Symmetries in Classical Field Theory

Classical and nonclassical symmetries of a generalized Boussinesq equation

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions $f(u)$ for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.

Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system

Q-conditional symmetries of the classical Lotka-Volterra system in the case of one space variable are completely described and a set of such symmetries in explicit form is constructed. The relevant non-Lie ansätze to reduce the classical Lotka-Volterra systems with correctly-specified coefficients to ODE systems and examples of new exact solutions are found. A possible biological interpretation of some exact solutions is presented.

On symmetries in Galilei classical mechanics

In the framework of Galilei classical mechanics (i.e., general relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of the geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure on spacetime and of its potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler-Lagrange morphism, Lagrangians. Then, we provide a covariant momentum map associated with a group of cosymplectic symmetries by using a covari…

Generalized conformal maps as classical symmetries of Yang-Mills fields

We show that a class of previously defined maps, called self-dual and causal morphisms, form classical symmetries of Yang-Mills fields in four complex dimensions. These maps generalize conformal transformations, and admit a nonlocal pullback connection that preserves the equations of the theory. First it is shown that self-dual morphisms form symmetries of the anti-self-dual Yang-Mills equations under this pullback. Then a supersymmetric generalization of causal morphisms is defined, which preserves solutions of the field equations for N=3 supersymmetric Yang-Mills theory. As a special case, t…

Classical Systems and Representations of (2+1) Newton-Hooke Symmetries

This paper investigates the classical and quantum elementary systems with Newton-Hoooke symmetry. A complete classification is given by explicit computation. In addition, we present an application example of quantization using the Moyal scheme.

Classical symmetries of monopole by group theoretic methods

We use group theoretic methods to obtain the extended Lie point symmetries of the equations of motion for a charged particle in the field of a monopole. Cases with certain model magnetic fields and potentials are also studied. Our analysis gives the generators and Lie algebras generating the inherent symmetries. The equations of motion of a scalar particle probing the near horizon structure of a black hole is also treated likewise. We have also found the generators of Krause's complete symmetry groups for some of the above examples.

Symmetries and classical quantization

A phenomenon of classical quantization is discussed. This is revealed in the class of pseudoclassical gauge systems with nonlinear nilpotent constraints containing some free parameters. Variation of parameters does not change local (gauge) and discrete symmetries of the corresponding systems, but there are some special discrete values of them which give rise to the maximal global symmetries at the classical level. Exactly the same values of the parameters are separated at the quantum level, where, in particular, they are singled out by the requirement of conservation of the discrete symmetries…

具有对称性的半经典磁薛定谔算子的隧穿

Tunneling for a semi-classical magnetic Schrödinger operator with symmetries

本文研究带对称性的半经典磁薛定谔算子基态在势阱外的衰减估计，主要针对大磁参数情形，并指出Agmon估计在隧穿极限下的应用。

对称性与热辐射：普朗克谱的一种经典推导

Symmetries and Thermal Radiation: A Classical Derivation of the Planck Spectrum

该研究基于相对论经典物理的波涨落推导普朗克谱，指出依赖惯性系无关的经典零点辐射热涨落的推导仅在洛伦兹协变经典理论中成立，为普朗克谱分析提供了纯经典替代方案。

Quantum restoration of broken symmetries

A certain non-linear non-local substitution is shown to transform the action of the self-interacting quantum field to the free one. The functional integrals in both theories are equal to each other. However, the integrations are performed over different functional spaces. The classical action and the classical limit of the corresponding quantum theory turn out to be different. And the symmetry originally broken in the classical theory is restored in the classical limit of the quantum theory.

Generating of additional force terms in Newton equation by twist-deformed Hopf algebras and classical symmetries

We compare two ways of force terms generating in the model of nonrelativistic particle moving in the presence of constant field force $\vec{F}$. First of them uses the twist-deformed acceleration-enlarged Newton-Hooke quantum space-times while the second one incorporates the doubly enlarged Newton-Hooke transformations of classical space. Particulary, we find the conditions for which the both treatments provide the same force terms.

Classical Noether's theory with application to the linearly damped particle

This paper provides a modern presentation of Noether's theory in the realm of classical dynamics, with application to the problem of a particle submitted to both a potential and a linear dissipation. After a review of the close relationships between Noether symmetries and first integrals, we investigate the variational point symmetries of the Lagrangian introduced by Bateman, Caldirola and Kanai. This analysis leads to the determination of all the time-independent potentials allowing such symmetries, in the one-dimensional and the radial cases. Then we develop a symmetry-based transformation o…

关于约化准经典自对偶Yang-Mills方程的完全非局部对称性的递推算子

On recursion operators for full-fledged nonlocal symmetries of the reduced quasi-classical self-dual Yang-Mills equation

本文提出为完全非局部对称性构造递推算子的方法，应用于约化准经典自对偶Yang-Mills方程，所得算子可视为无限维微分函数矩阵，并研究了其代数性质及优势。

Hidden Symmetries of Dynamics in Classical and Quantum Physics

This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description of physical systems as varied as non-relativistic, relativistic, with or without gravity, classical or quantum, and are related to the existence of conserved quantities of the dynamics and integrability. In recent years their study has grown intensively, due to the discovery of non-trivial examples that apply to different types of theories and different numb…

A Theorem on the Existence of Symmetries of Fractional PDEs

We propose a theorem that extends the classical Lie approach to the case of fractional partial differential equations (fPDEs) of the Riemann--Liouville type in (1+1) dimensions.

Symmetries and conservation laws in the Gunther k-symplectic formalism of field theory

This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.

Centre-of-mass and internal symmetries in classical relativistic systems

The internal symmetry of composite relativistic systems is discussed. It is demonstrated that Lorentz-Poincaré symmetry implies the existence of internal moments associated with the Lorentz boost, which are Laplace-Runge-Lenz (LRL) vectors. The LRL symmetry is thus found to be the internal symmetry universally associated with the global Lorentz transformations, in much the same way as internal spatial rotations are associated with global spatial rotations. Two applications are included, for an interacting 2-body system and for an interaction-free many-body system of particles. The issue of loc…

Noether对称群、局部守恒积分与经典力学中的动力学对称性

Noether symmetry groups, locally conserved integrals, and dynamical symmetries in classical mechanics

本文通过混合拉格朗日-哈密顿框架，探讨守恒积分与对称性的关系，通过三个例子展示局部Liouville可积性的推广，利用变分对称性得到局部守恒积分，并引入动作-角度变量实现局部时间内的显式积分。

Noether symmetries, energy-momentum tensors and conformal invariance in classical field theory

In the framework of classical field theory, we first review the Noether theory of symmetries, with simple rederivations of its essential results, with special emphasis given to the Noether identities for gauge theories. Will this baggage on board, we next discuss in detail, for Poincaré invariant theories in flat spacetime, the differences between the Belinfante energy-momentum tensor and a family of Hilbert energy-momentum tensors. All these tensors coincide on shell but they split their duties in the following sense: Belinfante's tensor is the one to use in order to obtain the generators of …

Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics

We review the fundamentals of coupling constant metamorphosis (CCM) and the Stäckel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the momenta to polynomials in the momenta. We study specializations of these actions which do preserve polynomials and also the structure of the symmetry algebras in both the classical and quantum cases. We give several examples of non-constant curvature 3rd and 4th order superinteg…

Polynomial symmetries of spherical Lissajous systems

In a previous work, both the constants of motion of a classical system and the symmetries of the corresponding quantum version have been computed with the help of factorizations. As their expressions were not polynomial, in this paper the question of finding an equivalent set of polynomial constants of motion and symmetries is addressed. The general algebraic relations as well as the appropriate Hermitian relations will also be found.

Structure results for higher order symmetry algebras of 2D classical superintegrable systems

Recently the authors and J.M. Kress presented a special function recurrence relation method to prove quantum superintegrability of an integrable 2D system that included explicit constructions of higher order symmetries and the structure relations for the closed algebra generated by these symmetries. We applied the method to 5 families of systems, each depending on a rational parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system. Here we work out the analogs of these constructions for all of the as…

On a kind of Noether symmetries and conservation laws in k-cosymplectic field theory

This paper is devoted to studying symmetries of certain kinds of k-cosymplectic Hamiltonian systems in first-order classical field theories. Thus, we introduce a particular class of symmetries and study the problem of associating conservation laws to them by means of a suitable generalization of Noether's theorem.

From Lagrangian to Quantum Mechanics with Symmetries

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schrödinger equation is the key that takes classical mechanics into quantu…

Heisenberg-type higher order symmetries of superintegrable systems separable in cartesian coordinates

Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $\mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $\hbar \to0$ limit.

Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

Are all classical superintegrable systems in two-dimensional space linearizable?

Several examples of classical superintegrable systems in two-dimensional spac are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [Phys. Lett. 13, 354 (1965)], and the more recent Tremblay-Turbiner-Winternitz system [J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable.

量子Mycielski图：对称性、孪生顶点与区分标号

Quantum Mycielskians: symmetries, twin vertices and distinguishing labelings

本文提出量子Mycielski图的构造，分析其量子对称性，引入孪生顶点的量子类比，并扩展区分数，研究其在量子构造下的性质。

Variational symmetries and pluri-Lagrangian systems

We analyze the relation of the notion of pluri-Lagrangian systems, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether.

Commensurate Harmonic Oscillators: Classical Symmetries

The symmetry properties of a classical N-dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurate oscillator possesses the same number of globally defined constants of motion as an isotropic oscillator. In both cases invariant phase-space functions form the algebra su(N) with respect to the Poisson bracket. In the isotropic case, the phase-space flows generated by the invariants can be integrated globally to a set of finite transformations isomorphic to the group SU(N). For a commensurate oscillator, however, the group SU(N) of symm…

(Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach

A classical deformation procedure, based on universal enveloping algebras, Casimirs and curvatures of symmetrical homogeneous spaces, is applied to several cases of physical relevance. Starting from the (3+1)D Galilei algebra, we describe at the level of representations the process leading to its two physically meaningful deformed neighbours. The Poincare algebra is obtained by introducing a negative curvature in the flat Galilean phase space (or space of worldlines), while keeping a flat spacetime. To be precise, starting from a representation of the Galilei algebra with both Casimirs differe…

非线性声学方程的保量律和精确解的古典对称约简

Conservation laws and exact solutions of a nonlinear acoustics equation by classical symmetry reduction

本文研究了广义韦斯特维尔特方程的对称性和保量律，通过约简发现点对称性和局部保量律，推导出潜在系统和非局部保量律，并研究了行波解导致的激波现象。

Symmetries and gauge symmetries in multisymplectic first and second-order Lagrangian field theories: electromagnetic and gravitational fields

Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conservation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the electromagnetic and the gravitational theory; in particular, the Einstein--Hilbert and the Einstein--Palatini approaches…

Quantization of the Szekeres spacetime through generalized symmetries

We present the effect of the quantum corrections on the Szekeres spacetime, a system important for the study of the inhomogeneities of the pre-inflationary era of the universe. The study is performed in the context of canonical quantisation in the presence of symmetries. We construct an effective classical Lagrangian and impose the quantum version of its classical integrals of motion on the wave function. The interpretational scheme of the quantum solution is that of Bohmian mechanics, in which one can avoid the unitarity problem of quantum cosmology. We discuss our results in this context.

Nonlocal Symmetries, Telescopic Vector Fields and $λ$-Symmetries of Ordinary Differential Equations

This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $λ$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the $λ$-coverings method. The $λ$-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknow…

Symmetries of noncommutative scalar field theory

We investigate symmetries of the scalar field theory with harmonic term on the Moyal space with euclidean scalar product and general symplectic form. The classical action is invariant under the orthogonal group if this group acts also on the symplectic structure. We find that the invariance under the orthogonal group can be restored also at the quantum level by restricting the symplectic structures to a particular orbit.

Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and non-uniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentu…

Classical Scaling Symmetry Implies Useful Nonconservation Laws

Scaling symmetries of the Euler-Lagrange equations are generally not variational symmetries of the action and do not lead to conservation laws. Nevertheless, by an extension of Noether's theorem, scaling symmetries lead to useful {\em nonconservation} laws, which still reduce the Euler-Lagrange equations to first order in terms of scale invariants. We illustrate scaling symmetry dynamically and statically. Applied dynamically to systems of bodies interacting via central forces, the nonconservation law is Lagrange's identity, leading to generalized virial laws. Applied to self-gravitating spher…

超对称振子的动力学对称性

Dynamical symmetries of supersymmetric oscillators

本文研究经典超对称振子的动力学对称性，通过广义泊松括号构造超荷并定义超对称变换，发现一维和二维哈密顿量分别具有U(1,1)和U(2,2)对称性，并探讨了相关振子的超对称推广。

经典对称性量子可积模型中的Bethe向量

Bethe Vectors in Quantum Integrable Models with Classical Symmetries

本文为经典李代数g=gl_n, o_{2n+1}, sp_{2n}, o_{2n}的g-不变可积模型给出了离壳Bethe向量的精确简单定义，并证明了其满足Bethe方程时成为在壳向量，进而推导了作用公式、矩形递推关系和余积公式等性质。

Lie Groupoids in Classical Field Theory I: Noether's Theorem

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. Here, we outline the basis of the program and, as a first step, show how to (re)formulate Noether's theorem about the connection between symmetries and conservation laws in this approach.

经典和量子振子中的离散对称性

Discrete symmetries in classical and quantum oscillators

研究通过经典和量子振子的本征函数揭示了离散对称性与波函数的关系，展示了量子谐振子本征函数在锥形空间中的定义及其与经典振子的联系。

Hidden symmetries of two-electron quantum dots in a magnetic field

Using a classical and quantum mechanical analysis, we show that the magnetic field gives rise to dynamical symmetries of a three-dimensional axially symmetric two-electron quantum dot with a parabolic confinement. These symmetries manifest themselves as near-degeneracies in the quantum spectrum at specific values of the magnetic field and are robust at any strength of the electron-electron interaction.

规范对称性、接触约化与奇异场论

Gauge Symmetries, Contact Reduction, and Singular Field Theories

本文通过De Donder-Weyl形式将全局尺度对称性的约化方法推广到奇异拉格朗日系统，并应用于经典广义相对论。

Applications of Classical Scaling Symmetry

Any symmetry reduces a second-order differential equation to a first-order equation: variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, but symmetries of only the equations of motion (exemplified by scale-invariant hydrostatics), yield first-order {\em non-conservation laws} between invariants. We obtain these conservation laws by extending Noether's Theorem to non-variational symmetries, and present a variational formulation of spherical adiabatic hydrostatics. For scale-invariant hydrostatics, we directly recover all the published properti…

Fluctuations around classical solutions for gauge theories in Lagrangian and Hamiltonian approach

We analyze the dynamics of gauge theories and constrained systems in general under small perturbations around a classical solution (background) in both Lagrangian and Hamiltonian formalisms. We prove that a fluctuations theory, described by a quadratic Lagrangian, has the same constraint structure and number of physical degrees of freedom as the original non-perturbed theory, assuming the non-degenerate solution has been chosen. We show that the number of Noether gauge symmetries is the same in both theories, but that the gauge algebra in the fluctuations theory becomes Abelianized. We also sh…