Dirac Quantum Fields on a Manifold

Abstract

Abstract. On globally hyperbolic Lorentzian manifolds we construct field operators which satisfy the Dirac equation and have a causal anticommutator. Ambiguities in the construction are removed by formulating the theory in terms of C* algebras of local observables. A generalized form of the Haag-Kastler axioms is verified.Introduction. A Lorentzian manifold is a four dimensional manifold TV/ together with a pseudo-Riemannian metric g of signature ( + , -, -, -). Such manifolds are widely used as models for space-time. In particular in the absence of boundaries and gravitational fields one uses M = R4 and g = tj = Minkowski metric which has components {r)ab} = diag(l, -1,-1,-1).We are interested in formulating quantum field theories on general Lorentzian manifolds. This means giving up many ideas familiar from the usual Minkowski space treatments. Thus there are no Poincaré transformations on M, no vacuum, no particle states, and so forth. All one is left with are the field equations. These are to be solved taking as data a representation of the CCR or CAR over a space-like hypersurface (CCR = canonical commutation relations, CAR = canonical anticommutation relations). For linear equations this quantum problem can be reduced to a corresponding classical problem for which solutions can be constructed in favorable cases. In the nonlinear case very singular formal solutions can be exhibited which one might hope to make rigorous.The procedure sketched above is fraught with ambiguities. What hypersurface should one take? what representation of the CCR/CAR? and so forth. In general there is no natural choice and one should show that all choices lead to the same theory. To do this in a fundamental way it seems necessary to abandon the field

The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes

“…in ref. [4]. The homogeneous and isotropic spacetimes in (N + 1) dimensions are of the form M κ = R × Σ κ , the spatial section Σ κ being the N -dimensional sphere S N for κ = +1, the Euclidean space R N for κ = 0 and the (real) hyperbolic space H N for κ = −1.…”

“…A bounded isotropic operator B on K t corresponds to an essentially bounded function b ks . The Dirac operator on a globally hyperbolic spacetime (such as RW-spacetimes) has unique retared and advanced fundamental solutions / S R and / S A , see [4], satisfying…”

“…The Dirac field on globally hyperbolic manifolds can be quantised in a straightforward manner. For convenience, we review the basic steps here, details can be found in [4], which we follow closely. As above, let K t = L 2 (Σ(t), DM ) and K ′ t its topological dual, identified with L 2 (Σ(t), D * M ) (with the inner product defined in a similar way as in (8)).…”

“…One has (see [4]), / S ↾ Σ × Σ = i/ n1 1 where 1 1 means the identity on the Cauchy surface. From [15,Thm.…”

The Reeh–Schlieder Property for Quantum Fields on Stationary Spacetimes

“…[Wal] and the references therein). The problem reduces to finding appropriate representations of the field algebra, which can be constructed straightforwardly on manifolds (see [Dim80,Dim82]). This is automatically obtained by specifying a vacuum state over the algebra.…”