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.
Orthogonal invariance of the CAR on Fock spacesIn this chapter we continue our study of the orthogonal invariance of the CAR.This invariance was already investigated in Chap.14.However, whereas in Chap.14 we used the representation-independent framework of CAR algebras, in this chapter we will consider Fock CAR representations in any dimensions.Therefore, to some extent, this chapter can be viewed as a continuation of Chap.13 about Fock CAR representations.Note also that this chapter is parallel to Chap.11 about the symplectic invariance of the CCR on a Fock space.
2026年8月,Howl、Cooper与Hackermüller在 Physical Review A 发表了一项突破性方案:利用两个平行的原子干涉仪,通过测量输出端原子数统计的互相关性来检测引力诱导纠缠,从而在实验室桌面尺度检验引力的量子本质。该方案的核心突破在于:引力诱导纠缠可以在不使用宏观叠加态的情况下产生,仅需相干的、近经典原子态即可实现。本文论证:引力诱导纠缠实验的核心机制——两个量子系统通过引力场中介产生纠缠——与互含范畴框架中投影条件f†∘f = P_X在量子引力尺度存在结构对应。引力场作为"关系网络的中介者",其是否能够产生纠缠,取决于它是否满足投影条件的非平凡性。如果引力是经典的(投影算子退化为恒等),则无法产生纠缠;如果引力是量子的(投影算子非平凡),则纠缠必然产生。本文进一步指出:引力诱导纠缠的相位与两个干涉仪间距的平方成反比、与质量的平方成正比——这一标度关系为反推投影算子P_X在引力尺度的具体形式提供了实验路径。本文提出下一步实验目标与具体方法,以进一步约束投影条件的参数空间。 关键词: 引力诱导纠缠;互含范畴;投影条件;原子干涉仪;量子引力