This paper develops a gauge-independent symmetry theory of non-relativistic quantum mechanical systems, in line with that previously considered in the context of classical mechanics. We first discuss at length the motivation for adopting the view that the invariance of a system K under a physical symmetry operation .<I' should be taken to mean invariance of the equation of motion of K under a certain gauge-independent unitary transformation U c( .<1'). The formal development of the theory is then carried through, and some detailed examples are presented. In particular, corresponding to every direction along which a system K happens to be translation invariant there exists a gauge-independent generator of translations which leaves K invariant but which is not, in general, a component of either the canonical or the kinetic momentum. The connexion between such invariant generators of translations and the so-called magnetic translation operators is referred to.
This paper develops a gauge-independent symmetry theory of non-relativistic quantum mechanical systems, in line with that previously considered in the context of classical mechanics. We first discuss at length the motivation for adopting the view that the invariance of a system K under a physical symmetry operation .<I' should be taken to mean invariance of the equation of motion of K under a certain gauge-independent unitary transformation U c( .<1'). The formal development of the theory is then carried through, and some detailed examples are presented. In particular, corresponding to every direction along which a system K happens to be translation invariant there exists a gauge-independent generator of translations which leaves K invariant but which is not, in general, a component of either the canonical or the kinetic momentum. The connexion between such invariant generators of translations and the so-called magnetic translation operators is referred to.
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.
We introduce a gauge-invariant Wigner operator (GIWO) and a gauge-independent Wigner function (GIWF) that allow for both quantized and classical electromagnetic fields. If only classical fields are present, a Weyl transform analogous to the one associated with ordinary Wigner functions can be defined for any operator; in the case of quantized fields this is at least possible for Weyl-ordered functions of the position and kinetic momentum operators. We show how the evolution of the operators having a Weyl transform can be followed with the aid of the GIWF defined as the Heisenberg-picture expectation value of the GIWO, and derive for the GIWO the Heisenberg equation of motion which only involves the physical electric and magnetic fields. Some aspects of the case of entirely classical fields are discussed in more detail. (i) The GIWF in conjunction with the postulate that physical observables can be measured without referring to the gauge permits a quantum-mechanical treatment of a full experimental run without the problem of relating the measured values to the gauge-dependent density operator. (ii) A closed equation of motion for the GIWF is obtained. (iii) In the dipole approximation quantum features of the dynamics are lost. (iv) Quantum corrections to dynamics are associated with recoil effects.
For a quantum mechanical particle in a classical electromagnetic field, the active and passive views of gauge invariance are compared. In the active point of view, both the wavefunction and the potentials are transformed so that the Schrödinger equation is form invariant. In the passive point of view, a change in the representation of the canonical momentum is made which just cancels the change in the gauge of the potentials. The condition that a gauge-invariant operator has the same expectation value in all gauges gives a functional equation, the solution of which shows that gauge-invariant operators depend only on the kinetic momentum or coordinates.
The wave function of quantum mechanics is not a boost invariant and gauge invariant quantity. Correspondingly, reference frame dependence and gauge dependence are inherited to most of the elements of the usual formulation of quantum mechanics (including operators, states and events). If a frame dependent and gauge dependent formalism is called, in short, a relative formalism, then the aim of the paper is to establish an absolute, i.e., frame and gauge free, reformulation of quantum mechanics. To fulfil this aim, we develop absolute quantities and the corresponding equations instead of the wave function and the Schrödinger equation. The absolute quantities have a more direct physical interpretation than the wave function has, and the corresponding equations express explicitly the independent physical aspects of the system which are contained in the Schrödinger equation in a mixed and more hidden form. Based on the absolute quantities and equations, events, states and physical quantities are introduced also in an absolute way. The formalism makes it possible to obtain some sharper versions of the uncertainty relation and to extend the validity of Ehrenfest's theorem. The absolute formulation allows wide extensions of quantum mechanics. To give examples, we discuss two known nonlinear extensions and, in close detail, a dissipative system. An argument is provided that the absolute formalism may lead to an explanation of the Aharonov-Bohm effect purely in terms of the electromagnetic field strength tensor. At last, on special relativistic and curved spacetimes, absolute quantities and equations instead of the Klein-Gordon wave function and equation are given, and their nonrelativistic limit is derived.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.
de totstandkordng van dit proefschriftThis thesis consists of four parts.Notes and references are given at the end of each part.In part I we will give a review of the various notions of "representation" and "equivalence" that are used in theoretical physics and we will give a precise definition of the concept of "gauge equivalence".There we will meet the notion of gauge matrix.A method will be developed for obtaining the gauge matrices up to equivalence, as this is necessary in order to find the gauge equivalence classes of representations.In this method we will use the concept of superequivalence of group exponents, generalised here to group multipliers.Part I has a rather abstract mathematical character.Its results will be applied to physically interesting symmetry groups in the subsequent parts, where the notation and terminology of part I will be used freely.In part II we will consider the Poincaré as well as the Galilei symmetry group of uniform parallel electromagnetic fields.We will see that there are representations of these groups that are projectively equivalent, but that describe particles which arc, indeed, not "equivalent in every physical aspect".Specifically, particles in different Landau levels are described by projectively equivalent representations.However, those particles are not at all equivalent from the physical point of view; a transition between different Landau levels is not a trivial transformation of the mathematical description but it is a physical process that involves emission or absorption of energy.This is one of the most striking examples showing that, in general, projective equivalence of representations does not correspond to equivalence of physical systems.On the other hand, the different -13 -Landau levels are inequivalent with respect to the gauge equivalence concept.Another example which shows that gauge equivalence corresponds better than projective equivalence to "equivalence in every physical aspect" is provided by particles with different masses in uniform parallel electromagnetic fields (with Ε φ 0) in the Poincaré theory.Surprisingly, these particles are described by projectively equivalent representations; these representations are, however, gauge inequivalent.As a by-product we will find in part II what might be called a group theoretical derivation of the minimal coupling of a spinless particle in uniform parallel electromagnetic fields.Traditionally, the minimal coupling in quantum mechanics is "derived" by correspondence with classical mechanics.A derivation has also been given by a pseudo-invariance argument using instantaneous transformations.Although the latter argument may be the best that is available for arbitrary electromagnetic fields (that break space-time symmetry) it is in fact a "waste" of invariance in those cases where a special field is present that preserves a part of the symmetry.Indeed, in part II we obtain the minimal coupling in the Schrödinger and Klein-Gordon equations from the (exact rather than pseudo) symmetry group of the uniform parallel fields, without the detour via the free particle equations and without the "trick" of the substitution ІЭ ->• ІЭ -eA.This may be seen as a contribution to the process of remodelling traditional quantum mechanics by modern (group) theoretical tools.In part III other uniform electromagnetic fields are considered which are particular, in the sense that their connected symmetry groups are not isomorphic to the connected symmetry group of the uniform parallel fields.These are the uniform crossed fields under the Poincaré group and the uniform electric fields under the Galilei group.The equations -14of motion, including the minimal coupling, will be derived from the symmetry, analogous to part II.We will find a "Galilean subdynamics" for a Poincaré particle in uniform crossed fields.The exponents of the symmetry groups will give rise to an unexpected parameter in the equations of motion, which can be interpreted in terms of an anisotropic mass, corresponding to the anisotropy of the physical system due to the external field.In part IV we apply the notion of gauge equivalence of locally operating representations to some cosmologica!symmetry groups, operating on curved universes.For these groups the notion of superequivalence of group exponents plays a non-trivial role in the determination of a complete set of inequivalent gauge matrices.We will find that in cosmologica!models with one space dimension a free particle and a particle in an external field are described by projective representations of the symmetry group that are projectively equivalent but gauge inequivalent.This shows once more that the concept of gauge equivalence of locally operating representations is a more natural concept for the description of physical systems than that of the simple projective equivalence.As a by-product we will find that the minimal coupling is not necessarily the correct description of the interaction of a charged particle in an external field in a (relativistic) cosmologica!space-time.