Quantum Action Research Articles

Renormalization of the nonprojectable Hořava theory

We present the proof of renormalization of the Hořava theory in the nonprojectable version. We obtain a form of the quantum action that exhibits a manifest Becchi-Rouet-Stora-Tyutin–symmetry structure. Previous analysis has shown that the divergences produced by irregular loops cancel completely between them. The remaining divergences are local. The renormalization is achieved by using the approach developed by Barvinsky with the background-field formalism. Published by the American Physical Society 2024

Three-loop renormalization of the quantum action for a five-dimensional scalar cubic model with the usage of the background field method and a cutoff regularization

Three-loop renormalization of the quantum action for a five-dimensional scalar cubic model with the usage of the background field method and a cutoff regularization

Conformal symmetry in quantum gravity

We study the problem of how to derive conformal symmetry in the framework of quantum gravity. We start with a generic gravitational theory which is invariant under both the general coordinate transformation (GCT) and Weyl transformation (or equivalently, local scale transformation), and then construct its BRST formalism by fixing the gauge symmetries by the extended de Donder gauge and scalar gauge conditions. These gauge-fixing conditions are invariant under global GL(4) and global scale transformations. The gauge-fixed and BRST invariant quantum action possesses a huge Poincaré-like IOSp(10|10) global symmetry, from which we can construct an extended conformal symmetry in a flat Minkowski background in the sense that the Lorentz symmetry is replaced with the GL(4) symmetry. Moreover, we construct the conventional conformal symmetry out of this extended symmetry. With a flat Minkowski background ⟨gμν⟩=ημν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle g_{\\mu \ u } \\rangle = \\eta _{\\mu \ u }$$\\end{document} and a non-zero scalar field ⟨ϕ⟩≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\phi \\rangle \ e 0$$\\end{document}, the GL(4) and global scale symmetries are spontaneously broken to the Lorentz symmetry, thereby proving that the graviton and the dilaton are respectively the corresponding Nambu–Goldstone bosons, and therefore they must be exactly massless at nonperturbative level. One of remarkable aspects in our findings is that in quantum gravity, a derivation of conformal symmetry does not depend on a classical action, and its generators are built from only the gauge-fixing and the FP ghost actions. Finally, we address a generalized Zumino theorem in quantum gravity.

Three-loop renormalization of the quantum action for a four-dimensional scalar model with quartic interaction with the usage of the background field method and a cutoff regularization

The paper studies the quantum action for the four-dimensional real ϕ4-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The absence of non-local singular contributions and the correctness of the renormalization R-operation on the example of separate three-loop diagrams are also discussed. The explicit forms of the first three coefficients for the renormalization constants and for the β-function are presented. Consistency with previously known results is shown.

Full three-loop renormalisation of an abelian chiral gauge theory with non-anticommuting γ5 in the BMHV scheme

In this work we present a complete three-loop renormalisation of an abelian chiral gauge theory within the Breitenlohner-Maison/’t Hooft-Veltman (BMHV) scheme of dimensional regularisation (DReg). In this scheme the γ5-matrix appearing in gauge interactions is a non-anticommuting object, leading to a breaking of gauge and BRST invariance. Employing an efficient method based on the quantum action principle, we obtain the complete three-loop counterterm action which serves both to render the theory finite and to restore gauge and BRST invariance. The UV singular counterterms involve not only higher order ϵ-poles but also new counterterm structures emerging at the three-loop level for the first time; the finite symmetry-restoring counterterms are restricted to the same structures as at lower loop orders, just with different coefficients, aligning with our expectations. Both the singular and the finite counterterms include structures which cannot be obtained by the standard multiplicative renormalisation. Our results demonstrate that a rigorous treatment of chiral gauge theories with γ5 defined in the BMHV scheme at the multi-loop level is possible and that the obtained counterterm action is suitable for computer implementations, allowing automated calculations without ambiguities caused by γ5.

Even-point multi-loop unitarity and its applications: exponentiation, anomalies and evanescence

We identify novel structure in newly computed multi-loop amplitudes and quantum actions for even-point effective field theories, including both the nonlinear sigma model (NLSM) and double-copy gauge theories such as Born-Infeld and its supersymmetric generalizations. We exploit special properties of all even-point theories towards establishing an efficient unitarity based amplitude construction. In doing so, we find evidence that the leading IR divergence of NLSM amplitudes exponentiates when the target space is CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{CP} $$\\end{document}1 ≅ SU(2)/U(1). We then systematically compute the two-loop anomalous behavior of Born-Infeld, and find that the counterterms needed to restore U(1) invariant behavior at loop-level can be constructed via a symmetric-structure double-copy. We also demonstrate that the divergent part of the one-minus (−+++) two-loop anomaly vanishes upon introducing an evanescent operator. In addition to these purely photonic counterterms, we verify through explicit calculation that the anomalous matrix elements that violate U(1) duality invariance can be alternatively cancelled by summing over internal N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 DBIVA superfields. Finally we find that N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Dirac-Born-Infeld-Volkov-Akulov (DBIVA) amplitudes admit double-copy construction through two-loop order by reproducing our unitarity based result with a double copy between color-dual N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills and our two-loop NLSM amplitudes. This result supports the possibility of color-dual representations for NLSM beyond one-loop. We conclude with an overview of how D-dimensional four-photon counterterms can be constructed in generality with the symmetric-structure double-copy, and outline a convenient way of counting evanescent operators using Hilbert series as generating functions.

Simulation of new waves in applied sciences via Schrödinger equations

The perturbed chiral nonlinear Schrö dinger equation (PCNLSE) reflects the quantum actions such as quantum pictures of Bohm potential and internal self-potential properties. Indeed, this equation introduces the basics of the hidden variable theory in quantum mechanics. Two unified solver methods and exp -expansion technique applied to PCNLSE to present many solitonic solutions in an explicit and effective way. The behaviour of these solutions is of qualitatively different structural natures, relying on physical coefficient parameters. The application of three mathematical techniques to our model system provides us with several possible physical property solutions that account for the majority of many phenomena the model under study attempts to depict. The reported bright explosive envelopes, explosive solitons, periodic blow up, bright periodic envelope and huge solitary waves are highly applicable in plasma and nuclear physics, optical communications, electro-magnetic propagations, superfluid and in a lot other applied sciences. The results of this system's solitary structures are consistent with the characteristics of the nonlinear Schrödinger equation systems used to study dispersive modes and higher-order perturbed systems. For more details about the physical dynamical representation of the presented solutions, we have illustrated them with profile pictures using Mathematica and Matlab 18, to obtain complete configurations. The proposed approach can be applied to several equations arising in all applied sciences.

Thinking, holograms, and the quantum brain

This article continues the development of the idea that all human behavior and thinking are innate. A model of thinking and functioning of the brain has been constructed, which is capable of explaining both the accuracy of molecular processes and the innateness of behaviors. The focus of the model is the phase of the wave function of the particle, which is an additional (free) parameter. It should also be emphasized that the phase of the wave function of a particle is inextricably linked with the quantum action S in the Feynman’s formulation of quantum mechanics (path integrals). A hypothesis is proposed: the set of particles that make up neurons and the brain is controlled by changing the phases from the outside (by a higher order system). Such a control system must be outside our world because our measurement methods do not allow us to determine the phase of an elementary particle. In a sense, it can be viewed as an extension of Bohm's ideas about the holographic brain and the holographic universe. Experiments are proposed that could confirm or disprove this model.

Evaluation of the Feynman Propagator by Means of the Quantum Hamilton-Jacobi Equation

It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different.Quanta 2023; 12: 22–26.

BRST symmetry and unitarity of the Hořava theory

We present an analysis on the BRST symmetry transformations of the Horava theory under the BFV quantization, both in the nonprojectable and projectable cases. We obtain that the BRST transformations are intimately related to a particular spatial diffeomorphism along one of the ghost vector fields. We show explicitly the invariance of the quantum action and the nilpotence of the BRST transformations, using this diffeomorphism largely. The BRST symmetry is verified in the whole phase space, inside and outside the constrained surface. When restricted to the constrained surface, the BRST transformations are completely local. The consistency of the BRST symmetry is a fundamental feature of a quantum field theory, specially for renormalization. The BFV quantization is independent of the chosen gauge-fixing condition. This allows us to show the unitarity of the quantum Horava theory, covering the gauge required for renormalization.

Safe reinforcement learning under temporal logic with reward design and quantum action selection

This paper proposes an advanced Reinforcement Learning (RL) method, incorporating reward-shaping, safety value functions, and a quantum action selection algorithm. The method is model-free and can synthesize a finite policy that maximizes the probability of satisfying a complex task. Although RL is a promising approach, it suffers from unsafe traps and sparse rewards and becomes impractical when applied to real-world problems. To improve safety during training, we introduce a concept of safety values, which results in a model-based adaptive scenario due to online updates of transition probabilities. On the other hand, a high-level complex task is usually formulated via formal languages, including Linear Temporal Logic (LTL). Another novelty of this work is using an Embedded Limit-Deterministic Generalized Büchi Automaton (E-LDGBA) to represent an LTL formula. The obtained deterministic policy can generalize the tasks over infinite and finite horizons. We design an automaton-based reward, and the theoretical analysis shows that an agent can accomplish task specifications with the maximum probability by following the optimal policy. Furthermore, a reward shaping process is developed to avoid sparse rewards and enforce the RL convergence while keeping the optimal policies invariant. In addition, inspired by quantum computing, we propose a quantum action selection algorithm to replace the existing varepsilon-greedy algorithm for the balance of exploration and exploitation during learning. Simulations demonstrate how the proposed framework can achieve good performance by dramatically reducing the times to visit unsafe states while converging optimal policies.

Derivation of a Unified Theory from the Holographic Principle

Holographic principle and the related self-similarity symmetry are observed widely in nature. Research in black hole, string theory and quantum theory of gravity suggests that holographic principle can be an important symmetry for solving challenging problems in theoretical physics, such as finding the grand unification theory (GUT). In this paper, we derive a quantum theory incorporating the holographic principle by introducing the new concept and elementary information. The derived quantum action incorporating holographic principle, holographic action, turns out to be the generalized action encompassing string theory, general relativity and thermodynamics. This holographic quantum theory indicates that phenomena and laws of physics emerge from the holograms represented by the holographic action. Specifically, it predicts the following: (1) Elementary particles, gravity and gauge interactions and the classical equations of motion are the emergence of the hologram due to Poincaré symmetry, diffeomorphic symmetry and Weyl symmetry, respectively. (2) Dark matter and dark energy are the vibrations on the horizon scale of the universe. (3) Cosmological constant is calculated to be [Formula: see text] in Planck unit, in agreement with the cosmological constant deduced from astrophysical observation (4) The observed space-time is negatively curved if its dimension is greater than 4, positively curved if its dimension is less than 4, and flat if its dimension is 4. (5) It gives the mathematical formula to derive the entropy of black hole and study the internal dynamics of black hole. (6) It provides the mathematical framework to study the dynamics of spacetime compactification and the large hierarchy between Planck scale and electroweak scale. One may conclude that the holographic quantum theory based on holographic principle may be not only a GUT but also able to tackle some of the problems impossible to be addressed before.

Quantum computation using action variables

In 2014, Lloyd and Montangero have made a brief research proposal on universal quantum computation in integrable systems. The main idea is to encode qubits into quantum action variables and build up quantum gates by the method of resonant control. We study this proposal to argue quantum computation using action variables as fault-tolerant quantum computation, whose fault tolerance is guaranteed by the quantum KAM theorem. Besides, we view the Birkhoff norm form as a mathematical framework of the extended harmonic oscillator quantum computation.

Drastic effect of weak interaction near special points in semiclassical multiterminal superconducting nanostructures

A generic semiclassical superconducting nanostructure connected to multiple superconducting terminals hosts a quasi-continuous spectrum of Andreev states. The spectrum is sensitive to the superconducting phases of the terminals. It can be either gapped or gapless depending on the point in the multi-dimensional parametric space of these phases. Special points in this space correspond to setting some terminals to the phase 0 and the rest to the phase of $\pi$. For a generic nanostructure, three distinct spectra come together in the vicinity of a special point: two gapped phases of different topology and a gapless phase separating the two by virtue of topological protection. In this paper, we show that a weak interaction manifesting as quantum fluctuations of superconducting phases drastically changes the spectrum in a narrow vicinity of a special point. We develop an interaction model and derive a universal generic quantum action that describes this situation. The action is complicated by incorporating a non-local in time matrix order parameter, and its full analysis is beyond the scope of the present paper. Here, we identify and address two limits: the semiclassical one and the quantum one, concentrating on the first-order interaction correction in the last case. In both cases, we find that the interaction squeezes the domain of the gapless phase in the narrow vicinity of the point so the gapped phases tend to contact each other immediately, defying the topological protection. We identify the domains of strong coupling where the perturbation theory does not work. In the gapless phase, we find the logarithmic divergence of the first-order corrections. This leads us to an interesting hypothesis: weak interaction might induce an exponentially small gap in the formerly gapless phase.

Gauge-independent description of the Aharonov-Bohm effect

The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region \textit{inaccessible} to the particle. Classically, such a non-local effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle, and the electromagnetic vector potential $\mathbf{A}$, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics $\mathbf{A}$ is in some sense more "fundamental" than $\mathbf {B}$ in spite of the former being gauge dependent, and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB-effect can be fully accounted for by considering only the gauge invariant $\mathbf{B}$ field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge invariant formulation is that we must give up locality -- the AB-phase for the particle will arise from the change in the action for the $\mathbf{B}$ field in the region inaccessible to the particle.

Universal deformation formula, formality and actions

In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak{g},r)$ on a smooth manifold $M$. Using the formality of $E$-polydifferential operators for Lie algebroids $E$ over $M$, we obtain a deformation quantization of $M$ together with a quantum group $\mathscr{U}{\rho\hbar}(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L\_\infty$-morphisms which generalizes the well-known definition given by Drinfeld.

Revisiting the Boltzmann Derivation of the Stefan Law

The Stefan–Boltzmann (SB) law relates the radiant emittance of an ideal black-body cavity at thermal equilibrium to the fourth power of the absolute temperature [Formula: see text] as [Formula: see text], with [Formula: see text][Formula: see text]W[Formula: see text]m[Formula: see text][Formula: see text]K[Formula: see text] the SB constant, first estimated by Stefan to within [Formula: see text] of the present theoretical value. The law is an important achievement of modern physics since, following Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563], its microscopic derivation implies the quantization of the energy related to the electromagnetic field spectrum. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By contrast, the Boltzmann derivation implies two assumptions not justified within a classical approach, namely: (i) the zero value of the chemical potential and (ii) the internal energy of the black body with a finite value and dependent from both temperature and volume. By using Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563] quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation free from the above assumptions that also provides the value of the SB constant on the basis of a set of universal constants including the quantum action constant [Formula: see text]. However, the successive consideration by Planck [Uber die Begründung des Gesetzes der schwarzen Strahlung [On the grounds of the law of black body radiation], Ann. Phys. 6 (1912) 642–656] concerning the zero-point energy contribution was found to be responsible of another divergence of the internal energy for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh–Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we would expect the divergence of the SB law (here called second Boltzmann paradox). Here, both the Boltzmann paradoxes are revised by accounting for both the quantum-relativistic photon gas properties, and the Casimir force.

Dissipation Effects Induced by Internal Relative Motion Between Two Nonlinear σ-Models

We studied the dissipation effect induced by the internal relative motion between two nonlinear [Formula: see text]-models (NLSMs). The internal degrees of freedom (DOF) of the two NLSMs coupled to each other via an interface potential. We performed the path integral of the vacuum persistence amplitude (VPA) and got the quantum action. Then we introduced the relative motion via a boost transformation with velocity parameter [Formula: see text]. Furthermore, we calculated the imaginary part of the boosted quantum action which was proportional to the vacuum decay probability. At last, we calculated the frictional force. The numerical results gave the velocity [Formula: see text] dependency of the vacuum decay probability, and the velocity [Formula: see text] dependency of the frictional force. There was a threshold of relative velocity [Formula: see text] for the vacuum decay probability and the frictional force. We gave a classical argument of the existence of the threshold, and the argument matched the mathematical results.

QFT with stubs

The BV-Laplacian ∆ in quantum field theory is singular, by construction, but can be regularized by deforming the classical BV-action. Taking inspiration from string theory we describe a non-local deformation of the latter by adding stubs to the interaction vertices while keeping classical BV-invariance manifest. This is achieved using a version of homotopy transfer resulting in a non-polynomial action for which the quantum master equation is now well defined and will be satisfied by adding additional vertices at loop level. The latter can be defined with the help of standard regularization schemes and is independent of the definition of ∆. In particular, the determination of anomalies reduces to the standard text-book calculation. Finally, we describe how the deformed (quantum) action can be obtained as a canonical transformation. As an example, we illustrate this procedure for quantum electrodynamics.

The sinh-Gordon model beyond the self dual point and the freezing transition in disordered systems

The S-matrix of the well-studied sinh-Gordon model possesses a remarkable strong/weak coupling duality b → 1/b. Since there is no understanding nor evidence for such a duality based on the quantum action of the model, it should be questioned whether the properties of the model for b > 1 are simply obtained by analytic continuation of the weak coupling regime 0 < b < 1. In this article we assert that the answer is no, and we develop a concrete and specific proposal for the properties when b > 1. Namely, we propose that in this region one needs to introduce a background charge Q∞ = b + 1/b − 2 which differs from the Liouville background charge by the shift of −2. We propose that in this regime the model has non-trivial massless renormalization group flows between two different conformal field theories. This is in contrast to the weak coupling regime which is a theory of a single massive particle. Evidence for our proposal comes from higher order beta functions. We show how our proposal correctly reproduces the freezing transitions in the multi-fractal exponents of a Dirac fermion in 2 + 1 dimensions in a random magnetic field, which provides a strong check since such transitions have several detailed features. We also point out a connection between a semi-classical version of this transition and the so-called Manning condensation phenomena in polyelectrolyte physics.