关于Noether对称性的两种理解

介绍了对Lagrange系统Noether对称性的两种理解，一种理解为Lagrange函数的不变性，另一种理解为作用量的不变性.研究表明，这两种理解是不同的.在一些条件下，Lagrange函数的不变性可以成为作用量的不变性，在另一些条件下，作用量的不变性可以成为Lagrange函数的不变性.将Noether对称性理解为作用量的不变性是合理的. 关键词： Lagrange系统 Noether对称性 作用量的不变性 Lagrange函数的不变性     

General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas

We show that the Lagrangian for interacting nonrelativistic particles can be coupled to an external gauge field and metric tensor in a way that exhibits a nonrelativistic version of general coordinate invariance. We explore the consequences of this invariance on the example of the degenerate Fermi gas at infinite scattering length, where conformal invariance also plays an important role. We find the most general effective Lagrangian consistent with both general coordinate and conformal invariance to leading and next-to-leading orders in the momentum expansion. At the leading order the Lagrangian contains one phenomenological constant and reproduces the results of the Thomas-Fermi theory and superfluid hydrodynamics. At the next-to-leading order there are two additional constants. We express various physical quantities through these constants.     

Discrete symmetrical perturbation and variational algorithm of disturbed Lagrangian systems

We investigate the perturbation to discrete conformal invariance and the adiabatic invariants of Lagrangian systems. A variational algorithm is proposed for a system subjected to the perturbation quantities. The discrete determining equations of the perturbations to conformal invariance are established. For perturbed Lagrangian systems, the condition of the existence of adiabatic invariant is derived from the discrete perturbation to conformal invariance. The numerical simulations demonstrate that the variational algorithm has the higher precision and the longer time stability than the standard numerical method.     

Lorentz invariance of gravitational Lagrangians in the space of reference frames

The recently proposed theories of gravitation in the space of reference framesS are based on a Lagrangian invariant with respect to the homogeneous Lorentz group. However, in theories of this kind, the Lorentz invariance is not a necessary consequence of some physical principles, as in the theories formulated in space-time, but rather a purely esthetic request. In the present paper, we give a systematic method for the construction of gravitational theories in the spaceS, without assuming a priori the Lorentz invariance of the Lagrangian. The Einstein-Cartan equations of gravitation are obtained requiring only that the Lagrangian is invariant under proper rotations and has particular transformation properties under space reflections and space-time dilatations     

Variational principle for theP(4) affine theory of gravitation and electromagnetism

We propose a Lagrangian for theP(4) theory of gravitation and electromagnetism which is a straightforward generalization of the Einstein Lagrangian. A constrained Palatini variation of this Lagrangian yields the geometrical Einstein-Maxwell affine field equations. We show that these results can be extended easily to include both electric and magnetic charges. Finally, we consider conservation laws arising from the invariance properties of the Lagrangian.     

间隔不变性与相对论拉格朗日力学

文章证明由时空和速度间隔不变性可以导出相对论哈密顿和拉格朗日力学，由此说明哈密顿或拉格朗日为洛仑兹不变量是时空和速度间隔不变性的必然结果。     

New theory of Lorentz violation from a general principle

We report that a general principle of physical independence of mathematical background manifolds brings a replacement of common derivative operators by co-derivative ones. Then we obtain a new Lagrangian for the ordinary minimal standard model with supplementary terms containing the Lorentz invariance violation information measured by a new matrix, denoted as the Lorentz invariance violation matrix. We thus provide a new fundamental theory to study Lorentz invariance violation effects consistently and systematically.     

Effective lagrangian and topological interactions in supersolids

We construct a low-energy effective Lagrangian describing zero temperature supersolids. Galilean invariance imposes strict constraints on the form of the effective Lagrangian. We identify a topological term in the Lagrangian that couples superfluid and crystalline modes. For small superfluid fractions, this interaction term is dominant in problems involving defects. As an illustration, we compute the differential cross section of scatterings of low-energy transverse elastic phonons by a superfluid vortex. The result is model independent.     

Cubic interaction vertex of higher-spin fields with external electromagnetic field

We fulfill the detailed analysis of coupling the charged bosonic higher-spin fields to external constant electromagnetic field in first order in external field strength. Cubic interaction vertex of arbitrary massive and massless bosonic higher-spin fields with external field is found. Construction is based on deformation of free Lagrangian and free gauge transformations by terms linear in electromagnetic field strength. In massive case a formulation with Stueckelberg fields is used. We begin with the most general form of deformations for Lagrangian and gauge transformations, admissible by Lorentz covariance and gauge invariance and containing some number of arbitrary coefficients, and require the gauge invariance of the deformed theory in first order in strength. It yields the equations for the coefficients which are exactly solved. As a result, the complete interacting Lagrangian of arbitrary bosonic higher-spin fields with constant electromagnetic field in first order in electromagnetic strength is obtained. Causality of massive spin-2 and spin-3 fields propagation in the corresponding electromagnetic background is proved.     

An independence theorem for Lagrangian systems on graded manifolds

Given a Lagrangian system on a graded manifold, we prove that the invariance of the action under independent reparametrizations of two subsystems implies the dynamical independence of those sybsystems.     

Boundary terms for globally supersymmetric actions

Supersymmetry is always broken on a manifold-with-boundary, since the supersymmetric variation of the Lagrangian is a total divergence which yields a boundary term when integrated. If attention is restricted to a subalgebra of generators which preserve the boundary, the invariance of the action can be restored by adding a boundary correction whose variation cancels that from the integrated Lagrangian. One can also impose boundary conditions which are invariant under this subalgebra.     

A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems： Adiabatic Invariants of Generalized Lutzky Type

For a Lagrangian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariant are presented in general infinitesimal transformation groups. On the basis of the invariance of disturbed Lagrangian systems under general infinitesimal transformations, the determining equations of Lie symmetries of the system are constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariant, i.e. generalized Lutzky adiabatic invariants, of a disturbed Lagrangian system are obtained by investigating the perturbation of Lie symmetries t＇or a Lagrangian system with the action of small disturbance. Finally, an example is given to illustrate the application of the method and results.     

Variational Equations and Symmetries in the Lagrangian Formalism; Arbitrary Vector Fields

We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the generalized Helmholtz equations (sometimes called the Anderson-Duchamp-Krupka equations). For the case of second-order equations and arbitrary vector fields we are able to establish a polynomial structure in the second-order derivatives. This structure is based on the some linear combinations of Olver hyper-Jacobians. We use as the main tools Fock space techniques and induction. This structure can be used to analyze Lagrangian systems with groups of Noetherian symmetries. As an illustration we analyze the case of Lagrangian equations with Abelian gauge invariance.     

Algebraic derivation of symmetry relations for disordered electronic systems

By means of superfields two time-reversal invariant disordered electronicn-orbital models one without, the other with a spin-dependent random potential can be described by the same Lagrangian except for the sign of an overall prefactor. Similarly two different treatments of a system which breaks time-reversal invariance yields the same Lagrangian but with opposite sign of the prefactor. Since this prefactor is proportional ton, identical saddle point expansions in powers of ±n ^–1 for the averaged Green's functions are obtained, relations first found diagrammatically by Oppermann and Jüngling. The invariance of the Lagrangian under unitary graded and unitary orthosymplectic transformations of the fields for systems without and with time-reversal invariance, respectively, is pointed out.Work supported in part through the Sonderforschungsbereich 123 Stochastic Mathematical Models by the Deutsche Forschungsgemeinschaft     

Birkhoff's theorem for ghost-free,tachyon-freeR + R 2 +Q 2 theories with torsion

We investigate the conditions under which the class of ghost-free, tachyon-freeR + R ² +Q ² theories with torsion satisfy Birkhoff's theorem. We prove a weakened Birkhoff theorem requiring an additional assumption of parity invariance for two Lagrangians one of which contains torsion squared terms in addition to curvature squared terms. For another Lagrangian, also containing torsion squared terms, a weakened Birkhoff theorem requiring the additional assumptions of parity invariance and constant scalar curvature is proven. A special case of this Lagrangian is shown to satisfy a weakened Birkhoff theorem requiring only the additional assumption of constant scalar curvature. In addition the explicit dependence of torsion on parity noninvariant quantities is displayed.     

Degrees of symmetry in quantum field theories

A hierarchy of possible symmetries in quantum field theory is defined, which reaches from a purely mathematical invariance to the conventional physical invariance, including the commonly discussed type of spontaneously broken symmetry (SBS). It is shown that one type of SBS, which is usually not considered, naturally leads to theories with an algebra of non-conserved currents and a non-linearly transforming phenomenological Lagrangian. An exactly solvable model is given and some general remarks are made.     

Electrodynamic Analysis of Dissipative Electromagnetic Materials Based on Fractional Derivative

The generalized Lagrangian is defined in a dissipative electromagnetic medium on the basis of the combination of dynamical analysis and fractional derivative. Lorentz medium models are obtained by formulating relevant Euler-Lagrange equations. The invariance is obtained subsequently by investigating the invariance of time variation in the system, and then the relation between the related Hamiltonian and electromagnetic energy density is investigated. Canonical equations are obtained eventually. The electrodynamic interpretation on dissipative electromagnetic systems is revealed.     

New invariance in electroweak standard model and conservation of helicity

The Lagrangian of the electroweak standard model is invariant under the chirality transformation of fermionic fields and reversal of the Higgs field. This invariance implies the conservation of helicity in weak interaction processes. The application to leptonic weak interaction is discussed in detail.