Gauge invariant approximation

The approximation can be tested for the ls-2s transition in Hydrogen. All p-states with ji > 2 are taken as uncharacterized, the average frequency v(K) is determined by fitting (14) to the exact transition amplitude Dj [Jq] from [2] at vi = 0.375, leading to v(K) = 0.0171. In Table 3 we compare at various frequencies the exact amplitude with the amplitudes [R, JQ 2] and Dj [J0,2], where only the 2p intermediate state is included. We notice that the error in Dj [R, J0 2] is less than 2%, and much smaller than the error in [J0, 2], The gauge invariant approximation partially overcomes the major difficulty encountered in any approximate calculation this scheme, however, relies on free parameters which must be determined independently and hence its application is not straightforward. 

In other words, the group-Born-Oppenheimer is a gauge invariant approximation. In the above equation, the dressed potential transforms as... 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

Now, we assume that the functions, tcoj, j = 1,. .., N are such that these uncoupled equations are gauge invariant, so that the various % values, if calculated within the same boundary conditions, are all identical. Again, in order to determine the boundary conditions of the x function so as to solve Eq. (53), we need to impose boundary conditions on the T functions. We assume that at the given (initial) asymptote all v / values are zero except for the ground-state function /j and for a low enough energy process, we introduce the approximation that the upper electronic states are closed, hence all final wave functions v / are zero except the ground-state function v /. ... 

In quantum electrodynamics (QED) the potentials asume a more important role in the formulation, as they are related to a phase shift in the wavefunction. This is still an integral effect over the path of interest. This manifests itself in the phase shift of an electron around a closed path enclosing a magnetic field, even though there are no fields (approximately) on the path itself (static conditions). As can be shown the result of such an experiment is gauge-invariant, allowing the use of various choices of the vector potential (all giving the same result). 

To an excellent approximation, the four Klein-Gordon equations (443) are d Alembert equations, which are locally gauge-invariant. 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

The strong-field approximation, on which the present formalism relies, is not gauge invariant see. e.g. [32]. This problem has been present not only for... 

Ab initio quantum mechanical methods are now able to provide reasonably good values for chemical shielding, but there are limitations to the accuracy that can be obtained. For the approximate wave functions that must be used, it is essential to use gauge-invariant methods to obtain meaningful shielding results. This require-... 

The semi-classical equations of motion obtained above involve only the transverse adiabatic vector potential which is, by definition, independent of the choice of gauge functions/(q) and g(q). The (Aj -f A2)/2M term in the potential is also independent of those two arbitrary functions. The locally quadratic approach to Gaussian dynamics therefore gives physically equivalent results for any choice of /(q) and g(q). The finding that the locally quadratic Hamiltonian approach developed here is strictly invariant with respect to choice of phases of the adiabatic electronic eigenstates supersedes the approximate discussion of gauge invariance given earlier by Romero-Rochin and Cina [25] (see also [40]). 

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