Gauge invariant resolution of the A⃗ · p⃗ versus r⃗ · E⃗ controversy

We delve into the problem of the choice of the interaction Hamiltonian in quantum transitions by using the gauge invariant formalism. We demonstrate that the gauge invariant transition amplitudes for the bare-bare transitions, which is the case in the majority of experiments, are identical to the conventional results with the velocity form interaction Hamiltonian. Under the electric dipole approximation, the gauge invariant formalism derives the modified length form interaction Hamiltonian in any gauge. We propose a simple experiment that one can perform to verify our prediction. On the other hand, the gauge invariant transition amplitudes for the dressed-dressed transitions, which is the case in the Lamb shift experiment, are identical to the conventional results with the length form interaction Hamiltonian. There is no preferred gauge in the calculations of transition amplitudes. The present investigation resolves the long-standing A ⃗ · p ⃗ versus r ⃗ · E ⃗ controversy and clarifies the prevailing misconception in the literature.


Introduction
The   · A p versus   · r E controversy is a long-standing problem in quantum physics. In the conventional formalism of nonrelativistic quantum mechanics, the time-independent HamiltonianĤ 0 is identified as the energy operator. . The wave function is expanded in terms of the eigenstates ofĤ 0 . The expansion coefficients, which are interpreted as the transition amplitudes, depend on the gauge. The transition amplitude in the Coulomb gauge and that in the electric field gauge differ by the ratio of the transition energy over the photon energy [1]. These considerations have a parallel in the relativistic theory. It has been shown [2] that in the nonrelativistic limit the transition amplitudes in the Coulomb gauge reduce to the velocity form amplitudes, while those in the Babushkin gauge reduce to the length form amplitudes. Relativistic transition amplitudes for one-electron system are gauge invariant only for resonant transitions.
A typical example of the length-velocity discrepancy was found in the Lamb shift measurement [3]. In the Lamb shift experiment, the hydrogen atoms in the metastable S 2 1 2 state were exposed to a rf field. This field couples the S 2 1 2 and P 2 1 2 states, thereby causes the metastable atoms to decay to the S 1 1 2 state. The decay curve depends on the matrix elements for the transition from S 2 1 2 to P 2 1 2 . Lamb found that it was the length form interaction Hamiltonian that yielded the correct decay curve. The use of the velocity form interaction Hamiltonian gave rise to a significant distortion of the decay curve. Lamb then asserted that the length form interaction Hamiltonian should be used under the electric dipole approximation.
In the mid-seventies of the twenty century, the problem was investigated in detail from the point of view of gauge invariance. Gauge invariance in quantum mechanics is elucidated in the textbook by Cohen-Tannoudji et al [4]. Yang [5] developed the gauge invariant interpretation of quantum mechanics. A summary of the publications about the gauge invariant interpretation can be found in [6]. Yang emphasized the importance of formulating the interpretation using only physical operators and physical states. Physical operators are defined as those operators whose expectation values are gauge invariant [1,4]. The eigenstates of physical operators are physical states. It is well known [1,4] that the time-independent HamiltonianĤ 0 is not a physical operator. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Consequently, the eigenstates ofĤ 0 are not physical states. This causes the gauge dependence of the transition amplitudes in the conventional formalism. Yang discriminated between the time-independent Hamiltonian and the energy operator. The energy operator for one-electron system e = - , which is just the sum of the electron's kinetic energy and potential energy, is a physical operator. Yang expanded the wave function in terms of the eigenstates of the energy operator. The expansion coefficients are interpreted by Yang as the gauge invariant transition amplitudes. Yang's transition amplitudes, with the exception of the electric dipole transition, differ from their counterparts in the conventional formalism. Under the electric dipole approximation wherein the magnetic field is neglected and the electric field varies slowly over atomic dimensions, Yang's gauge invariant transition amplitudes are identical to the conventional results with the length form interaction Hamiltonian. Yang's finding completely agree with the conclusion of Lamb [3]. Since then there has been a common assertion [1,[5][6][7][8] that the length form interaction Hamiltonian should be used in the conventional formalism of quantum mechanics.
The subject of gauge invariance in many-body system is more subtle and controversal because approximate wave functions must be used. Lin [9] claimed, in the conventional formalism of relativistic quantum mechanics, that the Hartree-Fock theory violates the gauge invariance even for resonant transitions because of the nonlocal potential. One of the elegant methods to treat the correlation effects in many-body system is the many-body perturbation theory. Chi and Chou [10] showed, also in the conventional formalism, that the relativistic manybody perturbation theory calculations are gauge invariant order by order for resonant transitions, provided that the calculations start from a local potential and include the contributions from the negative-energy states. Kobe [11] extended Yang's formulation , in the second-quantization formalism, to many-body system. The formulation was then applied to demonstrate that Hartree-Fock theory is gauge invariant in spite of being nonlocal. The only proviso is that the Hamiltonian for the many-body system is replaced by Yang's energy operator.
The gauge invariant interpretation of Yang has raised critical comments in the literature [12][13][14]. Feuchtwang et al [14] pointed out that the main difficulties of Yang's interpretation are as follows. First, Yang's energy operator has time dependent eigenvalues which cannot be measured precisely. Second, Yang's transition amplitudes, while gauge invariant, are neither measurable nor do they relate to probabilities of physically significant observations in the study of electromagnetic interactions with matter. The aim of this paper is to illustrate the confusion inherent in Yang's interpretation and to make a gauge invariant resolution of the longstanding   · A p versus   · r E controversy. We point out that the careful distinction between the bare-bare transitions and the dressed-dressed transitions is necessary for the choice of the interaction Hamiltonian. We obtain the rule over when to use   · A p and when to use   · r E.

Gauge transformation, gauge invariance, and gauge covariance
We consider an electron subject to a static potential  ( ) V r and a time-varying electromagnetic field   ( ) E r t , and   ( ) B r t , . The fields   ( ) E r t , and   ( ) B r t , can be written in the form g are the electromagnetic potentials in an arbitrary gauge. In this gauge, the Hamiltonian and the Schrödinger equation are given by The wave function of the electron in the Coulomb gauge is denoted by Y  ( ) r t , c . The electromagnetic potentials and the electron's wave function in other gauges can be obtained by the gauge transformation [1,4,7]:

Schrödinger equation in a manifestly gauge covariant form
The Schrödinger equation in (3) can be written in a manifestly gauge covariant form: , , In arriving at equation (10), we have used the identity is iÿ times the time component of the gauge covariant derivative. Note that eΦ g is an integral part ofÊ g . It cannot be interpreted as the interaction energy because it is not a physical operator. Another reason is that only static fields can be used to define a potential energy [5]. We shall derive the expression for the interaction energy operator in subsquent paragraphs.
A heuristic procedure to obtain the Schrödinger equation in (8) involves starting from the energymomentum relation = + The substitution (13) is Lorentz covariant, since it is a correspondence between two 4-vectors

Bare states
The electromagnetic potentials in the absence of the electromagnetic field can in general be chosen as , , Equation (14) indicates that the vector potential is longitudinal in the absence of the electromagnetic field. The energy operator in the absence of the electromagnetic field is called the bare energy operator. From equations (10) and (14) it follows that the bare energy operator is given by , , are the eigenfunctions ofĤ 0 with eigenvalues E i . It follows from (16), (17), , .

Dressed states
In the presence of the electromagnetic field, the vector potential contains the transverse component c . The energy operator in the presence of the electromagnetic field is called the dressed energy operator. It follows from (10) and (15)   are the dressed energies and the dressed states [16], which represent the energies and the eigenstates of the electron in the presence of the electromagnetic field. The eigenvalues  ( ) t i g is gauge invariant because of (12). Therefore we simply denote it by  ( ) t i . Note that  ( ) t i is in general time dependent. Thus we resolve the first difficulty of Yang's interpretation [5] by correctly identifying Yang's energy operator as the dressed energy operator.
Under the electric dipole approximation, the vector potential is approximated by In the study of the dressed state, the electric field gauge in which 0 , 2 8 ec c