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Molecular Physics

Gauge invariance of the S-matrix for atoms, molecules and electromagnetic radiation

R. Guy Woolley

Published Jun 20, 1998• 15 citations
Publisher

Abstract

The Hamiltonian for non-relativistic quantum electrodynamics can be expressed in terms of the usual dynamical variables for the field and charged particles and as a functional of a c-number vector field p(x;x') which is essentially arbitrary. This corresponds to the freedom to use the vector potential of he field in an arbitrary gauge since every p(x;x') determines a gauge condition for the potential. We study the S -matrix for two different choices of p(x;x') and calculate the difference. This is done directly in low-orders of perturbation theory and suggests a conjecture for the general formula for the difference, valid in any order of perturbation theory; we verify the validity of this conjecture by elementary means. On the energy shell the S- matrices are identical, i.e. gauge-invariant.

DOI
10.1080/00268979809482333

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The Hamiltonian for non-relativistic quantum electrodynamics can be expressed in terms of the usual dynamical variables for the field and charged particles and as a functional of a c-number vector field p(x;x') which is essentially arbitrary. This corresponds to the freedom to use the vector potential of he field in an arbitrary gauge since every p(x;x') determines a gauge condition for the potential. We study the S -matrix for two different choices of p(x;x') and calculate the difference. This is done directly in low-orders of perturbation theory and suggests a conjecture for the general formula for the difference, valid in any order of perturbation theory; we verify the validity of this conjecture by elementary means. On the energy shell the S- matrices are identical, i.e. gauge-invariant.
A novel Hamiltonian scheme for non–relativistic quantum electrodynamics in which the gauge arbitrariness of the field potential is kept explicit is used to study the gauge–dependence properties of various versions of perturbation theory when resonance and line–broadening effects are admitted. Time–dependent perturbation theory is shown to have severe gauge–dependence problems unless ad hoc modifications are made. A time–independent formulation of S–matrix theory is then studied. Far from resonance, the S–matrix is gauge invariant in all orders of perturbation theory due to a very precise cancellation of gauge–dependent terms which requires, among other things, complete sets of intermediate states; energy conservation also has a crucial role. However, an obvious separation of the S–matrix into a resonant (pole) and non–resonant background leads to incomplete cancellation of the gauge–dependent terms. The introduction of the Heitler damping matrix into an integral equation for the T–matrix leads to a gauge–invariant result. This provides the basis for a gauge invariant S–matrix theory of atoms and molecules interacting with electromagnetic radiation that encompasses resonance and damping effects.
The conventional perturbation theory in quantum mechanics for the interaction of electromagnetic radiation with matter, which is based on the time-dependent vector and scalar potentials, is shown to be dependent on the gauge. The problem is reformulated, using the approach of Yang, in a gauge-invariant way, in which the transitions between states are induced by the quantum-mechanical power operator. In order to use an unperturbed Hamiltonian which does not involve the electromagnetic field, the gauge is chosen to give the electric dipole interaction. This choice can be made if the magnetic field is negligible and the electric field is slowly varying over atomic dimensions. For a comparison of the gauge-invariant and conventional formulation, a two-level atom in a radiation field is considered. For the probability that an atom originally in the lower state will make a transition to the upper state when the field is not in resonance, the gauge-invariant formulation gives the correct result, whereas the gauge-dependent formulation gives an incorrect result.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
The minimal-coupling Hamiltonian for a non-relativistic bound system of two charges in the dipole approximation is gauge transformed to a Hamiltonian of a variable gauge which mixes the usual p · A and q · E interactions, and re-scales the charges. Various photon transition processes—single photon absorption and emission; two-photon scattering and absorption; the Lamb shift—are investigated by perturbation theory and found to be gauge independent, providing the transitions occur on the energy shell. It is demonstrated how a particular choice of gauge removes the counter-rotating terms and diagonalises the Hamiltonian of a coupled oscillator and two-level atom.
The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.
The case of a single charged particle with potential energy is considered when the classical electromagnetic field is zero. The vector and scalar potentials describing this situation are not necessarily zero. When the conventional approach to radiation problems using the interaction A⋅p plus p⋅A plus A2 plus A0 (constants omitted) is used, time-dependent probabilities are predicted if the vector potential A=−∇Λ≠0 even though the electromagnetic field is zero. The reason for this unphysical behavior is that eigenfunctions of the unperturbed Hamiltonian, which is not gauge invariant, are used to calculate the probability amplitudes. When a manifestly gauge-invariant procedure is applied to this case the probabilities are constant in all gauges. The reason that physical results are obtained is that eigenstates of the energy operator are used to calculate the probability amplitudes.