The Massless Modular Hamiltonian

We describe an analog model for quantum gravity effects in condensed matter physics. The situation discussed is that of phonons propagating in a fluid with a random velocity wave equation. We consider that there are random fluctuations in the reciprocal of the bulk modulus of the system and study free phonons in the presence of Gaussian colored noise with zero mean. We show that, in this model, after performing the random averages over the noise function a free conventional scalar quantum field theory describing free phonons becomes a self-interacting model.

In this manuscript, we discuss a remarkable phenomenon concerning nonlinear and nonintegrable field theories in ($3+1$) dimensions, living at finite density and possessing nontrivial topological charges and non-Abelian internal symmetries (both local and global). With suitable types of Ans\"atze, one can construct infinite-dimensional families of analytic solutions with nonvanishing topological charges (representing the baryonic number) labeled by both two integer numbers and by free scalar fields in ($1+1$) dimensions. These exact configurations represent ($3+1$)-dimensional topological solitons hosting ($1+1$)-dimensional chiral modes localized at the energy density peaks. First, we analyze the Yang-Mills-Higgs model, in which the fields depend on all the space-time coordinates (to keep alive the topological Chern-Simons charge), but in such a way to reduce the equations system to the field equations of two-dimensional free massless chiral scalar fields. Then, we move to the nonlinear sigma model, showing that a suitable Ansatz reduces the field equations to the one of a two-dimensional free massless scalar field. Then, we discuss the Skyrme model concluding that the inclusion of the Skyrme term gives rise to a chiral two-dimensional free massless scalar field (instead of a free massless field in two dimensions as in the nonlinear sigma model) describing analytically spatially modulated hadronic layers and tubes. The comparison of the present approach both with the instanton-dyon liquid approach and with lattice QCD is shortly outlined.

Using a recent proposal of circuit complexity in quantum field theories introduced by Jefferson and Myers, we compute the time evolution of the complexity following a smooth mass quench characterized by a time scale δt in a free scalar field theory. We show that the dynamics has two distinct phases, namely an early regime of approximately linear evolution followed by a saturation phase characterized by oscillations around a mean value. The behavior is similar to previous conjectures for the complexity growth in chaotic and holographic systems, although here we have found that the complexity may grow or decrease depending on whether the quench increases or decreases the mass, and also that the time scale for saturation of the complexity is of order δt (not parametrically larger).

We construct a consistent quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-times free of infrared divergences. We show that in such a quantum field theory (i) a continuous symmetry of (pseudo)scalar field translations is spontaneously broken, (ii) Goldstone bosons appear as quanta of a free massless (pseudo)scalar field and (iii) there is a non-vanishing spontaneous {magnetization}. In spite of the existence of a spontaneous {magnetization} the main inequality between vacuum expectation values of certain operators which have been used for the derivation of the Mermin–Wagner–Hohenberg theorem (C. Itzykson and J.-M. Drouffe, {Statistical field theory}, Vol. I, 1989, pp. 219–224) is fulfilled.

Conformal covariance, modular structure, and duality for local algebras in free massless quantum field theories

In this article, we review some illustrative results in the study of two related toy models for quantum gravity, namely cylindrical waves (which are cylindrically symmetric gravitational fields)and parametrized field theory (which is just free scalar field theory on a flat space–time in generally covariant disguise). In the former, we focus on the phenomenon of unexpected large quantum gravity effects in regions of weak classical gravitational fields and on an analysis of causality in a quantum geometry. In the latter, we focus on Dirac quantization, argue that this is related to the unitary implementability of free scalar field evolution along curved foliations of the flat space–time and review the relevant results for unitary implementability.

A free massive scalar field in inhomogeneous random media is investigated. The coefficients of the Klein-Gordon equation are taken to be random functions of the spatial coordinates. The case of an annealed-like disordered medium, modeled by centered stationary and Gaussian processes, is analyzed. After performing the averages over the random functions, we obtain the two-point causal Green's function of the model up to one-loop. The disordered scalar quantum field theory becomes qualitatively similar to a $\lambda\phi^{4}$ self-interacting theory with a frequency-dependent coupling.

Just as for non-relativistic quantum fields, the theory of free relativistic scalar quantum fields starts by taking as phase space an infinite dimensional space of solutions of an equation of motion. Quantization of this phase space proceeds by constructing field operators which provide a representation of the corresponding Heisenberg Lie algebra, using an infinite dimensional version of the Bargmann–Fock construction. In both cases, the equation of motion has a representation-theoretical significance: It is an eigenvalue equation for the Casimir operator of a group of space-time symmetries, picking out an irreducible representation of that group. In the non-relativistic case, the Laplacian \(\Delta \) was the Casimir operator, the symmetry group was the Euclidean group E(3), and one got an irreducible representation for fixed energy.

We expand on the investigation of the universal scaling properties in the\nearly time behaviour of fast but smooth quantum quenches in a general\n$d$-dimensional conformal field theory deformed by a relevant operator of\ndimension $\\Delta$ with a time-dependent coupling. The quench consists of\nchanging the coupling from an initial constant value $\\lambda_1$ by an amount\nof the order of $\\delta \\lambda$ to some other final value $\\lambda_2$, over a\ntime scale $\\delta t$. In the fast quench limit where $\\delta t$ is smaller\nthan all other length scales in the problem, $ \\delta t \\ll\n\\lambda_1^{1/(\\Delta-d)}, \\lambda_2^{1/(\\Delta-d)}, \\delta\n\\lambda^{1/(\\Delta-d)}$, the energy (density) injected into the system scales\nas $\\delta{\\cal E} \\sim (\\delta \\lambda)^2 (\\delta t)^{d-2\\Delta}$. Similarly,\nthe change in the expectation value of the quenched operator at times earlier\nthan the endpoint of the quench scales as $<{\\cal O}_\\Delta> \\sim \\delta\n\\lambda (\\delta t)^{d-2\\Delta}$, with further logarithmic enhancements in\ncertain cases. While these results were first found in holographic studies, we\nrecently demonstrated that precisely the same scaling appears in fast mass\nquenches of free scalar and free fermionic field theories. As we describe in\ndetail, the universal scaling refers to renormalized quantities, in which the\nUV divergent pieces are consistently renormalized away by subtracting\ncounterterms derived with an adiabatic expansion. We argue that this scaling\nlaw is a property of the conformal field theory at the UV fixed point, valid\nfor arbitrary relevant deformations and insensitive to the details of the\nquench protocol. Our results highlight the difference between smooth fast\nquenches and instantaneous quenches where the Hamiltonian abruptly changes at\nsome time.\n

We compute the Krylov Complexity of a light operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}L in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H is below or above the critical dimension hH = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS3 geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states.

In the companion paper Lang et al (2017 arXiv:1711.05685) we motivated a renormalisation flow on Osterwalder–Schrader data (OS-data) consisting of (1.) a Hilbert space, (2.) a cyclic vacuum and (3.) a Hamiltonian annihilating that vacuum. As the name suggests, the motivation was via the OS reconstruction theorem which allows to reconstruct the OS data from an OS measure satisfying (a subset of) the OS axioms, in particular reflection positivity. The guiding principle was to map the usual Wilsonian path integral renormalisation flow onto a flow of the corresponding OS data.We showed that this induced flow on the OS data has an unwanted feature which disqualifies the associated coarse grained Hamiltonians from being the projections of a continuum Hamiltonian onto vectors in the coarse grained Hilbert space. This motivated the definition of a direct Hamiltonian renormalisation flow which follows the guiding principle but does not suffer from the afore mentioned caveat.In order to test our proposal, we apply it to the only known completely solvable model, namely the case of free scalar quantum fields. In this paper we focus on the Klein Gordon field in two spacetime dimensions and illustrate the difference between the path integral induced and direct Hamiltonian flow. Generalisations to more general models in higher dimensions will be discussed in our companion papers.

We define a distinguished "ground state" or "vacuum" for a free scalar quantum field in a globally hyperbolic region of an arbitrarily curved spacetime. Our prescription is motivated by the recent construction of a quantum field theory on a background causal set using only knowledge of the retarded Green's function. We generalize that construction to continuum spacetimes and find that it yields a distinguished vacuum or ground state for a non-interacting, massive or massless scalar field. This state is defined for all compact regions and for many noncompact ones. In a static spacetime we find that our vacuum coincides with the usual ground state. We determine it also for a radiation-filled, spatially homogeneous and isotropic cosmos, and show that the super-horizon correlations are approximately the same as those of a thermal state. Finally, we illustrate the inherent non-locality of our prescription with the example of a spacetime which sandwiches a region with curvature in-between flat initial and final regions.

Covariant Lagrangian formulation for free bosonic massless fields of arbitrary mixed-symmetry type in (A)dS(d) space-time is presented. The analysis is based on the frame-like formulation of higher-spin field dynamics [1] with higher-spin fields described as p-forms taking values in appropriate modules of the (A)dS(d). The problem of finding free field action is reduced to the analysis of an appropriate differential complex, with the derivation Q associated with the variation of the action. The constructed action exhibits additional gauge symmetries in the flat limit in agreement with the general structure of gauge symmetries for mixed-symmetry fields in Minkowski and (A)dS(d) spaces.

The collisionless Boltzmann equation, also called free-streaming, is a convenient approximation. It is rather simple to implement numerically, and it is effective at reducing the irregularities of rough initial conditions. It can be obtained as a small $\ensuremath{\hbar}$ limit from a free scalar quantum field, namely, by neglecting the ${\ensuremath{\hbar}}^{2}$ term in the dynamical evolution of the Wigner distribution, the quantum precursor of the distribution function. In this work, the general form for the exact solutions of the Wigner distribution is presented, for a scalar field undergoing a ($1+1$)-dimensional expansion, namely, with the symmetry constraints of rotation and translation invariance in the transverse plane. It is very different from the (on-shell) free-streaming of classical particles. It is shown how to recognize the classical $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ limit from the general form. The numerical analysis for a specific example shows very large quantum corrections to the classical free-streaming.

An explicit construction of a free and massless Majorana quantum field theory, which exists on a conformal superworld M is presented. Emphasis is placed on the investigation of the action of an infinite-dimensional group G of spacetime symmetries on M. Starting with a one-particle theory, this action induces a strongly continuous representation of Diffapproximately +(S1) on the single-particle Hilbert space. After second quantization the group of implementers turns out to be a non-trivial central extension of Diffapproximately +(S1) by U(1), and a Schwinger term occurs, which gives rise to the anomalous transformation law of the energy-momentum tensor of the theory. This transformation law is studied, and an interesting connection to geometry is established.