Systems with outer constraints. Gupta-Bleuler electromagnetism as an algebraic field theory

Since 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 , :G Aut , _G Inn , and we find that the selection ofG-invariant states on is the same as the selection of states onM(G ) by (U _g)=1gG, whereU _g M (G )/ are the canonical elements implementing _g . These states are taken as the physical states, and this specifies the resulting algebraic structure of the physics inM(G ), and in particular the maximal constraint free physical algebra . A nontriviality condition is given for to exist, and we extend the notion of a crossed product to deal with a situation whereG is not locally compact. This is necessary to deal with the field theoretical aspect of the constraints. Next theC*-algebra of the CCR is employed to define the abstract algebraic structure of Gupta-Bleuler electromagnetism in the present framework. The indefinite inner product representation structure is obtained, and this puts Gupta-Bleuler electromagnetism on a rigorous footing. Finally, as a bonus, we find that the algebraic structures just set up, provide a blueprint for constructive quadratic algebraic field theory.     

The resolvent algebra: A new approach to canonical quantum systems

The standard C^∗-algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the formulation of physically interesting dynamical laws nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian. Here a novel C^∗-algebra of the canonical commutation relations is presented which does not suffer from such problems. It is based on the resolvents of the canonical operators and their algebraic relations. The resulting C^∗-algebra, the resolvent algebra, is shown to have many desirable analytic properties and the regularity structure of its representations is surprisingly simple. Moreover, the resolvent algebra is a convenient framework for applications to interacting and to constrained quantum systems, as we demonstrate by several examples.     

The quantum theory of second class constraints: Kinematics

The problem of second class quantum constraints is here set up in the context ofC*-algebras, utilizing the connection with state conditions as given by the heuristic quantization rules. That is, a constraint set is said to be first class if all its members can satisfy the same state condition, and second class otherwise. Several heuristic models are examined, and they all agree with this definition. Given then a second class constraint set, we separate out its first class part as all those constraints which are compatible with the others, and we propose an algebraic construction for imposition of the constraints. This construction reduces to the normal one when the constraints are first class. Moreover, the physical automorphisms (assumed as conserving the constraints) will also respect this construction. The final physical algebra obtained is free of constraints, gauge invariant, unital, and with the right choice, simple. ThisC*-algebra also contains a factor algebra of the usual observables, i.e. the commutator algebra of the constraints. The general theory is applied to two examples—the elimination of a canonical pair from a boson field theory, as in the two dimensional anomalous chiral Schwinger model of Rajaraman [14], and the imposition of quadratic second class constraints on a linear boson field theory.     

Group actions on C*-algebras, 3-cocycles and quantum field theory

We study group extensions , where acts on a C^*-algebraA. Given a twisted covariant representation ,V of the pairA, we construct 3-cocycles on with values in the centre of the group generated byV(). These 3-cocycles are obstructions to the existence of an extension of byV() which acts onA compatibly with . The main theorems of the paper introduce a subsidiary invariant which classifies actions of onV() and in terms of which a necessary and sufficient condition for the the cohomology class of the 3-cocycle to be non-trivial may be formulated. Examples are provided which show how non-trivial 3-cocycles may be realised. The framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories. We show how to interpret our results in that setting in two ways, one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups. In order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a Lie algebra version of the invariant and the 3-cocycle.     

Generalising group algebras

The main result in the above-mentioned paper, namely the existenceand uniqueness theorem for host algebras, is wrong. Received July 26, 2007; revised August 28, 2007;     

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.     

Localization via Automorphisms of the CARs: Local Gauge Invariance

The classical matter fields are sections of a vector bundle E with base manifold M, and the space L ²(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of C_b^￥(M){C_b^\infty(M)} on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum context, we show that this module action defines an automorphism group on the algebra of the canonical anticommutation relations, CAR(L ²(E)), with which we can perform the analogous localization. That is, the net structure of the CAR(L ²(E)) w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction maps, and as a corollary, we prove a well–known “folk theorem,” that the CAR(L ²(E)) contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E.     

Algebraic Supersymmetry: A Case Study

The treatment of supersymmetry is known to cause difficulties in the C^*–algebraic framework of relativistic quantum field theory; several no–go theorems indicate that super–derivations and super–KMS functionals must be quite singular objects in a C^*–algebraic setting. In order to clarify the situation, a simple supersymmetric chiral field theory of a free Fermi and Bose field defined on is analyzed. It is shown that a meaningful C^*–version of this model can be based on the tensor product of a CAR–algebra and a novel version of a CCR–algebra, the “resolvent algebra”. The elements of this resolvent algebra serve as mollifiers for the super–derivation. Within this model, unbounded (yet locally bounded) graded KMS–functionals are constructed and proven to be supersymmetric. From these KMS–functionals, Chern characters are obtained by generalizing formulae of Kastler and of Jaffe, Lesniewski and Osterwalder. The characters are used to define cyclic cocycles in the sense of Connes’ noncommutative geometry which are “locally entire”. Dedicated to Daniel Kastler on the occasion of his 80th birthday     

Generalising Group Algebras

We generalise group algebras to other algebraic objects withbounded Hilbert space representation theory; the generalisedgroup algebras are called ‘host’ algebras. The mainproperty of a host algebra is that its representation theoryshould be isomorphic (in the sense of the Gelfand–Raikovtheorem) to a specified subset of representations of the algebraicobject. Here we obtain both existence and uniqueness theoremsfor host algebras as well as general structure theorems forhost algebras. Abstractly, this solves the question of whena set of Hilbert space representations is isomorphic to therepresentation theory of a C*-algebra. To make contact withharmonic analysis, we consider general convolution algebrasassociated to representation sets, and consider conditions fora convolution algebra to be a host algebra.     

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.