From Special Relativity to Feynman Diagrams.pdf

We can of course expand the field at a certain time

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We can of course expand the field at a certain time, say t = 0 exactly as in equations ( 11.231 ) and ( 11.145 ), with creation and destruction operators obeying the same commutation rules as in the free case. However they are no longer eigenmodes of the Hamiltonian and hence we cannot interpret them as creation and destruction opera- tors of single particles. Indeed ( 11.300 ) imply that those operators evolve in time as c p , t = e i ˆ Ht c p , 0 e i ˆ Ht and analogously for the other operators. This means the entire apparatus of the free field theories for constructing the eigenmodes of the Hamiltonian breaks down and the exact solution of the coupled equations is unknown. Indeed interacting quantum theories are too complex to be solved exactly and we must resort to perturbative methods, to be developed in the next chapter. Let us here anticipate some concepts related to this issue. In the perturbative approach the quantum Lagrangian and Hamil- tonian are written in terms of the free field operators ψ( x ), A μ ( x ), evolving with H 0 = d 3 x ( H Dirac + H e . m . ), and acting on the Fock space of free-particle states. These are expressed as the tensor product of the electron/positron states and the photon states:
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11.8 Quantum Electrodynamics 429 |{ N e }; { N e + } 0 ⊗ |{ N γ } 0 , (11.301) where N e , N e + , N γ are the occupation numbers of electron, positron and photon states, the subscript “0” indicates that these states pertain to the free theory ( e = 0 ). Such states are constructed, as illustrated in this chapter, by acting on a vacuum state | 0 0 by means of creation operators. Also the free field operators are expressed in terms of creation and annihilation operators, and all terms in the quantum Lagrangian and Hamiltonian are written in normal ordered form. In particular the interaction term reads: ˆ H I = − e : ¯ ψγ μ ψ A μ :≡ − ˆ L I . (11.302) Writing everything (fields, states, Hamiltonian etc.) in terms of the solution to the free problem, corresponding the absence of interaction ( e = 0 ), is the lowest order approximation from which the perturbation analysis is developed, the perturba- tion parameter being the dimensionless fine structure constant α e 2 /( 4 π c ) 1 / 137 1 . All perturbative corrections, as we shall illustrate in next Chapter, are expressed in terms of a series expansion in powers of the interaction Hamiltonian: ˆ H I d 3 x ˆ H I , (11.303) (and thus in powers of the small constant α ), through the so called S - matrix . Let us stress that each term in this expansion is expressed in terms of free fields. Here we wish to comment on the issue of symmetries. So far we have defined symmetry transformations on free fields. The Lagrangian and Hamiltonian density operators ˆ L tot , ˆ H tot according to the above prescription, are obtained from their clas- sical expression in ( 11.284 ), ( 11.285 ), ( 11.289 ) and ( 11.290 ) by replacing the fields by their corresponding free quantum operators, and normal ordering the resulting expression. Let g be a symmetry transformation of the free theory ( e = 0 ), belong- ing to some symmetry group G , and let U ( g ) be the unitary operator which realizes it on the free-particle states. The invariance property is expressed in terms of the free
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