Hendrik Grundling

Obstruction results in quantization theory

SummaryQuantization is not a straightforward proposition, as demonstrated by Groenewolds and Van Hoves discovery, exactly fifty years ago, of an “obstruction” to quantization. Their “no-go theorems” assert that it is in principle impossible to consistently quantize every classical observable on the phase spaceR2n in a physically meaningful way. A similar obstruction was recently found forS2, buttressing the common belief that no-go theoremss should hold in some generality. Surprisingly, this is not so—it has also just been proven that there is no obstruction to quantizing a torus.In this paper we take first steps towards delineating the circumstances under which such obstructions will appear and understanding the mechanisms which produce them. Our objectives are to conjecture a generalized Groenewold-Van Hove theorem and to determine the maximal subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of classical systems and their representations. To these ends we include an exposition of both prequantization (in an extended sense) and quantization theory—formulated in terms of “basic sets of observables”—and review in detail the known results forR2n,S2, andT2. Our discussion is independent of any particular method of quantization; we concentrate on the structural aspects of quantization theory which are common to all Hilbert space-based quantization techniques.

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.

A Groenewold-Van Hove Theorem for ²

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}.

An obstruction to quantizing compact symplectic manifolds

We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.

A note on regular states and supplementary conditions

We show that linear Hermitian supplementary conditions can never be imposed in a representation associated with a regular state on the C*-algebra of the CCRs. Nevertheless, there is a well-defined method for imposing the constraints in an abstract C*-framework, which yields as its final physical algebra a CCR C*-algebra, on which one can again require its physical states to be regular. These states derive from states on the original C*-algebra which are ‘regular up to nonphysical quantities’.

Systems with Outer Constraints. Gupta-Bleuler Electromagnetism as an Algebraic Field Theory

AbstractSince there are some important systems which have constraints not contained in their field algebras, we develop here in aC*-context the algebraic structures of these. The constraints are defined as a groupG acting as outer automorphisms on the field algebra ℱ, α:G ↦ Aut ℱ, αG ⊄ Inn ℱ, and we find that the selection ofG-invariant states on ℱ is the same as the selection of states ω onM(G

On Quantizing T ∗ S 1

M(G\mathop \times \limits_\alpha F)

Algebraic structures of degenerate systems and the indefinite metric

ℱ) by ω(Ug)=1∨g∈G, whereUg ∈M (G