Quantum gauge invariance

Quantization of radiation field in terms of field intensity operators, 562 Quantum electrodynamics, 642 asymptotic condition, 698 gauge invariance in relation to operators inducing inhomogeneous Lorentz transformations, 678 invariance properties, 664 invariance under discrete transformations, 679... 

QUANTUM ELECTRODYNAMICS POTENTIALS, GAUGE INVARIANCE, AND ANALOGY TO CLASSICAL ELECTRODYNAMICS... 

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). 

Electron motion is more generally formulated in a form of the Schrodinger equation, including the spin in the presence of external fields known as the Pauli equation. This equation is gauge invariant in the sense that a transformation as in (5) also changes the quantum wavefunction as... 

So now we have the question poased in an interesting form. There are two quite different kinds of antennas, both of which produce electric dipole fields, but different Lorenz potentials, one emphasizing the vector potential and the other, the scalar potential. In a classical electromagnetic sense, one cannot distinguish these two cases by measurements of the fields (the measurable quantities) at distances away from the source region. The gauge invariance of QED implies the same in quantum sense. 

E. Baum, Vector and Scalar Potentials away from Sources, and Gauge Invariance in Quantum Electrodynamics, Physics Note 3 (1991). 

The derivation of Eq. (218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d Alembert wave equation is replaced by an 0(3) covariant derivative. The U(l) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically. The vacuum currents present in Eq. (218) can be computed from the right-hand side of the wave equation (212), and these vacuum currents follow from local gauge invariance. 

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 applications of continuum models to the study of solvent induced changes of the shielding constant are numerous. Solvent reaction field calculations differ mainly in the level of theory of the quantum mechanical treatment, the method used for the gauge invariance problem in the calculations of the shielding constants and the approaches used for the calculations of the charge interaction with the medium. 

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-... 

An additional source of chemical shielding anisotropies is that of ab initio theoretical calculations.20 25 There has been considerable progress in this area of molecular quantum mechanics, particularly with the use of gauge-invariant atomic orbitals within the framework of self-consistent-field (SCF) perturbation theory.26 In many cases the theoretical quantities have been extremely accurate and have served not only as a corroboration of experimental quantities but also as a reliable source of new data for molecules of second-row atoms (i.e., Li through F). 

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