Narrow Results

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.

We study the appearance of anomalies of the Master Ward Identity, which is a universal renormalization condition in perturbative QFT. The main insight of the present paper is that any violation of the Master Ward Identity can be expressed as a local interacting field; this is a version of the well-known Quantum Action Principle of Lowenstein and Lam. Proceeding in a proper field formalism by induction on the order in ħ, this knowledge about the structure of possible anomalies as well as techniques of algebraic renormalization are used to remove possible anomalies by finite renormalizations. As an example, the method is applied to prove the Ward identities of the O(N) scalar field model.

In this paper, we study symmetries, Newtonoid vector fields, conservation laws, Noether's theorem and its converse, in the framework of the k-symplectic formalism, using the Frölicher–Nijenhuis formalism on the space of k¹-velocities of the configuration manifold.

For the case k = 1, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's theorem and its converse. For the case k > 1, we provide a new proof for Noether's theorem, which shows that, in the k-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

We define symmetries in non-relativistic quantum electrodynamics, which have the physical interpretation of rotation, parity, and time reversal symmetry. We collect transformation properties related to these symmetries in Fock space representation as well as in the Schrödinger representation. As an application, we generalize and improve theorems about Kramers’ degeneracy in non-relativistic quantum electrodynamics.

It is well known that $k$-contact geometry is a suitable framework to deal with non-conservative field theories. In this paper, we study some relations between solutions of the $k$-contact Euler–Lagrange equations, symmetries, dissipation laws and Newtonoid vector fields. We review the $k$-contact Euler–Lagrange equations written in terms of $k$-vector fields and sections and provide new results relating the solutions in both approaches. We also study different kinds of symmetries depending on the structures they preserve: natural (preserving the Lagrangian function), dynamical (preserving the solutions), and $k$-contact (preserving the underlying geometric structures) symmetries. For some of these symmetries, we provide Noether-like theorems relating symmetries and dissipation laws. We also analyze the relation between $k$-contact symmetries and Newtonoid vector fields. Throughout the paper, we will use the damped vibrating string as our main illustrative example.

Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a method of reduction which enables us to remove gauge symmetries as well as non-gauge "rigid" symmetries at once. This method is compared with other step-by-step reduction procedures. As particular examples in this framework, we discuss the reduction of time-dependent dynamical systems with symmetry, the reduction of a mechanical model of field theories with gauge and non-gauge symmetries, and the gauge reduction of the system made of a conformal particle.

We give a characterization of modular curves by a single symmetry of Hecke type. In the proof, we use the theorem of André, which characterizes modular curves in terms of special points.

To find the discrete symmetries of a Hamilton operator Ĥ is of central importance in quantum theory. Here, we describe and implement a brute force method to determine the discrete symmetries given by permutation matrices for Hamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermi systems are considered as examples. A computer algebra implementation in SymbolicC++ is provided.

The symmetry algebra of a QFT in the presence of an external EM background (named "residual symmetry") is investigated within a Lie-algebraic, model-independent scheme. Some results previously encountered in the literature are extended here. In particular we compute the symmetry algebra for a constant EM background in D = 3 and D = 4 dimensions. In D = 3 dimensions the residual symmetry algebra, for generic values of the constant EM background, is isomorphic to , with the centrally extended two-dimensional Poincaré algebra. In D = 4 dimension the generic residual symmetry algebra is given by a seven-dimensional solvable Lie algebra which is explicitly computed. Residual symmetry algebras are also computed for specific non-constant EM backgrounds and in the supersymmetric case for a constant EM background. The supersymmetry generators are given by the "square roots" of the deformed translations.

We write down two E₁₁ invariant conditions which at low levels reproduce the known half BPS conditions for type II theories. These new conditions contain, in addition to the familiar central charges, an infinite number of further charges which are required in an underlying theory of strings and branes. We comment on the application of this work to higher derivative string corrections.

We review recent research in neutrino physics, including neutrino oscillations to test time reversal and CP symmetry violations, the measurement of parameters in the U matrix, sterile neutrino emission causing pulsar kicks, and neutrino energies in the neutrinosphere.

The nonlinear realization of the semi-direct product of $E_{11}$ with its vector representation leads to equation of motions for the field graviton, three-form, six-form, dual graviton and the level four fields which correctly describe the degree of freedom of 11-dimensional supergravity at the linearized level. The equations with one derivative generically hold as equivalence relations and are often duality relations. From these equations, by taking derivatives, one can derive equations that are equations of motion of the familiar kind. The entire hierarchy of equations is $E_{11}$ invariant and the construction does not require any steps beyond $E_{11}$. We review these past developments with an emphasis on the features that have been overlooked in hep-th:1703.01305, whose alternative approach we also comment on.

We determine the conformal algebra of Bianchi III and Bianchi V spacetimes or, equivalently, we determine all Bianchi III and Bianchi V spacetimes which admit a proper conformal Killing vector (CKV). The algorithm that we use has been developed in [M. Tsamparlis et al.Class. Quantum. Grav.15, 2909 (1998)] and concerns the computation of the CKVs of decomposable spacetimes. The main point of this method is that a decomposable space admits a CKV if the reduced space admits a gradient homothetic vector, the latter being possible only if the reduced space is flat or a space of constant curvature. We apply this method in a stepwise manner starting from the two-dimensional spacetime which admits an infinite number of CKVs and we construct step by step the Bianchi III and V spacetimes by assuming that CKVs survive as we increase the dimension of the space. We find that there is only one Bianchi III and one Bianchi V spacetime which admit at maximum one proper CKV. In each case, we determine the CKV and the corresponding conformal factor. As a first application in these two spacetimes, we study the kinematics of the comoving observers and the dynamics of the corresponding cosmological fluid. As a second application, we determine in these spacetimes generators of the Lie symmetries of the wave equation.

We discuss the application of point transformations in cosmological studies and the application of Lie symmetries as a selection rule for the constraint of the free parameters of the theories. We show that recent published results in Brans–Dicke theory are not correct while the main motivation of that applied is not mathematically inadequate.

We investigate the existence of Liouville integrable cosmological models in hybrid metric-Palatini theory. Specifically, we use the symmetry conditions for the existence of quadratic in the momentum conservation laws for the field equations as constraint conditions for the determination of the unknown functional form of the theory. The exact and analytic solutions of the integrable systems found in this study are presented in terms of quadratics and Laurent expansions.

This paper aims to investigate Conformal Vector Fields (CVFs) of Bianchi type-I spacetimes. A set of 10-coupled Partial Differential Equations (PDEs) is obtained from the conformal Killing equations. These equations are solved by using direct integration techniques to explore the components of CVFs. Utilizing these components, we get a system of three integrability conditions. Finally, we achieve CVFs along with conformal factors for unique possibilities of unknown metric functions from the solution of these conditions. From our results, it is examined that Bianchi type-I spacetimes admit five or fifteen CVFs for specific choices of metric functions.

We discuss the different possibilities of constructing the various energy–momentum tensors for noncommutative gauge field models. We use Jackiw's method in order to get symmetric and gauge invariant stress tensors — at least for commutative gauge field theories. The noncommutative counterparts are analyzed with the same methods. The issues for the noncommutative cases are worked out.

We argue that Poincaré symmetry can be implemented in noncommutative field theory (NCFT) if we allow the parameter of noncommutative deformation θ^μν to change as a two-tensor under the corresponding space–time symmetry. The implementation is consistent with the definition of θ^μν in terms of space–time coordinates and with the Moyal star product. Inspired by the standard definition of a variational symmetry we found a universal way to correct the implementation of the Poincaré symmetry by a term proportional to the variation of θ^μν in such a way that the new transformation define a symmetry of the theory. Finally we present as an example the case of NCYM theory and comment about the obstructions to implement generalized space–time symmetries in NCFT-like conformal or diffeomorphism transformations.

In the two-Higgs-doublet model (THDM), generalized-CP transformations ( where X is unitary) and unitary Higgs-family transformations (φ_i→U_ijφ_j) have recently been examined in a series of papers. In terms of gauge-invariant bilinear functions of the Higgs fields φ_i, the Higgs-family transformations and the generalized-CP transformations possess a simple geometric description. Namely, these transformations correspond in the space of scalar-field bilinears to proper and improper rotations, respectively. In this formalism, recent results relating generalized CP transformations with Higgs-family transformations have a clear geometric interpretation. We will review what is known regarding THDM symmetries, as well as derive new results concerning those symmetries, namely how they can be interpreted geometrically as applications of several CP transformations.

The kaon system has been a privileged observatory for the discovery and the study of the violations of fundamental discrete symmetries, elucidating the role of such phenomena within the present successful theory of particle physics. Its peculiar characteristics from an experimental and a theoretical point of view make this system an ideal one for the investigation of elusive asymmetries. An overview of recent measurements is presented, as well as a discussion of the rich future plans for high-precision measurements with several different approaches.