Abstract Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.
Abstract Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.
Sorkin (Expressing entropy globally in terms of (4D) field-correlations arXiv: 1205.2953 ) defines an entropy for a Gaussian scalar field ϕ in an arbitrary region of either a continuous spacetime or a causal set, given only the correlator within the region. The definition is global and independent of any choice of spacelike hypersurface. As a first application, we compute numerically the entanglement entropy in two cases where the asymptotic form is known or suspected from conformal field theory, finding excellent agreement when the required ultraviolet cutoff is implemented as a truncation on spacetime mode sums. We also show how the symmetry of entanglement entropy reflects the fact that RS and SR share the same eigenvalues, with R and S being arbitrary matrices.
Abstract In this work we provide a method to study the entanglement entropy for non-Gaussian states that minimize the energy functional of interacting quantum field theories at arbitrary coupling. To this end, we build a class of non-Gaussian variational trial wavefunctionals with the help of exact nonlinear canonical transformations. The calculability bonanza shown by these variational ansatze allows us to compute the entanglement entropy using the prescription for the ground state of free theories. In free theories, the entanglement entropy is determined by the two-point correlation functions. For the interacting case, we show that these two-point correlators can be replaced by their nonperturbatively corrected counterparts. Upon giving some general formulae for general interacting models we calculate the entanglement entropy of half space and compact regions for the ϕ4 scalar field theory in 2D. Finally, we analyze the rôle played by higher order correlators in our results and show that strong subadditivity is satisfied.
Abstract Entanglement entropy is now widely accepted as having deep connections with quantum gravity. It is therefore desirable to understand it in the context of causal sets, especially since they provide in a natural manner the UV cutoff needed to render entanglement entropy finite. Formulating a notion of entanglement entropy in a causal set is not straightforward because the type of canonical hypersurface-data on which its definition typically relies is not available. Instead, we appeal to the more global expression given in Sorkin (2012 (arXiv:1205.2953)) which, for a Gaussian scalar field, expresses the entropy of a spacetime region in terms of the field’s correlation function within that region (its ‘Wightman function’ W ( x , x ′ ) ). Carrying this formula over to the causal set, one obtains an entropy which is both finite and of a Lorentz invariant nature. We evaluate this global entropy-expression numerically for certain regions (primarily order-intervals or ‘causal diamonds’) within causal sets of 1 + 1 dimensions. For the causal-set counterpart of the entanglement entropy, we obtain, in the first instance, a result that follows a (spacetime) volume law instead of the expected (spatial) area law. We find, however, that one obtains an area law if one truncates the commutator function (‘Pauli–Jordan operator’) and the Wightman function by projecting out the eigenmodes of the Pauli–Jordan operator whose eigenvalues are too close to zero according to a geometrical criterion which we describe more fully below. In connection with these results and the questions they raise, we also study the ‘entropy of coarse-graining’ generated by thinning out the causal set, and we compare it with what one obtains by similarly thinning out a chain of harmonic oscillators, finding the same, ‘universal’ behaviour in both cases.
Entanglement entropy (EE) is a pivotal quantity in quantum field theory (QFT) that captures the nonlocal correlations between subsystems, offering deep insights into the quantum structure of spacetime, field dynamics, and phase transitions. This work derives EE in QFT using the covariance matrix formalism, establishing its foundations through the von Neumann entropy of reduced density matrices. We analytically compute EE for coupled harmonic oscillators as a prototypical model, extending the results to quantum fields via perturbative expansions. In the context of Maxwell QFT, we analyze entropy corrections induced by external perturbations, revealing their fractal dynamics through the emergence of Julia sets. The study further incorporates noncommutative geometry, where the deformation of spacetime modifies the covariance matrix, field strength tensors, and modular indices. Numerical simulations validate the derived scaling laws, entropy corrections, and noncommutative effects. These findings provide a mathematically rigorous framework to explore the interplay between EE, field theory, and geometric structures in high-energy physics.
Motivated by the necessity to UV-regularise entanglement entropy, we present a spectral method for calculating the entropy of quasifree states, for both bosonic and fermionic field theories. This construction is defined in spacetime rather than on a hypersurface, enabling the covariant regularisation of entropies, and its calculation in generic spacetime regions. We derive these formulae, which have previously appeared in the literature, in a new manner and highlight certain aspects of them, such as their connection to the density matrix and its eigenvalues. The spacetime nature of the formulation makes it particularly apt in the context of semiclassical and quantum gravity and in connection to black hole entropy. Another useful property of the formulation is its application to settings where no notion of a Cauchy surface exists, such as in the causal set theory approach to quantum gravity. We show example applications of the formulae which demonstrate their ability to reproduce known results. We also show a calculation in a causal set in 1+1 dimensions which makes use of several of the unique and useful features of the formalism. In this last example, we obtain a novel result of a slightly modified entanglement entropy scaling coefficient, giving a possible signature of spacetime discreteness.
For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv:1509.00113 .