ABSTRACT
Components of a field function are functions of coordinates and time and satisfy some definite differential equations, which are called field equations. Field equations can be given a very general variational or Lagrangian form, if one accepts, that a field as the dynamic system must be characterized by a definite Lagrangian function. For a mechanical system this function is written down as the sum over all the material points of the system. For a continuous system like a wave field, Lagrangian function is expressed by the space integral from Lagrangian function density:
L(t) =
∫ Ld3x. (4.1)
However, the variational principle deals not with the Lagrangian function but with an action S obtained from it by means of integration over x0 = ct
S = c
∫ L(t)dt. (4.2)
Therefore, noncovariant expression (4.1) in Lagrangian formalism is in fact intermediate and it is sufficient to consider the Lagrangian function density L, which depends on the four space-time coordinates x in an implicit way. Hereinafter, we shall simply call L by the Lagrangian. Analogously, the Hamiltonian function density H will be named the Hamiltonian. The Lagrangian depends on components of a field wavefunction ψk and on their first derivatives with respect to the space-time coordinates ∂ψk/∂x
ν . In so doing we assign the meaning of generalized field coordinates to ψk, and the meaning of generalized field velocities to ∂ψk/∂x
ν . The demand of lacking derivatives with respect to field functions above the first order in
the Lagrangian is caused by the following reason. Dynamics of all known quantum fields is described either by equations of the first or the second order. On the one hand, these equations follow from the principle of a least action:
δS = 0, (4.3)
and under action variation the order of derivatives, entering L, is raised by one. The Lagrangian must be a relativistically covariant quantity, namely, a relativistic scalar L′(x′) = L(x). Thanks to the invariance under the space-time translations, we conclude that the Lagrangian must not manifestly contain any space-time coordinates x. Since dynamical observables (four-dimensional vector of the energy-momentum, the moment of momentum tensor, and so on) are real quantities, then the Lagrangian must be also a real quantity. We shall also impose the locality condition on the Lagrangian, that is, we assume that the Lagrangian depends only on the state of fields in the infinitely small vicinity of the point
Particles, Fields, and Quantum
x. So:
L(x) = L(ψk, ∂ψk ∂xν
) = L (ψk(x), ψk;ν (x)) . General variation of an action, related with varying both field wave functions and of
integration domain boundaries, is equal to:
δS = δ
∫ L(x)dx =
∫ L(x′)dx′ −
∫ L(x)dx, (4.4)
where dx = dx0dr. For L′(x′) we have the expression:
L′(x′) = L (ψ′k(x′), ψ′k;ν(x′)) = L(x) + δL(x) = L(x) + δL(x) + dLdxλ δxλ, (4.5) In Eq. (4.5) we denote a form variation of corresponding quantities by the symbol δ. This way δL(x) represents varying the Lagrangian only on the account of the variation of field functions and field function derivatives:
δL(x) = ∂L ∂ψk
δψk + ∂L ∂ψk;ν
δψk;ν , (4.6)
and δψk(x) does a variation of a field function form:
δψk(x) = ψ ′ k(x) − ψk(x).
