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Journal of the London Mathematical Society

Pure Symplectic Spinors in the Fock Representation

Paul Lee Robinson

Published Oct 1, 1992• 0 citations
Publisher

Abstract

We describe explicitly the space of smooth vectors and its antidual space of symplectic spinors for the Fock representation of a separable complex Hilbert space. Further, we show that a strongly positive polarization annihilates a complex line of pure symplectic spinors if and only if it satisfies a Hilbert-Schmidt condition. Throughout, we use the complex-wave model of the Fock representation due to Segal.

DOI
10.1112/jlms/s2-46.2.250

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We describe explicitly the space of smooth vectors and its antidual space of symplectic spinors for the Fock representation of a separable complex Hilbert space. Further, we show that a strongly positive polarization annihilates a complex line of pure symplectic spinors if and only if it satisfies a Hilbert-Schmidt condition. Throughout, we use the complex-wave model of the Fock representation due to Segal.
The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.
CAR on Fock spacesThis chapter is devoted to the study of Fock representations of the canonical anticommutation relations, a basic tool of quantum many-body theory and quantum field theory.It is parallel to Chap. 9, where Fock CCR representations were studied.The basic framework is almost the same as in Chap.9. Throughout this chapter Z is a Hilbert space, called the one-particle space.We will consider the Fock CAR representation acting on the fermionic Fock space Γ a (Z).As in Sect.1.3, we introduce the space Y = Re(Z ⊕ Z) := {(z, z) : z ∈ Z}, which will serve as the dual phase space of our system.Recall that in the bosonic case we equipped Y with the structure of the Kähler space consisting of the anti-involution j, the Euclidean scalar product • and the symplectic form ω: j(z, z) := (iz, iz), (13.1) (z, z) • (w, w) := 2Re(z|w), (13.2) (z, z)•ω(w, w) := 2Im(z|w) = -(z, z) • j(w, w).(13.3)In our presentation of the fermionic case, we will need the Kähler anti-involution j.The symplectic form ω will not be used.Instead of the scalar product (13.2) we will use another scalar product,Again we will avoid identifing Z with Y. CY is identified with Z ⊕ Z by the mapY # , the space dual to Y, is canonically identified with Re(Z ⊕ Z) by using the scalar product (9.2), and CY # is identified with Z ⊕ Z. Fock CAR representationConsider the fermionic Fock space Γ a (Z).Recall that, for z ∈ Z, a * (z), resp.a(z) denote the corresponding creation, resp.annihilation operators defined in Sect.3.4.
A state σ on the Weyl algebra A(V, Ω) over a real symplectic vector space (V, Ω) is polarized when the vectors v for which the absolute value of σ on the corresponding Weyl generator δ_v is unity constitute a maximal Ω -integral additive subgroup of V ; such a state is necessarily pure, but has inseparable carrier space and is not regular. We determine fundamental properties of such states: in particular, we decide precisely when the GNS representations associated to a pair of polarized states are unitarily equivalent and decide precisely when a given symplectic automorphism is unitarily implemented in the GNS representation associated to a given polarized state.
We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physical sound meaning of some of the direct sum structures showing up in the context of the so-called black-hole/qubit correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self-conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of Spin(16,C)\text{Spin}(16,\mathbb{C})Spin(16,C) invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with eight modes. Here we find an algebraically independent generating set of the generalized stochastic local operations and classical communication invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group E8{E}_{8}E8​ is playing in these considerations.