ABSTRACT

Having used the principle of the local gauge invariance with respect to the U(1)-group in Section 4.5 we have established the form of the Lagrangian to describe the interaction between the electromagnetic and electron-positron fields:

L = i 2

[ ψ(x)γµDµψ(x) −D†µψ(x)γµψ(x)

] −mψ(x)ψ(x) − 1 4 Fµν(x)F

µν(x), (18.1)

where Dµ = ∂µ + ieAµ(x) and Fµν(x) = ∂µAν(x) − ∂νAµ(x). Here the operators of the electromagnetic and electron-positron fields are denoted by a bold font to stress that they are set in the Heisenberg representation. Attention is drawn to the fact that the local gauge invariance is kept only when an electromagnetic field quanta mass is identical equal to zero. If the photons had a mass m, then the Lagrangian of a free electromagnetic field would include the mass term −m2AµAµ that is gauge noninvariant. From the Lagrangian (18.1) follows the basic equations of the QED:

[γµ(i∂µ − eAµ(x)) −m]ψ(x) = 0, (18.2)

[γµT (i∂µ + eAµ(x)) +m]ψ(x) = 0, (18.3)

∂µFµν(x) = jν(x). (18.4)

The electromagnetic current is defined by the expression:

jν(x) = eψ(x)γνψ(x), (18.5)

which formally coincides with that for the current in the case of the free electron-positron field. However, it is not exactly the same since this expression contains the quantities ψ(x) and ψ(x) to obey the equations for interacting fields rather than those for free fields. The

Particles, Fields, and Quantum

field operators act on the state vectors in the particles number space Φ. Since we work in the Heisenberg representation, then the following relation:

∂t Φ = 0

is valid. In Eq. (18.4) one may cross to the electromagnetic potentials:

Aν(x) = jν(x), (18.6)

In so doing, we should supplement this equation, as in the case of free fields, with the condition on the state vectors: (

∂xµ A+µ (x)

) Φ = 0. (18.7)

Thanks to the current continuity, the operator ∂µAµ(x) satisfies one and the same equation both in a free case and in the presence of interaction:

∂Aµ(x)

∂xµ = 0. (18.8)

Therefore, the additional condition (18.7) is relativistically invariant. Eqs. (18.2), (18.3), and (18.6) follow from the Lagrangian that one conveniently writes

down as a sum of the Lagrangians of free and interacting fields:

L = L(e)0 + L(γ)0 + Lint, (18.9)

where

L(e)0 = i

[ ψ(x)γµ∂µψ(x) − ∂µψ(x)γµψ(x)

]−mψ(x)ψ(x), (18.10) L(γ)0 = −

2 ∂µA

ν∂µAν , (18.11)

Lint = −jµ(x)Aµ(x), (18.12) Recall, that the Lagrangians (18.1) and (18.9) differ from each other by an insignificant

term having the four-dimensional divergence form. The interaction Lagrangian Lint does not contain derivatives of the field operators. We speak of such interactions as couplings without derivatives. In this case the interaction Hamiltonian density Hint is simply equal to −Lint and the interaction Hamiltonian is given by:

Hint(t) = − ∫ Lintd3x. (18.13)

A system of differential equations (18.2)–(18.4) contains only a part of the information about the interacting fields. These equations should be amplified by permutation relations (PR) between the field operators. As evolution of the operators in time is defined by equations of motion, the PR in the case of interacting fields may be given for the initial instant of time only. It is clear that the determination of the PR at some time is equivalent to their calculation at the coincident instants of time. Having realized what PR we should find, we can easily guess the way of their searching. We remember that precisely the simultaneous PR are at the base of the canonical quantization procedure. Then, with the

help of the canonical formalism one may find the following nonvanishing simultaneous PR (see, the formulae (15.257) and (17.51) ):

[Aµ(x), ∂0Aν(x ′)]t0=t′0 = igµνδ

(3)(r− r′). (18.14)

{ψ(x),ψ†(x′)}t0=t′0 = δ(3)(r− r′). (18.15) As it seems, a change to the PR valid for any time intervals may be realized in the same way as for free fields. So, for example, in the free case for the electromagnetic field the solution Aν(x) on arbitrary space-like hypersurface σ0 could be determined by application of the Green function D(x) in terms of the solution on any more early space-like hypersurface

Aν(x) =

dσµ(x′) [ ∂D(x− x′)

∂x′µ Aν(x

′)−D(x − x′)∂Aν(x ′)

∂x′µ

] . (18.16)

Further, selecting the hypersurface σ0 by the corresponding way, one can obtain the PR sought. If the formula similar to (18.16) existed in the case of the interacting fields, then we would manage to define the PR at arbitrary instants of time. However to derive such a formula demands solutions knowledge of Eqs. (18.2), (18.3), and (18.6), which represent the partial differential inhomogeneous equations system. It is obvious that the task of solving this system is rather complicated. Moreover, in the majority of cases solutions may be obtained by means of approximate methods only. Because approximate PR do not suit us, we have to look for other ways to solve this problem.