Commutation relations field theory

When quantizing the theory, the commutation relations for the complex field u become... 

The basic expression for the quantization of the electromagnetic field is the expansion Eq(54). In the quantized theory the numbers Ck,, C x become operators of the creation C x and the annihilation Ck,x of photons. These operators are acting on the state vector < ) that is defined in the Fock space (occupation number space). The C xt Ck, operators satisfy the commutation relations ... 

In the first step one has to quantize the classical field theory. The standard canonical quantization via equal-time commutation relations for the fermion field operator % yields... 

What has already been said about space and time coordinates in the preceding chapters suggests the obvious question for which coordinates the Pauli principle is valid. Do we need to apply the pair permutation to only spatial coordinates or to space-time coordinates The permutation is to be applied to the spatial coordinates only since in quantum field theory the commutators are understood as equal-time commutation relations. Moreover, in nonrela-tivistic quantum mechanics this problem does not show up and we will later refer to space-spin coordinates that need to be exchanged for pair permutation. The situation will become more clear in section 8.6.5 once we have introduced the theoretical tools and background needed. 

This makes Maxwell s equations look simple. In quantum field theory things are more complicated because equations like (2.4.1), if interpreted literally as equations for the field operators, sometimes contradict the fundamental commutation relations that the fields must satisfy. Another common choice in QED is to have... 

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