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arXiv:2405.07888 · Communications in Mathematical Physics

The Fermionic Massless Modular Hamiltonian

Francesca La Piana, Gerardo Morsella

Published Mar 5, 2025• 1 citation
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Abstract

We provide an explicit expression for the modular hamiltonian of the von Neumann algebras associated to the unit double cone for the (fermionic) quantum field theories of the 2-component Weyl (helicity 1/2) field, and of the 4-component massless Dirac and Majorana fields. To this end, we represent the one particle spaces of these theories in terms of solutions of the corresponding wave equations, and obtain the action of the modular group on them. As an application, we compute the relative entropy between the vacuum of the massless Majorana field and one particle states associated to waves with Cauchy data localized in the spatial unit ball.

DOI
10.1007/s00220-025-05253-1

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The Tomita-Takesaki modular operator for local algebras plays an important role in quantum field theory, and more recently in the study of relative entropy. However, the explicit expression of this operator, except for the case of wedges, is difficult to describe mathematically. We have obtained instead numerical results for the form of the modular Hamiltonian for a double cone in a massive scalar free field in (1+1)- and (3+1)-dimensional Minkowski space, which shows how it differs from the wedge case, in particular regarding the dependence of the modular Hamiltonian on the mass of the field.
We investigate Bell-CHSH correlations for a chiral massless Majorana field localized in a double cone. The conformal modular flow of the associated light-ray interval is converted into translations by a rapidity-like coordinate, and the one-particle scalar product is derived in modular momentum space. Particular attention is paid to the fact that this scalar product carries a Fermi-Dirac weight. Requiring the modular conjugation to be antiunitary with respect to that weighted scalar product fixes the Fourier-space realization of the modular objects. The resulting Tomita-Takesaki operator and its adjoint generate Alice and twisted-dual Bob directions for which all four crossed CAR orthogonality relations follow automatically from modular localization. The Bell problem then reduces to a simple one-function functional. For a Gaussian modular-momentum profile hσ(k)=exp⁡[−k2/(2σ2)]h_σ(k)=\exp[-k^2/(2σ^2)]hσ​(k)=exp[−k2/(2σ2)] the Bell parameter is larger than the classical bound for a broad range of widths and approaches the Tsirelson value analytically, lim⁡σ→0+B(σ)=22 \lim_{σ\to 0^+}{\it B}(σ)=2\sqrt2 limσ→0+​B(σ)=22​ Thus near-maximal Bell violation is obtained from an explicit sequence of smooth rapidly decreasing profiles concentrated around zero modular momentum, corresponding to the modular spectral value λ=1λ=1λ=1.
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We study the relative entropy, in the sense of Araki, for the representation of a self-dual CAR algebra ASDC(H,Γ)\mathfrak{A}_{SDC}(\mathcal{H},Γ)ASDC​(H,Γ). We notice, for a specific choice of f∈Hf \in \mathcal{H}f∈H, that the associated element in ASDC(H,Γ)\mathfrak{A}_{SDC}(\mathcal{H},Γ)ASDC​(H,Γ) is unitary. As a consequence, we explicitly compute the relative entropy between a quasifree state over ASDC(H,Γ)\mathfrak{A}_{SDC}(\mathcal{H},Γ)ASDC​(H,Γ) and an excitation of it with respect to the abovely mentioned unitary element. The generality of the approach, allows us to consider H\mathcal{H}H as the Hilbert space of solutions of the classical Dirac equation over globally hyperbolic spacetimes, making our result, a computation of relative entropy for a Fermionic Quantum Field Theory. Our result extends those of Longo and Casini et al. for the relative entropy between a quasifree state and a coherent excitation for a free Scalar Quantum Field Theory, to the case of fermions. As a first application, we computed such a relative entropy for a Majorana field on an ultrastatic spacetime.