The equivalence of representations of symmetry groups operating upon wavefunctions in configuration space is studied with regard to the (intuitive) notion of physical equivalence. A refinement of the usual projective equivalence relation is introduced, called gauge equivalence, for which the allowed unitary equivalence transformations are gauge transformations. For a Euclidean as well as for a Newton-Hooke symmetry group the gauge equivalence classes of unitary multiplier representations are determined. These examples support the assertion that equivalence from a physical viewpoint corresponds better to this new gauge equivalence concept than to the usual notion of projective equivalence.
The equivalence of representations of symmetry groups operating upon wavefunctions in configuration space is studied with regard to the (intuitive) notion of physical equivalence. A refinement of the usual projective equivalence relation is introduced, called gauge equivalence, for which the allowed unitary equivalence transformations are gauge transformations. For a Euclidean as well as for a Newton-Hooke symmetry group the gauge equivalence classes of unitary multiplier representations are determined. These examples support the assertion that equivalence from a physical viewpoint corresponds better to this new gauge equivalence concept than to the usual notion of projective equivalence.
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.
Symmetry in quantum mechanicsRoughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is.Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map u : H → H satisfying one and hence all of the following equivalent conditions:• u is invertible and preserves the inner product, i.e., uϕ, uψ = ϕ, ψ (ϕ, ψ ∈ H).The discussion of symmetries in quantum physics is based on the above idea, but the mathematically obvious choices need not be the physically relevant ones.Even in elementary quantum mechanics, where A = B(H), i.e., the C*-algebra of all bounded operators on some Hilbert space H, the concept of a symmetry is already diverse.The main structures whose symmetries we shall study in this chapter are:1.The normal pure state space P 1 (H), i.e., the set of one-dimensional projections on H, with transition probability τ : P 1 (H) × P 1 (H) → [0, 1] defined by (2.44).2. The normal state space D(H), i.e. the convex set of density operators ρ on H. The self-adjoint operators B(H)sa on H, seen as a Jordan algebra (see below).4. The effects E (H) = [0, 1] B(H) , seen as a convex partially ordered set (poset). 5.The projections P(H) on H, seen as an orthocomplemented lattice.6.The unital commutative C*-subalgebras C (B(H)) of B(H), seen as a poset.Each of these structures comes with its own notion of a symmetry, but the main point of this chapter will be to show these notions are equivalent, corresponding in all cases to either unitary or-surprisingly-anti-unitary operators, both merely defined up to a phase.The latter subtlety will open the world of projective unitary group representation to quantum mechanics (without which the existence of spin-1 2 particles such as electrons, and therewith also of ourselves, would be impossible).
There is a natural equivalence relation on representations of the states of a given quantum system in a Hilbert space, two representations being equivalent iff they are related by a unitary transformation. There are two equivalence classes, with members of opposite classes being related by a conjugate-unitary (anti-unitary) transformation. These two conjugacy classes are related in much the same way as are the two imaginary units of a complex field, and there is a priori no basis on which to prefer one over the other in any individual case. This is potentially problematic in that the choice of conjugacy class of a representation determines the sign of energy and other quantities defined as generators of continuous symmetries of the system in question, so that it would appear that principles like conservation of energy for a compound system may hold or fail depending on relative choices of conjugacy class of representations of its subsystems. We show that for any finite set of quantum systems there are exactly two ways of choosing conjugacy classes of representations consistent with the usual tensor-product construction for representing the compound system composed of these. Each is obtained from the other by reversing the conjugacy of all the representations at once. The relation of unitary equivalence for representations of a single system is therefore uniquely extendible to representations of all systems that can interact with it.
The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations. We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.
In classical theory, a physical state is an equivalence class under gauge transformations. Is the same true in quantum theory? The physical quantum states are the solutions of Dirac's quantum constraint equation. They cannot be constructed as equivalence classes under the ``simple'' gauge transformations generated by the Dirac constraints. However, we show here that they can be constructed as equivalence classes under suitably defined ``complete'' gauge transformations. The complete gauge transformations are generated by the action of the quantum constraints on arbitrary individual components of the state.