Gauge field unitary transformation

The vanishing of the YM field intensity tensor can be shown to follow from the gauge transformation properties of the potential and the field. It is well known (e.g., Section II in [67]) that under a unitary transformation described by the matrix... 

In the case of quantum field theory the section determines the Hilbert space of states under a certain gauge. This choice of gauge then determines the unitary representation of the Hilbert space. We may then replace the section with the fermion field /, which acts on the Fock space of states. It is then apparent that a gauge transformation A t > A t + 84 is associated with a unitary transform of the fermion field v / > v / I 8 /. The unitary transformation of the fermion... 

This demonstrates the association between the unitary transformation of the fermion field and the gauge theory. 

If the physical results are to remain unchanged under a unitary transformation, it is necessary to transform the operators as well as the state functions (eqns (8.28) and (8.29)). A gauge transformation has no effect on a coordinate operator but the momentum operator p is changed into p + [e/c) x- Thus, to maintain gauge invariance in properties determined by the momentum operator, p must be replaced by a new operator. In the presence of an electromagnetic field or just a magnetic field, the momentum operator is replaced by the expression... 

In a different gauge, it is possible to construct the multipolar Hamiltonian which is obtained by applying a unitary transformation to the minimal coupling Hamiltonian [75-77,106]. In the multipolar Hamiltonian, it is the transverse electric field, and the magnetic field, B(r) (satisfying Maxwell s equation, V x Et = - f), that appear, rather than the vector potential. Now, the interaction is written as... 

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