QUANTUM CYLINDRICAL WAVES AND PARAMETRIZED FIELD THEORY

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).

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.

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.

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.

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 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

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.

Black hole physics and Liouville theory

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 consider the algebras generated by observables in quantum field theory localized in regions in the null plane. For a scalar free field theory, we show that the one-particle structure can be decomposed into a continuous direct integral of lightlike fibres and the modular operator decomposes accordingly. This implies that a certain form of QNEC is valid in free fields involving the causal completions of half-spaces on the null plane (null cuts). We also compute the relative entropy of null cut algebras with respect to the vacuum and some coherent states.

The definition of a length operator in quantum cosmology is usually influenced by a quantum theory for gravity considered. The semiclassical limit at the Planck age must meet the requirements implied in present observations. The features of a semiclassical wave-functional state are investigated, for which the modern measure(ment)s is consistent. The results of a length measurement at present times are compared with the same measurement operation at cosmological times. By this measure, it is possible to discriminate, within the same Planck-length expansion, the corrections to a Minkowski flat space possibly due to classicalization of quantum phenomena at the Planck time and those due to possible quantum-gravitational manifestations of present times. This analysis and the comparison with the previous literature can be framed as a test for the verification of the time at which anomalies at present related to the gravitational field, and, in particular, whether they are ascribed to the classicalization epoch. Indeed, it allows to discriminate not only within the possible quantum features of the quasi (Minkowski) flat spacetime, but also from (possibly Lorentz violating) phenomena detectable at high-energy astrophysical scales. The results of two different (coordinate) length measures have been compared both at cosmological time and as a perturbation element on flat Minkowski spacetime. The differences for the components of the corresponding classical(ized) metric tensor have been analyzed at different orders of expansions. The results of the expectation values of a length operator in the universe at the Planck time must be comparable with the same length measurements at present times, as far as the metric tensor is concerned. The comparison of the results of (straight) length measures in two different directions, in particular, can encode the pertinent information about the parameters defining the semiclassical wavefunctional for (semiclassicalized) gravitational field.

In this work, we construct a noncommutative version of the Friedmann equations in the framework of effective loop quantum cosmology, extending and applying the ideas presented in a previous proposal by some of the authors. The model under consideration is a flat FRW spacetime with a free scalar field. First, noncommutativity in the momentum sector is introduced. We establish the noncommutative equations of motion and obtain the corresponding exact solutions. Such solutions indicate that the bounce is preserved, in particular, the energy density is the same as in the standard LQC. We also construct an extension of the modified Friedmann equations arising in effective LQC which incorporates corrections due to noncommutativity, and argue that an effective potential is induced. This, in turn, leads us to investigate the possibility of an inflationary era. Finally, we obtain the Friedmann and the Raychaudhuri equations when implementing noncommutativity in the configuration sector. In this case, no effective potential is induced.

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.

In this article we extend the test of Hamiltonian renormalisation proposed in this series of articles to the D-dimensional case using a massive free scalar field. The concepts we introduce are explicitly computed for the D = 2 case but transfer immediately to higher dimensions. In this article we define and verify a criterion that monitors, at finite resolution defined by a cubic lattice, whether the flow approaches a rotationally invariant fixed point.

Motivated by a recent proposal (by Koslowski-Sahlmann) of a kinematical representation in loop quantum gravity (LQG) with a nondegenerate vacuum metric, we construct a polymer quantization of the parametrized massless scalar field theory on a Minkowskian cylinder. The diffeomorphism covariant kinematics is based on states that carry a continuous label corresponding to smooth embedding geometries, in addition to the discrete embedding and matter labels. The physical state space, obtained through the group averaging procedure, is nonseparable. A physical state in this theory can be interpreted as a quantum spacetime, which is composed of discrete strips and supersedes the classical continuum. We find that the conformal group is broken in the quantum theory and consists of all Poincar\'e translations. These features are remarkably different compared to the case without a smooth embedding. Finally, we analyze the length operator whose spectrum is shown to be a sum of contributions from the continuous and discrete embedding geometries, being in perfect analogy with the spectra of geometrical operators in LQG with a nondegenerate vacuum geometry.