Unitary gauge

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 transfoiination described by the matrix... 

In accordance with refs.(camarda 2003,abreu),we apply tranverse unitary gauge and express the partition function as... 

The double trace term is due to the absence of the condition for the vanishing of the trace for the broken generator X5. It emerges naturally in the non linear realization framework at the same order in derivative expansion with respect to the single trace term. In the unitary gauge these two terms correspond to the five gluon masses [431. 

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. 

This defines the fermion contribution to an isovector gauge current density. Although the Euler-Lagrange equation is gauge covariant by construction, this fermion gauge current is not invariant, because the matrix r does not commute with the 5(7(2) unitary transformation matrices. It will be shown below that the... 

This distinction is largely formal, owing to the substantial identity of the unitary time-dependent transformation (8)-(9) with the gauge transformations of the Hamiltonian and its eigenfunctions [21-22]. However, alterna-... 

In quantum mechanics the unitary transformation (8) leads to different gauges for the Hamiltonian, which are sometimes also referred to as different formalisms. An alternative first-order Hamiltonian can be defined from (8)... 

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... 

The scalar potential can be removed by moving to the A p gauge. We perform a unitary transformation ... 

Now we can see that the change of the gauge origin corresponds to a unitary transformation of the Hamiltonian. Although the wave function is affected by the unitary transformation, the eigenvalues remain unchanged and the Schrodinger equation is solved exactly. 

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... 

Thus, the generated fnuc,n+ is a unitary transformation of mid,n+ This means that the FC function is always a gauge-including function if its initial function is gauge including. Because this can be extended to a general Hamiltonian and initial... 

Note that is the vector matrix of derivative couplings in the adiabatic electronic basis and the gauge transformation (R) is the unitary transformation matrix connecting the adiabatic and diabatic basis sets. In the above example of two real electronic states, Eq. (32) is identical to Eq. (30a) where x is set to zero ... 

The gauge transformations form a group. It is Abelian, i.e. diflferent transformations of the group commute with each other, and it is onedimensional, i.e. the transformations are specified by one parameter 0. This group is C/(l), the group of unitary transformations in one dimension. We say that 17(1) is a symmetry group of , and that the functions form a one-dimensional representation of U 1). 

If we take the form of C given in (3.2.9) as our quantum Lagrangian and we attempt to do perturbation theory, all propagators will look sensible, with poles corresponding only to the physical particles. For this reason the chosen gauge is called a unitary gauge U gauge). 

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