Phase Transitions for the Growth Rate of Linear Stochastic Evolutions

We consider a discrete-time stochastic growth model on d-dimensional lattice. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of binary contact path process and the voter model. We study the phase transition for the growth rate of the “total number of particles” in this framework. The main results are roughly as follows: If d≥3 and the system is “not too random”, then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, d=1,2, or the system is “random enough”, then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the model with proper normalization. Supported in part by JSPS Grant-in-Aid for Scientific Research, Kiban (C) 17540112.     

Superspace Ward identities in supersymmetric gauge theories

In superspace formulation of supersymmetric gauge theories, gauge invariance requires an infinite set of identities between the infinite set of renormalization constants. Using Ward identities in superspace, the same is derived. These identities at one loop level are also demonstrated.     

Poisson-Lie T-duality and complex geometry in N = 2 superconformal WZNW models

Poisson-Lie T-duality in N = 2 superconformal WZNW models on the real Lie groups is considered. It is shown that Poisson-Lie T-duality is governed by the complexifications of the corresponding real groups endowed with Semenov-Tian-Shansky symplectic forms, i.e. Heisenberg doubles. Complex Heisenberg doubles are used to define on the group manifolds of the N = 2 superconformal WZNW models the natural actions of the isotropic complex subgroups forming the doubles. It is proved that with respect to these actions N = 2 superconformal WZNW models admit Poisson-Lie symmetries. The Poisson-Lie T-duality transformation maps each model onto itself but acts non-trivially on the space of classical solutions.     

Chiral Ring Structure in the N = 2 Superconformal Field Theory Based on the Coset Construction with SU(1,1)/U(1)

In this paper, we carefully analyze the unitarity of the simplest Kazama-Suzuki model based on the noncompact group. The chiral ring structure in this N = 2 theory is clarified. In general, it is infinitely dimensional and innumerable. The primary chiral states belonging to the ring can be constructed by means of the direct products of the highest weight states of the coset SU(1,1)/ U(1) and the scalar representation of U(1) for the fermions. The primary chird and anti-chiral states Jive individually in the positive discrete series D⁺ and negative discrete series D^- of SU(1,1), respectively. With appropriate restrictions, this theory can be regarded as one at the fixed point of a Landau-Ginzburg theory, in which the superpotential is given by S₁₂ in the fourteen exceptional singularities.     

The Stochastic Limit for the Hydrogen Atom in Interaction with the EM Field

In the stochastic limit the resonances play a fundamental role because they determine the generalized susceptivities which are the building blocks of all the physical information which survives in this limit. There are two sources of possible divergences, one related to the singularities of the form factor, another to the chaoticity of the spectrum. The situation will be illustrated starting from the example of the discrete part of the hydrogen atom in interaction with the electromagnetic field.     

Quantum supersymmetric Toda–mKdV hierarchies

In this paper we generalize the quantization procedure of Toda–mKdV hierarchies to the case of arbitrary affine (super)algebras. The quantum analogue of the monodromy matrix, related to the universal R-matrix with the lower Borel subalgebra represented by the corresponding vertex operators is introduced. The auxiliary L-operators satisfying RTT-relation are constructed and the quantum integrability condition is obtained. General approach is illustrated by means of two important examples.     

Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.     

量子场论中的自旋算符

从量子场论的角度对相对论粒子的运动自旋概念作了进一步深入研究.构造了场量子自旋以及场系统运动自旋两个新算符.给出了场量子自旋动量空间的显式表达式以及用Poincar啨群生成元表示的场系统运动自旋的显式表达式.借助这两个算符,可以干净地解决有关场自旋的问题,表明它们才是场自旋的恰当的算符.     

Types of two-dimensionalN = 4 superconformal field theories

Various types ofN = 4 superconformal symmetries in two dimensions are considered. It is proposed that apart from the well-known cases ofSU (2)and SU(2) × SU(2) ×U (1), their Kac-Moody symmetry can also be SU(2) × (U (1))⁴. Operator product expansions for the last case are derived. A complete free field realization for the same is obtained     

On Field Theory Methods in Singular Perturbation Theory

Singular and supersingular finite rank perturbations of self-adjoint operators are studied using methods from renormalization theory for quantum fields. It is shown that the ideas from dimensional and Pauli–Villars regulatizations can be applied to determine uniquely certain finite rank supersingular perturbations. Approach is based on the regularization of homogeneous singular quadratic forms.     

Quantum Field Theory and Dense Measurement

We show, using quantum field theory (QFT), that performing a large number of identical repetitions of the same measurement does not only preserve the initial state of the wave function (the Zeno effect), but also produces additional physicaleffects. We first discuss the Zeno effect in the framework of QFT, that is, as a quantum field phenomenon. We then derive it from QFT for the general case in which the initial and final states are different. We use perturbation theory and Feynman diagrams and refer to the measurement act as an external constraint upon the system that corresponds to the perturbative diagram that denotes this constraint. The basic physical entities dealt with in this work are not the conventional once-perfomed physical processes, but their n times repetition where n tends to infinity. We show that the presence of these repetitions entails the presence of additional excited state energies, and the absence of them entails the absence of these excited energies.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

Topological Field Theory and Computing with Instantons

It is well known that dynamical systems may be employed as computing machines. However, not all dynamical systems offer particular advantages compared to the standard paradigm of computation, in regard to efficiency and scalability. Recently, it was suggested that a new type of machines, named digital –hence scalable– memcomputing machines (DMMs), that employ non‐linear dynamical systems with memory, can solve complex Boolean problems efficiently. This result was derived using functional analysis without, however, providing a clear understanding of which physical features make DMMs such an efficient computational tool. Here, we show, using recently proposed topological field theory of dynamical systems, that the solution search by DMMs is a composite instanton. This process effectively breaks the topological supersymmetry common to all dynamical systems, including DMMs. The emergent long‐range order – a collective dynamical behavior– allows logic gates of the machines to correlate arbitrarily far away from each other, despite their non‐quantum character. We exemplify these results with the solution of prime factorization, but the conclusions generalize to DMMs applied to any other Boolean problem.     

Effective Field Theory for Fermi Systems in a Large N Expansion

A system of fermions with short-range interactions at finite density is studied using the framework of effective field theory. The effective action formalism for fermions with auxiliary fields leads to a loop expansion in which particle-hole bubbles are resummed to all orders. For spin-independent interactions, the loop expansion is equivalent to a systematic expansion in 1/N, where N is the spin-isospin degeneracy g. Numerical results at next-to-leading order are presented and the connection to the Bose limit of this system is elucidated.     

Finsler Geometry and Relativistic Field Theory

Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of gauge transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.     

Information Field Theory and Artificial Intelligence

Information field theory (IFT), the information theory for fields, is a mathematical framework for signal reconstruction and non-parametric inverse problems. Artificial intelligence (AI) and machine learning (ML) aim at generating intelligent systems, including such for perception, cognition, and learning. This overlaps with IFT, which is designed to address perception, reasoning, and inference tasks. Here, the relation between concepts and tools in IFT and those in AI and ML research are discussed. In the context of IFT, fields denote physical quantities that change continuously as a function of space (and time) and information theory refers to Bayesian probabilistic logic equipped with the associated entropic information measures. Reconstructing a signal with IFT is a computational problem similar to training a generative neural network (GNN) in ML. In this paper, the process of inference in IFT is reformulated in terms of GNN training. In contrast to classical neural networks, IFT based GNNs can operate without pre-training thanks to incorporating expert knowledge into their architecture. Furthermore, the cross-fertilization of variational inference methods used in IFT and ML are discussed. These discussions suggest that IFT is well suited to address many problems in AI and ML research and application.     

Field Theory Approaches to Relativistic Hydrodynamics

Just as non-relativistic fluids, oftentimes we find relativistic fluids in situations where random fluctuations cannot be ignored, with thermal and turbulent fluctuations being the most relevant examples. Because of the theory’s inherent nonlinearity, fluctuations induce deep and complex changes in the dynamics of the system. The Martin–Siggia–Rose technique is a powerful tool that allows us to translate the original hydrodynamic problem into a quantum field theory one, thus taking advantage of the progress in the treatment of quantum fields out of equilibrium. To demonstrate this technique, we shall consider the thermal fluctuations of the spin two modes of a relativistic fluid, in a theory where hydrodynamics is derived by taking moments of the Boltzmann equation under the relaxation time approximation.     

Group Contraction in Quantum Field Theory

Group contraction plays a relevant rôle in spontaneously broken symmetry theories. Its physical meaning in connection with Bose condensation and the origin of macroscopic quantum systems is discussed.     

Effective Field Theory of Random Quantum Circuits

Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such features have already been experimentally demonstrated in noisy intermediate-scale quantum (NISQ) devices. On the theory side, properties of random quantum circuits have been studied on a case-by-case basis and for certain specific systems, and a hallmark of quantum chaos—universal Wigner–Dyson level statistics—has been derived. This work develops an effective field theory for a large class of random quantum circuits. The theory has the form of a replica sigma model and is similar to the low-energy approach to diffusion in disordered systems. The method is used to explicitly derive the universal random matrix behavior of a large family of random circuits. In particular, we rederive the Wigner–Dyson spectral statistics of the brickwork circuit model by Chan, De Luca, and Chalker [Phys. Rev. X 8, 041019 (2018)] and show within the same calculation that its various permutations and higher-dimensional generalizations preserve the universal level statistics. Finally, we use the replica sigma model framework to rederive the Weingarten calculus, which is a method of evaluating integrals of polynomials of matrix elements with respect to the Haar measure over compact groups and has many applications in the study of quantum circuits. The effective field theory derived here provides both a method to quantitatively characterize the quantum dynamics of random Floquet systems (e.g., calculating operator and entanglement spreading) and a path to understanding the general fundamental mechanism behind quantum chaos and thermalization in these systems.