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the structure of (3-96) is formally identical to the ordinary propagator. We observe, however, that the denominator
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never vanishes. The reader should realize that (3-95) and (3-96) represent very different quantities relative to unrelated problems.
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The above thermodynamical example leads us at once to the radiation field. Indeed, Planck's blackbody spectrum was historically the first instance of field quantization. We have simply to generalize the preceding construction to the degrees of freedom described by the potential AI'(x), Since we insist upon the local character of the theory, we cannot make use of the gauge-invariant tensor F I'V as the fundamental dynamical variable. We have to face the gauge dependence of AI' already discussed at length in Chap. 1. Various devices have been used to circumvent this difficulty. We shall not try to quote all of them here and present the Gupta-Bleuler indefinite metric quantization.
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We recall that the lagrangian (3-97) where F is expressed in terms of A, is unsuited for canonical quantization since the conjugate momentum of Ao vanishes. If we were not to worry about manifest Lorentz covariance we could content ourselves with A by constraining A 0 and using a condition of the type div A = O. This of course is physically reasonable, but does not fit our goals. We therefore use the procedure of modifying the equations of motion at the price of restoring Maxwell's theory by appropriate constraints on the physical states, since we shall introduce spurious degrees of freedom. The price will seem at first rather heavy, for not only will the Hilbert space of quantization be too large but instead of carrying a positive metric it shall allow for negative norm states! Thus the usual probabilistic interpretation of quantum mechanics will only emerge when we will have restricted ourselves to the physical quanta: the photons with only two polarizations. We also recall that to preserve gauge invariance, the principle which justifies the above procedure, we will have to make sure that the current remains conserved when introducing interactions. We therefore modify (3-97) to read, with an arbitrary constant A, =1= 0,
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il'= -"4F -2'(ooA)
(3-98)
so that Maxwell's equations are replaced by (3-99) and the conjugate momenta to the four components of A are
7T;P = - - - = po - A,gpO(o A)
ail' o(ooAp)
(3-100)
In particular nO no longer vanishes.
QUANTUM FIELD THEORY
Whether the right-hand side of Eq. (3-99) vanishes, as it does in this free case, or not if we replace it by a conserved current, we find that
0(0' A)
(3-101)
so that o A is a free scalar field. Classically we argued that appropriate conditions could be imposed on the boundary data in order that o A vanishes everywhere as a consequence of Eq. (3-101), hence restoring Maxwell's theory in the Lorentz gauge, o A = O. This cannot be achieved in the quantum case. Indeed, we shall assume the canonical commutation rules (3-102) The operator equation o A = 0 is then inconsistent, since up to a constant factor this is nO, and the latter has nonvanishing commutators with AO at equal times. We see at once that quantization according to (3-102) has overdone its job by adding unphysical features which will have to be eliminated. Even though all calculations can be pursued, keeping A. arbitrary, in the interest of simplicity we shall henceforth set A. = 1, in which case Eq. (3-99) reduces to
DAI'=O
A.=1
(3-103)
In the quantum case the choice of A. is called by abuse of language a choice of gauge, and A. = 1 is referred to as the Feynman gauge.
The price for simplicity is that we will not directly verify that our results are 2 independent. Within the framework of the free field we leave it as an exercise to the reader. He or she may as well discuss alternative forms of (3-98), where the added term might be taken as (2/2)[8' A - <p(x)Y with <p(x) a c-number classical field or (2/2) [n(x)' AY and n(x) a classical vector field (perhaps constant).
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