(x\):

\[ \gamma_{\lambda}(x)\bar{\psi}(x)=\bar{\psi}(x)\gamma_{5}, \qquad \gamma_{\lambda}(x)\bar{\psi}(y)\ldots\psi(x)=\bar{\psi}(y)\ldots\gamma_{\lambda}\psi(x). \]

In the Heisenberg representation one cannot introduce amplitudes by a formula of type (4.1), but with Heisenberg operators and state vectors, since the Heisenberg operators \(\psi\), \(\bar{\psi}\), and \(\varphi\) cannot be divided in a relativistically invariant way into positive- and negative-frequency parts. The relativistic functions whose construction is simplest are the matrix elements of the chronological \(T\)-products of the field operators themselves*):

\[ T_{nml}(x_{1}\ldots x_{n}\mid y_{1}\ldots y_{m}\mid z_{1}\ldots z_{l})= \]

\[ =(\Psi_{0},T[\psi(x_{1})\ldots\psi(x_{n})\bar{\psi}(y_{1})\ldots\bar{\psi}(y_{m})\varphi(z_{1})\ldots\varphi(z_{l})]\Psi), \tag{4.5} \]

where \(\Psi\) is the state vector in the Heisenberg representation, and \(\Psi_{0}\) denotes the physical vacuum; the functions \(T_{nml}(x\ldots\mid y\ldots\mid z\ldots)\) are symmetric with respect to the variables \(z\) and antisymmetric with respect to the variables \(x\ldots y\ldots\).

In the functions \(T_{nml}(x\ldots\mid y\ldots\mid z\ldots)\) the coordinates and times of the points \(x\ldots y\ldots z\ldots\) are not restricted by the requirement that they lie on some space-like surface.

With the aid of the functions \(T_{nml}\) one can find the energy of stationary states and the matrix elements of transitions. Let \(\Psi_{a}\) describe a state belonging to the eigenvalue \(p_{\mu}^{(a)}\) of the energy-momentum operator \(P_{\mu}\) of the interacting fields:

\[ P_{\mu}\Psi_{a}=p_{\mu}^{(a)}\Psi_{a}. \tag{4.6} \]

*) In the chronological product the order of the operators is such that the times of the operators increase from right to left. The \(T\)-product differs from the chronological product by a factor \((-1)^n\), where \(n\) is the number of interchanges of Fermi-field operators necessary for chronologizing the product.\(^{41}\)

Then the eigenvalues \(p_\mu^{(a)}\) can be determined from the equation

\[ \begin{aligned} p_\mu^{(a)} T_{nml}(x\ldots|y\ldots|z\ldots) &=(\Psi_0,T[\psi(x)\ldots \bar\psi(y)\ldots \varphi(z)\ldots]P_\mu\Psi_a)\\ &=(\Psi_0,T[-i\,{\partial\psi(x)\over \partial x_\mu}\ldots \bar\psi(y)\ldots \varphi(z)]\Psi_a)+\cdots\\ &\quad+\left(\Psi_0,T\left[\psi(x)\ldots -i\,{\partial\bar\psi(y)\over \partial y_\mu}\ldots \varphi(z)\right]\Psi_a\right)+\cdots\\ &\quad+\left(\Psi_0,T\left[\psi(x)\ldots \bar\psi(y)\ldots -i\,{\partial\varphi(z)\over \partial z_\mu}\right]\Psi_a\right)+\cdots\\ &=-i\sum_{nml}\left({\partial\over \partial x_\mu}+\cdots+{\partial\over \partial y_\mu}+\cdots+{\partial\over \partial z_\mu}\right) T_{nml}(x\ldots|y\ldots|z\ldots). \end{aligned} \tag{4.7} \]

Let us consider the connection of the functions \(T_{nml}(x\ldots|y\ldots|z\ldots)\) with transition amplitudes in the case when there are no bound states and at \(t=\pm\infty\) there are only free particles. Then as \(t\to-\infty\) the field operators \(\psi,\bar\psi\) asymptotically go over into the operators \(\psi_{in},\bar\psi_{in}\), and as \(t\to+\infty\) into the operators \(\psi_{out},\bar\psi_{out}\), describing free particles. If \(\Psi_a^{in}\) and \(\Psi_{a'}^{out}\) are state vectors in the Heisenberg representation, defined by means of the “in” and “out” field operators, then the transition amplitude \(U_{aa'}\) is related to the matrix element of the \(S\)-matrix by the relation

\[ U_{aa'}=(\Psi_{a'}^{out},\Psi_a^{in})=(\Psi_{a'}^{in},S\Psi_a^{in}). \tag{4.8} \]

Since \(\Psi_{a'}^{out}\) and \(\Psi_a^{in}\) describe the state of the field with free particles, for the representation of \(\Psi_a^{in}\) one may use formula (2.30), writing it in the form

\[ \Psi_{nml}^{in}=(n!m!l!)^{-1/2}a_{in}^{+}(\mathbf p_1)\ldots a_{in}^{+}(\mathbf p_n) b_{in}^{+}(\mathbf q_1)\ldots b_{in}^{+}(\mathbf q_m) c_{in}^{+}(\mathbf k_1)\ldots c_{in}^{+}(\mathbf k_l)\Psi_0. \tag{4.9} \]

and analogously for \(\Psi_{nml}^{out}\). As an example, let us analyze meson scattering by a nucleon. Then

\[ \Psi_a^{in}=\Psi_{101}^{in}=a_{in}^{+}(\mathbf p)c_{in}^{+}(\mathbf k)\Psi_0, \qquad \Psi_{a'}^{out}=\Psi_{101}^{out}=a_{out}^{+}(\mathbf p')c_{out}^{+}(\mathbf k')\Psi_0. \]

Substituting into (4.8) those expressions for the creation operators \(a^+\) and \(c^+\) which are obtained from (1.13) and (1.14), we find:

\[ \begin{aligned} U(\mathbf{pk};\mathbf{p'k'}) &=\\ &=\int f_{p'k'}(xz')\, i{\partial\over \partial z'_0}\,d^3z'\,\gamma_4(x)\,d^3x\, (\Psi_0,\varphi(z')\psi(x)\bar\psi(y)\varphi(z)\Psi_0) \times\\ &\qquad\qquad\times i{\partial\over \partial z_0}\,d^3z\,\gamma_4(y)\,f_{pk}(yz), \end{aligned} \tag{4.10} \]

\[ x_0,z'_0\to+\infty;\qquad y_0,z_0\to-\infty, \]

where \(f_{pk}(yz)\) and \(f_{p'k'}(xz')\) are the wave functions of the incident and scattered particles, which we shall assume to be orthonormal. The vacuum mean of the field operators in (4.10) can be expressed through the vacuum function \(T^{0}_{112}\equiv \tau_{112}\):

\[ (\Psi_0,\varphi(z')\psi(x)\bar\psi(y)\varphi(z)\Psi_0) =T^{0}_{112}(x|y|z'z)\equiv \tau^{0}_{112}(x|y|zz'), \tag{4.11} \]

since in (4.10) \(x_0,z'_0\to+\infty;\ y_0,z_0\to-\infty\). The three-dimensional integrals in (4.10) can be transformed into four-dimensional ones if in (4.10) the function \(\tau_{112}\) from (4.11) is introduced, since then for \(x_0,z'_0\to-\infty\) or \(y_0,z_0\to+\infty\) the quantity \(U(\mathbf{pk};\mathbf{p'k'})\) vanishes by virtue of the definition of the vacuum. After the transformation

the expression (4.10) is equal to

\[ U(p,k;p'k')= \int \bar f_{p'k'}(xz')\{K(z')K(z)D(-y)D(x)\tau_{112}(x|\;y|\;zz')\} \times \]
\[ \times f_{pk}(yz)d^4x\,d^4y\,d^4z\,d^4z'. \]

In the general case, for various states \(\Psi_{\alpha'}^{out}\) and \(\Psi_\alpha^{in}\), we obtain:

\[ U_{\alpha\alpha'}=(-1)^{\alpha+\alpha'}\int \bar f_{\alpha'}(y'\ldots|x'\ldots|z'\ldots)\{D(-y')\ldots D(+x')\ldots K(z')\ldots \]
\[ \ldots D(x)\ldots D(-y)\ldots K(z)\ldots \times \tau_{\alpha+\alpha'}(x'\ldots x\ldots|y'\ldots y\ldots|z'\ldots z\ldots)\} \times \]
\[ \times f_\alpha(x\ldots|y\ldots|z\ldots)d^4x\ldots d^4x'\ldots d^4y\ldots d^4y'\ldots d^4z\ldots d^4z'\ldots, \tag{4.12} \]

where \(f_\alpha\) and \(f_{\alpha'}\) are the wave functions of the incident and scattered particles. Thus, for the calculation of transition amplitudes it is sufficient to know only the vacuum functions \(\tau(x\ldots|y\ldots|z\ldots)^{32,39,40}\).

In order to describe the field, one must in general know the entire collection of functions \(T_{nml}(x\ldots|y\ldots|z\ldots)\); \(n,m,l=0,1,2,\ldots,\infty\). Just as, instead of the collection of probability amplitudes \(\Psi_{nml}^{-}(p\ldots|q\ldots|k\ldots)\), one may consider the generating functional \(\Omega\{a,\bar b,c\}\) (§ 2, item 3), so also, instead of the collection of functions \(T_{nml}\), one may consider the corresponding generating functional \(^{32,47,50}\). Since the functions \(T_{nml}(x\ldots|y\ldots|z\ldots)\) depend on space-time coordinates, in order to construct the generating functional it is necessary to introduce auxiliary quantities \(\eta(x)\), \(\bar\eta(y)\), and a function \(I(z)\), which also depend on space-time coordinates and are proportional to the unit matrix in the occupation-number space for the nucleon and meson fields. This functional \(Z\{\eta,\bar\eta,I\}\) is conveniently written in the form

\[ \left. \begin{aligned} Z\{\eta,\bar\eta,I\} &=\sum_{nml}^{\infty}\frac{i^{\,n+m+l}}{n!\,m!\,l!}\, Z_{nml}\{\eta,\bar\eta,I\},\\[6pt] Z_{nml}\{\eta,\bar\eta,I\} &=\\ &=\int \bar\eta(x_n)\ldots\bar\eta(x_1) T_{nml}(x_1\ldots x_n|y_1\ldots y_m|z_1\ldots z_l)\times\\ &\quad \times \eta(y_m)\ldots\eta(y_1)I(z_1)\ldots I(z_l) \,d^4x_1\ldots d^4x_n \times\\ &\quad \times d^4y_1\ldots d^4y_m\,d^4z_1\ldots d^4z_l . \end{aligned} \right\} \tag{4.13} \]

From the antisymmetry of \(T_{nml}(x\ldots|y\ldots|z\ldots)\) with respect to the variables \(x\ldots, y\ldots\), it follows that the quantities \(\eta(y)\), \(\bar\eta(x)\), moreover, must anticommute with one another,

\[ \left. \begin{aligned} \{\eta(x),\eta(x')\}&=0, & \{\eta(x),\bar\eta(y)\}&=0,\\ \{\bar\eta(y),\bar\eta(y')\}&=0, & [\eta(x),I(x')]&=0, \end{aligned} \right\} \tag{4.14} \]

whence it follows that \(\eta\) and \(\bar\eta\) are proportional to external sources of the nucleon field (see item 3, § 2). The factors in the expansion (4.13) are chosen so that \(\eta\), \(\bar\eta\), and \(I\) have the meaning of external sources. Indeed, from the definition (4.5) for the functions \(T_{nml}\) it is clear that (4.13) can be symbolically represented as an expansion in a series of the quantity:

\[ Z\{\eta,\bar\eta,I\}=(\Psi_0, S\{\eta,\bar\eta,I\}\Psi), \tag{4.15} \]

where the operator \(\tau\{\eta,\bar\eta,I\}\) is

\[ \tau\{\eta,\bar\eta,I\} = T\exp\left\{i\int\left[\bar\eta(x)\psi(x)+\bar\psi(x)\eta(x)+I(x)\varphi(x)\right]\,d^4x\right\} = \]

\[ = T\exp\left[-i\int_{-\infty} H'(t)\,dt\right] \]

and where \(H'(t)\) is the part of the interaction-energy operator that depends on the external sources. If one regards \(\tau\{\eta,\bar\eta,I\}\) as the limit, as \(t\to+\infty\), of the operator

\[ \tau\{\eta,\bar\eta,I;t\} = T\exp\left[-i\int_{-\infty}^{t} H'(t')\,dt'\right], \tag{4.16} \]

then the state vector \(\Psi'=\tau\{\eta,\bar\eta,I;t\}\Psi\) will change with time owing to the interaction with the external sources:

\[ i\frac{\partial\Psi'}{dt}=H'\Psi', \tag{4.17} \]

In all formulas here it is assumed that the operators \(\psi,\bar\psi\), and \(\varphi\) do not depend on the external sources (see equations (4.2)—(4.4)).

If the functional \(Z\{\eta,\bar\eta,I\}\) is known, then the function

\[ T_{nml}(x\ldots|y\ldots|z\ldots) \]

is expressed through the \((n+m+l)\)-fold functional derivative of \(Z\{\eta,\bar\eta,I\}\):

\[ T_{nml}(x_1\ldots x_n|y_1\ldots y_m|z_1\ldots z_l)= \]

\[ =(-i)^{n+m+l} \left. \frac{\delta^{\,n+m+l}Z\{\eta,\bar\eta,I\}} {\delta\bar\eta(x_1)\ldots\delta\bar\eta(x_n)\,\delta\eta(y_1)\ldots\delta\eta(y_m)\,\delta I(z_1)\ldots\delta I(z_l)} \right|_{\eta=\bar\eta=I=0}, \tag{4.18} \]

where, after the calculation, one must set \(\eta=0,\ \bar\eta=0\), and \(I=0\). With the aid of (4.18) one can obtain the following useful relations. If \(Z'\{\eta,\bar\eta,I\}=\eta(\xi)Z\{\eta,\bar\eta,I\}\), then for the corresponding functions \(T'_{nml}\) and \(T_{nml}\) the equality holds

\[ T'_{nml}(x_1\ldots|y_1\ldots y_m|z\ldots) = -i\sum_{1}^{m}\delta^4(\xi-y_1)\, T_{nm-1\,l}(x\ldots|y_2\ldots y_m|z\ldots), \tag{4.19} \]

the right-hand side of which is antisymmetrized with respect to the variables \(y\). If

\[ Z''\{\eta,\bar\eta,I\} = \frac{\delta}{\delta\eta(\xi)}Z\{\eta,\bar\eta,I\}, \]

then the functions \(T''_{nml}\) and \(T_{nml}\) are related by

\[ T''_{nml}(x_1\ldots x_n|y\ldots|z\ldots) = iT_{n+1\,m\,l}(\xi x_1\ldots x_n|y\ldots|z\ldots)(-1)^{n+m}. \tag{4.20} \]

If

\[ Z'''\{\eta,\bar\eta,I\} = \frac{\delta}{\delta J(z)}Z\{\eta,\bar\eta,I\}, \]

then for the functions \(T'''_{nml}\) and \(T_{nml}\) we obtain the equality

\[ T'''_{nml}(x\ldots|y\ldots|z_1\ldots z_l) = iT_{nml+1}(x\ldots|y\ldots|zz_1\ldots z_l) \quad\text{and so on.} \tag{4.21} \]

Equations for the generating functional \(Z\{\eta,\bar\eta,I\}\) can be obtained from the equations for the functions \(T_{nml}(x\ldots|y\ldots|z\ldots)\). In the simplest case of the function

\[ T_{110}(x|y|-)=\left(\Psi_0,T[\psi(x)\bar\psi(y)]\Psi\right) \]

we find from equation (4.2) for

of the operator \(\Psi(x)\) and the definition of the \(T\)-product:

\[ D(x)T_{110}(x|y|-)=D(x)\left(\Psi_0,\frac{1}{2}[\psi(x),\bar{\psi}(y)]\Psi\right)+ \]

\[ + D(x)\varepsilon(x,y)\left(\Psi_0,\frac{1}{2}\{\psi(x),\bar{\psi}(y)\}\Psi\right)= \]

\[ = g\gamma_5'(x)\left(\Psi_0,T[\psi(x)\bar{\psi}(y)\varphi(x)]\Psi\right)+\delta^4(x-y), \]

since \(\partial\varepsilon(x)/\partial x_0=2\delta(x_0)\), and for equal times \(x_0=y_0\) the permutation
\(\{\psi(x),\bar{\psi}(y)\}=\gamma_4\delta^3(x-y)\). Thus, \(T_{110}(x|y|-)\) satisfies the equation

\[ D(x)T_{110}(x|y|-)=g\gamma_5'(x)T(x|y|x)+\delta^4(x-y). \]

In the general case the chronological product of the operator \(\psi(x)\) and \(n\) other operators
\(A(x_1)\ldots A(x_n)\) (\(A\) may be equal to \(\psi,\bar{\psi}\), or \(\varphi\)) can be represented in the form

\[ T[\psi(x),A(x_1),\ldots,A(x_n)] =\sum(-1)^P\psi(x)T[A(x_1),\ldots,A(x_n)] \times \]

\[ \times \theta(x-x_1),\ldots,\theta(x-x_n), \tag{4.22} \]

where the summation is over all permutations of \(x\) and \(x_k\); \(P\) is equal to the number of permutations of
\(\psi(x)\) with the operators \(\psi(x_k)\) and \(\bar{\psi}(x_l)\);
\(\theta(x)=\frac{1}{2}[1+\varepsilon(x)]\). Since at equal times \(x_0=y_0\) one has
\(\{\psi(x),\bar{\psi}(y)\}=[\psi(x),\varphi(y)]=0\), according to (4.2) and (4.22) the function
\(T_{nml}(x\ldots|y\ldots|z\ldots)\) must satisfy the following equation with respect to the coordinate \(x_1\):

\[ D(x_1)T_{nml}(x_1\ldots|y_1\ldots|z_1\ldots) =g\gamma_5'(x_1)T_{nml+1}(x_1\ldots|y_1\ldots|x_1z_1\ldots)+ \]

\[ +\sum_i^m \delta^4(x_1-y_i)T_{n-1\,m-1\,l}(x_2\ldots|y_2\ldots|z_1\ldots). \tag{4.23} \]

From expression (4.18) for the functions \(T_{nml}\) and formulas (4.19)—(4.21) it follows that equation (4.23) can also be written in the form

\[ \left\{ D(x)\frac{\delta}{\delta\eta(x_1)} -ig\gamma_5\frac{\delta}{\delta\eta(x_1)}\frac{\delta}{\delta I(x_1)} +\eta(x_1) \right\} \times \]

\[ \times \frac{\delta^{n+m+l-1}Z\{\eta,\bar{\eta},I\}} {\delta\eta(x_1)\ldots\delta\eta(y_1)\ldots\delta I(z_1)\ldots} =0 \tag{4.24} \]

for \(\eta=\bar{\eta}=I=0\).

In view of the arbitrariness of the numbers \(n,m,l\), this equation will be satisfied if \(Z\{\eta,\bar{\eta},I\}\) satisfies the equation

\[ \left\{ D(x)\frac{\delta}{\delta\eta(x)} -ig\gamma_5\frac{\delta}{\delta\eta(x)}\frac{\delta}{\delta I(x)} \right\} Z\{\eta,\bar{\eta},I\} = -\eta(x)Z\{\eta,\bar{\eta},I\}. \tag{4.25} \]

This is the first variational Schwinger equation for the generating functional
\(Z\{\eta,\bar{\eta},I\}\). Two other variational equations can be obtained in the same way from the equations for
\(T_{nml}(x\ldots|y\ldots|z\ldots)\) with respect to the variables
\(y\ldots\) and \(z\ldots\), which in turn are equivalent to the operator equations
(4.3) and (4.4). We have:

\[ \left\{ D(-y)-ig\gamma_5(y)\frac{\delta}{\delta I(y)} \right\} \frac{\delta}{\delta\bar{\eta}(y)} Z\{\eta,\bar{\eta},I\} = \bar{\eta}(y)Z\{\eta,\bar{\eta},I\} \tag{4.26} \]

and

\[ K(z)\frac{\delta Z\{\eta,\bar{\eta},I\}}{\delta I(z)} = \left\{ I(z)+ig\frac{\delta}{\delta\eta(z)}\gamma_5\frac{\delta}{\delta\bar{\eta}(z)} \right\} Z\{\eta,\bar{\eta},I\}. \tag{4.27} \]

To the system of equations (4.25), (4.26), (4.27) it is necessary to add boundary conditions. If, for example, \(\Psi=\Psi_a\) refers to a state with the value of the energy-momentum \(p_\mu^{(a)}\), \(a\ne 0\), then for \(\eta=\bar{\eta}=l=0\) one of the conditions has the form

\[ Z\{0,0,0\}=0; \tag{4.28a} \]

whereas if \(\Psi=\Psi_0\), then, according to (4.15), one must have

\[ Z\{0,0,0\}=1. \tag{4.28b} \]

The remaining conditions are likewise determined by the meaning of the state \(\Psi\); if, for example, \(\Psi=\Psi_0\) is the physical vacuum, then for \(\eta=\bar{\eta}=l=0\) one must have

\[ \frac{\delta Z}{\delta \eta} = \frac{\delta Z}{\delta \bar{\eta}} = \frac{\delta Z}{\delta l} =0. \tag{4.29} \]

The equations with functional derivatives (4.26), (4.27), and (4.28) for \(Z\{\eta,\bar{\eta},l\}\) can be solved in general form \(^{42-46}\). This is the advantage of the functional method, since the successive application of the other methods is in one way or another connected with the use of perturbation theory.

The solution of the equations for \(Z\{\eta,\bar{\eta},l\}\) and the questions connected with this will be discussed in §§ 5 and 6.

2. Feynman amplitudes and the generating functional. Let us consider the question of constructing relativistic wave functions that are the four-dimensional analogue of the ordinary wave functions of a system of particles (probability amplitudes) in configuration space \(\psi_{nml}(x\ldots|y\ldots|z\ldots)\) (formula (4.1)). A direct generalization of the definition (4.1) to the case of Heisenberg operators is impossible, since, as was already noted in § 4, the Heisenberg operators cannot be invariantly divided into positive- and negative-frequency parts. In any case, the four-dimensional wave functions \(f_{nml}(x\ldots|y\ldots|z\ldots)\) must have the following properties:

a) The functions \(f(x\ldots|y\ldots|z\ldots)\) must be antisymmetric with respect to the nucleon and antinucleon variables \(x\ldots, y\ldots\) and symmetric with respect to the meson variables \(z\ldots\).

b) The equations for the functions \(f_{nml}(x\ldots|y\ldots|z\ldots)\) must have no singularities at coincident variables (or at least at coincident times) of any two particles. In this case the interval between two points \(x\) and \(y\) may be arbitrary.

c) In the absence of interaction the function \(f_{nml}(x\ldots|y\ldots|z\ldots)\), up to a constant factor, must be equal to the probability amplitude \(\psi_{nml}(x\ldots|y\ldots|z\ldots)\).

d) For stationary states the relation

\[ -i \sum^{n,m,l} \left\{ \frac{\partial}{\partial x_\mu} +\cdots+ \frac{\partial}{\partial y_\mu} +\cdots+ \frac{\partial}{\partial z_\mu} +\cdots \right\} f_{nml}(x\ldots|y\ldots|z\ldots) = p_\mu^{(a)} f_{nml}(x\ldots|y\ldots|z\ldots), \tag{4.30} \]

must hold, where \(p_\mu^{(a)}\) is the eigenvalue of the energy-momentum vector in the stationary state.

As is seen from definition (4.5) and equation (4.23), the functions \(T_{nml}(x\ldots|y\ldots|z\ldots)\) do not possess properties b) and c) and therefore cannot serve as four-dimensional wave functions.

If there is no interaction, then the relation between the amplitudes \(\Psi_{nml}\) and the functions \(T_{nml}\) is easily found with the aid of Wick’s formula for \(T\)-products.

and \(N\)-products (see, for example,\(^{41}\)), since the probability amplitude \(\Psi_{nml}\) is nothing other than a matrix element of an \(N\)-product:

\[ \Psi_{nml}(x\ldots|y\ldots|z\ldots) = (\Lambda_0,\, N[\psi(x)\ldots \bar{\psi}(y)\ldots \varphi(z)\ldots]\Phi), \tag{4.1′} \]

and, in the absence of interaction, the intervals between the points \(x,\ldots, y,\ldots, z,\ldots\) may be arbitrary. Taking into account the definitions (4.1′) and (4.5) for the functions \(\Psi^0_{nml}\) and \(T_{nml}\), and Wick’s formula, we find that in the absence of interaction

\[ \begin{aligned} T^0_{nml}(x\ldots|y\ldots|z\ldots) ={}& \Psi^0_{nml}(x\ldots|y\ldots|z\ldots) \\ &+\sum \Delta_F(z_i-z_k)\, \Psi^0_{nml-2}(x\ldots|y\ldots|zz_i^{-1}z_k^{-1}) \\ &-\sum S_F(x_i-y_k)\, \Psi^0_{n-1\,m-1\,l}(x\ldots x_i^{-1}|y\ldots y_k^{-1}\ldots|z\ldots) +\ldots , \end{aligned} \tag{4.31} \]

where \(x_i^{-1}\) denotes that \(\Psi^0_{n-1\,ml}(x\ldots x_i^{-1}|y\ldots|z\ldots)\) does not depend on the variable \(x_i\); \(\Delta_F\) and \(S_F\) are the Feynman Green functions for free fields\(^*\), and \(T^0_{nml}\) is the function \(T_{nml}\) in the absence of interaction.

From (4.31) it is clear that the singularities of the functions \(T^0_{nml}(x\ldots|y\ldots|z\ldots)\) have the same character as the singularities of the Green functions \(\Delta_F\) and \(S_F\).

In the case when \(T_{nml}\) is defined with the aid of the operators of interacting fields, formula (4.31) may be used as an indication of the character of the relation between \(T_{nml}\) and \(f_{nml}\). Conditions a), b), c), and d) do not determine this relation uniquely, since the subtraction of singularities may be carried out in various ways.

If one assumes that, in the presence of interaction, the singularities of the functions \(T_{nml}(x\ldots|y\ldots|z\ldots)\) have the same character as in the absence of interaction, then for interacting fields the functions \(f(x\ldots|y\ldots|z\ldots)\) can be defined by replacing, in formula (4.31), the functions \(\Psi^0_{nml}\) and \(T^0_{nml}\) for free fields by the functions \(f_{nml}\) and \(T_{nml}\):

\[ \begin{aligned} T_{nml}(x\ldots|y\ldots|z\ldots) ={}& f_{nml}(x\ldots|y\ldots|z\ldots) \\ &+\sum \Delta_F(z_i-z_k)\, f_{nml-2}(x\ldots|y\ldots|z\ldots z_i^{-1}z_k^{-1}\ldots) \\ &-\sum S_F(x_i-y_k)\, f_{n-1\,m-1\,l}(x\ldots x_i^{-1}\ldots|y\ldots y_k^{-1}\ldots|z\ldots) +\ldots , \end{aligned} \tag{4.32} \]

or, solving (4.32) with respect to \(f_{nml}\):

\[ \begin{aligned} f_{nml}(x\ldots|y\ldots|z\ldots) ={}& T_{nml}(x\ldots|y\ldots|z\ldots) \\ &-\sum \Delta_F(z_i-z_k)\, T_{nml-2}(x\ldots|y\ldots|z\ldots z_i^{-1}\ldots z_k^{-1}) \\ &+\sum S_F(x_i-y_k)\, T_{n-1\,m-1\,l}(x\ldots x_i^{-1}\ldots|y\ldots y_k^{-1}|z\ldots) +\ldots . \end{aligned} \tag{4.33} \]

Definition (4.33) satisfies conditions a), b), c), d). Thus, the functions \(f(x\ldots|y\ldots|z\ldots)\) are defined in works \(^{28,40,49}\). However, as Lehmann showed,\(^{23}\) the singularities of the Green functions \(\Delta'_F\) and \(S'_F\) for interacting fields cannot be less than the singularities of \(\Delta_F\) and \(S_F\). The same conclusion can also be drawn concerning the character of the singularities of the functions \(T_{nml}\) in comparison with the singularities of the functions \(T^0_{nml}\). Therefore, generally speaking, definition (4.33) may not ensure the fulfillment of condition b). From this point of

\(^*\) The Green functions

\[ S_F(x-y)=(\Lambda_0,\, T[\psi(x)\bar{\psi}(y)]\Lambda_0) \quad\text{and}\quad \Delta_F(x-y)=(\Lambda_0,\, T[\varphi(x)\varphi(y)]\Lambda_0) \]

satisfy the equations

\[ D(x)S_F(x-y)=\delta^4(x-y),\qquad K(x)\Delta_F(x-y)=-\delta^4(x-y). \]

from the point of view of convenience, instead of \(\Delta_F\) and \(S_F\), to define the functions \(f'(x\ldots \mid y\ldots \mid z\ldots)\) by means of the functions \(\Delta'_F\) and \(S'_F\):

\[ \begin{aligned} f'_{nml}(x\ldots \mid y\ldots \mid z\ldots) &=T_{nml}(x\ldots \mid y\ldots \mid z\ldots)- \\ &\quad-\sum \Delta'_F(z_i-z_k)\,T_{nml-2}(x\ldots \mid y\ldots \mid z\ldots z_i^{-1}\ldots z_k^{-1})+ \\ &\quad+\sum S_F(x_i-y_k)\,T_{n-1\,m-1\,l}(x\ldots x_i^{-1}\ldots \mid y\ldots y_k^{-1}\ldots \mid z\ldots)-\ldots , \end{aligned} \tag{4.34} \]

which, however, leads to more cumbersome equations for \(f'_{nml}\). The time development of the functions \(T_{nml}\), \(\Delta_F\), \(S_F\), \(\Delta'_F\), \(S'_F\) occurs in a “Feynman” manner: waves with positive energy propagate forward in time, and waves with negative energy propagate backward in time. It follows from definitions (4.32)—(4.34) for the functions \(f_{nml}(x\ldots \mid y\ldots \mid z\ldots)\) that these functions develop in time in the same way. The functions \(f_{nml}(x\ldots \mid y\ldots \mid z\ldots)\) are sometimes called Feynman amplitudes.

The generating functional \(S\{\eta,\bar\eta,I\}\) for the four-dimensional wave functions \(f_{nml}\):

\[ \left. \begin{aligned} S\{\eta,\bar\eta,I\} &=\sum_{nml}^{\infty}\frac{i^{\,n+m+l}}{n!\,m!\,l!}\,S_{nml}\{\eta,\bar\eta,I\},\\ S_{nml}\{\eta,\bar\eta,I\} &=\int \bar\eta(x_n)\ldots \bar\eta(x_1)\, f_{nml}(x_1\ldots x_n \mid y_1\ldots y_m \\ &\quad\times \mid z_1,\ldots,z_l)\,\eta(x)\ldots \eta(x)\, I(z_1)\ldots I(z_l)\,d^4x_1\ldots d^4y_1\ldots d^4z, \end{aligned} \right\} \tag{4.35} \]

is related to the functional of external sources \(Z\{\eta,\bar\eta,I\}\) by the transformation

\[ S\{\eta,\bar\eta,I\} =e^{\frac12 I\Delta_F I+\bar\eta S_F\eta}\,Z\{\eta,\bar\eta,I\}, \tag{4.36} \]

where the notation has been introduced

\[ \left. \begin{aligned} I\Delta_F I&=\int d^4x\,d^4y\,I(x)\Delta_f(x-y)I(y),\\ \bar\eta S_F\eta&=\int d^4x\,d^4y\,\bar\eta(x)S_F(x-y)\eta(y). \end{aligned} \right\} \tag{4.37} \]

To verify the validity of formula (4.35), one must expand both sides of formula (4.36) in functional power series in \(\eta,\bar\eta\), and \(I\), and then use the definitions of the generating functionals \(Z\{\eta,\bar\eta,I\}\) and \(S\{\eta,\bar\eta,I\}\).

Under the transformation (4.36), the operators of functional derivatives are transformed as follows:

\[ e^{\bar\eta S_F\eta}\frac{\delta}{\delta\eta(x)}e^{-\bar\eta S_F\eta} =\frac{\delta}{\delta\eta(x)}-\int S_F(x-y)\eta(y)\,d^4y =i\chi(x), \tag{4.38} \]

\[ e^{\bar\eta S_F\eta}\frac{\delta}{\delta\bar\eta(x)}e^{-\bar\eta S_F\eta} =\frac{\delta}{\delta\bar\eta(x)}+\int \bar\eta(y)S_F(y-x)\,d^4y =\frac{1}{i}\bar\chi(x), \tag{4.39} \]

\[ e^{\frac12 I\Delta_F I}\frac{\delta}{\delta I(x)}e^{-\frac12 I\Delta_F I} =\frac{\delta}{\delta I(x)}-\int \Delta_F(x-y)I(y)\,d^4y =\frac{1}{i}\Phi(x). \tag{4.40} \]

The equations for the functional \(S\{\eta,\bar\eta,I\}\) are obtained from equations (4.25), (4.26), and (4.27) for the functional \(Z\{\eta,\bar\eta,I\}\). Substituting into equation (4.25) the expression (4.36) for \(Z\{\eta,\bar\eta,I\}\) and using formulas (4.38) and

(4.40) for the transformation of the functional derivatives, we obtain the equation for \(S\{\eta,\bar\eta,l\}\):

\[ \{D(x)-g\gamma_5(x)\Phi(x)\}\chi(x)S\{\eta,\bar\eta,l\}=+i\eta(x)S\{\eta,\bar\eta,l\}. \tag{4.41} \]

In an analogous way the remaining equations for \(S\{\eta,\bar\eta,l\}\) are derived:

\[ \{D(-x)-g\gamma_5(x)\Phi(x)\}\bar\chi(x)S\{\eta,\bar\eta,l\}=i\bar\eta(x)S\{\eta,\bar\eta,l\}, \tag{4.42} \]

\[ \{K(x)\Phi(x)+g[\bar\chi(x)\gamma_5\chi(x)]\}S\{\eta,\bar\eta,l\}=i l(x)S\{\eta,\bar\eta,l\}. \tag{4.43} \]

The boundary conditions for the functional \(S\{\eta,\bar\eta,l\}\) can be found from the conditions for \(Z\{\eta,\bar\eta,l\}\). According to formula (4.28), for \(l=\eta=\bar\eta=0\) it must be, for the stationary state \(a\),

\[ S^{(a)}\{0,0,0\}=\delta_{(a)(0)}. \tag{4.44a} \]

For the generating functional \(S^{(0)}\{\eta,\bar\eta,l\}\) of the vacuum functions, condition (4.29) corresponds to

\[ \frac{\delta}{\delta\eta}S=0;\qquad \frac{\delta}{\delta\bar\eta}S=0;\qquad \frac{\delta}{\delta l}S=0 \tag{4.44б} \]

when \(\eta=\bar\eta=l=0\).

The transition amplitude \(U_{aa'}\) can be computed by means of the Feynman amplitudes \(f_{nml}(x\ldots|y\ldots|z\ldots)\) according to formula (4.12), if in it the function \(\tau_{nml}\) is replaced by the function \(f_{nml}\), since the additional terms which appear in this case do not contribute to \(U_{aa'}\).

In the limiting case, when all times \(x_0\ldots y_0\ldots z_0\ldots\) are equal and tend to \(+\infty\) (then the fields \(\psi,\bar\psi\), and \(\varphi\) are equal to the free fields \(\psi_{\mathrm{out}},\bar\psi_{\mathrm{out}}\), and \(\varphi_{\mathrm{out}}\)), the Feynman amplitude \(f_{nml}(x\ldots|y\ldots|z\ldots)\), according to (4.31), is equal to the probability amplitude \(\Psi_{nml}(x\ldots|y\ldots|z\ldots)\).

3. Functions \(\rho^{51}\). Introduce a new functional \(R\{\eta,\bar\eta,l\}\), representing the functional \(Z^0\) in the form

\[ Z^0\{\eta,\bar\eta,l\}=\exp R\{\eta,\bar\eta,l\}. \tag{4.45} \]

The functional \(R\{\eta,\bar\eta,l\}\) will be the generating functional for the functions \(\rho_{nml}(x\ldots|y\ldots|z\ldots)\):

\[ R=\sum i^{n+m+l}(n!m!l!)^{-1}\int \rho_{nml}(x\ldots|y\ldots|z\ldots)\times \]

\[ \times \eta(x)\ldots \bar\eta(y)\ldots l(z)\ldots d^4x\,d^4y\,d^4z. \tag{4.46} \]

The relation between the functions \(\rho_{nml}\) and the functions \(\tau_{nml}\) can be established by expanding the right-hand side of (4.45) in a series and using definitions (4.13) and (4.46). For the first functions \(\rho\) and \(\tau\), the relations have the form

\[ \left. \begin{aligned} \tau_{200}(x_1x_2|-|-) &=\rho_{100}(x_1|-|-)\rho_{100}(x_2|-|-) +\rho_{200}(x_1x_2|-|-),\\ \tau_{011}(-|y|z)&=\rho_{011}(-|y|z),\\ \tau_{111}(x|y|z) &=\rho_{100}(x|-|-)\rho_{011}(-|y|z)+\rho_{111}(x|y|z),\\ \tau_{211}(x_1x_2|y|z) &=\rho_{100}(x_1|-|-)\rho_{100}(x_2|-|-)\rho_{011}(-|y|z)\\ &\quad+\rho_{100}(x_1|-|-)\rho_{111}(x_2|y|z) +\rho_{100}(x_2|-|-)\rho_{111}(x_1|y|z)\\ &\quad+\rho_{200}(x_1x_2|-|-)\rho_{011}(-|y|z) +\rho_{211}(x_1x_2|y|z). \end{aligned} \right\} \tag{4.47} \]

From formulas (4.47) it is seen that the functions \(\rho\) are equal to zero if the values of the coordinates \(x\ldots y\ldots z\ldots\) make it possible to represent the corresponding function \(\tau\) in the form of a product of \(\tau\)-functions with a smaller number of variables. The function \(\tau\) decomposes into products of functions of lower order in the case when the coordinates of a definite group of variables in the \(\tau\)-function are separated by a large space-like or time-like interval (for example, \(|x,x'|\gg \left(\frac{h}{mc}\right)^2\)). This means that the function \(\rho\) will be large only if the intervals between all the coordinates in it are small. This property of the functions \(\rho\) may be useful for considering scattering at small energies or large angular momenta. We obtain the equations for the functional \(R\{\eta,\bar\eta,l\}\) by substituting in (4.25)—(4.27) the expression (4.45) for \(Z^0\{\eta,\bar\eta,l\}\):

\[ \left. \begin{aligned} \left\{D(x)-ig\gamma_5\frac{\delta}{\delta l(x)}\right\} \frac{\delta}{\delta\bar\eta(x)}R &= -\eta(x)+ig\gamma_5\frac{\delta R}{\delta l(x)} \frac{\delta R}{\delta\bar\eta(x)},\\ \left\{K(x)\frac{\delta}{\delta l(x)} -ig\frac{\delta}{\delta\eta(x)}\gamma_5 \frac{\delta}{\delta\bar\eta(x)}\right\}R &= l(x)+ig\frac{\delta R}{\delta\eta(x)}\gamma_5 \frac{\delta R}{\delta\bar\eta(x)} . \end{aligned} \right\} \tag{4.48} \]

Thus, equations (4.48) for \(R\{\eta,\bar\eta,l\}\) are nonlinear. From equations (4.48) there follows a system of “coupled” equations for the functions \(\rho\). Here, in contrast to the system of equations for the functions \(\tau\), a \(\delta\)-function occurs only in the equations for the functions \(\rho\) with two coordinates.

§ 5. Space-time treatment of quantum field theory and functionals

1. Basic equations for the four-dimensional state vector.

The space-time description has certain essential features in comparison with the usual “three-dimensional” description, which may prove important for the further development of quantum field theory. One may suppose that the space-time treatment will serve as a formal basis for the ideas of a future field theory. Therefore we shall consider here in detail the apparatus of such a treatment in connection with the method of functionals \(^{26,33—35,37}\).

The usual “three-dimensional” formulation of quantum field theory is based on equations for field operators and canonical commutation relations. The solution of the field equations leads to infinite expressions, which are eliminated in the course of calculations by introducing renormalization constants. In the usual formulation one has to use the concept of masses and charges of “bare” particles, and only at the end of the calculations, after the divergences have been removed, do only the experimental masses and charges remain in the formulas. In other words, using the equations and commutation relations for field operators, it is impossible to avoid divergences and the introduction of the concept of a “bare” particle.

The space-time description is not connected with the canonical formalism; it contains no canonical commutation relations and no equations for field operators. This makes it possible to hope that, on the basis of the space-time treatment, it will be possible to construct a theory that contains no divergences and uses only the concepts of physical mass and physical charge.

This conclusion can be explained with the aid of a visual representation of an elementary particle as a system consisting of a “bare” particle surrounded by a “cloud” of other particles. The total mass, charge, and spin of this system must be equal to the experimental values of the mass, charge, and spin of the elementary particle. But the same values of these quantities can be obtained with different “clouds” around the “bare” particle. If we consider the development in time, then, generally speaking (depending on the interaction), the “clouds”

in an elementary particle may be different for different instants of time and, consequently, at different times different clouds will be assigned to the elementary particle. Therefore, in the three-dimensional treatment one cannot define, for all times, a “cloud” around the particle. In the four-dimensional treatment, however, the development in time is not considered and, consequently, such a problem does not arise at all.

As we shall see below, functionals in the space-time treatment do not differ from the functionals of external sources considered in § 4. The difference between the four-dimensional treatment and the theory with external sources lies in the approach to the description of the field. The theory with external sources is developed on the basis of the usual “three-dimensional” apparatus of quantum field theory; the external sources in it are auxiliary quantities. The space-time treatment can be developed independently of the usual apparatus of field theory, without using the operators and state vectors of the “three-dimensional” theory^34,35,26. In the space-time treatment one can from the very beginning introduce four-dimensional state vectors and one’s own operators. The external sources in it serve to represent these operators, just as in Fock’s method of functionals (§ 2) the quantities \(\bar a(k)\), \(\bar b(q)\), \(\bar c(p)\) serve to represent the operators \(\psi\), \(\bar\psi\), \(\varphi\). From this point of view, the external sources in the space-time treatment need not be regarded as auxiliary quantities.

Let us turn to the construction of the apparatus of the space-time treatment of field theory in connection with the method of functionals. In the “three-dimensional” treatment the state vector is defined on a space-like hypersurface; in the space-time treatment the state vector \(\Omega\) must be defined in the entire four-dimensional volume. This can be done if, as the basic field operators, one chooses not the usual operators \(\psi\), \(\bar\psi\), \(\varphi\), but other operators \(\chi(x)\), \(\bar\chi(y)\) (nucleon field) and \(\Phi(z)\) (meson field), which anticommute or commute for arbitrary intervals between the points \(x, y, z\)*:

\[ \left. \begin{aligned} \{\chi(x), \bar\chi(y)\} &= 0, \qquad [\Phi(z), \Phi(z')] = 0,\\ \{\chi(x), \chi(y)\} &= 0, \qquad [\chi(x), \Phi(z)] = 0 \ \text{etc.} \end{aligned} \right\} \tag{5.1} \]

The operators \(\chi\), \(\bar\chi\), and \(\Phi\) were proposed to be called causal operators^35. Owing to relations (5.1), by means of the operators \(\chi\), \(\bar\chi\), and \(\Phi\) one can construct a complete system of mutually commuting operators \(\hat\xi\) referring to the four-dimensional volume. Then, as basis vectors, one may choose the eigenvectors \(\Omega(\xi)\) of the operators \(\hat\xi\). The four-dimensional state vector

\[ \Omega=\int C(\xi)\Omega(\xi)\,d\xi \tag{5.2} \]

will be defined if the expansion coefficients \(C(\xi)=(\Omega(\xi),\Omega)\) are known. The equations for the coefficients \(C(\xi)\) follow from the action principle**).

The action principle in quantum theory was developed in detail by Feynman and Schwinger^27,28. As applied to the space-time treatment considered here, the action principle can be expressed by the formula

\[ \hat\delta C(\xi)=i(\Omega(\xi),\delta W\cdot\Omega), \tag{5.3} \]

* The operators \(\chi\) and \(\bar\chi\) are not adjoint, although, as we shall see below (§ 6), the operators \(\chi\) and \(\bar\chi\) in the three-dimensional treatment can be associated with the operators \(\psi\) and \(\bar\psi\).

** An attempt to obtain equations without relying on the Lagrangian formalism was undertaken in work^32.

where \(\delta C(\xi)\) is an infinitesimal change of \(C(\xi)\) caused by the variation \(\xi\) in the four-dimensional volume; \(W\) is the action operator. Since \(\delta C(\xi)=(\delta\Omega(\xi),\Omega)\), then, introducing the operators of the infinitesimal transformation \(G_\xi\) by means of the relation

\[ (\delta\Omega(\xi),\Omega)=(\Omega(\xi),G_\xi\Omega), \]

one can represent the principle of action (5.3) in the form

\[ \delta\Omega=G_\xi\Omega=i\delta W\cdot\Omega . \tag{5.4} \]

The variation \(\delta\Omega\) in (5.4) is caused by the variation of the coefficients \(C(\xi)\).

We shall assume that the action \(W\) has the same form as in the ordinary “three-dimensional” treatment, but is composed of the operators \(\chi\), \(\bar\chi\), and \(\Phi\). Then equation (5.4) can easily be solved in symbolic form in the representation where the operators \(\chi\), \(\bar\chi\), and \(\Phi\) are operators of multiplication by the quantities \(\chi'\), \(\bar\chi'\), and the function \(\Phi'\), while the four-dimensional state vector \(\Omega\) is a functional of \(\chi'\), \(\bar\chi'\), and \(\Phi'\). In this representation the operator \(W\) will be composed only of multiplication operators and, consequently, equation (5.4) has the solution

\[ \Omega\{\chi',\bar\chi',\Phi'\}=e^{iW'}\cdot\frac{1}{N}, \tag{5.5} \]

where \(W'\) is composed of \(\chi'\), \(\bar\chi'\), and \(\Phi'\), and the constant \(N^{-1}\) is \(\Omega\) for \(\chi'=\bar\chi'=\Phi'=0\).

It is convenient to represent the action \(W\) in the form

\[ W=\int L(x,y)\,d^4x\,d^4y, \]

\[ L(x,y)=\frac{1}{2}\,i\delta^4(x-y)\{\bar\chi(y)[D(x)-g\Phi(x)\gamma_5]\chi(x)- \]

\[ -\chi(y)[D(-x)-g\Phi(x)\gamma_5(x)]\bar\chi(x)-\Phi(x)K(y)\Phi(y)\}. \tag{5.6} \]

From (5.5) one can obtain symbolic solutions for other representations.

If the operators \(\hat{\xi}\) are constructed from the operators \(\chi\), \(\bar\chi\), and \(\Phi\), then we can express the variations \(\delta\hat{\xi}\) formally through \(\delta\chi\), \(\delta\bar\chi\), and \(\delta\Phi\) (the meaning of the variations of the Fermi field operators \(\chi\) and \(\bar\chi\) we shall consider in § 6). The variations \(\delta\chi\), \(\delta\bar\chi\), and \(\delta\Phi\) satisfy the same commutation relations (5.1) as the operators \(\chi\), \(\bar\chi\), \(\Phi\). To construct the operator of an infinitesimal transformation \(G_\xi\), we introduce the operators \(\pi\), \(\bar\pi\), and \(\Pi\), which we define by means of the commutation relations:

\[ \left. \begin{aligned} \{\pi(x),\chi(y)\}&=-i\delta^4(x-y),\qquad [\Pi(x),\Phi(y)]=-i\delta^4(x-y),\\ \{\bar\pi(x),\bar\chi(y)\}&=i\delta^4(x-y). \end{aligned} \right\} \tag{5.7} \]

The operators \(\pi\), \(\bar\pi\), and \(\Pi\) are four-dimensional analogues of the canonically conjugate momenta of the “three-dimensional” theory. In the representation with the functional \(\Omega\{\chi',\bar\chi',\Phi'\}\) (formula (5.5)), \(\pi\), \(\bar\pi\), and \(\Pi\) are operators of functional derivatives.

The fundamental equation (5.4) can then be written in the form

\[ \left[G_\chi+G_{\bar\chi}+G_\Phi\right]\Omega =-i(\delta_\chi W+\delta_{\bar\chi}W+\delta_\Phi W)\Omega, \tag{5.8} \]

where the operators \(G_\chi\), \(G_{\bar\chi}\), and \(G_\Phi\) have the properties

\[ [G_\chi,\chi]=\delta\chi;\qquad [G_{\bar\chi},\bar\chi]=\delta\bar\chi;\qquad [G_\Phi,\Phi]=\delta\Phi, \]

and \(\delta_\chi W=[G_\chi,W]\), etc. The operators of the infinitesimal transformation \(G_l\), \(G_{\bar\chi}\), and \(G_\Phi\) have the form

\[ \left. \begin{aligned} G_l&=i\int \delta\chi(x)\pi(x)\,d^4x,\\ G_{\bar\chi}&=-i\int \delta\bar\chi(x)\bar\pi(x)\,d^4x,\\ G_\Phi&=i\int \delta\Phi(x)\Pi(x)\,d^4x . \end{aligned} \right\} \tag{5.9} \]

Owing to the independence of the variations \(\delta\chi\), \(\delta\bar\chi\), and \(\delta\Phi\), from formula (5.8) we obtain three equations:

\[ \left. \begin{aligned} \{D(x)-g\gamma_5\Phi(x)\}\chi(x)\Omega&=i\pi(\bar\chi)\Omega,\\ \{D(-x)-g\gamma_5(x)\Phi(x)\}\bar\chi(x)\Omega&=i\bar\pi(x)\Omega,\\ \{K(x)\Phi(x)+g\bar\chi(x)\gamma_5\chi(x)\}\Omega&=i\Pi(x)\Omega . \end{aligned} \right\} \tag{5.10} \]

Let us note that equations (5.10) have been obtained here without recourse to the usual “three-dimensional” apparatus of field theory. Let us now consider the representation in which the operators \(\pi\), \(\bar\pi\), and \(\Pi\) are multiplication operators; then \(\chi\), \(\bar\chi\), and \(\Phi\), according to (5.6), will be operators of functional differentiation. Then, comparing (5.10) with (4.25)—(4.27), it is not difficult to establish that equations (5.10) are nothing other than the equations for the functional of external sources \(Z\{\eta,\bar\eta,I\}\).

Thus, we have found a correspondence between the space-time and the usual three-dimensional treatments: the “four-dimensional” state vector \(\Omega\) in the representation in which multiplication by the functions of external sources corresponds to the operators \(\pi\), \(\bar\pi\), and \(\Pi\) (here \(\bar\eta=\eta^*\gamma_4\)):

\[ \pi(x)\Omega=\eta(x)\Omega;\qquad \bar\pi(x)\Omega=\bar\eta(x)\Omega;\qquad \Pi(x)\Omega=I(x)\Omega, \tag{5.11} \]

is proportional to the Schwinger functional of external sources \(Z\{\eta,\bar\eta,I\}\).

The reasons for such a correspondence can be understood by again considering the action principle (5.4) or (5.3). The action principle in this form assumes that variations of the field quantities (for example, the variations \(\delta\chi\), \(\delta\bar\chi\), and \(\delta\Phi\)) at any point of the four-dimensional volume can affect the state vector \(\Omega\). In other words, in the space-time treatment the principle of stationarity of the action is rejected; from it, in the usual “three-dimensional” treatment, the equations for the field operators follow. A consequence of this, in particular, is the absence of equations of motion for the operators \(\chi\), \(\bar\chi\), and \(\Phi\) (there is only the condition determining the possible functionals—the equation (5.4)). Since in the four-dimensional treatment the principle of stationarity of the action is rejected, i.e. variations of the field quantities that violate the principle of stationarity of the action are allowed, the apparatus of the four-dimensional treatment is equivalent to such an apparatus of the usual “three-dimensional” theory in which variations violating the principle of stationarity of the action are also considered. Such variations are the variations of the field quantities caused by variations of the external sources or external parameters. Therefore the “three-dimensional” treatment with external sources corresponds to the space-time treatment.

2. Generalized Fock functional. The correspondence between the four-dimensional formalism and the apparatus of the usual “three-dimensional” theory can be presented in a more convenient and transparent form if one passes to another representation, where the four-dimensional state vector will be the generalized Fock functional. Recall that the Fock functional is the generating functional for probability amplitudes and at the same time the state vector in the representation in which the creation operators \(a^+(p)\), \(b^+(q)\), and \(c^+(k)\) are multiplication operators.

To construct the generalized Fock functional in the space-time treatment, we introduce “four-dimensional” creation operators \(a_\rho^+(x)\), \(b_\lambda^+(y)\) (for the nucleon field) and \(c^+(z)\) (for the meson field). (\(\lambda,\rho\) are spinor indices.) In contrast to the usual commutation relations, the permutation of “four-dimensional” creation operators and Hermitian-conjugate annihilation operators \(a_{\rho'}(x')\), \(b_{\lambda'}(y)\), \(c(z)\) contains on the right-hand side a four-dimensional \(\delta\)-function

\[ \left. \begin{aligned} \{a_\rho(x),a_\sigma^+(y)\}&=\delta_{\rho\sigma}\delta^4(x-y),\\ \{b_\rho(x),b_\lambda^+(y)\}&=\delta_{\rho\lambda}\delta^4(x-y), \end{aligned} \qquad [c(z),c^+(z')]=\delta^4(z-z'). \right\} \tag{5.12} \]

Relying on the definition of the “four-dimensional” creation and annihilation operators (5.12), we can transfer formally to the space-time theory all the mathematical results of the method of Fock functionals (§ 2). We choose a representation in which \(a^+(x)\), \(b^+(y)\), \(c^+(z)\) are operators of multiplication by the anticommuting quantities \(\bar a(x)\), \(\bar b(y)\) and the function \(\bar c(z)\). The four-dimensional state vector \(F[\bar a,\bar b,\bar c]\), which is a functional of \(\bar a,\bar b,\bar c\), can then be represented in the form of an expansion in eigenfunctionals of the operators of the “number” of nucleons \(\int a^+(x)a(x)\,d^4x\), the “number” of antinucleons \(\int b^+(x)b(x)\,d^4x\), and mesons \(\int c^+(x)c(x)\,d^4x\):

\[ F=\sum F_{nml} =\sum_{nml}(n!m!l!)^{-1}\int \bar a(x_n)\ldots f_{nml}(x_1\ldots x_n|y_1\ldots y_m|z_1\ldots z_l) \times \]

\[ {}\times \bar b(y_m)\ldots \bar c(z_1)\ldots d^4x_1\ldots d^4y_1\ldots d^4z_1\ldots . \tag{5.13} \]

The operators \(a\), \(b\), \(c\) with respect to the functional (5.13) are operators of functional differentiation:

\[ a(x)=\frac{\delta}{\delta \bar a(x)};\qquad b(y)=\frac{\delta}{\delta \bar b(y)};\qquad c(z)=\frac{\delta}{\delta \bar c(z)}. \tag{5.14} \]

The functions \(f_{nml}\) for which the generalized Fock functional is the generating one may be regarded as four-dimensional analogues of probability amplitudes. As we shall see below, \(f_{nml}\) is the Feynman amplitude. The functional (5.13) was introduced by Kester \({}^{37}\).

Let us consider the energy-momentum vector. In the usual “three-dimensional” treatment, the energy-momentum vector can be found if the Lagrange function is known, while the properties of the energy-momentum vector as a displacement operator are a consequence of the commutation relations. In the “four-dimensional” treatment of field theory there is no variational principle; consequently there are no canonical commutation relations and equations for the field operators. Therefore we define the energy-momentum vector \(P_\mu\) \((iP_0=P_4)\) as a quantity possessing the properties of a displacement operator:

\[ -i[P_\mu,a(x)]=\frac{\partial a(x)}{\partial x_\mu},\quad \text{etc.} \tag{5.15} \]

with mutually commuting components:

\[ [P_\mu,P_\nu]=0. \]

The expression for \(P_\mu\) has the form

\[ P_\mu=-i\int\left\{ a^+(x)\frac{\partial a(x)}{\partial x_\mu} +b^+(x)\frac{\partial b(x)}{\partial x_\mu} +c^+(x)\frac{\partial c(x)}{\partial x_\mu} \right\}d^4x. \tag{5.16} \]

Application of the operator \(P_\mu\) to the state vector (5.13) is equivalent to differentiating the amplitudes \(f(x\ldots|y\ldots|z\ldots)\): if \(P_\mu F=F'\) and

\(f'_{nml}(x \ldots \mid y \ldots \mid z \ldots)\)—the amplitudes in the functional \(F'\), then

\[ f'_{nml}(x \ldots \mid y \ldots \mid z \ldots) = \]

\[ = - i \sum \left( \frac{\partial}{\partial x_\mu} + \ldots + \frac{\partial}{\partial y_\mu} + \ldots + \frac{\partial}{\partial z_\mu} + \ldots \right) f_{nml}(x \ldots \mid y \ldots \mid z \ldots). \tag{5.17} \]

Let us define the vector of the “vacuum” state \(F_0\). In complete analogy with the three-dimensional treatment we set

\[ a(x)F_0=0;\qquad b(x)F_0=0;\qquad c(z)F_0=0;\qquad (F_0,F_0)=1. \tag{5.18} \]

It is obvious that \(P_\mu F_0=0\), and the state \(F_0\) will have the lowest energy if all amplitudes \(f_{nml}(x \ldots \mid y \ldots \mid z \ldots)\) contain only positive frequencies.

Up to this point, only the permutation relations (5.24) have been used to construct the generalized Fock functional. In order to establish the correspondence with the functional \(\Omega\) (see § 5, 1), it is necessary to express the causal operators \(\chi,\bar{\chi}\), and \(\Phi\) through the creation operators \(a^{+}, b^{+}, c^{+}\) and the annihilation operators \(a,b,c\). For this it is necessary to assume that, in the representation with the state vector (5.13), the field operators \(\chi,\bar{\chi}\), and \(\Phi\) can be divided into creation and annihilation parts, with all the creation parts and all the annihilation parts separately commuting or anticommuting. Formulas (5.1) and (5.12) will be satisfied only when the permutations between the creation parts \(\chi^c,\bar{\chi}^c,\Phi^c\) and the annihilation parts \(\chi^a,\bar{\chi}^a,\Phi^a\) are equal to certain functions \(\sigma_F\) and \(d_F\):

\[ \begin{gathered} \{\chi^a(x),\bar{\chi}^c(y)\}=\sigma_F(x,y);\\ \{\bar{\chi}^a(x),\chi^c(y)\}=-\sigma_F(y,x); \end{gathered} \qquad \left. [\Phi^a(x),\Phi^c(y)]=d_F(x,y)=d_F(y,x). \right\} \tag{5.19} \]

This means that, with respect to the generalized Fock functional \(F\), the operators \(\chi,\bar{\chi}\), and \(\Phi\) can be represented in the form

\[ \left. \begin{aligned} \chi(x)&=a(x)-\int \sigma_F(x,y)b^{+}(y)\,d^4y,\\ \bar{\chi}(x)&=b(x)+\int a^{+}(y)\sigma_F(y,x)\,d^4y,\\ \Phi(x)&=c(x)+\int d_F(x,y)c^{+}(y)\,d^4y. \end{aligned} \right\} \tag{5.20} \]

Now one can establish the relation of the generalized Fock functional \(F\) to the functional of external sources \(\Omega\) (see formulas (5.10) and (5.11)), with respect to which the operators \(\chi,\bar{\chi}\), and \(\Phi\) are functional-derivative operators. The functionals \(F\) and \(\Omega\) are related by the transformation

\[ F=R\Omega, \tag{5.21a} \]

where

\[ R=\exp\left\{\int\left[a^{+}(x)\sigma_F(x,y)b^{+}(y)-\right.\right. \]

\[ \left.\left. -\frac{1}{2}c^{+}(x)d_F(x,y)c^{+}(y)\right]d^4x\,d^4y\right\}, \tag{5.21b} \]

with \(\pi=-ia^{+};\ \bar{\pi}=ib^{+};\ \Pi=ic^{+}\).

The transformation (5.21) corresponds to the transformation (4.36) from the generating functional for \(T\)-functions to the generating functional for Feynman amplitudes, if \(\sigma_F\) and \(d_F\) are the Feynman Green functions \(S_F\) and \(\Delta_F\), which, incidentally, also follows from (5.19).

The correspondence between the four-dimensional formalism and the apparatus of the ordinary theory can now be expressed by the equality[^37]

\[ T_{nml}(x_1\ldots x_n \mid y_1,\ldots y_m \mid z_1\ldots z_l)\equiv \]

\[ \equiv (\Psi_0,T[\psi(x_1)\ldots\psi(x_n)\overline{\psi}(y_1)\ldots \overline{\psi}(y_m)\varphi(z_1)\ldots\varphi(z_l)]\Psi)= \]

\[ =(F_0,\chi(x_1)\ldots\chi(x_n)\overline{\chi}(y_1)\ldots \overline{\chi}(y_m)\Phi(z_1)\ldots,\Phi(z_l)F\{a,\overline b,c\}), \tag{5.22} \]

where \(\Psi_0,\Psi\) are state vectors, and \(\psi,\overline{\psi}\), and \(\varphi\) are field operators in the Heisenberg representation; on the right-hand side stands the matrix element of the “four-dimensional” theory. If we use the transformation (5.21), we find another formula connecting the matrix elements of the four-dimensional and three-dimensional interpretations:

\[ (\Psi_0,T[\psi(x_1)\ldots\psi(x_n)\overline{\psi}(y_1)\ldots \overline{\psi}(y_m)\varphi(z_1)\ldots\varphi(z_l)]\Psi)= \]

\[ =(F_0,\chi(x_1)\ldots\chi(x_n)\overline{\chi}(y_1)\ldots \overline{\chi}(y_m)\Phi(z_1)\ldots\Phi(z_l)\Omega\{\eta,\overline\eta,I\}). \tag{5.23} \]

In (5.22) the operators \(\chi,\overline{\chi},\Phi\) must be represented in the form (5.20); in (5.23) the action of \(\chi,\overline{\chi}\), and \(\Phi\) on \(\Omega\) and \(F_0\) is determined by formulas (5.7), (5.11), and (5.216).

3. Functional Fourier transformation. In Sec. 1 § 5 an expression (5.5) was found for the state vector \(\Omega\{\chi',\overline{\chi}',\Phi'\}\) in the case when the operators \(\chi,\overline{\chi}\), and \(\Phi\) are functions, while \(\pi,\overline{\pi}\), and \(\Pi\) are operators of functional derivatives. The real case, as was established there, however, corresponds to a representation in which \(\chi,\overline{\chi}\), and \(\Phi\) are not functions but operators of functional derivatives; the multiplication operators here are \(\pi,\overline{\pi}\), and \(\Pi\) (formula (5.11)). Therefore the general solution of the problem of interacting fields, i.e. the solution of equation (5.10) with condition (5.11), can be obtained by means of the functional Fourier transformation from the functional (5.5) found earlier.

Let \(F\{I\}\) be a functional of the function \(I(x)\), depending on the space-time variable \(x\). Let us divide all space and time into \(n\) cells of equal volume and, instead of the function \(I(x)\), consider the system of its mean values \(I_1,\ldots,I_k,\ldots,I_n\) in the cells (\(k\) is the cell number). Then the functional \(F\{I\}\) will be a function \(F(I_1\ldots I_n)\) of the values \(I_1\ldots I_n\). The Fourier transform for \(F(I_1\ldots I_n)\) will have the form

\[ F(I_1\ldots I_n)=\int e^{-i\sum I_k\Phi'_k}F(\Phi'_1\ldots \Phi'_n)\frac{d\Phi'_1}{\sqrt{2\pi}}\ldots\frac{d\Phi'_n}{\sqrt{2\pi}}, \]

where \(\Phi'_k\) also refers to the \(k\)-th cell. Passing to the limit when each cell contains only one space-time point, we obtain for \(F\{I\}\) the representation by means of a continual integral

\[ F\{I\}=\int e^{-i\int I(x)\Phi'(x)d^4x}F\{\Phi'\}\,d(\Phi'), \tag{5.24} \]

where in

\[ d(\Phi')=\prod_k\frac{d\Phi'_k}{\sqrt{2\pi}} \]

the index \(k\) runs over all points of space-time.

Application of the operator

\[ \Phi(x)=i\frac{\delta}{\delta I(x)} \]

to the functional \(F\) leads to the appearance of the factor \(\Phi'\) under the integral. We

we may say that \(F\{\Phi'\}\) is a functional in a representation in which the multiplication operator is \(\Phi(x)\), while \(F\{I\}\) is connected with the representation in which \(\Phi\) is the operator of functional differentiation with respect to \(I\).

Similarly, for the functional \(\Omega\{\eta,\bar{\eta},I\}\) the Fourier transform will be the functional \(\Omega\{\chi',\bar{\chi}',\Phi'\}\), which is a four-dimensional state vector in the representation in which \(\chi,\bar{\chi}\), and \(\Phi\) are multiplication operators. Therefore the general solution of equations (5.10) for the functional \(\Omega\{\eta,\bar{\eta},I\}\) can be represented in the form of the following continual integral:

\[ \Omega\{\eta,\bar{\eta},I\} = \int e^{-i\int(\eta\bar{\chi}'+\chi'\bar{\eta}+\Phi'I)\,d^4x}\, \Omega\{\chi',\bar{\chi}',\Phi'\}\, d(\chi')\,d(\bar{\chi}')\,d(\Phi'). \tag{5.25} \]

Substituting for \(\Omega\{\chi',\bar{\chi}',\Phi'\}\) the expression (5.5), we find that

\[ \Omega\{\eta,\bar{\eta},I\} = \frac{1}{N} \int e^{-i\int(\eta\bar{\chi}'+\chi'\bar{\eta}+\Phi'I)\,d^4x}\, e^{iW'}\,d(\chi')\,d(\bar{\chi}')\,d(\Phi'), \tag{5.26} \]

where the constant \(N^{-1}\), equal to \(\Omega\{\chi',\bar{\chi}',\Phi'\}\) for

\[ \chi'=\bar{\chi}'=\Phi'=0 \]

(see (5.5)), is determined from the normalization condition.

We are interested in the generating functional for the vacuum \(T\)-functions. If the corresponding four-dimensional state vector is denoted by \(\Omega^0\), then from the correspondence formulas (5.23) we find the normalization condition for \(\Omega^0\):

\[ (F_0,\Omega^0\{\eta,\bar{\eta},I\})=1. \tag{5.27a} \]

Taking into account the definition (5.18) of the “vacuum” state vector \(F_0\), which, by virtue of (5.216) and (5.11), is equivalent to the equalities \((F_0,\pi(x)\Omega')=(F_0,\bar{\eta}(x)\Omega')=0\), etc., for an arbitrary functional \(\Omega'\), we may write the normalization condition (5.27a) in the same form as the boundary condition for the functional \(Z\{\eta,\bar{\eta},I\}\) (see § 4, 1):

\[ \Omega^0\{0,0,0\}=1. \tag{5.27б} \]

This means that the functional \(\Omega^0\) coincides with \(Z\).

From condition (5.27б) we find the normalization constant:

\[ N=\int e^{iW'}\,d(\chi')\,d(\bar{\chi}')\,d(\Phi'). \tag{5.28} \]

For the \(T\)-functions we then obtain from (5.23) and (5.25) an expression in the form of a continual integral:

\[ T(x\ldots \mid y\ldots \mid z\ldots) = (F_0,\chi(x)\ldots \bar{\chi}(y)\ldots \Phi(z)\ldots \Omega\{\eta,\bar{\eta},I\}) = \]

\[ = \frac{1}{N} \int \chi'(x)\ldots \bar{\chi}'(y)\ldots \Phi'(z)\ldots e^{iW'}\,d(\chi')\,d(\bar{\chi}')\,d(\Phi'), \tag{5.29} \]

which may be interpreted as the averaging over the Fermi and Bose fields of the product
\([\chi'(x)\ldots \bar{\chi}'(y)\ldots \Phi'(z)\ldots]\). The exponential with the action then plays the role of a weight function.

In formulas of the type (5.28) and (5.29) one can easily carry out either the integration over the Fermi field or the integration over the Bose field.

If the function \(T\) does not depend on the mesonic coordinates, then, introducing instead of \(\Phi'\) a new variable \(\Phi_1\), according to the formula

\[ \Phi'(x)=\Phi_1(x)+\frac{g}{2}\int \Delta_F(x-y)\,\bar{\chi}(y)\gamma_5\chi(y)\,d^4y, \tag{5.30} \]

we find:

\[ T(x\ldots|y|\ldots|-)=\frac{1}{N}\int \chi'(x)\ldots \chi'(y)\ldots e^{iW_1}\,d(\chi')\,d(\bar{\chi}'), \tag{5.31} \]

where

\[ W_1=i\int \bar{\chi}(x)D(x)\chi(x)\,d^4x -\frac{ig^2}{2}\int \bar{\chi}(x)\gamma_5\chi(x)\times \]

\[ \times \Delta_F(x-y)\bar{\chi}(y)\gamma_5\chi(y)\,d^4x\,d^4y. \]

The derivation of formulas (5.29) and (5.33) did not rely on perturbation theory and therefore their applicability is not connected with the magnitude of the interaction constant. This is the value of formulas of analogous type. Such formulas may be used as starting points in attempts to develop an approximation method different from the usual perturbation method*).

In doing so, however, it should be borne in mind that the integrals over the Fermi field \(\chi'\) and \(\bar{\chi}'\) have here a symbolic character, since \(\chi'\) and \(\bar{\chi}'\) are anticommuting functions. Therefore the question of integration over a Fermi field requires a special investigation, which will be carried out in § 6.

§ 6. Variation of an operator and functional integration in the case of a Fermi field

The symbolic character of integration over a Fermi field in the solution (5.25) is closely connected with the symbolic character of equations (5.10), which contain variational derivatives with respect to anticommuting functions. Below it will be considered how integration over a Fermi field can be reduced to ordinary functional integration and how equations (5.10) can be written without derivatives with respect to anticommuting functions. To clarify these questions it is necessary to consider the variation of a Fermi-field operator.

In the case of a Bose field, the definition of the variation of a field operator presents no difficulty, since there exists a representation in which the field operator is the operator of multiplication by some (auxiliary) function. In the case of a Bose field, the variation of the field \(\delta \Phi'\) may be represented, for example, in the following way: if we expand the Bose field \(\Phi'(x)\) in a series in a system of basis functions \(\varphi_n^{53}\):

\[ \Phi'(x)=\sum_n \gamma_n\varphi_n(x), \]

then

\[ \delta\Phi'(x)=\sum_n \delta\gamma_n\varphi_n(x), \]

and integration over \(\Phi'(x)\) will thus be replaced by integration over \(\gamma_n\).

In the case of a Fermi field a difficulty arises, since the variation of an operator here must anticommute with the operator itself. Therefore one can proceed in an analogous way in the case of a Fermi field only if it is possible to define a complete system of anticommuting (basis) functions \(\chi_n(x)\), \(\bar{\chi}_n(x)\). Then for an arbitrary Fermi field \(\chi\) and \(\bar{\chi}\) the expansion will take place

\[ \chi(x)=\sum_n \alpha_n\chi_n(x);\qquad \bar{\chi}(x)=\sum_n \beta_n\bar{\chi}_n(x) \tag{6.1} \]

*) It is possible that in the modern quantum field theory there is altogether no domain of solutions for the meson field\({}^{20}\).

(\(\bar\chi_n(x)\), generally speaking, is not conjugate to \(\chi_n(x)\); see the footnote to formula (5.1)). If the representation (6.1) is possible for \(\chi\) and \(\bar\chi\), then the variations \(\delta\chi\) and \(\delta\bar\chi\) are expressed through variations of the numbers \(a_n\) and \(\beta_n\):

\[ \delta\chi(x)=\sum \delta a_n\,\chi_n(x);\qquad \delta\bar\chi(x)=\sum \delta\beta_n\,\bar\chi_n(x), \tag{6.2} \]

and functional integration “over anticommuting quantities” \(\chi(x)\) and \(\bar\chi(x)\) reduces to integration over the numbers \(a_n\) and \(\beta_n\). Thus the question reduces to the definition of the basic anticommuting functions \(\chi_n(x)\) and \(\bar\chi_n(x)\).

Let us introduce a complete system of orthogonal functions \(\Psi_n(x)\), normalized according to the condition

\[ \int \bar\Psi_n(x) Z(x,y)\Psi_m(y)\,d^4x\,d^4y=\delta_{nm} \tag{6.3} \]

(here \(\bar\Psi_n=\Psi_n^+\gamma_4\)). As \(Z(x,y)\) we shall use either the Dirac operator \(-i\delta^4(x-y)D(y)\), or the Dirac operator with an external meson field

\[ D(x,y,\Phi)=-i\delta^4(x-y)[D(y)-g\gamma_5\Phi(y)], \]

since we are interested in the case when the action for the nucleon field is different from zero. In addition, we introduce the anticommuting operators \(A_n\) and \(B_n\)

\[ A_n=a_n+b_n^+;\qquad B_m=b_m-a_m^+, \tag{6.4} \]

where \(a_n^+\), \(b_n^+\) are “creation” operators, and \(a_n\) and \(b_n\) are (“four-dimensional”) annihilation operators:

\[ \{a_n,a_m^+\}=\{b_n,b_m^+\}=\delta_{nm}, \]

so that

\[ a_nF_0=b_nF_0=0 \]

(\(F_0\) is the “vacuum functional,” see (5.18)). We compose the operators

\[ \chi_n(x)=A_n\Psi_n(x);\qquad \bar\chi_n(x)=B_n\bar\Psi_n(x), \tag{6.5} \]

which, as we shall see below, can be used as basic anticommuting functions. In order that \(\chi_n\) and \(\bar\chi_n\) may be regarded as basic functions and that the expansion (6.1) be valid, the quantities \(\chi_n\) and \(\bar\chi_n\) (or \(A_n\) and \(B_n\)) must be connected by a normalization condition, which is possible only if we can restrict ourselves to considering only a certain class of matrix elements from the operators \(\chi_n\) and \(\bar\chi_n\).

To clarify the question of normalization for \(\chi_n\) and \(\bar\chi_n\) (or for \(A_n\) and \(B_n\)), let us show that an arbitrary functional \(F\) can be expressed through a functional \(F^0\), which is determined by the equations of motion (with allowance for interaction) and by the conditions \((F_0,F^0)=1\), \((F_0,a(x)F^0)=(F_0,b(x)F^0)=(F_0,c(x)F^0)=0\). Equations (5.10) have a formal solution in the form

\[ \Omega(g)=S\Omega(0), \]

where the functional for noninteracting fields \(\Omega(0)\) satisfies (5.10) for \(g=0\), while the operator \(S\) is an analogue of the scattering matrix

\[ S=S_0\exp\left\{g\int N[\bar\chi(x)\gamma_5\chi(x)]\Phi(x)\,d^4x\right\}, \tag{6.6} \]

where \(S_0\) is a normalization constant. Since \(F(g)\) is connected with \(\Omega(g)\) by the transformation (5.21), \(F=R\Omega\), the generalized Fock functional \(F(g)\) is expressed in terms of the functional \(F(0)\) in the absence of interaction by the formula\(^{37,35,38}\)

\[ F(g)=RSR^{-1}F(0)=S'F(0). \tag{6.7} \]

It follows from (5.21) and (5.10) that the equations for \(F(0)\) have the form

\[ \begin{aligned} D(x)\chi(x)F(0)&=-b^{+}(x)F(0),\\ D(-x)\bar{\chi}(x)F(0)&=a^{+}(x)F(0),\\ K(x)\Phi(x)F(0)&=-c^{+}(x)F(0), \end{aligned} \tag{6.8} \]

i.e., the amplitudes \(f^0(x\ldots \mid y\ldots \mid z\ldots)\) in \(F(0)\) satisfy the equations for free fields. In particular, the generating functional \(F^0(g)\) for vacuum Feynman amplitudes is

\[ F^0(g)=S'F^0(0)=S'F_0. \tag{6.9} \]

Let us now consider, as an example, \(F(g)=S'F_{110}(0)\), where

\[ F_{110}(0)=\int f_0(x\mid y\mid -)\,a^{+}(x)b^{+}(y)\,d^4x\,d^4y\,F_0. \]

Then from (6.8), (6.9) and the commutativity of \(S'\) with \(\bar{\chi}\) and \(\chi\) it follows that

\[ \begin{aligned} F(g)=S'F_{110}(0) &=-\int f^0(x\mid y\mid -)\,D(-x)\bar{\chi}(x) \\ &\qquad =D(y)\chi(y)\,d^4x\,d^4y\,F^0(g). \end{aligned} \tag{6.10} \]

The generalization of (6.10) presents no difficulties. Formulas of the type (6.10) reduce the problem of matrix elements \((F_0,\chi(x)\ldots\chi(y)\ldots\Phi(z)\ldots F(g))\) for an arbitrary \(F(g)\) to the problem of matrix elements between the states \(F_0\) and \(F^J(g)\). Since \(F^J(g)\) is the result of applying to the “vacuum” functional \(F_0=F^0(0)\) the operator \(S'\), which depends on \(\chi,\bar{\chi},\Phi\), it follows that all the variety of matrix elements of \(\chi_n,\bar{\chi}_m\) that arise in the computation of \(T\)-functions or functionals \(F\) and \(\Omega\) can be reduced to vacuum matrix elements of products of the operators \(\chi_n\) and \(\bar{\chi}_m\):

\[ \langle \bar{\chi}_{m_1}(y_1)\bar{\chi}_{m_2}(y_2)\ldots \chi_{n_2}(x_2)\chi_{n_1}(x_1)\rangle_0, \]

or else

\[ \langle B_{m_1}B_{m_2}\ldots A_{n_2}A_{n_1}\rangle_0, \]

if we denote

\[ \langle M\rangle_0=(F_0,MF_0). \]

The vacuum average of products of the operators \(B_m\) and \(A_n\) can be represented in the form of a determinant composed of \(\langle B_m A_n\rangle_0\):

\[ \langle B_{m_1}\ldots B_{m_r}A_{n_r}\ldots A_{n_1}\rangle_0 = \left| \begin{array}{cccc} \langle B_{m_1}A_{n_1}\rangle_0 & \langle B_{m_1}A_{n_2}\rangle_0 & \ldots & \langle B_{m_1}A_{n_r}\rangle_0\\ \langle B_{m_2}A_{n_1}\rangle_0 & \langle B_{m_2}A_{n_2}\rangle_0 & \ldots & \langle B_{m_2}A_{n_r}\rangle_0\\ \cdot & \cdot & \cdot & \cdot\\ \langle B_{m_r}A_{n_1}\rangle_0 & \langle B_{m_r}A_{n_2}\rangle_0 & \ldots & \langle B_{m_r}A_{n_r}\rangle_0 \end{array} \right|. \tag{6.11} \]

Since \(\langle B_m A_n\rangle_0=\delta_{nm}\), i.e., in computing the vacuum average the operator \(B_nA_n\) may be replaced by the identity operator, the result (6.11) can also be obtained by means of the rule that in vacuum averages of produc-

... of the operators \(B_{m_1}B_{m_2}\ldots A_{n_1}A_{n_2}\ldots\)

\[ B_n A_n=-A_n B_n \to 1 \tag{6.12} \]

— the product of two anticommuting operators \(B_n\) and \(A_n\) with identical values of \(n\) is equivalent to the identity operator.

Formula (6.12) may be called the normalization condition for the operators \(A_n\) and \(B_n\), since (6.3) can now be written in the form

\[ \int \overline{\chi}_n(x) Z(x,y)\chi_m(y)\,d^4x\,d^4y \to \delta_{nm}. \tag{6.13} \]

The normalization condition (6.13) was proposed by Matthews and Salam\({}^{36}\). This condition should likewise be understood not literally, but in the same sense as (6.12).

Now, to derive equations of the type (5.10), it is not necessary to introduce anticommuting functional derivatives. Let us construct the operators of infinitesimal transformations \(G_\chi\) and \(G_{\bar\chi}\) (see § 5.1, formula (5.9)). If only functions of \(\chi\), \(\bar\chi\), and \(\Phi\) are considered, then \(G_\chi\) and \(G_{\bar\chi}\) can be represented in the form

\[ \left. \begin{aligned} G_\chi&=\sum \delta\alpha_n\,\frac{\partial}{\partial \alpha_n};\\ G_{\bar\chi}&=\sum \delta\beta_n\,\frac{\partial}{\partial \beta_n}, \end{aligned} \right\} \tag{6.14} \]

where only derivatives with respect to numbers enter. From the action principle (5.8) we then obtain, instead of the first two equations (5.10), equations with ordinary derivatives:

\[ \left. \begin{aligned} i\,\frac{\partial}{\partial \alpha_n}\Omega &=\sum_m \beta_m Q_{mn}\Omega,\\[4pt] i\,\frac{\partial}{\partial \beta_n}\Omega &=\sum_m \alpha_m Q_{nm}\Omega, \end{aligned} \right\} \tag{6.15} \]

where

\[ Q_{nm}=\int \overline{\chi}_n(x)D(x,y,\Phi')\chi_m(y)\,d^4x\,d^4y. \tag{6.16} \]

The solution of equations (6.15) and of the third (meson) equation (5.10) is the functional

\[ \Omega\{\alpha,\beta,\Phi'\} =\frac{1}{N}\exp\left\{-i\sum_{nm}\beta_m Q_{mn}\alpha_n+iW_M\right\}, \tag{6.17} \]

where \(W_M\) is the action for the meson field; \(N\) is a normalization constant. The functional \(\Omega\{\alpha,\beta,\Phi'\}\) corresponds to the state vector \(\Omega\{\chi',\bar\chi',\Phi'\}\) (formula (5.5)) in a representation in which \(\chi\), \(\bar\chi\), \(\Phi\) are multiplication operators, for \(\alpha\) and \(\beta\) are regarded as numbers. The formal solution of the problem of interacting fields, § 5.3, was obtained by means of a Fourier transform of the functional \(\Omega\{\chi',\bar\chi',\Phi'\}\), depending on the anticommuting functions \(\chi'(x)\), \(\bar\chi'(x)\), and the function \(l'(x)\). Now, for the solution of the same problem it is necessary to perform a Fourier transform of the functional \(\Omega\{\alpha,\beta,\Phi'\}\), depending on the numbers \(a_n\), \(\beta_n\), and the function \(\Phi'(x)\). Since the quantity \(Q_{nm}\), according to (6.13), may be regarded as real, the Fourier transform of \(\Omega\{\alpha,\beta,l'\}\) will be possible if

\[ \alpha_n=\beta_n^* \tag{6.18} \]

and it is assumed that in the operators \(D(x)\) and \(K(x)\) the masses \(m\) and \(\mu^2\) are replaced by \(m-i\varepsilon\) and \(\mu^2-i\varepsilon\), since only then will the functional integrals over \(\alpha,\beta,I(x)\) converge.

As a result, instead of the functional \(\Omega\{\eta,\bar\eta,I\}\) of the anticommuting functions \(\eta(x)\) and \(\bar\eta(x)\), we now find a functional depending on the function \(I(x)\) and the numbers \(\zeta_n,\zeta_n^*\):

\[ \omega\{\zeta^*,\zeta,I\} = \int e^{-i\sum\left(\zeta_n^*\beta_n^*+\zeta_n\beta_n\right)-i\int I(x)\Phi(x)\,d^4x} \Omega\{\beta^*,\beta,\Phi\}\,d(\beta)\,d(\Phi), \tag{6.19} \]

\[ \Omega\{\beta^*,\beta,\Phi\} = \frac{1}{N}\exp\left[-i\sum_{nm}Q_{nm}\beta_n\beta_m^*+iW_M\right], \tag{6.20} \]

which differs from (5.25) in that in (6.19) the integration is carried out over the complex numbers \(\beta_n\):

\[ d(\beta)=\prod_n \frac{d\beta_n}{\sqrt{2\pi}}\,\frac{d\beta_n^*}{\sqrt{2\pi}} . \]

The normalization condition (5.276) for the vacuum functional \(\Omega^0\) requires that \(\omega^0\{\zeta^*,\zeta,I\}=1\) when \(\zeta^*=\zeta=I=0\). Hence

\[ N= \int \exp\left[-i\sum_{nm}\beta_n Q_{nm}\beta_m^*+iW_M\right]\,d(\beta)\,d(\Phi). \tag{6.21} \]

Now the calculation of the integrals over the Fermi field, i.e. over \(\beta_n^*\) and \(\beta_n\), presents no special difficulties in comparison with the integrals over the Bose field. The transition from (6.17) to (6.19) differs from the functional Fourier transformation from (5.5) to (5.25) in that in (6.19) no operator Fourier transformation was performed for the Fermi field\(^*\).

The operators \(\chi,\bar\chi\) with respect to the functional (6.19) will have the form

\[ \left. \begin{aligned} \chi(x)&=\sum_n \chi_n(x)\left(i\frac{\partial}{\partial \zeta_n^*}\right);\\ \bar\chi(x)&=\sum_n \bar\chi_n(x)\left(i\frac{\partial}{\partial \zeta_n}\right). \end{aligned} \right\} \tag{6.22} \]

Applying the operators \(\chi\) and \(\bar\chi\) to the functional (6.17), according to (6.22), introduces under the integral additional factors \(\beta_n^*\) and \(\beta_m\). From the form of \(\Omega\{\beta^*,\beta,\Phi'\}\) it follows that, when \(\zeta=\zeta^*=0\), the integrals over \(\beta_n^*\) and \(\beta_m\) of products of the coefficients \(\beta_n^*\) and \(\beta_m\) with \(\Omega\{\beta^*,\beta,\Phi'\}\) will be different from zero only if these coefficients enter in pairs \(\beta_n\beta_n^*\). This corresponds to the fact that in the \(T\)-function (5.22) each operator \(\chi\) must occur only together with \(\bar\chi\).

Using condition (6.12) (or (6.13)), we may now omit writing the vacuum functional \(F_0\) in the definition of the \(T\)-function:

\[ T(x\ldots|y\ldots|z\ldots) = [\chi(x)\ldots\bar\chi(y)\ldots\Phi(z)\ldots]\, \omega\{\zeta^*,\zeta,I\}, \tag{6.23} \]

where

\[ \Phi(z)=i\frac{\delta}{\delta I(z)}, \]

\[ \text{}^* \]
Detailed calculations of \(\Omega\{\eta,\bar\eta,I\}\) by means of such an operator transformation are given in the work\(^{33}\).

\(\chi(x)\) and \(\bar{\chi}(y)\) are determined by formulas (6.22), and after differentiation in (6.23) one must put \(\zeta=\zeta^*=I=0\). Formulas (6.19)—(6.23) and condition (6.13) are the starting point for calculating \(T\)-functions.

Let us consider integration over the Fermi field. As an example, let us calculate the exact nucleon Green’s function, which according to (6.23) is equal to

\[ S'_F(x,y)=-\chi(x)\bar{\chi}(y)\omega\{\zeta^*,\zeta,0\}\big|_{\zeta=\zeta^*=0}. \tag{6.24} \]

Introduce the notation

\[ N(\Phi')=N\int \Omega\{\beta^*,\beta,\Phi'\}\,d(\beta);\qquad \int N(\Phi')\,d(\Phi')=N. \tag{6.25} \]

Then the expression for \(S'_F\) can be represented in the form

\[ S'_F(x,y)=\frac{1}{N}\int S_F(x,y,\Phi')\,N(\Phi')\,d(\Phi'), \tag{6.26} \]

where, as we shall verify below, \(S_F(x,y,\Phi')\) is the Green’s function for the nucleon in the external meson field \(\Phi'\). Indeed, for \(S_F(x,y,\Phi')\) from formulas (6.22) and (6.26) there follows the expression

\[ S_F(x,y,\Phi')=\sum_{l,m}\Psi_l(x)\overline{\Psi}_m(y)A_lB_m \int \beta_l^*\beta_m \exp\left[-i\sum_n \beta_n\beta_n^*\right]\times \]

\[ \times \prod_n \frac{d\beta_n\,d\beta_n^*}{2\pi} = -i\sum_n \chi_n(x)\bar{\chi}_n(y) = -iD^{-1}(x,y,\Phi'), \tag{6.27} \]

where in the integration we have put \(Q_{nm}=\delta_{nm}\).

Thus, the exact nucleon Green’s function is obtained from the Green’s function in the external field \(S_F(x,y,\Phi')\) by averaging over all external fields with the weight function \(N(\Phi')\).

The quantity \(N(\Phi')\) is also transformed comparatively simply with the aid of the method analyzed above. According to (6.25),

\[ N(\Phi')=\int \exp\left[-i\sum_{nm}Q_{nm}\beta_n\beta_m^*+iW_M\right]\,d(\beta). \tag{6.28} \]

Choosing in the normalization condition (6.13), as \(Z(x,y)\), the Dirac operator \(-i\delta^4(x-y)D(y)\) without the external field, we find that

\[ Q_{nm}=-i\int \bar{\chi}_n(x)\left[D(x)-g\gamma_5\Phi'(x)\right]\chi_m(x)\,d^4x= \]

\[ =\delta_{nm}+ig\int \bar{\chi}_n(x)\gamma_5\Phi'(x)\chi_m(x)\,d^4x = \delta_{nm}+igq_{nm}B_nA_m. \tag{6.29} \]

If \(B_n\) and \(A_m\) were not anticommuting operators, but numbers, i.e., if the operators \(\chi\) and \(\bar{\chi}\) belonged to the Bose field, then we would obtain

\[ N_B^0(\Phi')=e^{iW_M}N(\Phi) = \int \exp\left[-i\sum_{nm}\beta_n\beta_m^*(\delta_{nm}+igq_{nm})\right]\,d(\beta) = \]

\[ = \int e^{-i\sum_n \lambda_n\lambda_n^*}\, \frac{\partial(\beta,\beta^*)}{\partial(\lambda,\lambda^*)}\,d(\lambda). \tag{6.30} \]

The transformation from the variables \(\beta,\beta^*\) to the variables \(\lambda,\lambda^*\) is the transformation that diagonalizes the quadratic form \(\sum \beta_n \beta_m^*(\delta_{nm}+igq_{nm})\). The Jacobian of this transformation is equal to the reciprocal of the determinant composed of the coefficients of this quadratic form,

\[ \frac{\partial(\beta,\beta^*)}{\partial(\lambda,\lambda^*)} = \left|\left|\delta_{nm}+igq_{nm}\right|\right|^{-1} \tag{6.31} \]

and does not depend on \(\lambda,\lambda^*\); consequently,

\[ N_B^0=\left|\left|\delta_{nm}+igq_{nm}\right|\right|^{-1}. \]

In the case when \(\chi\) and \(\bar\chi\) refer to the Fermi field, one must diagonalize the form \(\sum \beta_n\beta_m^*(\delta_{nm}+igq_{nm}B_nA_m)\), and in analogous fashion we obtain

\[ N^0(\Phi')=\left|\left|\delta_{nm}+igq_{nm}B_nA_m\right|\right|^{-1}. \]

In the case of infinite determinants of Fredholm type, the reciprocal of the determinant \(\left|\left|\delta_{nm}+gG_{nm}\right|\right|\) is equal to the permanent of the form \((\delta_{nm}-gG_{nm})\), i.e., to a quantity which is calculated by the same rules as the determinant, but without changing signs—only with positive signs:

\[ N^0(\Phi')=\operatorname{Perm}(\delta_{nm}-igq_{nm}B_nA_m). \]

If now, in the expansion for \(\operatorname{Perm}\), one rearranges the operators \(B_n\) and \(A_m\) so as to use the formula \(B_nA_n\to 1\), then this will lead to a change of signs, which just compensates for the change of signs in passing from the permanent to the determinant, \(^{45}\)

\[ N^0(\Phi')=\left|\left|\delta_{nm}+igq_{nm}\right|\right|. \tag{6.32} \]

This same result can be obtained directly from formula (6.28) after substituting \(Q_{nm}\) from (6.29), expanding

\[ \exp\left[-g\sum_{nm}q_{nm}\beta_n\beta_m^*B_nA_m\right] \]

in a series, and carrying out termwise integration using the equality \(B_nA_n=1\).

The meaning of (6.32) is easy to clarify if the matrix \(q_{nm}\) is reduced to diagonal form, where

\[ q_{nm}=\delta_{nm}q_n; \]

\[ N^0=\prod_n(1+ig\hat q_n), \]

or

\[ \ln N^0=\operatorname{Sp}\ln(1+ig\hat q). \]

Since

\[ q_{nm}=\int \bar\psi_n(x)\gamma_5\Phi'(x)\psi_m(x)\,d^4x \]

and, by virtue of (6.13),

\[ \sum \Psi_n(x)\overline{\Psi}_n(y)=-iS_F(x-y), \]

then \(\ln N^0(\Phi')\) can also be represented in the form

\[ \ln N^0(\Phi')=\operatorname{Sp}\ln(1-gS_F\gamma_5\Phi), \tag{6.33} \]

if \(S_F(x,y)\) is regarded as the matrix elements of the operator \(S_F\). Further-

further transformation (6.33) leads (see, for example, \(^{3}\)) to the expression

\[ N^{0}(\Phi')\exp\left\{-g\,\operatorname{Sp}\gamma_{5}\int_{0}^{1} d\lambda \int S_{F}(x,x,\lambda\Phi')\Phi'(x)\,d^{4}x\right\}. \tag{6.34} \]

Formula (6.26) for \(S'_{F}\), as well as formula (6.34), was derived by various methods \(^{43,44,45}\). The derivation presented illustrates a method of direct integration over the Fermi field. Usually integration over the Fermi field is carried out first, while approximate methods of functional integration are associated with the study of integrals over the Bose field \(^{45}\). The possibility of reducing integrals over anticommuting functions to integrals over ordinary numbers makes it possible to treat these integrals on an equal footing with integrals over the Bose field, which may prove convenient in investigating the difficulties of the modern theory.

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1. Reference number as printed in the source. ↩↩