A group theoretical derivation of the minimal coupling in elementary quantum mechanics

Henk Hoogland

The equation of motion of a spinless quantum particle in an external uniform parallel electromagnetic field is derived from the symmetry of the field by group theoretical techniques. The minimal coupling is obtained directly, without a detour via the free particle equation. This derivation, which is valid in a Galilean as well as in a Poincare framework, is a contribution to the recasting process of quantum mechanics in the spirit of a paper by Levy-Leblond (1975).

I.G. KAPLAN

Henk Hoogland

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.

James Lindesay

Paul O’Hara

With the advent of general relativity and quantum mechanics, physics has in many cases shifted the emphasis from dynamics to symmetry principles associated with different types of mathematical groups. We speak of the group structure of crystals, the group invariants of the equations of motion, the group classification of elementary particles, and also of the group homeomorphisms of fractals. The objective of this article is to show how the group theoretical structure of entanglement can be used to classify atomic and molecular structure. The article begins by establishing a working definition of entanglement and then using it to derive the Pauli-exclusion principle by showing that Fermi-Dirac statistics can be considered as an entangled state invariant under the action of the SL(n, C)With the advent of general relativity and quantum mechanics, physics has in many cases shifted the emphasis from dynamics to symmetry principles associated with different types of mathematical groups. We speak of the group structure of crystals, the group invariants of the equations of motion, the group classification of elementary particles, and also of the group homeomorphisms of fractals. The objective of this article is to show how the group theoretical structure of entanglement can be used to classify atomic and molecular structure. The article begins by establishing a working definition of entanglement and then using it to derive the Pauli-exclusion principle by showing that Fermi-Dirac statistics can be considered as an entangled state invariant under the action of the $SL(n, C)$ group. Based on this, a group classification scheme of entangled states is proposed to distinguish atomic and molecular structures associated with chemical bonds. group. Based on this, a group classification scheme of entangled states is proposed to distinguish atomic and molecular structures associated with chemical bonds.