From Special Relativity to Feynman Diagrams.pdf

The hamiltonian operator ˆ h for instance should be

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observables, should be defined as normal ordered products of the field operators. The Hamiltonian operator ˆ H , for instance, should be defined as follows: ˆ H = V k : | ˆ E k | 2 := k α ω k 2 : ( a k a k + a k a k ) = k α ω k ( a k a k ) = k α ω k ˆ N k . (6.61) Similarly, the correct definition of the momentum operator is: ˆ P = 2 V kc k k : | ˆ E k | 2 := k α k ˆ N k . (6.62) Let us note that using the normal ordering in the definition of ˆ H and ˆ P amounts to subtracting to their eigenvalues the infinite unphysical contribution associated with their ground state in the previous definitions ( 6.48 ), ( 6.51 ). Having set the energy and momentum of the ground state |{ 0 } , t to zero, the energy and momentum of a generic state |{ N k } , t of the electromagnetic field is now simply given by the sum of quanta ω k and k : E = k α N k ω k , P = k α N k k , (6.63)
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176 6 Quantization of the Electromagnetic Field We associate with each oscillator ( k , α) , i.e. with each plane wave, a state of a particle called photon and denoted by the symbol γ , carrying the quantum of momentum, k , and of energy, ω k , and having polarization α . The state | N k , t of the ( k , α) - oscillator is then then interpreted as describing N k photons in the state ( k , α) . Its energy and momentum are N k ω k and N k k respectively, namely the sum of the N k quanta of the two quantities associated with each photon. The state |{ N k } , t of the whole electromagnetic field then describes N k photons in each state ( k , α) and its energy and momentum, as given in ( 6.63 ), is the sum of the energy and momenta of the photons in the various states. A photon with energy E = ω k and momentum p = k has a rest mass given by: m 2 γ = 1 c 4 E 2 1 c 2 | p | 2 = 1 c 4 2 2 k c 2 | k | 2 ) = 0 , (6.64) where we have used the definition of ω k . As was anticipated in Chap.5 , the photon is therefore a massless particle. Its momentum four-vector p μ is thus times the wave number four-vector k μ associated with the corresponding plane wave and defined in ( 5.14 ). The action of a k or of a k on a state amounts to “creating” or “destroying” a ( k , α) -photon since they increase or decrease the energy and momentum of the corresponding oscillator state by one quantum respectively. This can be seen by recalling, from elementary quantum mechanics, the following relations which hold for the ( k , α) -oscillator: a k | N k , t = N k + 1 | N k + 1 , t , a k | N k , t = N k | N k 1 , t . (6.65) Expressing the canonical operators in terms of a k , a k , ˆ P k = − i ω k 2 a k a + k , ˆ Q k = 2 ω k a k + a + k , (6.66) we find the following relations: N k | ˆ Q k | N k 1 = N k 1 | ˆ Q k | N k = N k 2 ω k , N k | ˆ P k | N k 1 = − N k 1 | ˆ P k | N k = i ω k N k 2 . (6.67) The representation of the states of the electromagnetic field in terms of occupation number eigenstates associated with the constituent harmonic oscillators, is called
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6.2 Quantization of the Electromagnetic Field 177 occupation number representation or second quantization . In this construction each state is obtained by applying the a k operators to the ground state.
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