Schroedinger's equation with scalar and vector potentials is shown to describe "nothing but" hopping of a quantum particle on a lattice; any spatial variation of the hopping amplitudes acts like an external electric and/or magnetic field. The main point of the argument is the superposition principle for state vectors; Lagrangians, path integrals, or classical Hamiltonians are not (!) required. Analogously, the Hamiltonian of the free electromagnetic field is obtained as a twofold continuum limit of unitary hopping in Z(N) link configuration space, if gauge invariance and C and P symmetries are imposed.
Schroedinger's equation with scalar and vector potentials is shown to describe "nothing but" hopping of a quantum particle on a lattice; any spatial variation of the hopping amplitudes acts like an external electric and/or magnetic field. The main point of the argument is the superposition principle for state vectors; Lagrangians, path integrals, or classical Hamiltonians are not (!) required. Analogously, the Hamiltonian of the free electromagnetic field is obtained as a twofold continuum limit of unitary hopping in Z(N) link configuration space, if gauge invariance and C and P symmetries are imposed.
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as “minimal coupling” and consists of replacing the momentum operators in the Hamiltonian by the modified ones with an added vector potential. In lattice systems, it is not so clear how to do this because there is no continuous translation symmetry, and hence, there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, but only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighbourhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.
It was pointed out in a previous paper that although neither the Klein-Gordon equation nor the Dirac Hamiltonian produces sound solitary free-particle relativistic quantum mechanics, the natural square-root relativistic Hamiltonian for a nonzero-mass free particle does achieve this. Failures of the Klein-Gordon and Dirac theories are reviewed: the solitary Dirac free particle has, inter alia, an invariant speed well in excess of c and staggering spontaneous Compton acceleration, but no pathologies whatsoever arise from the square-root relativistic Hamiltonian. Dirac's key misapprehension of the underlying four-vector character of the time-dependent, configuration-representation Schroedinger equation for a solitary particle is laid bare, as is the invalidity of the standard "proof" that the nonrelativistic limit of the Dirac equation is the Pauli equation. Lorentz boosts from the particle rest frame point uniquely to the square-root Hamiltonian, but these don't exist for a massless particle. Instead, Maxwell's equations are dissected in spatial Fourier transform to separate nondynamical longitudinal from dynamical transverse field degrees of freedom. Upon their decoupling in the absence of sources, the transverse field components are seen to obey two identical time-dependent Schroedinger equations (owing to two linear polarizations), which have the massless free-particle diagonalized square-root Hamiltonian. Those fields are readily modified to conform to the attributes of solitary-photon wave functions. The wave functions' relations to the potentials in radiation gauge are also worked out. The exercise is then repeated without the considerable benefit of the spatial Fourier transform.
This chapter constructs path integrals for general Hamiltonians with potentials linear in the velocities, like Hamiltonians of particles in a magnetic field. Two examples are considered: a quantum system coupled to a magnetic field, and diffusion as described by the Fokker-Planck equation. In both examples, the Hamiltonian contains products of the position and momentum operators. A quantization problem then arises since these operators do not commute, and the correspondence principle is no longer sufficient to determine the quantum Hamiltonian. The order between quantum operators is determined by additional conditions, such as hermiticity or conservation of probabilities. The calculation of the corresponding path integral then suffers from ambiguities, directly related to this quantization problem. The continuum limit is no longer unique, but depends on the limiting process. This chapter also considers a situation where space has a nontrivial topology, in this case a circle, and shows how this influences the calculation of the path integral.
We employ here the enhancement of the vector potential amplitude quantization at a single photon state. The analysis of the general solution of the vector potential, obtained by resolving Maxwell’s equations, implies that the amplitude is proportional to the angular frequency. The photon vector potential function αkλ(r,t) can be written in the plane wave representation satisfying the classical wave propagation equation, Schrödinger’s equation for the energy with the relativistic massless field Hamiltonian and a linear time-dependent equation for the vector potential amplitude operator. Thus, the vector potential αkλ(r,t) with the quantized amplitude may play the role of a real wave function for the photon in a nonlocal representation that can be suitably normalized. We then deduce that the amplitudes of the electric and magnetic fields, respectively, of a single free photon are proportional to the square of the angular frequency. This might open perspectives for the development of nondestructive photon detection methods based on the influence of the electric and/or magnetic fields of photons on the energy levels of atoms and molecules.