From Special Relativity to Feynman Diagrams.pdf

V d 3 x e 2 b 2 v 2 k e k e k e k e k c c b k b k b k

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V d 3 x | E | 2 + | B | 2 = V 2 k ( E k · E k + E k · E k + c . c .) + ( B k · B k + B k · B k + c . c .) , (6.26) where c . c . denotes the complex conjugate of the previous terms. The terms E k · E k and B k · B k cancel since: B k · B k = ( n k × E k ) · ( n k × E k ) = − i j n j k E k ipq n p k E q k = −| n k | 2 E k · E k + ( n k · E k )( n k · E k ) = − E k · E k , where we have set n k = − n k and have used the transversality condition n k · E ± k = 0 and the contraction properties of two i jk symbols (see Sect.4.5 ). We also find: | B k | 2 = | E k | 2 = ω 2 k c 2 ( A k · A k ), (6.27) which allows us to rewrite ( 6.26 ) in the form: E = 2 V k | E k | 2 = 2 V c 2 k ω 2 k ( A k · A k ). (6.28) Let us now introduce the following variables 2 : Q k = V c ( A k + A k ) ; P k = − i ω k V c ( A k A k ). (6.29) Taking into account the time-dependence of A k ( t ) , see ( 6.14 ), it is straightforward to verify that P k = ˙ Q k . Equations ( 6.29 ) can be easily inverted to express A k and A k in terms of Q k , P k : A k = c 2 ω k V ( i P k + ω k Q k ) ; A k = c 2 ω k V ( i P k + ω k Q k ), (6.30) Using the above relations we can rewrite the energy in the new variables: E = H = 1 2 k ( | P k | 2 + ω 2 k | Q k | 2 ) = k E k , (6.31) 2 Notice that the P k here have dimension ( Energy ) 1 2 .
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170 6 Quantization of the Electromagnetic Field where we have identified the energy E with the Hamiltonian H of the system of infinitely many degrees of freedom, described by Q k = ( Q i k ) , each labeled by a wave number vector k and a polarization index i (as a consequence of the transversality condition, not all these polarizations are independent, as we shall discuss below). We can easily verify that P k , Q k are indeed the canonical variables corresponding to the Hamiltonian H by showing that they satisfy Hamilton’s equations 3 : ˙ Q i k = H P i k = P i k , ˙ P i k = − H Q i k = − ω 2 k Q i k , where, as usual, the dot represents the time-derivative. These equations can also be written in the second order form: ¨ Q i k + ω 2 k Q i k = 0 , (6.32) which, given the relation ( 6.29 ), are equivalent to the Maxwell equations ( 6.12 ) for each component A k . We realize that the above equation in the variable Q i k , for each polarization component i and wave-number vector k , is the equation of motion of a harmonic oscillator with angular frequency ω k . Note now that the vectors P k and Q k are orthogonal to k in virtue of the transversality property A k and A k : k · P k = k · Q k = 0 . (6.33) This allows us, for a given direction of propagation n k , to decompose P k and Q k along an ortho-normal basis u k , α = 1 , 2 on the plane transverse to n : P k = 2 1 P k α u k , Q k = 2 1 Q k α u k . The index α labels the two polarizations of the plane wave. Taking into account the ortho-normality of the ( u k ) we can write the Hamiltonian as follows: H = 1 2 k α = 1 , 2 ( P 2 α k + ω 2 k Q 2 α k ) = k H k = k E k = k α = 1 , 2 H k , (6.34) which describes a system of infinitely many, decoupled, harmonic oscillators, each described by the conjugate variables P α k , Q α k and Hamiltonian function H k .
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