Modular Operator for Null Plane Algebras in Free Fields

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 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

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.

We investigate the first law of complexity proposed in Bernamonti et al (2019 Phys. Rev. Lett. 123 081601), i.e., the variation of complexity when the target state is perturbed, in more detail. Based on Nielsen’s geometric approach to quantum circuit complexity, we find the variation only depends on the end of the optimal circuit. We apply the first law to gain new insights into the quantum circuits and complexity models underlying holographic complexity. In particular, we examine the variation of the holographic complexity for both the complexity = action and complexity = volume conjectures in perturbing the AdS vacuum with coherent state excitations of a free scalar field. We also examine the variations of circuit complexity produced by the same excitations for the free scalar field theory in a fixed AdS background. In this case, our work extends the existing treatment of Gaussian coherent states to properly include the time dependence of the complexity variation. We comment on the similarities and differences of the holographic and QFT results.

We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massless Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the higher dimensional Legendre differential operator. The quadratic form of the massless modular Hamiltonian is expressed in terms of an integral of the energy density with the parabolic distribution. We then get the formula for the local entropy of a 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 provide the passivity characterisation of the modular Hamiltonian within the standard subspace setup.

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.

To treat the front-form Hamiltonian approach to quantum field theory, called light cone quantum field theory, in a mathematically rigorous way, the existence of a well-defined restriction of the corresponding free fields to the hypersurface {x0+x3=0} in Minkowski space is of an essential necessity. However, even in the situation of a real scalar free field such a restriction does canonically not exist; this is called the restriction problem. Furthermore, since the beginning of light cone quantum field theory there is the problem of nonexistence of a well-defined Fock space expansion of a free quantum field in terms of light cone momenta which is called the zero-mode problem. In this paper we present solutions to these long outstanding problems where the study of the zero-mode problem (of the corresponding classical field) will lead us to a solution of the restriction problem. We introduce a new function space of “squeezed” smooth functions which can canonically be embedded into the Schwartz space S(R3). The restriction of the free field to {x0+x3=0} is canonically definable on this function space and we show that the covariant field is uniquely determined by this “tame” restriction.

The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition, which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $⟨{T}_{kk}{⟩}_{p}\ensuremath{\ge}{\mathrm{lim}}_{A\ensuremath{\rightarrow}0}(\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}A}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}})$, where ${S}_{\mathrm{out}}$ is the entanglement entropy restricted to one side of a codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$, which is deformed in the null direction about a neighborhood of point $p$ with area $A$. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.

Black hole physics and Liouville theory

This thesis presents various examples of the application of quantum-mechanical methods to the understanding of the structure of space-time. It focuses on noncommutative geometry and the gauge/gravity duality as intermediaries between quantum mechanics and classical geometry. First, we numerically calculate entanglement entropy and mutual information for a massive free scalar field on commutative and noncommutative (fuzzy) spheres. To define a subregion with a well-defined boundary in the noncommutative geometry, we use the symbol map between elements of the noncommutative algebra and functions on the sphere. We show that the UV-divergent part of the entanglement entropy on a fuzzy sphere does not follow an area law. In agreement with holographic predictions, it is extensive for small (but fixed) regions. This is true even in the limit of small noncommutativity. Nonetheless, we find that mutual information (which is UV-finite) is the same in both theories. This suggests that nonlocality at short distances does not affect quantum correlations over large distances in a free field theory. Previous work has shown that a surface embedded in flat ℝ³ can be associated to any three Hermitian matrices. By constructing coherent states corresponding to points in the emergent geometry, we study this emergent surface when the matrices are large. We find that the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes. Finally, we use the gauge/gravity correspondence to translate the positivity of relative entropy on the boundary into constraints on allowable space-time metrics in the bulk. Using the Einstein equations, we interpret these constraints as energy conditions. For certain three-dimensional bulks, we obtain strict constraints coming from the positivity of relative entropy with a thermal reference state which turn out to be equivalent to a version of the weak energy condition near the boundary. In higher dimensions, we use the canonical energy formalism to obtain similar but weaker constraints.

We numerically compute the Rényi entropy for four-dimensional free scalar field theory with a spherical entangling surface. As is well known, the Rényi entropy as a function of the boundary area exhibits linear dependence in the leading order. The coefficient of the subleading logarithmic term from our numerical data, as a function of the Rényi order q, agrees nicely with the general prediction of conformal field theory computation. The motivation of this work is also partly to see how the efficiency of numerical computation changes as a function of q. For q<1 the summation over eigenvalues of reduced density matrix takes longer since the series converges more slowly than for q=1. For q>1 the convergence is faster, but the relative error becomes large as a general trend.

Could a conformal Galileon be describing a gauge mode of a broader diffeomorphism invariant theory? We answer this question affirmatively in 3D by using a coset construction for a nonlinearly realized conformal symmetry. In particular, we show that the conformal Galileon emerges as a Stückelberg field of a local Weyl symmetry. The coset construction gives us a diffeomorphism and Weyl invariant 3D theory containing first, second, and third powers of the curvatures, their couplings to certain tensors made of the Galileon and its derivatives, and the conformal Galileon terms. In a theory with a boundary additional surface terms are required for the boundary Weyl anomaly cancellation. When gravity is switched off, the theory reduces to the conformal Galileon plus a boundary term. On the other hand, it reduces to a generalization of “New Massive Gravity” by A. Sinha, if the Weyl symmetry is gauge-fixed in a particular way. Last but not least, there exists a parameter space for which the theory has no propagating ghosts. Thus we show by an explicit construction that “New Massive Gravity” (and its generalization) is a unique scalar ghost free low energy effective field theory non-linearly realizing conformal symmetry in 3D.

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 order to determine the radiated sound power of an underwater source below the Schroeder cut-off frequency in a non-anechoic tank, a low-frequency extension measuring technique is proposed. This technique is based on a unique relationship between the transmission characteristics of the enclosed field and those of the free field, which can be obtained as a correction term based on previous measurements of a known simple source. The radiated sound power of an unknown underwater source in the free field can thereby be obtained accurately from measurements in a non-anechoic tank. To verify the validity of the proposed technique, a mathematical model of the enclosed field is established using normal-mode theory, and the relationship between the transmission characteristics of the enclosed and free fields is obtained. The radiated sound power of an underwater transducer source is tested in a glass tank using the proposed low-frequency extension measuring technique. Compared with the free field, the radiated sound power level of the narrowband spectrum deviation is found to be less than 3 dB, and the 1/3 octave spectrum deviation is found to be less than 1 dB. The proposed testing technique can be used not only to extend the low-frequency applications of non-anechoic tanks, but also for measurement of radiated sound power from complicated sources in non-anechoic tanks.