C. A. Hurst

Automorphisms of the canonical anticommutation relations and index theory

Let H be a complex Hilbert space, P+ an orthogonal projection on H, and P− the complementary projection. If G is any symmetrically normed ideal in the ring of bounded operators on H, then we consider the group of unitary operators on H such that P+ UP−and P− U P+ lie in G. When G is the Hilbert-Schmidt class, these unitaries define automorphisms of the C∗-algebra b of the canonical anticommutation relations over H which are implementable in the representation of b determined by P−. We investigate the structure of the group U, proving in particular that it has infinitely many connected components, Uk, labelled by the Fredholm index of P+ U P+. The connected component of the identity, U0, is generated by unitaries of the form exp(iA), with A self-adjoint and P+ A P− in G. Finally we consider an application of these results to two dimensional field theory, showing in particular that the charge and chiral charge quantum numbers arise as the Fredholm indices of P± UP± for certain unitary U on L2(R, C2)

Algebraic Quantization of Systems with a Gauge Degeneracy

Systems with a gauge degeneracy are characterized either by supplementary conditions, or by a set of generators of gauge transformations, or by a set of constraints deriving from Diracs canonical constraint method. These constraints can be expressed either as conditions on the field algebra ℱ, or on the states on ℱ. In aC*-algebra framework, we show that the state conditions give rise to a factor algebra of a subalgebra of the field algebra ℱ. This factor algebra, ℛ, is free of state conditions. In this formulation we show also that the algebraic conditions can be treated in the same way as the state conditions. The connection between states on ℱ and states on ℛ is investigated further within this framework, as is also the set of transformations which are compatible with the set of constraints. It is also shown that not every set of constraints can give rise to a nontrivial system. Finally as an example, the abstract theory is applied to the electromagnetic field, and this treatment can be generalized to all systems of bosons with linear constraints. The question of dynamics is not discussed.

The Boson Calculus for the Orthogonal and Symplectic Groups

The boson calculus as used for U(n) cannot be applied directly to O(n) and Sp(n). Modifications are made to boson operators, in order to obtain new operators which enable explicit states of irreducible representations of O(n) and Sp(n) to be constructed. Calculations for O(3) and O(4) show how these operators permit a greater simplicity, and reveal more fully the group structure than has previously been the case.

A note on the boson-fermion correspondence and infinite-dimensional groups

AbstractWe show how to construct irreducible projective representations of the infinite dimensional Lie group Map (S1,

Are wave functions uniquely determined by their position and momentum distributions

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New Approach to the Ising Problem

), by embedding it into the group of Bogoliubov automorphisms of the CAR. Using techniques of G. Segal for extending certain representations of Map (S1, SU(2)) we show that our representations extend to give representations of a certain infinite dimensional superalgebra. We relate our work to the well known boson-fermion correspondence which exists in 1+1 dimensions.

The geometry of state space

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S2 which is irreducible on the su(2) subalgebra generated by the components {S1, S2, S3} of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra P consisting of polynomials in {S1, S2, S3}. Furthermore, we show that the maximal Poisson subalgebra of P containing {1, S1, S2, S3} that can be so quantized is just that generated by {1, S1, S2, S3}.

A note on regular states and supplementary conditions

Abstract The problem of determining a square integrable function from both itsmodulus and the modulus of its Fourier transform is studied. It is shownthat for a large class of real functions the function is uniquely determinedfrom this data. We also construct fundamental subsets of functions that arenot uniquely determined. In quantum mechanical language, bound states areuniquely determined by their position and momentum distributions but, ingeneral, scattering states are not. 1. Introduction The Pauli problemIn a footnote to hi Handbuchs der Physik article on the general principles of wavemechanics [4], Pauli raised the question of whether the ijs(q) wav wae functios nuniquely determined by the probability densities | ^{q)f and |^(p)| 2 in configurationand momentum space. In this article we show that there are fundamental subsetswhere the answer is no and there is a large and interesting class of functions (q) eL 2 (R) for which the answer is yes.The question was motivated by the fact that if the wave function was uniquelydetermined by its position and momentum probability densities then we have, atleast in principle, a method of determining the wave function from experiment.Nevertheless, we should point out that the mathematical problem is of interest inother contexts, such as control theory and crystallography. We will commentfurther on the physical meaning o thif s problem and our results in the final section.The mathematical problem is that of determining the phase of a complex-valuedsquare integrable function given both the amplitude of the function and the