UDC 517.945.51

THE SYMMETRY GROUP OF SOLUTIONS OF THE KLEIN–GORDON EQUATION IN AN EXTERNAL VECTOR FIELD

A. A. GRIB

In modern quantum field theory, the question of dynamical symmetry groups of elementary particles is acquiring ever greater importance. It is of interest to clarify the reason for the appearance of these symmetries, as well as what the general methods are for studying them. In this connection, the study of the group properties of differential equations of field theory by means of constructing the determining equations of groups admitted by these equations is of considerable interest. The method of constructing determining equations goes back to Lie and has been developed at present by L. V. Ovsyannikov in his work [1]. The value of this method for applied problems consists in the fact that, by solving the determining equations, we can find symmetry groups which cannot be guessed directly from the form of the equation.

In the present paper, using the simple example of the Klein–Gordon equation in an external vector field, which describes charged scalar particles in this field, it is shown that this equation admits not only the inhomogeneous Lorentz group but also, depending on the conditions on the external vector field, a certain infinite-dimensional Lie group replacing the translation subgroup of the inhomogeneous Lorentz group.

1. Suppose there is a Lagrangian [2] of a charged scalar (pseudoscalar) field interacting with an external vector (axial-vector) field, with density

\[ L(x)=g^{\mu\nu}\frac{\partial \bar u}{\partial x^\mu}\frac{\partial u}{\partial x^\nu} +m^2\bar u u+ \left(g\bar u b^\mu\frac{\partial u}{\partial x^\mu} -g\frac{\partial \bar u}{\partial x^\mu}b^\mu u\right), \]

\[ g^{\mu\nu}= \begin{pmatrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix} \quad (\mu,\nu=0,1,2,3), \tag{1} \]

\[ \partial_\mu b^\mu \ne 0, \]

where \(g\) is a certain coupling constant; the fields are not quantized. From (1) we obtain the Euler equation for \(u(x)\):

\[ \square u-2g b^\mu \partial_\mu u-g\partial_\mu b^\mu u-m^2u=0. \tag{2} \]

The equation of the free field \(u(x)\) is

\[ (\square-m^2)u=0. \tag{3} \]

This equation admits the inhomogeneous Lorentz group. What is the group of transformations admitted by equation (2)? We shall obtain the answer to this question by considering the determining equations of the Lie group corresponding to equation (2), by the method set forth in [1].

We seek the generator of the group in the form

\[ X=\xi^i(x)\frac{\partial}{\partial x^i}+\sigma(x)u\frac{\partial}{\partial u}\quad (i=0,\ 1,\ 2,\ 3), \tag{4} \]

where \(\xi^i(x)\), \(\sigma(x)\) are certain unknown functions; \(u(x)\) is a solution of equation (2). The determining equations of the group for equation (2) have the form

\[ g^{ik}\frac{\partial \xi^j}{\partial x^k}+g^{jk}\frac{\partial \xi^i}{\partial x^k}=\mu g^{ij}, \]

\[ 2g^{ik}\frac{\partial \sigma}{\partial x^k} = g^{jk}\frac{\partial^2 \xi^i}{\partial x^j\partial x^k} -2gb^k\frac{\partial \xi^i}{\partial x^k} +2g\frac{\partial b^i}{\partial x^k}\xi^k +2g\mu b^i, \tag{5} \]

\[ g^{ij}\frac{\partial^2\sigma}{\partial x^i\partial x^j} -2gb^i\frac{\partial\sigma}{\partial x^i} = g\xi^k\frac{\partial}{\partial x^k}(\partial_\lambda b^\lambda) +\mu(g\partial_\lambda b^\lambda+m^2). \]

In these equations the unknowns are the functions \(\xi^i(x)\), \(\sigma(x)\), and \(\mu(x)\); the Greek and Latin indices run over the same values \((0,\ 1,\ 2,\ 3)\).

The first equation of system (5) is an equation only for the functions \(\xi^i(x)\). In the case \(\mu=0\), this equation is the Killing equation, determining the group of motions in a space with metric \(g^{ij}\). In the case \(\mu\ne 0\), it determines the group of conformal transformations of the space with this metric.

From system (5) it is clear that

\[ \frac{\partial\sigma}{\partial x^k}\ne 0 \]

only in the case when \(gb^k\ne 0\), or when there is a conformal group, when \(\mu\ne 0\) and \(\Box \xi^i\ne 0\).

Thus it is seen that the term \(2gb^\mu\partial_\mu u\) in equation (2) is essential for the occurrence of a group of transformations

\[ \sigma(x)u\frac{\partial}{\partial u}, \]

where

\[ \frac{\partial\sigma}{\partial x^k}\ne 0. \]

Let us find the solution of system (5). First we consider the simplified case based on the assumption that the conformal group is absent, i.e. \(\mu=0\). Then the \(\xi^i(x)\), being solutions of the first equation of system (5), correspond to the inhomogeneous Lorentz group, and, if the solutions of system (5) corresponded to

\[ X=\xi^i\frac{\partial}{\partial x^i}, \tag{6} \]

i.e. \(\sigma(x)=0\), then the inhomogeneous Lorentz group would be admitted by equation (2). However, in the general case \(b^\mu(x)\ne 0\), this will not be so: a transformation of the inhomogeneous Lorentz group must be accompanied by the transformation

\[ \sigma(x)u\frac{\partial}{\partial u}. \]

From the accepted assumption there follow the following equations of system (5):

\[ g^{ik}\frac{\partial \xi^j}{\partial x^k} + g^{jk}\frac{\partial \xi^i}{\partial x^k}=0, \]

\[ 2g^{ik}\frac{\partial \sigma}{\partial x^k} = -2g\left( b^l\frac{\partial \xi^i}{\partial x^l} - \frac{\partial b^i}{\partial x^l}\xi^l \right), \tag{7} \]

\[ g^{ij}\frac{\partial^2\sigma}{\partial x^i\partial x^j} - 2gb^i\frac{\partial\sigma}{\partial x^i} = g\xi^k\frac{\partial}{\partial x^k}(\partial_\mu b^\mu). \]

Substituting the second equation of system (7) into the third, we obtain the following equations for \(\sigma(x)\):

\[ 2g^{ik}\frac{\partial\sigma}{\partial x^k} = -2g\left( b^l\frac{\partial \xi^i}{\partial x^l} - \frac{\partial b^i}{\partial x^l}\xi^l \right), \]

\[ \Box\sigma = -2g^2\left[ b^1b^k\frac{\partial \xi^1}{\partial x^k} - b^1\frac{\partial b^1}{\partial x^k}\xi^k + b^2b^k\frac{\partial \xi^2}{\partial x^k} - \right. \tag{8} \]

\[ \left. - b^2\frac{\partial b^2}{\partial x^k}\xi^k + b^3b^k\frac{\partial \xi^3}{\partial x^k} - b^3\frac{\partial b^3}{\partial x^k}\xi^k - b^0b^k\frac{\partial \xi^0}{\partial x^k} + \right. \]

\[ \left. + b^0\frac{\partial b^0}{\partial x^k}\xi^k \right] + g\xi^k\frac{\partial}{\partial x^k}(\partial_\mu b^\mu). \]

If, as \(\xi^i(x)\), one takes solutions corresponding to the homogeneous Lorentz group, then it is easy to see that \(\dfrac{\partial\sigma}{\partial x^k}=0\), i.e. \(\sigma=\mathrm{const}\), and this constant is arbitrary and may be zero; that is, the homogeneous Lorentz group is admissible. Indeed, in this case \(\xi^i=c_{ik}x^k\), where \(c_{ik}=-c_{ki}\) \((i,k\ne 0)\); \(c_{0k}=c_{k0}\) \((k\ne0)\). From the second equation (7) it is seen that

\[ 2g^{ik}\frac{\partial\sigma}{\partial x^k}=0. \tag{9} \]

Here we have taken into account that \(b^i(x)\) transforms as a vector under transformations of the homogeneous Lorentz group (i.e. \(\xi^l\dfrac{\partial b^i}{\partial x^l}=c_{il}b^l\)). The third equation (7) has the form

\[ \Box\sigma=0, \tag{10} \]

since \(\partial_\mu b^\mu\) is a scalar.

Now let us see into what the generators of translations

\[ \xi^i\frac{\partial}{\partial x^i}=e^i\frac{\partial}{\partial x^i}, \]

where \(e^i\) is the unit vector in the direction of translation, are transformed. System (8) in this case has the form

\[ 2g^{ik}\frac{\partial\sigma}{\partial x^k} = 2g\frac{\partial b^i}{\partial x^l}e^l, \]

\[ \Box\sigma = g^2e^k\frac{\partial}{\partial x^k}(b^\mu b^\mu) + ge^k\frac{\partial}{\partial x^k}(\partial_\mu b^\mu). \tag{11} \]

It is clear from (11) that \(\dfrac{\partial \sigma}{\partial x^k} \ne 0\) and \(\sigma(x)=0\) is not a solution of the system in the case \(g\dfrac{\partial b^i}{\partial x^l}e^l \ne 0\), i.e., the translations \(e^k\dfrac{\partial}{\partial x^k}\) pass into some other operators.

Solutions of the system (11) exist if the compatibility conditions are satisfied

\[ \frac{\partial^2 \sigma}{\partial x^i \partial x^k} = \frac{\partial^2 \sigma}{\partial x^k \partial x^i}, \tag{12} \]

which leads to the condition on \(b^i(x)\)

\[ e^m \frac{\partial}{\partial x^m} \left( \frac{\partial b^k}{\partial x^i} - \frac{\partial b^i}{\partial x^k} \right) =0, \tag{13} \]

which will be satisfied if \(b^i=\dfrac{\partial \psi}{\partial x^i}\).

In addition to (12), it is also necessary to satisfy the following compatibility condition ([1], p. 220):

\[ \xi^i \frac{\partial}{\partial x^i} H+\mu H=0. \tag{14} \]

In our case \(\mu=0\), and

\[ H=2\left(g\partial_\mu b^\mu+m^2\right)-2g\partial_\mu b^\mu+2g^2b^\mu b^\mu =2m^2+2g^2b^\mu b^\mu, \tag{15} \]

and (14) leads to the condition

\[ 2g^2 e^i \frac{\partial}{\partial x^i}\left(b^\mu b^\mu\right)=0. \tag{16} \]

Then from the second equation of the system (11) we obtain

\[ \sigma(x)=g\int D(x-x')e^k\frac{\partial}{\partial x'^k} \left(\partial_\mu' b^\mu\right)dx', \tag{17} \]

where \(D(x-x')\) is the Green’s function of this equation. In the case of incompatibility of equations (11), \(\sigma(x)\) does not exist, i.e., the only admissible group will be the homogeneous Lorentz group.

Consider the Lie group associated with the generators

\[ X=e^k\frac{\partial}{\partial x^k} +g\int D(x-x')e^k\frac{\partial}{\partial x'^k} \left(\partial_\mu' b^\mu\right)dx'\cdot u\frac{\partial}{\partial u}. \tag{18} \]

For the translation generators of the inhomogeneous Lorentz group the relations

\[ \left[ \frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^k} \right]=0. \tag{19} \]

hold. In our case (18) these relations are replaced by

\[ \begin{aligned} \Bigg[ &\frac{\partial}{\partial x^i} +g\int D(x-x')\frac{\partial}{\partial x'^i} \left(\partial_\mu' b^\mu\right)dx'\cdot u\frac{\partial}{\partial u}, \\ &-\frac{\partial}{\partial x^k} +g\int D(x-x')\frac{\partial}{\partial x'^k} \left(\partial_\mu' b^\mu\right)dx'\cdot u\frac{\partial}{\partial u} \Bigg] = \end{aligned} \]

\[ = g \int \frac{\partial}{\partial x^{i}} D(x-x') \frac{\partial}{\partial x'^{k}}(\partial'_{\mu} b^{\mu})\,dx' \cdot u\frac{\partial}{\partial u} \]

\[ - g \int \frac{\partial}{\partial x^{k}} D(x-x') \frac{\partial}{\partial x'^{i}}(\partial'_{\mu} b^{\mu})\,dx' \cdot u\frac{\partial}{\partial u}. \tag{20} \]

In the general case of the replacement
\[ \frac{\partial}{\partial x^{i}} \to \frac{\partial}{\partial x^{i}}+\sigma_i u\frac{\partial}{\partial u} \]
we have the relations

\[ \left[ \frac{\partial}{\partial x^{i}}+\sigma_i u\frac{\partial}{\partial u},\, \frac{\partial}{\partial x^{k}}+\sigma_k u\frac{\partial}{\partial u} \right] = F_{ik}(x)u\frac{\partial}{\partial u}, \tag{21} \]

where
\[ F_{ik}(x)=\frac{\partial}{\partial x^{i}}\sigma_k(x)-\frac{\partial}{\partial x^{k}}\sigma_i(x), \qquad F_{ik}=-F_{ki}. \]

The properties of the Lie algebra obtained in (20) or (21) depend essentially on the differential properties of the quantities \(F_{ik}(x)\). If the \(F_{ik}(x)\) have arbitrarily many nonzero derivatives, then the algebra obtained will be infinite-dimensional, and the corresponding Lie group will also be infinite-dimensional.

The replacement
\[ \frac{\partial}{\partial x^{i}} \to \frac{\partial}{\partial x^{i}}+\sigma_i(x)u\frac{\partial}{\partial u} \]
can be illustrated in another way. Equation (2) in the case
\[ b^{i}=\frac{\partial \psi}{\partial x^{i}} \]
can be transformed by multiplying the solution by some function into an equation not containing the term
\[ b^{\mu}\frac{\partial u}{\partial x^{\mu}}, \]
but in this case the term \(\Box u\) is replaced by another one, in which the metric tensor \(g^{ik}\) passes into some \(g^{ik}(x)\), which leads to the replacement of the ordinary derivative by the covariant one, and, solving the corresponding Killing equations, we can find the desired group.

1. Finally, let us consider the general case \(\mu \ne 0\). First of all, let us note that to the solutions \(\xi^{i}(x)\) one can always add \(\xi^{i}_{2}(x)\), corresponding to the homogeneous Lorentz group. In this case we change nothing in the equations for \(\sigma(x)\), and the homogeneous Lorentz group remains a solution as before. For \(\mu \ne 0\), condition (14) has the form

\[ \xi^{i}\frac{\partial}{\partial x^{i}}(2g^{2}b^{\nu}b_{\nu}) = -\mu(2g^{2}b^{\nu}b_{\nu}+2m^{2}). \tag{22} \]

Instead of (13) there will be the condition (see [1], p. 217)

\[ \frac{\partial}{\partial x^{i}} \left[ \left( \frac{\partial b^{i}}{\partial x^{l}}-\frac{\partial b^{l}}{\partial x^{i}} \right)\xi^{l} \right] = \frac{\partial}{\partial x^{i}} \left[ \left( \frac{\partial b^{j}}{\partial x^{l}}-\frac{\partial b^{l}}{\partial x^{j}} \right)\xi^{l} \right]. \tag{23} \]

To find the solutions of system (5) under conditions (22) and (23), let us first consider the solutions of the first equation of the system. They have the form

\[ \xi^{i} = A_i \left[ (x^{i})^{2}-\varepsilon_i \sum_{s=0}^{3\,\prime} \varepsilon (x^{s})^{2} \right] + 2\sum_{s=0}^{3\,\prime} A_s x^{s}x^{i} + \]

\[ + Bx^{i}+\sum_{s=0}^{3\,\prime} c_{is}x^{s}+C_i \qquad (i=0,1,2,3), \tag{24} \]

\[ \begin{gathered} \varepsilon_s=1\quad (s=1,2,3),\qquad \varepsilon_s=-1\quad (s=0);\\ c_{is}=-c_{si}\quad (i\ne s\ne 0),\qquad c_{i0}=c_{0i}\quad (i\ne 0). \end{gathered} \tag{24} \]

In (24) no summation is performed over the index \((i)\); the prime on the summation sign indicates that the term corresponding to \(s=i\) is to be omitted. The constants \(A_i, B, c_{is}, C_i\) are arbitrary. Analogously to the case considered in [1], pp. 232–234, we obtain

\[ \mu=4A_k x^k+2B. \tag{25} \]

From (22) it is seen that \(C_i\equiv 0\), if \(C_i\dfrac{\partial}{\partial x^i}(b^\mu b^\mu)\ne 0\). Substituting (24) and (25) into (5), we obtain equations for \(\sigma(x)\):

\[ \begin{aligned} \frac{\partial\sigma}{\partial x^i} ={}&-2A_i-gb^k\frac{\partial}{\partial x^k} \left\{ A_i\left[(x^i)^2-\varepsilon_i\sum_{s=0}^{3}{}' \varepsilon_s(x^s)^2\right] +2\sum_{s=0}^{3}{}' A_s x^s x^i+Bx^i \right\} \\ &+g\frac{\partial b^i}{\partial x^k} \left\{ A_k\left[(x^k)^2-\varepsilon_k\sum_{s=0}^{3}{}' \varepsilon_s(x^s)^2\right] +2\sum_{s=0}^{3}{}' A_s x^s x^k+Bx^k+C_k \right\} \\ &+g(4A_kx^k+2B)b^i, \end{aligned} \tag{26} \]

\[ \begin{aligned} \square\sigma ={}&2gb^i\frac{\partial\sigma}{\partial x^i} +g\left\{ A_k\left[(x^k)^2-\varepsilon_k\sum_{s=0}^{3}{}'\varepsilon_s(x^s)^2\right]\right.\\ &\left. +2\sum_{s=0}^{3}{}' A_s x^s x^k+Bx^k+\sum_{s=0}^{3}{}' c_{ks}x^s+C_k \right\} \frac{\partial}{\partial x^k}(\partial_\mu b^\mu)\\ &+(4A_kx^k+2B)(g\partial_\mu b^\mu+m^2). \end{aligned} \]

and, substituting the expression for \(\dfrac{\partial\sigma}{\partial x^i}\) from the first equation of the system (26) into the second, we obtain, analogously to (17), an expression for \(\sigma(x)\). The resulting group will also be infinite-dimensional.

Finally, in the case when conditions (22), (23) or (13), (16) are not satisfied, for \(\mu=0\) the equation under consideration admits only the homogeneous Lorentz group.

The main reason for the appearance of the infinite-dimensional Lie group is the term \(2gb^\mu\partial_\mu u\) in equation (2).

1. However, if instead of the interaction Lagrangian (1) we took

\[ L_I=gu\bar b^\mu\partial_\mu u, \tag{27} \]

then we would obtain a group of transformations with generators \(\dfrac{\partial}{\partial x^i}+\sigma_i u\dfrac{\partial}{\partial u}\), where

\[ \sigma_i = g^2 \int D(x - x') e^i \frac{\partial}{\partial x'^i} (b^\mu b^\mu)\,dx', \qquad \mu = 0, \tag{28} \]

and the field \(b^\mu(x)\) satisfies the conditions

\[ b^\mu(x) = \frac{\partial \psi}{\partial x^\mu}, \qquad e^i \frac{\partial}{\partial x^i} \left( \partial_\mu b^\mu + \frac{1}{2} b^\mu b^\mu \right) = 0. \tag{29} \]

Thus, our main conclusion is the fact that the violation of translational invariance under the replacement of the translation generators considered above is a consequence of a definite differential structure of the interaction. Lagrangians of the type considered may arise in the theory of weak interaction. For example, to explain the decay of the \(K_2^0\)-meson, \(K_2^0 \to \pi^+ \pi^-\), the Lagrangian [3]

\[ L_{\mathrm{вз}} = \int \frac{1}{2} i g_s [\overline{K}\partial_\mu K - \partial_\mu \overline{K}K] S_\mu\,dx. \tag{30} \]

was proposed.

This Lagrangian belongs precisely to the type considered by us.

Depending on the conditions (equations) imposed on the field \(S_\mu\), we obtain three different symmetries of the solutions of the equation for \(K\)-mesons. However, the case \(\mu \ne 0\) requires the fulfillment of too rigid relations (22) for the field \(S_\mu\) and, apparently, has no physical meaning.

The method for investigating symmetries considered in this paper is, of course, applicable also to any field equation. The physical conclusions of the present work can naturally be regarded only as suggestive, since the entire theory concerned unquantized fields.

The author is grateful to Yu. V. Novozhilov and L. V. Ovsyannikov for a number of valuable comments relating to the work.

References

1. Ovsyannikov L. V. Group properties of differential equations. Publishing House of the Academy of Sciences of the USSR, 1962.

2. Bogolyubov N. N., Shirkov D. V. Introduction to the Theory of Quantized Fields. GITTL, 1957.

3. Levy M., Nauenberg M. Phys. Letters, 12, 155, 1964.

Received by the editors
July 13, 1965

Leningrad State University
named after A. A. Zhdanov