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UDC 539.12
PHYSICS
F. I. FEDOROV
ELEMENTARY PARTICLES IN THE FIELD OF A PLANE ELECTROMAGNETIC WAVE
(Presented by Academician V. A. Fock on 30 VI 1966)
The equation for a particle in an electromagnetic field has the form
\[ (\hat{\nabla}-i\hat{A}+m)\psi=0 \tag{1} \]
(the charge is included in \(A\)). Here \(\nabla=(\nabla_l)=\left(\dfrac{\partial}{\partial x_l}\right)\); \(x_4=ix_0=ict\); \(A=(A_l)\) is the vector potential of the electromagnetic field; \(\hat{A}=A_l\gamma^l\); \(\gamma^l\) are the basic matrices of the equation. Exact solutions of equation (1) have so far been obtained only in a few cases. For particles with a single value of the rest mass (see \((^1)\))
\[ \hat{q}^{\,n}(\hat{q}^{\,2}-q^2)=0, \tag{2} \]
where \(q\) is an arbitrary (including operator) 4-vector with mutually commuting components. Denote \(H=\hat{\nabla}+m\), \(U=-i\hat{A}\), and let \(\psi=\psi_0 e^{ipx}\) be the wave function of a free particle \((px=p_lx_l,\ p^2+m^2=0)\). Then \(H\psi=(i\hat{p}+m)\psi=0\), while equation (1) takes the form \((H+U)\psi_1=0\). Formally the latter equation may be written in the form \(H(1+H^{-1}U)\psi_1=0\). Comparing with \(H\psi=0\) (see (2)), we conclude that \((1+H^{-1}U)\psi_1=\psi\), whence
\[ \psi_1=(1+H^{-1}U)^{-1}\psi = \left[1-H^{-1}U+(H^{-1}U)^2-\ldots\right]\psi . \tag{3} \]
As shown in \((^2)\), the operator inverse to \(H\), owing to (2), can be represented in the form \(H^{-1}=P_1(\hat{\nabla})+(m^2-\square)^{-1}P_2(\hat{\nabla})\), where \(P_1\) and \(P_2\) are certain polynomials. Obviously, \((m^2-\square)^{-1}e^{iqx}\), and the action of \((m^2-\square)^{-1}\) on a function of a more general form is determined from this by expanding the latter in a series or Fourier integral. Let
\[ A=af(\varphi),\qquad \varphi=kx=k_lx_l,\qquad k^2=ka=0, \tag{4} \]
i.e. the electromagnetic field has the character of a plane wave. From (3), (4) it is clear that the expression for \(\psi_1\) will have the form
\[ \psi_1=\chi(\varphi)e^{ipx},\qquad \chi(\varphi)=\Phi(k,p,\varphi)\psi_0, \tag{5} \]
where \(\Phi(k,p,\varphi)\) is a matrix whose elements depend on \(x_l\) only through \(\varphi=k_lx_l\). The preceding arguments, despite their formal character, have the significance that they lead to the conclusion that it is possible to seek a solution of equation (1) for the field (4) in the form (5) in the case of a particle with arbitrary spin. Substituting (4), (5), and (1), we obtain
\[ \hat{k}\chi' + (i\hat{b}+m)\chi=0,\qquad \chi'=d\chi/d\varphi,\qquad b=p-af. \tag{6} \]
Thus, in the general case, the problem is reduced to solving a system of ordinary first-order differential equations for the function \(\chi=\chi(\varphi)\).
The matrix \(\hat k\) is singular, since according to (2), (4) \(\hat k^{n+2}=k^2\hat k^n=0\). Therefore the differential equation (6) contains additional algebraic conditions on the function \(\chi\). Their extraction is complicated by the fact that the matrix \(\hat k\) is not reducible to diagonal form, since its minimal equation \(\hat k^{n+2}=0\) has multiple zero roots. This makes it difficult to apply the method of projective operators \({}^{(4)}\). Therefore we multiply equation (6) by the matrices of the invariant bilinear form \(\eta\). We obtain the equation
\[ \alpha\chi' + \eta(i\hat b+m)\chi=0,\quad (\alpha=\eta\hat k=-\alpha^+), \tag{7} \]
which is completely equivalent to equation (6), since \(\eta^2=1\). However, in (7) the matrix \(\alpha\) will be anti-Hermitian \({}^{(4)}\), as a result of which its minimal equation will not contain multiple roots and it can be written in the form \(aP_0(a)=0\), with \(P_0(0)\ne0\). Therefore \(P_0(a)/P_0(0)\) will be a projective operator selecting, without exception, all vectors \(\chi\) for which \(a\chi=0\) \({}^{(1)}\). Multiplying (7) by \(P_0(a)\), we obtain \(P_0(a)\eta(i\hat b+m)\chi=0\), whence it follows that \(\eta(i\hat b+m)\chi=\alpha\chi_1=\eta\hat k\chi_1\), where \(\chi_1\) is an arbitrary vector. Hence, in turn, it follows that
\[ \chi=(i\hat b+m)^{-1}\hat k\chi_1. \tag{8} \]
The operator \((i\hat b+m)^{-1}\) is not difficult to find with the aid of the minimal equation (2); it has the form (see \({}^{(2)}\))
\[ (i\hat b+m)^{-1}=\frac{1}{m}\sum_{k=0}^{n-1}\left(-\frac{i}{m}\hat b\right)^k+\left(-\frac{i}{m}\right)^n \frac{\hat b^n(m-i\hat b)}{b^2+m^2}, \tag{9} \]
The function \(\chi_1\) in (8), generally speaking, may depend on \(\varphi\) in an arbitrary manner. The simpler case is that in which the dependence on \(\varphi\) is concentrated in a scalar factor \(F(\varphi)\) with a constant vector \(\chi_0\). Taking (9) into account, in order to get rid of the denominator \(b^2+m^2=a^2f^2-2apf\), which depends on \(\varphi\), it is convenient to include explicitly in \(F\) the factor \(b^2+m^2\) and to seek the solution in the form
\[ \chi=F(\varphi)(b^2+m^2)(i\hat b+m)^{-1}\hat k\chi_0. \tag{10} \]
We shall apply the general approach described above to particles with the lowest spins \(1/2, 0\), and 1.
- Spin \(1/2\). In this case the Dirac algebra holds:
\[ \hat p\hat q+\hat q\hat p=2pq,\quad \hat q^2=q^2, \tag{11} \]
\[ \hat k^2=\hat k\hat a+\hat a\hat k=\hat k\hat a\hat k=0,\quad (b^2+m^2)(i\hat b+m)^{-1}=m-i\hat b. \]
Substituting \(\chi=F(m-i\hat b)\hat k\chi_0\) into (6), we obtain
\[ \left(F'-\frac{b^2+m^2}{2ikp}F\right)\hat k\chi_0=0, \tag{12} \]
whence it follows that
\[ F=\exp\left(-\frac{i}{2kp}\int(b^2+m^2)\,d\varphi\right). \tag{13} \]
Thus, the solution of equation (1) has the form (see (5))
\[ \psi=(m-i\hat b)\hat k\chi_0\exp i\left[px-\frac{1}{2kp}\int(b^2+m^2)\,d\varphi\right], \tag{14} \]
where \(\chi_0\) is an arbitrary bispinor. This solution was obtained in another way and in another form by D. M. Volkov \({}^{(5)}\).
2. Spins 0 and 1
In this case the Duffin–Kemmer algebra is valid
\[ \hat p\hat q\hat r=\hat r\hat p\hat q=pq\cdot r+rp\cdot q,\qquad \hat q^3=q^2\hat q,\qquad \hat q\hat r\hat q=qr\cdot \hat q, \tag{15} \]
\[ \hat k^3=\hat k\hat a\hat k=\hat a\hat k\hat a=\hat k^2\hat a+\hat a\hat k^2=0, \]
\[ (b^2+m^2)(i\hat b+m)^{-1}=\frac{1}{m}\,[b^2+m^2+i\hat b(i\hat b-m)]. \tag{16} \]
Substituting (10), (16) into (6), with the same value of \(F\) (13), after some transformations, taking (15) into account, we obtain
\[ \{F'(i\hat p+m)\hat k^2(i\hat p+m)+(Ff)'[\hat p,\hat k^2\hat a]+(Ff^2)'\hat k^2\hat a^2\}\chi_0=0. \tag{17} \]
In order that this relation be satisfied for constant \(\chi_0\), it is necessary that
\[ (i\hat p+m)\chi_0=0,\qquad \hat k^2\hat a^2\chi_0=0,\qquad [\hat p,\hat k^2\hat a]\chi_0=0. \tag{18} \]
The conditions \(\hat k^2\hat a^2\chi_0=0\) and \(\hat k^2\hat a\chi_0=0\) follow one from the other as a result of multiplication by \(\hat a\). Therefore the third equation (18) is a consequence of the first two. We shall now consider separately the cases of spin 0 and 1.
a) Spin 0. In this case \(\hat k^2\hat a=0\), and only the condition is imposed on the function \(\chi_0\) that it be a solution of the equation for a free particle,
\[ (i\hat p+m)\chi_0=0. \]
b) Spin 1. In this case the wave function has the form \(\psi=(\psi_k,\psi_{ln})\), where \(\psi_k\) is a 4-vector and \(\psi_{ln}\) is an antisymmetric tensor. The operator \(\hat p\) acts on \(\psi\) in the following way:
\[ \hat p\psi=\hat p(\psi_k,\psi_{ln})=(p_l\psi_{kl},\;p_n\psi_l-p_l\psi_n). \tag{19} \]
The first of equations (18) leads to the relations \(p_k\psi_k=0\), \(\psi_{ln}=\dfrac{i}{m}(p_l\psi_n-p_n\psi_l)\), while from the second it follows that \(k_n\psi_n=0\) and \(a_n k_l\psi_{nl}=-kp\cdot a_n\psi_n=0\). Thus, the conditions (18) are satisfied if
\[ p\psi_l=k\psi_l=0,\qquad \psi_l=C\varepsilon_{lmnr}p_mk_na_r, \tag{20} \]
where \(\varepsilon_{lmnr}\) is the Levi-Civita symbol.
Thus, the exact solution of the problem of a particle in the field of a plane electromagnetic wave can be written in a single general form at once for three different particles with spins \(1/2,0,1\),
\[ \psi=(b^2+m^2)(i\hat b+m)^{-1}\hat f\chi_0 \exp i\left(px-\int\frac{b^2+m^2}{2kp}\,d\varphi\right), \tag{21} \]
where the function \(\chi_0\), independent of \(\varphi\), is arbitrary in the case of spin \(1/2\), for spin 0 is subject to the condition \((i\hat p+m)\chi_0=0\), and for spin 1 has the form (20). Let us note that this solution is valid not only in the case of a monochromatic electromagnetic wave, but also in the case of an arbitrary function \(f(\varphi)\). Obviously, the method set forth here is in principle applicable also to particles with higher spins, although the absence or complexity of the algebra of the matrices \(\gamma_\nu^l\), naturally, will make the calculations difficult.
Institute of Physics
Academy of Sciences of the BSSR
Received
28 VI 1966
REFERENCES
- F. I. Fedorov, DAN, 79, 787 (1951).
- F. I. Fedorov, DAN, 65, 813 (1949).
- F. I. Fedorov, Dokl. AN BSSR, 4, 454 (1960).
- F. I. Fedorov, ZhETF, 35, 495 (1958).
- D. W. Volkow, Zs. Phys., 94, 250 (1935).