QUANTUM FIELD THEORY in .NET framework

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QUANTUM FIELD THEORY
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it is impossible to abstract a charged particle from its radiation field. However, as stressed in Chap. 1, it is meaningless to count the number of emitted soft photons, and the only physically measurable quantity is the emitted energy. Mathematically, the situation is really catastrophic. Indeed, when fi --+ 00, we learn from Eqs. (4-21) and (4-26) that 1<0 out 10 in)1
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Every matrix element between in- and out-states vanishes. Clearly, it is impossible to construct the "out" Fock space from the "in" space, nor to find a unitary operator S. This is exactly the same situation as in the simple model discussed at the beginning of this section. For a system with an infinite number of degrees of freedom, and under certain circumstances (here fi = (0), inequivalent representations of the canonical commutation relations may exist. It is no wonder that we get into trouble here, since we try to describe the final states as superpositions of states of a finite number of photons, while we know that their actual number is infinite. Physically, we may cut off a part of phase space, i.e., we decide to observe only final photons within a certain range of energy momentum. This indeed corresponds to an experimental limitation. Every photon detector has a given finite resolution, and photons of energy lower than this resolution are therefore unobservable. Let R be the unobserved region of phase and C R its complement. The total probability of emission in R only, i.e., to detect no photon
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is non vanishing since
is finite, while each term of the sum vanishes as fi --+ 00. The observed radiated energy g(C R ) = g - g(R) is as close to g as desired, for small enough R, and the probability of detecting at least a photon with momentum outside R, 1 - exp (- fiCR)' is finite.
There is an alternative possibility to avoid the slippery subtleties of the previous treatment of infrared divergences, at least of its mathematical difficulties. We give the photon a small mass Jl and use the Stueckeiberg gauge of Chap. 3 to quantize such a field. This will cut off the low-energy region since now kO > Jl and therefore remove the infrared divergence. However, we have to verify that the extra degrees of freedom do not introduce any spurious effect, namely, that observable quantities are not affected in the limit Jl ---> O. We thus consider a massive photon field coupled to a conserved current. In the massless case, we recall that the indefinite metric state has played no role; only transverse degrees of freedom have been excited. Similarly, here, in the Stueckelberg gauge, the conservation law of the current implies that only the transverse field
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is coupled to the current
Therefore the negative metric states disappear. However, the longitudinal polarization state stilI contributes. We may compute fi, for instance, from Eqs. (4-16) and (3-148):
(4-36) Typically for an accelerated charge, J. J* ~ kmax. we get a potentially divergent fi:
1/k6, as we recalled above.
dk k
Cutting off integrals at some
Ikl =
m ..
+ J12)3 / 2
~ In
k max
while the emitted energy
has a smooth behavior as J1-> O. We also find that the contribution fiL of longitudinal photons of given momentum k is vanishingly small with respect to that of transverse photons fiT:
fiL fiT
IJzl2 - iJol 2
jJ,,1 2
(4-37)
where current conservation has been used to write iJzI = (ko/lkl)jJol. The introduction of this small photon mass has provided a convenient regularization of the infrared divergence. However, the existence of a third state of polarization might modify the blackbody radiation spectrum by a factor 3/2. This difficulty is bypassed by the assumption that the equilibrium time for the third mode may be so large that the effect is totally unobservable.
It must be clear from this discussion that the main features of the infrared divergences of quantum electrodynamics are essentially classical, and depend on the nature of the external current and on the experimental resolution. However, quantum effects manifest themselves, e.g., in the fluctuation of the number of emitted photons. Assuming ii to be finite, we compute Lln 2
== n2
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