From Special Relativity to Feynman Diagrams.pdf

Operators ˆ a k in terms of the canonical operators

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operators ˆ A k in terms of the canonical operators ˆ P k , ˆ Q k using the same relations ( 6.30 ): ˆ A k c 2 ω k V i ˆ P k + ω k ˆ Q k . (6.41) Next we expand ˆ A k along the two transverse directions and define the dimensionless operators a k as follows: ˆ A k = c 2 ω k V 2 α = 1 a k u ( k , α). (6.42) where a k α = 1 2 ω k i ˆ P k + ω k ˆ Q k . (6.43) As it is well known, when passing from classical quantities to quantum operators, the complex conjugation operation is replaced by hermitian conjugation. The hermitian conjugate of a k α is: a k = 1 2 ω k i ˆ P k + ω k ˆ Q k . (6.44) Using ( 6.40 ), we find that the operators a k α , a k α satisfy the following commutation relations: [ a k , a k ] = [ a k , a k ] = 0 , [ a k , a k ] = δ k , k δ α,α . (6.45) The operator ˆ A ( x , t ) associated with the vector potential of the electromagnetic field is then expressed by the Fourier series: ˆ A ( x , t ) = k ( ˆ A k ( t ) e i k · x + ˆ A k ( t ) e i k · x ) = 2 α = 1 k c ω k V ω k ˆ Q k α cos ( k · x ) ˆ P k α sin ( k · x ) u k .
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6.2 Quantization of the Electromagnetic Field 173 We can now use the expansions ( 6.17 ) and ( 6.19 ) as well as ( 6.18 ) and ( 6.20 ) to define the electric and magnetic field operators ˆ E and ˆ B in terms of the operators ˆ A k , A k and thus of a k α , a k α : ˆ E = k ˆ E k e i k · x + ˆ E k e i k · x ; ˆ B = k ˆ B k e i k · x + ˆ B k e i k · x . (6.46) with: ˆ E k = 2 α = 1 ˆ E k α u k = 2 α = 1 i ω k 2 V a k α u k ; ˆ B k = 2 α = 1 ˆ B k α u k = n k × ˆ E k . (6.47) We are now able to compute the Hamiltonian operator ˆ H following the same deriva- tion as in the classical case. Care, however, has to be used in deriving the operator versions of equations ( 6.27 ) and ( 6.28 ) from ( 6.26 ) since, as opposed to the corre- sponding classical quantities which were just numbers, the operators ˆ E k and ˆ E k , as well as their magnetic counterparts, no longer commute. As a consequence of this, in writing the expression for the Hamiltonian, we should keep the order of factors in each product and thus, instead of a sum over ˆ A k ˆ A k , we would find a sum over 1 2 ( ˆ A k ˆ A k + ˆ A k ˆ A k ) . The Hamiltonian operator then reads: ˆ H = k α ω k 2 a k a k α + a k α a k α = k α ˆ N k + 1 2 ω k , (6.48) where ˆ N k a k a k , (6.49) The operator ˆ H , in terms of the canonical operators, has the same form as in ( 6.34 ): ˆ H = 1 2 k α ( ˆ P k ) 2 + ω k ( ˆ Q k ) 2 . (6.50) Equations ( 6.48 ) and ( 6.50 ) describe the Hamiltonian operator associated with the system of infinitely many quantum harmonic oscillators ( k , α) defined in the previous section. The quantities a k and a k are indeed nothing but the annihilation and creation operators associated with the quantum oscillator ( k , α) , which are useful in constructing the corresponding quantum states. It is now straightforward to determine the expression for the momentum operator, by using ( 6.38 ) and ( 6.48 ): ˆ P = 1 c V d 3 xE × B = k α k ˆ N k + 1 2 . (6.51) Both H and ˆ P are expressed in terms of the occupation number operators ˆ N k ( 6.49
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