Local operators in massive quantum field theories

Roberto Longo, Gerardo Morsella

We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massive Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the massive Legendre differential operator and the Green integral for the Helmholtz operator. The quadratic form of the massive modular Hamiltonian is expressed in terms of an integral of the energy density with parabolic distribution and of a Yukawa potential, that here appears intrinsically. We then get the formula for the local entropy of a Klein-Gordon wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free scalar QFT. Among other points, we have a criterion for a Bogoliubov automorphism to be locally weakly inner and we provide the passivity characterisation of the modular Hamiltonian within the standard subspace set up.

Гарий Владимирович Ефимов

Daniele Colosi, Robert Oeckl

Abstract The Newton-Wigner states and operator are widely accepted to provide an adequate notion of spatial localization of a particle in quantum field theory on a spacelike hypersurface. Replacing the spacelike with a timelike hypersurface, we construct one-particle states of massive Klein-Gordon theory that are localized on the hypersurface in the temporal as well as two spatial directions. This addresses the longstanding problem of a “time operator" in quantum theory. It is made possible by recent advances in quantization on timelike hypersurfaces and the introduction of evanescent particles. As a first application of time-localized states, we consider the time-of-arrival problem. Our results are in accordance with semiclassical expectations of causal propagation of massless and massive particles. As in the Newton-Wigner case, localization is not perfect, but apparent superluminal propagation is exponentially suppressed.

W. N. Mathews, Mark A. Esrick

Karan Tiwana, Edoardo Lauria, Antoine Tilloy

Abstract We propose a method to compute expectation values in 1+1-dimensional massive Quantum Field Theories (QFTs) with line defects using Relativistic Continuous Matrix Product State (RCMPS). Exploiting Euclidean invariance, we use a quantization scheme where (imaginary) time runs perpendicularly to the defect. With this choice, correlation functions of local operators in the presence of the defect can be computed as expectation values of extended operators in the no-defect vacuum, which can be approximated by a homogeneous RCMPS. We demonstrate the effectiveness of this machinery by computing correlation functions of local bulk and defect operators in ϕ 4 theory with a magnetic line defect, in perturbative, strong coupling, critical, and symmetry-broken regimes.

Vladimir Naroditsky