From Special Relativity to Feynman Diagrams.pdf

The argument given above relies on the possibility in

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The argument given above relies on the possibility, in high energy processes, for particles to be created and destroyed. This fact, as anticipated earlier, is at odds with the Schroedinger’s formulation of quantum mechanics, which is based on the notion of single particle state, or, in general of multi-particle states with a fixed number of particles. Such description is no longer appropriate in a relativistic theory. In order to have a more quantitative understanding of this state of affairs let us go back to the quantum description of the electromagnetic field given in Chap.6 . We have seen that in the Coulomb gauge ( A 0 = 0 , · A = 0 ) , the classical field A ( x , t ) satisfies the Maxwell equation: A = 1 c 2 2 t 2 A − ∇ 2 A = 0 . (10.3) Suppose that we do not quantize the field as we did in Chap.6 , but consider the Maxwell equation as the wave equation for the classical field A ( x , t ) , just as the Schroedinger equation is the wave equation of the classical field ψ( x , t ). We may ask whether a solution A ( x , t ) to Maxwell’s equations can be consistently given the sameprobabilisticinterpretationasasolution ψ( x , t ) totheSchroedingerequation.In other words, does the quantity | A μ ( x , t ) | 2 d 3 x make sense as probability of finding a photon with a given polarization in a small neighborhood d 3 x of a point x at a time t ? To answer this question we consider the Fourier expansion of the classical field A ( x , t ) given in ( 6.15 ): A ( x , t ) = k 1 k 2 k 3 k e ik · x + k e ik · x , (10.4)
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306 10 Relativistic Wave Equations where k can be written as in ( 6.42 ) k = c 2 ω k V 2 α = 1 a k u k , (10.5) but with the operators a , a replaced by numbers a , a since we want to consider A ( x , t ) as a classical field. If Maxwell’s propagation equation could be regarded as a quantum wave equation, then, according to ordinary quantum mechanics, the (complex) component of the Fourier expansion of A ( x , t ) A k ( x ) c 2 ω k V a k u k e ik · x , can be given the interpretation of eigenstate of the four momentum operator ˆ P μ , describing a free particle with polarization u k , energy E = ω and momentum p = k , respectively and satisfying the relation E / c = | p | . This would imply that A k ( x ) represents the wave function of a photon with definite values of energy and momentum. Consequently it would seem reasonable to identify the four potential A μ ( x , t ) as the photon wave function expanded in a set of eigenstates, so that the Maxwell equation for the vector potential would be the natural relativistic general- ization of the non-relativistic Schroedinger’s equation. Wenotehowever that, whiletheSchroedinger’s equationis of first order inthetime derivatives , the Maxwell equation, being relativistic and therefore Lorentz invariant, contains the operator 1 / c 2 2 t 2 which is of second order both in time and in spatial coordinates . This makes a great difference as far as the conservation of prob- ability is concerned since the proof ( 10.1 ) of the continuity ( 10.2 ) makes use of the Schroedinger equation ( 9.78 ). More specifically such proof strongly relies on the fact
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