ON THE MATHEMATICAL STRUCTURE OF RENORMALIZABLE LEE FIELD THEORY*

G. Källén and W. Pauli

INTRODUCTION

In a recently published paper[^1], T. D. Lee proposed a very interesting variant of renormalizable field theory. This variant is sufficiently simple to obtain a more or less explicit solution, yet it retains essential features inherent in any serious theory. In this variant one carries out the renormalization of the mass of particles of one kind, as well as the renormalization of the interaction constant \(g\), describing the interaction between the particles. In the exact solution found by Lee, the ratio of the square of the renormalized interaction constant \(g\) to the square of the unrenormalized interaction constant \(g_0\) has the form

\[ \frac{g^2}{g_0^2}=1-Ag^2, \tag{1} \]

where \(A\) is a divergent integral. Thus the ratio (1) is equal to \(-\infty\). This is a very remarkable result, in contradiction with very general principles[^2], according to which the value of this ratio must lie between unity and zero.

The aim of the present article is, first, to clarify the mathematical reasons for the occurrence of result (1); second, it will be shown that the violation of the fundamental principles expressed by relation (1) leads to a number of consequences, since the \(S\)-matrix of the theory turns out to be nonunitary. In order to avoid operations with divergent integrals, we shall introduce a cutoff factor into the interaction. It will then be found that the anomalous value of relation (1) is also obtained for a finite value of the cutoff and that it

* Dan. Mat. Fys. Medd. 30, issue 7 (1955).

is not directly connected with infinities in the initial assumptions.

For completeness of exposition we begin with an outline of the basic propositions of Lee’s theory and a presentation of the renormalization method used in this theory.

1. RENORMALIZATION IN LEE’S THEORY

Consider a system consisting of three kinds of particles, which, following Lee, we shall call \(V\)-, \(N\)-, and \(\theta\)-particles. To each kind of particle there corresponds its own field, which we shall denote respectively by \(\psi_V\), \(\psi_N\), and \(a\). The system is described by the unrenormalized Hamiltonian:

\[ H = H_0 + H_{\mathrm{int}}, \tag{2} \]

\[ H_0 = \sum_{\mathbf p} E_V(\mathbf p)\psi_V^*(\mathbf p)\psi_V(\mathbf p) + \sum_{\mathbf p} E_N(\mathbf p)\psi_N^*(\mathbf p)\psi_N(\mathbf p) + \]

\[ + \sum_{\mathbf k}\omega(\mathbf k)a^*(\mathbf k)a(\mathbf k), \tag{3} \]

\[ H_{\mathrm{int}} = -\frac{g_0}{\sqrt V} \sum_{\mathbf p=\mathbf p' + \mathbf k} \frac{f(\omega)}{\sqrt{2\omega}} \left(\psi_V^*(\mathbf p)\psi_N(\mathbf p')a(\mathbf k)+\right. \]

\[ \left. +a^*(\mathbf k)\psi_N^*(\mathbf p')\psi_V(\mathbf p)\right). \tag{4} \]

The operators in (3) and (4) may be written both in the \(p\)- and in the \(x\)-representations. This theory is not invariant with respect to the Lorentz group, and therefore there is no need to use the complicated variants of relativistic field theory. The energies \(E_V(\mathbf p)\), \(E_N(\mathbf p)\), and \(\omega(\mathbf k)\) may in principle be arbitrary functions of the momenta under consideration; the theory may also be considered for such arbitrary functions. Nevertheless, it suffices for us to consider the special case

\[ E_V(\mathbf p)=E_N(\mathbf p)=m \quad \text{(independent of \(\mathbf p\))}, \tag{5} \]

\[ \omega(\mathbf k)=\sqrt{\mathbf k^2+\mu^2}. \tag{6} \]

Equation (5), in particular, greatly simplifies the formal results, while preserving all the details that are of interest to us.

If desired, such a choice of the energy as a function of the momentum may be regarded as a description of the interaction between very heavy \(V\)- and \(N\)-particles of equal mass with light relativistic \(\theta\)-particles. The function \(f(\omega)\) in (4) is a cutoff function, simpli-

mentioned above; its purpose is to make the sums that arise later convergent. \(V\) is the magnitude of the spatial periodicity volume.

The field operators obey the following commutation and anticommutation rules:

\[ \{\psi_V^*(\mathbf{p}),\,\psi_V(\mathbf{p}')\} = \{\psi_N^*(\mathbf{p}),\,\psi_N(\mathbf{p}')\} = \delta_{\mathbf{p},\,\mathbf{p}'}, \tag{7} \]

\[ \{\psi_V(\mathbf{p}),\,\psi_V(\mathbf{p}')\} = \{\psi_V(\mathbf{p}),\,\psi_N(\mathbf{p}')\} = \ldots = 0, \tag{8} \]

\[ [a(\mathbf{k}),\,a^*(\mathbf{k}')] = \delta_{\mathbf{k},\,\mathbf{k}'}, \tag{9} \]

\[ [a(\mathbf{k}),\psi_V(\mathbf{p})] = [a(\mathbf{k}),\psi_N(\mathbf{p}')] = \ldots = 0. \tag{10} \]

With the help of these rules one can establish a representation in Hilbert space in which each state is characterized by the number of particles present in it. Further, each state in this representation is an eigenstate of the free-particle Hamiltonian \(H_0\) (3), but not of the complete Hamiltonian (2). We denote these states as

\[ |n_V,\,n_N,\,n_k\rangle, \tag{11} \]

where \(n_V\), \(n_N\), and \(n_k\) are the numbers of the available “free” \(V\)-, \(N\)-, and \(\theta\)-particles*).

Using (7)—(10), it is easy to verify that the following two operators commute with the complete Hamiltonian:

\[ Q_1=\sum_p \psi_V^*(\mathbf{p})\psi_V(\mathbf{p}) + \sum_p \psi_N^*(\mathbf{p})\psi_N(\mathbf{p}), \tag{12} \]

\[ Q_2=\sum_p \psi_N^*(\mathbf{p})\psi_N(\mathbf{p}) - \sum_k a^*(\mathbf{k})a(\mathbf{k}), \tag{13} \]

\[ [H,Q_i]=0,\qquad i=1,2. \tag{14} \]

Since each state (11) is simultaneously an eigenstate of the operators \(Q_i\), the eigenstates of the complete Hamiltonian can be represented as linear combinations

*) The “states of free particles” introduced in this way coincide completely with the known states of free particles, for example, in the Tamm—Dancoff method; however, these are quite different from the so-called “in” (or out) states of free particles used in relativistic field theories. As will be seen below, the Tamm—Dancoff method gives an exact solution for our case.

of the states (11) belonging to one and the same eigenvalue \(q_i\). This circumstance substantially simplifies the problem of diagonalizing the full Hamiltonian, and in some cases even makes it possible to obtain an exact solution. As an example we point out that there is only one state (11) in which \(q_1=q_2=0\), namely the state \(|0,0,0\rangle\), or the “free vacuum” (“vacuum of free particles”).

Consequently, this state is also an eigenstate of the full Hamiltonian, and a simple calculation gives zero as the eigenvalue of this operator. Thus, in this model the “physical vacuum” is at the same time also the “vacuum of free particles.” Similarly, one can show that the physical states of \(N\)-particles and \(\vartheta\)-particles are identical with the corresponding states of free particles, but that the states of free \(V\)-particles are not eigenstates of the full Hamiltonian. In order to obtain the eigenstates of the full Hamiltonian in this case, it is necessary to consider a linear combination of the states \(|1_V,0,0\rangle\) and \(|0,1_N,1_\theta\rangle\). We shall return to this somewhat later. For the moment we note only that, under these conditions, there is no need to renormalize the masses of the \(N\)- and \(\vartheta\)-particles. In Lee’s theory the mass renormalization is carried out by adding to the Hamiltonian the term

\[ \delta H=-\delta m \sum_{\mathbf p}\psi_V^*(\mathbf p)\psi_V(\mathbf p), \tag{15} \]

which leaves the conservation equations (14) unchanged. The constant \(\delta m\) in (15) must, if possible, be chosen so that the states corresponding to \(V\)-particles have mass \(m\), entering into \(H_0\). Following the rules of quantum electrodynamics, we perform the renormalization of the interaction constant \(g_0\) and of the field operator \(\psi_V\) by a factor \(N\) in the following way:

\[ g=g_0N, \tag{16} \]

\[ \psi_V(\mathbf p)=\psi_V(\mathbf p)\frac{1}{N}. \tag{17} \]

It is essential to emphasize that the constant \(N\) in (16) and (17) can be chosen real, since the field operators contain an arbitrary phase factor. The choice of a real multiplier \(N\) merely imposes a relation between the phases of \(\dot{\psi}_V\) and \(\dot{\psi}_V^{*}\), but does not lead to any physical consequences. The quantity \(N\) is determined from the condition\(^3\)

\[ \langle 0|\dot{\psi}_V(\mathbf p)|V\rangle=1. \tag{18} \]

The state \(|V\rangle\) in (18) corresponds to the physical state of a \(V\)-particle, and the state \(|0\rangle\) to the state of the physical vacuum. In what follows we omit the prime on the renormalized \(\psi_V\)-operator, since the corresponding unrenormalized operator will not be used at all. Thus, if one uses the renormalized expressions for the Hamiltonian and the canonical commutation rules, one may write:

\[ H=H_0+H_{\mathrm{int}}+\delta H, \tag{19} \]

\[ H_0=mN^2\sum_{\mathbf p}\psi_V^*(\mathbf p)\psi_V(\mathbf p) +m\sum_{\mathbf p}\psi_N^*(\mathbf p)\psi_N(\mathbf p)+ \]

\[ +\sum_{\mathbf k}\omega(\mathbf k)a^*(\mathbf k)a(\mathbf k), \tag{20} \]

\[ H_{\mathrm{int}}=-\frac{q}{\sqrt V}\sum_{\mathbf p=\mathbf p'+\mathbf k} \frac{f(\omega)}{\sqrt{2\omega}} \left(\psi_V^*(\mathbf p)\psi_N(\mathbf p')a(\mathbf k)+\right. \]

\[ \left.+a^*(\mathbf k)\psi_N^*(\mathbf p')\psi_V(\mathbf p)\right), \tag{21} \]

\[ \delta H=-\delta m\,N^2\sum_{\mathbf p}\psi_V^*(\mathbf p)\psi_V(\mathbf p), \tag{22} \]

\[ \{\psi_V^*(\mathbf p),\psi_V(\mathbf p')\}=\frac{1}{N^2}\delta_{\mathbf p,\mathbf p'} \tag{23} \]

(the remaining commutation rules do not change).

Equations (19)—(23) are fundamental for the subsequent discussion.

II. PHYSICAL STATES OF THE \(V\)-PARTICLE AND STATES DESCRIBING SCATTERING OF \(N\)- AND \(\theta\)-PARTICLES

We shall seek eigenstates of the Hamiltonian having the form

\[ |z\rangle=|1_V,\ 0,\ 0\rangle+\sum_{\mathbf k}\Phi(\mathbf k)|0,\ 1_N,\ 1_k\rangle. \tag{24} \]

In this expression all terms have one and the same total momentum. In the following formulas the factor expressing conservation of the total three-dimensional momentum will often be omitted. Denoting the eigenvalues of the state Hamiltonian (24) by \(m+\omega_0\) and using (19)—(23), we obtain, after straightforward

calculations:

\[ \omega_0+\delta m=-\frac{g}{N\sqrt V}\sum_{\mathbf k}\frac{\Phi(\mathbf k) f(\omega)}{\sqrt{2\omega}}, \tag{25} \]

\[ (\omega-\omega_0)\Phi(\mathbf k)=-\frac{g}{N\sqrt V}\frac{f(\omega)}{\sqrt{2\omega}}. \tag{26} \]

Eliminating the quantity \(\Phi(\mathbf k)\) from (25) and (26), we obtain the equation determining the eigenvalue \(\omega_0\):

\[ \omega_0+\delta m+\frac{g^2}{2N^2V}\sum_{\mathbf k}\frac{f^2(\omega)}{\omega}\frac{1}{\omega-\omega_0}=0. \tag{27} \]

The constant \(\delta m\) is determined from the condition that \(\omega_0=0\) is one of the solutions of (27). The corresponding eigenstate of the Hamiltonian (24), with the corresponding normalization, is the physical state of the \(V\)-particle. We obtain, in the end:

\[ \delta m=-\frac{g^2}{2V}\frac{1}{N^2}\sum_{\mathbf k}\frac{f^2(\omega)}{\omega^2}, \tag{28} \]

\[ |V\rangle=C\left[\,|1_V,0,0\rangle+\frac{g}{N\sqrt{2V}}\sum_{\mathbf k}\frac{f(\omega)}{\omega^{3/2}}|0,1_N,1_{\mathbf k}\rangle\,\right], \tag{29} \]

\[ C^{-2}=1+\frac{g^2}{2VN^2}\sum_{\mathbf k}\frac{f^2(\omega)}{\omega^3}. \tag{30} \]

Next, using equation (18), we obtain:

\[ C=N \tag{31} \]

or

\[ |V\rangle=N|1_V,0,0\rangle+\frac{g}{\sqrt{2V}}\sum_{\mathbf k}\frac{f(\omega)}{\omega^{3/2}}|0,1_N,1_{\mathbf k}\rangle, \tag{32} \]

\[ N^2=1-\frac{g^2}{2V}\sum_{\mathbf k}\frac{f^2(\omega)}{\omega^3}. \tag{33} \]

The results obtained so far in this section correspond exactly to Lee’s results. In particular, from equations (33) and (16) follows relation (1), obtained by Lee, if the form factor is set equal to unity for all values of \(\omega\). However, if a finite cutoff is used, equation (33) takes the form

\[ N^2=1-\frac{g^2}{g_{\mathrm{crit}}^2}; \tag{34} \]

where

\[ g_{\mathrm{crit}}^{-2}=\frac{1}{2V}\sum_{\mathbf{k}}\frac{f^2(\omega)}{\omega^3}. \tag{34a} \]

The value \(N^2\), determined by relation (34), is, as was to be expected, confined between unity and zero only in the case where the renormalized interaction constant \(g\) is smaller than a certain critical value \(g_{\mathrm{crit}}\), defined by (34a) and depending on the cutoff function. If no cutoff is introduced, then the critical value of the interaction constant is equal to zero. On the other hand, if no renormalization of the interaction constant has been carried out and all quantities are expressed in terms of the initial interaction constant \(g_0\), then in all the formulas given above the quantity \(g^2\) should be replaced according to the equality

\[ g^2=\frac{g_0^2\cdot g_{\mathrm{crit}}^2}{g_0^2+g_{\mathrm{crit}}^2}. \tag{35} \]

Relation (35) indicates that the renormalized interaction constant is always smaller than the critical interaction constant, provided that the Hamiltonian is Hermitian, i.e., that the quantity \(g_0\) is real. As Lee has already emphasized, it is also of some interest to study the case in which the renormalized interaction constant is greater than the critical value and the Hamiltonian is non-Hermitian. The decisive question in such a consideration is to determine whether such a violation of the usual concepts of quantum mechanics will lead to any noticeable undesirable consequences, or whether in this way one can nevertheless obtain an at least partially satisfactory theory.

We now return to the study of the other solutions of the eigenvalue problem defined by equation (27). This equation, taking into account (28) and (33), may be rewritten somewhat differently, namely:

\[ h(\omega_0)\equiv \omega_0\left[1+\frac{g^2}{2V}\sum_{\mathbf{k}}\frac{f^2(\omega)\,\omega_0}{\omega^3(\omega-\omega_0)}\right]=0. \tag{36} \]

The second factor in (36) has poles at \(\omega_0=\omega_i\), where \(\omega_i\) are the eigenvalues of the unperturbed Hamiltonian \(H_0\). Since the derivative of the last factor in (36) with respect to \(\omega_0\) is always positive, this factor vanishes once and only once in each interval \((\omega_i,\omega_{i+1})\). The corresponding eigenstates of the Hamiltonian (24) describe the scattering of one \(N\)-particle and one \(\theta\)-particle. After formal transformations these states...

can be represented in the form

\[ |N,\vartheta\rangle=|0,1_N,1_k\rangle+\sum_{\mathbf{k}'}\alpha(\mathbf{k},\mathbf{k}')\,|0,1_{N'},1_{k'}\rangle+ \]
\[ +\beta(\mathbf{k})\,N\,|1\nu,0,0\rangle, \tag{37} \]

\[ \alpha(\mathbf{k},\mathbf{k}')= \frac{g}{\sqrt{2V}}\, \frac{\beta(\mathbf{k})f(\omega')}{\sqrt{\omega'}} \left\{ P\frac{1}{\omega'-\omega} +i\pi\delta(\omega'-\omega) \right\}, \tag{38} \]

\[ \beta(\mathbf{k})= -\frac{g f(\omega)}{\sqrt{2V}\,\omega^{3/2}}\times \]
\[ \times\left[ 1+\frac{g^2\omega}{2V}\sum_{\mathbf{k}} \frac{f^2(\omega')}{\omega'^3} \left( P\frac{1}{\omega'-\omega} +i\pi\delta(\omega'-\omega) \right) \right]^{-1}. \tag{39} \]

In relations (38) and (39) the passage to the limit \(V\to\infty\) is provided for, and the equations themselves contain a prescription as to how their denominators are to be handled when integrating over \(\mathbf{k}'\). This prescription ensures the presence of only outgoing waves in the second term of (37). Only particles with momentum \(\mathbf{k}\) enter these states. From the formulas given above one can calculate the part of the \(S\)-matrix responsible for the scattering of \(N\)- and \(\vartheta\)-particles by one another. This part is a unitary matrix

\[ \langle N,\vartheta|S|N',\vartheta'\rangle =\delta_{\mathbf{k},\mathbf{k}'} +\frac{i\pi g^2}{V}\frac{f^2(\omega)}{\omega} \frac{\delta(\omega'-\omega)} {h(\omega)+i\frac{g^2}{4\pi}|k|f^2(\omega)} . \tag{40} \]

From (40) we find the differential cross section

\[ \frac{d\tau}{d\Omega}=\frac{1}{|k|^2}\sin^2\delta, \tag{41} \]

where

\[ \tg\delta=\frac{g^2}{4\pi}\frac{|k|f^2(\omega)}{h(\omega)}. \tag{42} \]

And these results coincide exactly with the results obtained by Lee. In the last three formulas the limiting transition \(V\to\infty\) has been carried out, and the integral arising in \(h(\omega)\) [see (36)] is taken in the sense of the principal value.

It remains to clarify an important circumstance: whether the states (32) and (37) obtained by us form a complete system, or whether there exist other possible states of the system whose Hamiltonian is given in the form (24), which are also eigenstates of the full Hamiltonian. If such states exist, then they must correspond to other solutions of the eigenvalue problem, i.e. to other solutions of equation (36). Therefore one should begin with a more detailed investigation of this equation. Earlier

the preceding considerations made it possible to go through all the roots of equation (36) lying in the region \(\omega_0>\mu\) ). In the region \(\omega_0<\mu\) we found that the second factor in (36) still retains a positive derivative and that for large values of \(|\omega_0|\) it tends to the value
\[ N^2=1-\frac{g^2}{g_{\mathrm{crit}}^2}. \]
If the interaction constant is less than the critical value, then equation (36) has no additional roots and the states considered form a complete system. However, if the interaction constant is greater than the critical constant, there exists one additional root of equation (36) in the region \(\omega_0<\mu\). The corresponding eigenstate is no longer a scattering state, but represents another state of the \(V\)-particle *). This state can be obtained directly from our formalism and has the form
\[ |V_{-\lambda}\rangle=\frac{1}{\sqrt{|h'(-\lambda)|}}\times \]
\[ \times\left[ N\,|1_V,0,0\rangle +\frac{g}{\sqrt{2V}}\sum_{\mathbf{k}} \frac{f(\omega)}{\sqrt{\omega}}\, \frac{1}{\omega+\lambda}\, |0,1_N,1_{\mathbf{k}}\rangle \right], \tag{43} \]
\[ h(-\lambda)=0;\qquad \lambda>0 . \tag{44} \]

The choice of normalization of the state (43) will be explained in the following section.

In Appendix I it will be shown that equation (36) has no other roots besides the real ones.

III. INTRODUCTION OF AN INDEFINITE METRIC IN HILBERT SPACE

The negative sign of the quantity \(N^2\) in (34) in the case when \(g\) is greater than \(g_{\mathrm{crit}}\) evidently causes difficulties in normalizing the physical state of the \(V\)-particle (32). If one tries to correct the normalization of this state by multiplying by a suitable factor, this will entail a change in the renormalization that we have performed,

\[ \text{*)} \]
If the cutoff function is equal to zero for \(\omega\) greater than some value \(\Omega\), the region \(\omega_0>\Omega\) requires special investigation, since the arguments given after formula (36) no longer apply here. Indeed, one can show that in this case there is one more root in the region \(\omega_0>\Omega\), if \(g\) is smaller than the critical value \(g_{\mathrm{crit}}\). To avoid unnecessary complications in the reasoning, we shall consider cutoff functions with large tails, such as, for example, \(f(\omega)=e^{-\omega/\Omega}\); in this case such difficulties do not arise.

\[ \text{**)} \]
In Lee’s paper, in note 4, the possibility of another stable state of the \(V\)-particle is mentioned in passing, but its properties are not investigated. For us this state is of primary importance.

since we can no longer use one and the same multiplier in (16) and (17) for renormalizing the interaction constant and the field operator \(\psi_V\). In this case one has to introduce special factors into the Hamiltonian (21), which describes the interaction, and it is easy to see that in this way it is impossible to make the theory mathematically consistent. The only possibility of preserving the normalization of the state (32) is to define the norm of the state \(a(n_V,n_N,n_k)\) as \(|a|^2(-1)^{n_V}\). Since in our case the quantity \(N^2\) is real and negative, such an indefinite metric will be a suitable framing of Lee’s model\(^4\). The introduction of such a metric does not alter most of the formal calculations carried out above, and in particular the scattering states (37) and the \(S\)-matrix (40), which remain unchanged. At the same time, the norm of the state (32) will now be what it should be according to the new metric. The norm of the state (43) will be:

\[ \frac{1}{|h'(-\lambda)|} \left[ N^2+\frac{g^2}{2V}\sum_{\mathbf{k}}\frac{f^2(\omega)}{\omega(\omega+\lambda)^2} \right] = \]

\[ = \frac{1}{|h'(-\lambda)|} \left[ 1+\frac{g^2}{2V}\sum_{\mathbf{k}}\frac{f^2(\omega)}{\omega} \left[ \frac{1}{(\omega+\lambda)^2}-\frac{1}{\omega^2} \right] \right] = \]

\[ = \frac{1}{|h'(-\lambda)|}\frac{g^2}{2V} \sum_{\mathbf{k}}\frac{f^2(\omega)}{\omega} \left[ \frac{1}{(\omega+\lambda)^2}-\frac{1}{\omega^2} +\frac{\lambda}{\omega^2(\omega+\lambda)} \right] = \]

\[ = \frac{h'(-\lambda)}{|h'(-\lambda)|}=-1. \tag{45} \]

Consequently, the norm of the state \(|V-\lambda\rangle\) is negative and is normalized in (43) to \(-1\).

In order to make the formal discussion as simple as possible, it is convenient to introduce\(^4\) here a “metric operator” \(\eta\), which has the following matrix elements for the free-particle states (11):

\[ \langle n_V,n_N,n_k|\eta|n'_V,n'_N,n'_k\rangle = \delta_{n_V n'_V}\delta_{n_N n'_N}\delta_{n_k n'_k}(-1)^{n_V}. \tag{46} \]

For the physical states considered earlier, we have:

\[ \langle V|\eta|V\rangle=\langle N,\vartheta|\eta|N,\vartheta\rangle=1, \tag{47} \]

\[ \langle V-\lambda|\eta|V-\lambda\rangle=-1. \tag{48} \]

All off-diagonal elements of \(\eta\) between these states are equal to zero. The conditions imposed on the operator \(F\) in order to have real expectation values are no longer

in the Hermiticity of the operator, but in the condition of “self-adjointness,” which has the following meaning:

\[ F=F^+\equiv \eta F^* \eta . \tag{49} \]

A detailed examination of the preceding calculations shows that the introduction of an indefinite metric corresponds mathematically to replacing the operators \(\psi_V^*, \psi_N^*\) and \(a^*\) in equations (23) by the operators \(\psi_V^+, \psi_N^+\) and \(a^+\). Such a replacement also makes the Hamiltonian self-adjoint. On the other hand, the right-hand side of (23) now has no definite sign, but negative values of these \(c\)-numbers will no longer contradict the foundations of the theory. The special case of the observed values of these anticommutators is investigated in Appendix I.

If the transformation leading from the states of the free particles \(|n\rangle\) to the physical states \(|P\rangle\) is specified in the form of a matrix \(U\)

\[ |P\rangle=\sum_{|n\rangle} |n\rangle \langle n|U|P\rangle , \tag{50} \]

then this matrix is not unitary, but has the property

\[ U^+U=\eta U^*\eta U=1 . \tag{51} \]

It is important to know whether the \(S\)-matrix of the theory is unitary or whether it likewise has the property (51). The presence of the property (51) would not contradict the result (40), since for all the physical states under consideration the operator \(\eta\) has the single matrix element \(+1\). Equation (51) could have nontrivial consequences only if physical states with nonpositive norms were involved.

The simplest process of this kind is the scattering of \(\vartheta\)-particles by \(V\)-particles in the normal state, or else in the state \(|V_{-\lambda}\rangle\). In the first case one may expect that transitions of \(V\)-particles into new states take place and that these transitions occur with “negative probability.” The next section is devoted to consideration of this question.

IV. SCATTERING OF \(\vartheta\)-PARTICLES BY \(V\)-PARTICLES

We shall investigate the eigenvectors of the total Hamiltonian, taken in the form

\[ |z\rangle=\sum_{\mathbf{k}} \Phi_1(\mathbf{k})N|1_V,0,1_{\mathbf{k}}\rangle +\sum_{\mathbf{k},\mathbf{k}'} \Phi_2(\mathbf{k},\mathbf{k}')|0,1_N,1_{\mathbf{k}},1_{\mathbf{k}'}\rangle . \tag{52} \]

If the eigenvalues are still denoted by \(m+\omega_0\), direct calculations lead to the following equations

for the coefficients entering into (52):

\[ \Phi_1(\mathbf{k})(\omega-\omega_0-\delta m) = \frac{1}{N^2}\,g\sqrt{\frac{2}{V}}\sum_{\mathbf{k}'} \Phi_2(\mathbf{k},\mathbf{k}')\frac{f(\omega')}{\sqrt{\omega'}}, \tag{53} \]

\[ \Phi_2(\mathbf{k},\mathbf{k}')(\omega+\omega'-\omega_0) = \frac{g}{\sqrt{2V}}\,\frac{1}{2} \left[ \Phi_1(\mathbf{k})\frac{f(\omega')}{\sqrt{\omega'}} + \Phi_1(\mathbf{k}')\frac{f(\omega)}{\sqrt{\omega}} \right]. \tag{54} \]

In the present case we are not at all interested in the complete system of states of the Hamiltonian (52), but shall seek only those states which correspond to the scattering of a $\vartheta$-particle by a $V$-particle in the normal state. In other words, we seek the solution of (53) and (54) when $\Phi_1(\mathbf{k})$ is given in the form

\[ \Phi_1(\mathbf{k},\mathbf{k}_0)=\delta_{\mathbf{k},\mathbf{k}_0}+\psi(\mathbf{k},\mathbf{k}_0), \tag{55} \]

where the outgoing waves are contained only in $\psi(\mathbf{k},\mathbf{k}_0)$ and $\Phi_2(\mathbf{k},\mathbf{k}')$. The latter condition makes it possible to obtain:

\[ \Phi_2(\mathbf{k},\mathbf{k}',\mathbf{k}_0) = \frac{g}{\sqrt{2V}}\,\frac{1}{2} \left[ \Phi_1(\mathbf{k},\mathbf{k}_0)\frac{f(\omega')}{\sqrt{\omega'}} + \Phi_1(\mathbf{k}',\mathbf{k}_0)\frac{f(\omega)}{\sqrt{\omega}} \right] \left[ P\frac{1}{\omega+\omega'-\omega_0} +i\pi\delta(\omega+\omega'-\omega_0) \right], \tag{56} \]

or, using (28) and (33),

\[ \Phi_1(\mathbf{k},\mathbf{k}_0)\,h(\omega_0-\omega) = \]

\[ = \frac{g^2}{2V}\frac{f(\omega)}{\sqrt{\omega}} \sum_{\mathbf{k}'} \frac{f(\omega')\Phi_1(\mathbf{k}',\mathbf{k}_0)}{\sqrt{\omega'}} \left[ P\frac{1}{\omega+\omega'-\omega_0} +i\pi\delta(\omega+\omega'-\omega_0) \right]. \tag{57} \]

In contrast to what we had in Section II, we cannot find an explicit solution of equation (57). But this is not at all necessary for our purposes, since it is sufficient for us to investigate the properties of the $S$-matrix. The investigation can be carried out by a method very similar to Møller’s method$^{5}$ for proving the unitarity of the $S$-matrix in the case of a Hermitian Hamiltonian. Following Møller, we introduce:

\[ U(\mathbf{k},\mathbf{k}_0)= \]

\[ = i\,\frac{g^2}{2V}\frac{f(\omega)}{\sqrt{\omega}} \sum_{\mathbf{k}'} \frac{f(\omega')\Phi_1(\mathbf{k}',\mathbf{k}_0)}{\sqrt{\omega'}} \left[ P\frac{1}{\omega+\omega'-\omega_0} +i\pi\delta(\omega+\omega'-\omega_0) \right]. \tag{58} \]

From equation (57) we derive:

\[ \sum_{\mathbf{k}}\Phi_1^*(\mathbf{k},\mathbf{k}_0)\,U(\mathbf{k},\mathbf{k}_0)= \]

\[ = i\,\frac{g^2}{2V}\sum_{\mathbf{k},\mathbf{k}''} \frac{\Phi_1^*(\mathbf{k},\mathbf{k}_0)f(\omega)}{\sqrt{\omega}}\, \frac{f(\omega'')\Phi_1(\mathbf{k}'',\mathbf{k}_0')}{\sqrt{\omega''}} \times \]

\[ \times\left[ P\,\frac{1}{\omega+\omega''-\omega_0} +i\pi\delta(\omega+\omega''+\omega_0') \right], \tag{59} \]

\[ \sum_{\mathbf{k}''}U^*(\mathbf{k}'',\mathbf{k}_0)\Phi_1(\mathbf{k}'',\mathbf{k}_0')= \]

\[ = i\,\frac{g^2}{2V}\sum_{\mathbf{k},\mathbf{k}''} \frac{\Phi_1^*(\mathbf{k},\mathbf{k}_0)f(\omega)}{\sqrt{\omega}}\, \frac{f(\omega'')\Phi_1(\mathbf{k}'',\mathbf{k}_0')}{\sqrt{\omega''}} \times \]

\[ \times\left[ P\,\frac{1}{\omega+\omega''-\omega_0} -i\pi\delta(\omega+\omega''-\omega_0) \right]. \tag{60} \]

The sums in (59) and (60) vanish only under the condition \(\omega_0<2\mu\), exactly as do the corresponding sums in Møller’s work. In this case, the expression \(\omega+\omega''-\omega_0\) nowhere vanishes in the physical frequency interval \(\omega,\omega''\;(\mu,\infty)\), and the transitions \(V+\vartheta \to N+\vartheta'+\vartheta''\) do not occur.

In the case when \(\omega_0>2\mu\), these transitions somewhat complicate the picture and we obtain:

\[ \delta(\omega_0-\omega_0') \left[ \sum_{\mathbf{k}}\Phi_1^*(\mathbf{k},\mathbf{k}_0)U(\mathbf{k},\mathbf{k}_0') + \sum_{\mathbf{k}}U^*(\mathbf{k},\mathbf{k}_0)\Phi_1(\mathbf{k},\mathbf{k}_0') \right] = \]

\[ = -\frac{\pi g^2}{V}\,\delta(\omega_0-\omega_0') \sum_{\mathbf{k},\mathbf{k}''} \Phi_1^*(\mathbf{k},\mathbf{k}_0) \frac{f(\omega)f(\omega'')}{\sqrt{\omega\omega''}} \times \]

\[ \times \Phi_1(\mathbf{k}'',\mathbf{k}_0')\, \delta(\omega+\omega''-\omega_0). \tag{61} \]

With the aid of (55), (57), and (58), bearing in mind that \(h(0)=0\), one can obtain:

\[ \psi(\mathbf{k},\mathbf{k}_0)\,h(\omega_0-\omega)=i\,U(\mathbf{k},\mathbf{k}_0). \tag{62} \]

We shall write the solution of equation (62) symbolically in the form

\[ \psi(\mathbf{k},\mathbf{k}_0)= i\,\frac{U(\mathbf{k},\mathbf{k}_0)}{h(\omega_0-\omega)_+}, \tag{63} \]

where the plus sign indicates that the outgoing waves must be chosen at the zeros of the function \(h(\omega_0-\omega)\). Using this solution, one can rewrite (61) as follows:

\[ \delta(\omega_0-\omega'_0)\,[U(\mathbf{k}_0,\mathbf{k}'_0)+U^*(\mathbf{k}'_0,\mathbf{k}_0)]+ \]

\[ +i\delta(\omega_0-\omega'_0)\sum_{\mathbf{k}}U^*(\mathbf{k},\mathbf{k}_0)U(\mathbf{k},\mathbf{k}'_0) \left[\frac{1}{h(\omega_0-\omega)_+}-\frac{1}{h(\omega_0-\omega)_-}\right]+ \]

\[ +\frac{\pi g^2}{V}\delta(\omega_0-\omega'_0) \sum_{\mathbf{k},\mathbf{k}''} \Phi_1^*(\mathbf{k},\mathbf{k}_0)\frac{f(\omega)f(\omega'')}{\sqrt{\omega\omega''}} \Phi_1(\mathbf{k}'',\mathbf{k}'_0)\delta(\omega+\omega''-\omega_0)=0 . \tag{64} \]

In the last expression the square bracket of the second term can be transformed as follows:

\[ \frac{1}{h(\omega_0-\omega)_+}-\frac{1}{h(\omega_0-\omega)_-} =-2\pi i\sum_{\rho_i}\frac{1}{h'(\rho_i)}\delta(\omega_0-\omega-\rho_i), \tag{65} \]

where the summation is over all roots of the equation \(h(x)=0\).

To simplify the notation, we introduce the matrices

\[ \langle V,\vartheta \mid R^{(1)} \mid V',\vartheta'\rangle =2\pi\delta(\omega-\omega')U(\mathbf{k},\mathbf{k}'), \tag{66} \]

\[ \langle V-\lambda,\vartheta \mid R^{(2)} \mid V,\vartheta'\rangle =2\pi\delta(\omega+\lambda-\omega')\frac{U(\mathbf{k},\mathbf{k}')}{\sqrt{-h'(-\lambda)}}, \tag{67} \]

\[ \langle N,\vartheta',\vartheta'' \mid R^{(3)} \mid V,\vartheta\rangle = \]

\[ =2\pi\delta(\omega'+\omega''-\omega)\frac{g}{\sqrt{2V}}\frac{1}{2} \left[ \Phi_1(\mathbf{k}',\mathbf{k})\frac{f(\omega'')}{\sqrt{\omega''}} +\Phi_1(\mathbf{k}'',\mathbf{k})\frac{f(\omega')}{\sqrt{\omega'}} \right]. \tag{68} \]

It can be shown that the sum over all roots in (65) corresponding to the scattering states of Section II, and the last term in (64), can be expressed through the matrix \(R^{(3)}\). Thus, (64) can be rewritten in the form

\[ \langle V,\vartheta \mid R^{(1)}+R^{(1)*}+R^{(1)*}R^{(1)} \mid V',\vartheta'\rangle - \]

\[ -\langle V,\vartheta \mid R^{(2)*}R^{(2)} \mid V',\vartheta'\rangle +\langle V,\vartheta \mid R^{(3)*}R^{(3)} \mid V',\vartheta'\rangle=0. \tag{69} \]

It follows from this that the \(S\)-matrix in Lee’s theory, which for the states considered in the present section is represented in the form

\[ S=1+R^{(1)}+R^{(2)}+R^{(3)}, \tag{70} \]

is not unitary, since the probability of the transitions \(V+\vartheta\to V_{-\lambda}+\vartheta'\) turns out, according to (69), to be negative. As expected, the matrix \(S\)

satisfies the relation

\[ \eta S^* \eta S = 1, \tag{71} \]

if the diagonal elements of \(\eta\) belonging to the states \(|V_{-\lambda}, \vartheta\rangle\) are set equal to \(-1\). It can also be shown that the same result is obtained if one considers transitions from the states \(|V_{-\lambda}, \vartheta\rangle\). The nonunitarity of the transformation (50), which relates the states of free particles to the physical states, is closely connected with the nonunitarity of the \(S\)-matrix and makes the theory unacceptable on physical grounds.

Here it is natural to ask whether one can give another interpretation to the formalism of the theory, using the arguments of the theory of holes in quantum electrodynamics. One might, for example, call the states \(|V_{-\lambda}\rangle\) the vacuum, and call the states which here have been called the vacuum “antiparticle” states. However, it is easy to see that in this way the formalism of the theory cannot be improved, since no new interpretation of this kind changes the nonunitary properties of the \(S\)-matrix in (69).

Thus, we have shown that Lee’s theory agrees with the physical concept of probability only in the case where a cutoff is introduced into the theory and where the renormalized interaction constant is less than the critical interaction constant defined by relation (34a).

In this case the value of the constant \(N^2\) lies between zero and unity, as it should according to general considerations.\(^2\) If no cutoff is introduced, the critical value of the interaction constant is equal to zero.

APPENDIX I

In this appendix we shall show by direct calculation how an indefinite metric may be responsible for the negative sign on the right-hand side of the anticommutator

\[ \{\psi_V^\dagger(p), \psi_V(p')\} = \delta_{\mathbf p,\mathbf p'} \frac{1}{N^2}. \tag{D.1} \]

We calculate the mean value of this quantity for the vacuum under the conditions \(g > g_{\mathrm{crit}},\ p = p'\), and obtain:

\[ \langle 0 | \{\psi_V^\dagger(p), \psi_V(p)\} | 0 \rangle = \sum_{|z\rangle} |\langle 0 | \psi_V(p) | z \rangle|^2 \langle z | \eta | z \rangle . \tag{D.2} \]

In formula (D.2) the summation is performed over a complete system of states. One may, for example, sum over all physical states and take into account the contribution from the physical states of \(V\)-particles, the states \(|V_{-\lambda}\rangle\), and the scattering states \(|N,\vartheta\rangle\). According to the results

of Section II the contribution of these states can be represented in the form

\[ \langle 0| \{\psi_V^{+}(\mathbf p), \psi_V(\mathbf p)\} |0\rangle = 1+\sum_{\mathbf k}|\beta(\mathbf k)|^2-\frac{1}{|h'(-\lambda)|} = \]

\[ =1+\sum_{\mathbf k}|\beta(\mathbf k)|^2+\frac{1}{h'(-\lambda)} . \tag{D.3} \]

If an indefinite metric is not introduced, the right-hand side is positive and exceeds unity. This, incidentally, is the usual proof that \(N^2\) is a positive number less than unity.\(^2\) In the case under consideration, the last term is negative, and it does not follow from any general considerations that the right-hand side of (D.3) must have a definite sign. We shall give a rigorous proof that this quantity gives the correct value, as determined by (33). The proof rests essentially on the fact that the function \(h(z)\), defined by (36) and continued into the complex plane in the form

\[ h(z)=z\left[1+\frac{g^2}{2V}\sum_{\mathbf k} \frac{f^2(\omega)z}{\omega^3(\omega-z)}\right], \tag{D.4} \]

has zeros only on the real axis. Indeed, putting \(z=x+iy\), we obtain

\[ \operatorname{Im}\frac{h(z)}{z} = \frac{g^2}{2V}\operatorname{Im}\sum_{\mathbf k} \frac{f^2(\omega)z}{\omega^3(\omega-z)} = \frac{g^2}{2V}\sum_{\mathbf k} \frac{f^2(\omega)y}{\omega^2[(\omega-x)^2+y^2]} . \tag{D.5} \]

The last expression is always different from zero if \(y\ne 0\).

Moreover, after passing to the limit \(V\to\infty\), the function \(h(z)\) is transformed into an analytic function of the form

\[ h(z)=z\left[ 1+\gamma z\int_{\mu}^{\infty} f^2(\omega)\frac{\sqrt{\omega^2-\mu^2}\,d\omega}{\omega^2(\omega-z)} \right] \tag{D.4a} \]

(where the notation \(\gamma=\dfrac{g^2}{4\pi^2}\) has been introduced), single-valued in the complex plane cut along the real axis from the point \(\mu\) to positive infinity. The imaginary part of the function \(h(z)\) on this part of the real axis has a discontinuity, since in the upper and lower half-planes \(h(z)\) has opposite signs, whereas its real part is continuous. The two-valuedness of \(h(z)\) corresponds to the circumstance that the point \(z=\mu\) is a branch point for square roots of the type \(h(z)\) (compare the explicit form of the function \(h(z)\) for the special form of \(f(\omega)\) given in Appendix II).

These properties of the function \(h(z)\) make it possible to estimate in two ways the integral

\[ \frac{1}{2\pi i}\int_C \frac{dz}{h(z)}, \]

taken along the path shown in the figure. First of all, we note that

\[ \sum_{\mathbf{k}}|\beta(\mathbf{k})|^2 = \gamma\left\{ \int_{\mu}^{\infty} f^2(\omega)\sqrt{\omega^2-\mu^2}\,d\omega \left[ h^2(\omega)+ \left( \frac{\pi\gamma}{\omega}f^2(\omega)\sqrt{\omega^2-\mu^2} \right)^2 \right]^{-1} \right\} = \frac{1}{\pi}\lim_{\varepsilon\to 0}\operatorname{Im} \int_{\mu}^{\infty}\frac{d\omega}{h(\omega-i\varepsilon)} . \tag{D.6} \]

Next, we divide the path \(C\) into two parts. One of them, \(C_1\), begins at the point \(z=R-i\varepsilon\), where \(R\) is arbitrarily large and positive and \(\varepsilon\) is arbitrarily small, then goes below the real axis at a distance \(\varepsilon\) from it, describes a semicircle of radius \(\varepsilon\) about the point \(z=\mu\) in the negative direction, continues above the real axis at a distance \(\varepsilon\), and ends at the point \(z=R+i\varepsilon\). The second part of the path, \(C_R\), is a circle of sufficiently large radius \(R\), a small part of which near the real axis is omitted.

Passing to the limit \(\varepsilon\to 0\), in which the contribution to \(C_1\) from the semicircle becomes arbitrarily small, we obtain:

\[ \lim_{\varepsilon\to 0}\int_{C_1}\frac{dz}{h(z)} = -2i\lim_{\varepsilon\to 0}\operatorname{Im} \int_{\mu}^{\infty}\frac{dz}{h(z-i\varepsilon)} = -2\pi i\sum_{\mathbf{k}}|\beta(\mathbf{k})|^2 . \tag{D.7} \]

Under this limiting transition, the second part of \(C\), i.e. \(C_R\), is transformed into the full circle \(C_R\). The corresponding integral is easily estimated by means of the asymptotic representation of the function \(h(z)\) (compare the remarks made before writing equation (43)); the estimate gives:

\[ \int_{C_R} \frac{dz}{h(z)} = 2\pi i\,\frac{1}{N^2}. \tag{Д.8} \]

Consequently, we have obtained:

\[ \frac{1}{2\pi i}\int_C \frac{dz}{h(z)}+\sum_{\mathbf{k}}|\beta(\mathbf{k})|^2=\frac{1}{N^2}. \tag{Д.9} \]

On the other hand, the fact that the function \(h(z)\) has only real zeros, and the knowledge of the residues of the function \(h(z)^{-1}\) at the poles \(z=0\) and \(z=-\lambda\), make it possible to take the integral directly:

\[ \frac{1}{2\pi i}\int_C \frac{dz}{h(z)}=1+\frac{1}{h'(-\lambda)}. \tag{Д.10} \]

Consequently,

\[ 1+\sum_{\mathbf{k}}|\beta(\mathbf{k})|^2+\frac{1}{h'(-\lambda)}=\frac{1}{N^2}. \tag{Д.11} \]

Equations (Д.11) and (Д.3), taken together, give the required result (Д.1). If the interaction constant is smaller than its critical value, the integrand in (Д.9) has no pole at \(z=-\lambda\), and the last term in (Д.10) must be omitted. The remaining matrix elements, both of the commutators and of the anticommutators, can be considered in an analogous way.

APPENDIX II

In the particular case of absence of cutoff \(f(\omega)=1\), \(1/N=0\), and the function \(h(z)\) (compare (Д.4a)) can be represented in the compact form:

\[ h(\omega \pm i\varepsilon) = \omega+\gamma\left[ \omega+\frac{\pi\mu}{2} - \sqrt{\omega^2-\mu^2} \left( \log\frac{\omega+\sqrt{\omega^2-\mu^2}}{\mu} \mp i\pi \right) \right], \tag{Д.12} \]

if \(\omega>\mu\) and \(\varepsilon>0\),

\[ h(-\lambda)=-\lambda+ \]

\[ +\gamma\left[-\lambda+\frac{\mu\pi}{2}+\sqrt{\lambda^2-\mu^2}\log\frac{\lambda+\sqrt{\lambda^2-\mu^2}}{\mu}\right], \tag{D.13} \]

if \(\lambda>\mu\).

Disregarding the imaginary part entering formula (D.12), both of these cases can be represented by identical formulas if, for the expression standing under the logarithm sign, one takes its absolute value. For the third interval of the real axis we have:

\[ h(\omega)=\omega+ \]

\[ +\gamma\left[\omega-\sqrt{\mu^2-\omega^2}\arcsin\frac{\omega}{\mu} +\frac{\pi}{2}\frac{\omega^2}{\mu+\sqrt{\mu^2-\omega^2}}\right], \tag{D.14} \]

if \(-\mu<\omega<\mu\).

From the last equations one can find the roots of the equation

\[ h(-\lambda)=0 \tag{D.15} \]

both in the case of weak and in the case of strong coupling. For weak coupling, from (D.13) one can find that

\[ \lambda\simeq \frac{\mu}{2}e^{1/\gamma},\quad \text{if}\quad \gamma\ll 1, \tag{D.16} \]

and in this case the possibility of an expansion into a power series of any kind is excluded.*) In the case of strong coupling, use of equation (D.14) leads to the following expression for the root:

\[ -\omega=\lambda\simeq \frac{4}{\pi}\frac{\mu}{\gamma},\quad \text{if}\quad \gamma\gg 1, \tag{D.17} \]

and here there is a possibility of expansion in a power series in \(\gamma^{-1}\).

*) This circumstance is of some interest in connection with the fact that a number of attempts to obtain a power series with a finite radius of convergence in certain renormalizable field theories by means of perturbation theory ended unsuccessfully. See, for example, C. A. Hurst, Proc. Cambr. Phil. Soc. 48, 625 (1952); W. Thirring, Helv. Phys. Acta 26, 33 (1953); A. Petermann, Phys. Rev. 89, 1160 (1953); R. Utiyama and T. Imamura, Progr. Theor. Phys. 9, 431 (1953).

References

1. T. D. Lee, Phys. Rev. 95, 1329 (1954).

2. This was first pointed out by Schwinger (unpublished), and subsequently by several other authors. Compare H. Umezawa and S. Kamefuchi, Progr. Theor. Phys. 6, 543 (1951); G. Källén, Helv. Phys. Acta 25, 417 (1952); H. Lehmann, Nuovo Cimento 11, 342 (1954); M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954). The proof of this theorem is contained in Lee’s paper (Appendix II).

3. G. Källén, Helv. Phys. Acta 25, 417 (1952).

4. An indefinite metric had already long been used in quantum field theory. P. A. M. Dirac, Proc. Roy. Soc. A 180, 1 (1942), in a case very close to ours. See also W. Pauli, Rev. Mod. Phys. 15, 175 (1943). The result of Feynman (Phys. Rev. 76, 749 (1949), especially p. 756) implicitly presupposes the use of an indefinite metric. Compare W. Pauli, Progr. Theor. Phys. 5, 526 (1950). An indefinite metric was also used in quantum electrodynamics in considering scalar photons. See S. N. Gupta, Proc. Phys. Soc. 53, 681 (1950) and K. Bleuler, Helv. Phys. Acta 23, 567 (1950).

5. C. Møller, Dan. Mat. Fys. Medd. 23, No. 1 (1945); 22, No. 19 (1946).