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.
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.
The relative entropy between two states is a key concept in quantum information theory and quantum field theory. In the setting of quantum field theory, its computation requires the handling of relative modular Hamiltonians, which are typically very difficult to compute explicitly. In this paper, we exploit locality properties of general algebraic QFTs to estimate relative modular Hamiltonians between two states,
ω and
ω~, and hence also their relative entropy, in terms of the modular Hamiltonian of a reference state
ω^, which might be better understood. For suitable pairs of states we can estimate the relative modular Hamiltonian for the algebra of a region
V2 from above, resp. from below, in terms of the modular Hamiltonian of
ω^ on a larger region
V3, resp. a smaller region
V1. Pairs of states and choices of regions which are susceptible to our scheme are related to the presence of superluminal signalling in the sense of Sorkin's paradox. If
ω=ω^, then there exists a unitary that maps
ω to
ω~ on
V3 and that does not allow superluminal signalling from the spacelike complement
V3′ to
V2, resp. from
V1 to
V2′, if our upper, resp. lower, bound applies. To investigate the strength of our estimates we consider coherent states for CCR systems, focussing particularly on free scalar fields. Our estimates apply even if the relative modular Hamiltonian cannot be computed exactly. For sufficiently regular excitations we recover an exact result by squeezing. Our method thus yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian cannot be computed exactly. For massless fields we establish the analogous result also for double cone regions. These results indicate that our estimates do not lose too much information.
Abstract We consider the generalization of the Araki–Uhlmann formula for relative entropy to Petz–Rényi relative entropy. We compute this entropy for a free scalar field in the Minkowski wedge between the vacuum and a coherent state, as well as for the free chiral current in a thermal state. In contrast to the relative entropy which in these cases only depends on the symplectic form and thus reduces to the classical entropy of a wave packet, the Petz–Rényi relative entropy also depends on the symmetric part of the two-point function and is thus genuinely quantum. We also consider the relation with standard subspaces, where we define the Rényi entropy of a vector and show that it admits an upper bound given by the entropy of the vector.
Building on the established one-particle formula for the Araki relative entropy of coherent states, we study how its value acquires a nontrivial dependence on the mass of the scalar field. For a localized vector
h belonging to the standard subspace
Hm of the one-particle Hilbert space, the known quadratic-form expression is:
SHm(h)=−⟨h,logδHmh⟩. Our contribution is to construct explicitly a mass-indexed family of vectors of the wedge standard subspace on which this expression is evaluated. The mass-shell map
hm=Emf organizes four structural conditions on rapidity representatives---on-shell dependence, a controlled massless boundary value, decay for large real rapidity, and Bisognano--Wichmann strip analyticity---and we exhibit an entire rapidity wave function, built from a doubled light-cone phase, two sinc factors, and a Gaussian pair, that satisfies them together with the sharp localization criterion: Hardy-type
L2 control throughout the Bisognano--Wichmann strip and the exact Tomita boundary relation. The family therefore belongs to $H_m(\W_R)$ for every
m>0, and its Araki relative entropy is finite and strictly positive, with an exact spectral representation that makes positivity manifest. The entropy is strongly suppressed at large mass, attains a maximum at intermediate mass in
1+1 dimension, and converges to a finite value along the modular flow as
m→0+. The construction extends fiberwise to
1+d dimensions through the transverse mass.
We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral
U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.