CLASSICAL THEORY in VS .NET

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CLASSICAL THEORY
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the local character of the theory. This is why we shall not use this gauge with its instantaneous Coulomb potential. Locality is a deeply rooted physical principle emerging from the nineteenth century formulation of field theory. Its implications underlie most of the developments of relativistic field theory, and its verification down to very small space-time intervals (via dispersion relations, for instance) is unquestioned up to now. The Lagrange function should therefore be expressed as a space integral over a density, the so-called lagrangian 2'(x), which in turn should depend only on the fields and on finitely many of their derivatives. We do not really claim that these fields are directly measurable quantities; this is obviously not the case for the gauge-dependent potential AJl' Locally measurable quantities should, however, be expressed in terms oflocal combinations of the dynamical variables. We write
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4 d x2'(x)
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Most of the time we shall not specify the boundary conditions of this integral. Let us assume that it extends throughout space and that the fields vanish sufficiently fast at infinity to justify integration by parts. We assume AJl(x) to transform as a four-vector field and the lagrangian as a scalar density in order for the action to be a Lorentz invariant. We require, furthermore, that 2' depends only on the fields and their first derivatives and be modified at most by a divergence under a gauge transformation (1-40). This is a natural generalization of the analogous property of the Lagrange function and ensures the gauge invariance of the equations of motion. In general, if a lagrangian depends on fields <Pi(X) and their gradients 0Jl<Pi(X), then under an infinitesimal variation b<Pi(X) of the fields the action will change according to
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b<Pi(X)
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O[~~:~~)] b[OJl<Pi(X)] }
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{02'(X) 02'} d x 0<pi(X) - 0Jl o [OJl<Pi(X)] b<Pi(X)
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Requiring the action to be stationary leads to the generalized Euler-Lagrange equations ~ = o2'(x) _ 0 o2'(x) = 0 (1-44)
b<Pi(X) 0<Pi(X) Jl o [OJl<Pi(X)]
According to the rules set before, we have thus in the electromagnetic case to adjust the coefficients a, b, c, d, e in the lagrangian
2'(x)
i[aoJlAVoJlAv
+ bOJlAvovAJl + c(OJlAJl)2 + dAJlAJl + eAJljJl]
in such a way that Eqs. (1-44) will coincide with (1-41). Up to an overall coefficient, a simple calculation yields
2'(x)
= -iFJlvpv - jJlAJl + ~ [(oJlAJl - oJlAvovAJl]
QUANTUM FIELD THEORY
where FIlV is used here as a shorthand notation for the expression of the electromagnetic tensor in terms of the potential. The coefficient c remains arbitrary and multiplies a divergence as was expected:
(0IlAIl)2 - 0IlAvovAIl
= 01l[Av(gIlVopAP - OVAIl)]
This last term can therefore be omitted and we are led to the action (1-45)
In the presence of a given external current this lagrangian is not gauge invariant. Under a gauge transformation (1-40) it picks an additional contribution which is a divergence by virtue of current conservation:
jllOllcP
= OIl(cPjll)
This fact explains the invariance of Maxwell's equations. We conclude that current conservation is a necessary and sufficient condition for the gauge invariance of the theory. Since this question is of the utmost importance in the quantum case, it is interesting to present another point of view. Up to now we did not restrict the arbitrariness on All. The structure of Eq. (1-41) naturally suggests the Lorentz condition (1-46) in which case (1-41) reduces to (1-47) This is, of course, compatible with (1-46) and restricts severely the gauge arbitrariness, since now only those transformations All -+ All + OllcP with
DcP = 0
(1-48)
are allowed. The Lorentz constraint can be incorporated in the formalism using a Lagrange multiplier A to add a term AS d4 xi(0 . A to the action. In this way Maxwell's equations are modified and the new equations read
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