Gauge transformations and quantum mechanics I. Gauge invariant interpretation of quantum mechanics

Abstract

Quantum mechanical operators are interpreted according to their equations of motion. Operators representing physical quantities which have classical analog are constructed by requiring that the quantum and the classical (i.e., Newtonian) equations of motion have a term by term correspondence. Of special importance to the interpretation of quantum mechanics is the particle's energy operator. In the presence of a time-varying electric field E, the particle's energy operator is constructed so that its time derivative is the power operator J · E ( J being the current operator). This interpretation of operators, such as the particle's energy operator, is gauge invariant despite the possible explicit dependence on electromagnetic potentials of the operators concerned. A gauge invariant interpretation of quantum mechanics is obtained by expanding the wave-function (in an arbitrary gauge) in the orthonormal set of eigenfunctions of the particle's energy operator (in the same gauge) and by interpreting the resulting expansion coefficients as probability amplitudes. This formulation possesses all the traditional gauge freedom and contains no gauge ambiguity. (Here, by gauge invariance we also mean that the dependence on paths in the DeWitt-Mandelstam formalism and on the procedures for path averaging in the Belinfante-Rohrlich-Strocchi formalism does not occur.) In particular, probability amplitudes and transition matrix elements are gauge invariant, and the transition matrix elements between states of different energies are proportional to the corresponding matrix elements of J · E, rather than J μA μ . Lamb found experimental evidence that led to the conclusion that “the usual interpretation of probability amplitudes” was gauge dependent and was correct only in the gauge in which the interaction Hamiltonian was of the form of the electric dipole interaction −er · E(0, t), instead of the usual −eA(0, t) ·p/ mc. It is shown here that the gauge invariant formulation for bound systems derives the electric dipole interaction in any arbitrary gauge as the result of the long wavelength and lowest order approximation of fields. For a quantum system interacting with a precessing magnetic field, the Güttinger-Schwinger procedure of quantizing the system along the instantaneous magnetic field has been known to yield the correct transition probabilities during the interaction. This quantization procedure follows directly from the gauge invariant formulation. The electric and the magnetic multipole interactions appearing in the gauge invariant formulation directly correspond to terms in the classical Poynting theorem. The gauge invariant magnetic multipole interactions differ from their counterparts in the conventional formalism. For example, the gauge invariant magnetic dipole interaction involves the time derivative of the magnetic field. This result is shown to be consistent with the Poynting theorem. Although the gauge invariant interpretive scheme proposed here is formulated for a nonrelativistic, spinless charged particle, the extension to the Dirac equation is straightforward.