Symmetries and conservation laws in non-Hermitian field theories

Physics Letters B, 2005

Some quantum field theories described by non-Hermitian Hamiltonians are investigated. It is shown that for the case of a free fermion field theory with a γ 5 mass term the Hamiltonian is PT -symmetric. Depending on the mass parameter this symmetry may be either broken or unbroken. When the PT symmetry is unbroken, the spectrum of the quantum field theory is real. For the PT -symmetric version of the massive Thirring model in two-dimensional space-time, which is dual to the PT -symmetric scalar Sine-Gordon model, an exact construction of the C operator is given. It is shown that the PT -symmetric massive Thirring and Sine-Gordon models are equivalent to the conventional Hermitian massive Thirring and Sine-Gordon models with appropriately shifted masses.

Journal of Geometry and Physics, 2008

We present new results on the correspondence between symmetries, conservation laws and variational principles for field equations in general non-abelian gauge theories. Our main result states that second order field equations possessing translational and gauge symmetries and the corresponding conservation laws are always derivable from a variational principle. We also show by the way of examples that the above result fails in general for third order field equations.

Arxiv preprint arXiv: …, 2010

The recently-developed techniques of Noether analysis of the quantum-group spacetime symmetries of some noncommutative field theories rely on the ad hoc introduction of some peculiar auxiliary transformation parameters, which appear to have no role in the structure of the quantum group. We here show that it is possible to set up the Noether analysis directly in terms of the quantum-group symmetry transformations, and we therefore establish more robustly the attribution of the conserved charges to the symmetries of interest. We also characterize the concept of "time independence" (as needed for conserved charges) in a way that is robust enough to be applicable even to theories with space/time noncommutativity, where it might have appeared that any characterization of time independence should be vulnerable to changes of ordering convention.

2009

The symmetry properties of a proposal to go beyond relativistic quantum field theory based on a modification of the commutation relations of fields are identified. Poincaré invariance in an auxiliary spacetime is found in the Lagrangian version of the path integral formulation. This invariance is contrasted with the idea of Doubly (or Deformed) Special Relativity (DSR). This analysis is then used to go from the free theory of a complex field to an interacting field theory.

International Journal of Geometric Methods in Modern Physics, 2004

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.

Journal of Physics A: Mathematical and Theoretical, 2008

The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, describing the evolution of the associated component of the momentum map. Keeping up with the underlying geometric philosophy, a small modification of the derivation of the momentum lemma allows us to treat also generalized nonholonomic symmetries, which are vector fields along a projection. Such symmetries arise for example in practical examples of nonholonomic field theories such as the Cosserat rod, for which we recover both energy conservation (a previously known result), as well as a modified conservation law associated with spatial translations.

Physical review, 1973

This paper discusses the role played by conserved currents in fixing the structure of currently popular renormalizable theories of strong electromagnetic and weak interactions. The major objective of this work is to show that these theories correspond to another kind of symmetry-which we call a Higgs-type symmetry-and to clarify the relation of this scheme to the already familiar normal and Goldstone symmetries. In order to do this, we introduce a language * which makes no reference to any specific Lagrangian formalism and so avoids questions of whether or not hadrons are composite and whether or not the Goldstone bosons (massless particles of these theories) necessarily-have massive partners, For pedagogical reasons, we discuss the original Weinberg model of leptons and a model coupling leptons and hadrons from the current algebra point of view.

Modern Physics Letters A, 2004

We study space-time symmetries in Non-Commutative (NC) gauge theory in the (constrained) Hamiltonian framework. The specific example of NC CP (1) model, posited in [9], has been considered. Subtle features of Lorentz invariance violation in NC field theory were pointed out in . Out of the two -Observer and Particle -distinct types of Lorentz transformations, symmetry under the former, (due to the translation invariance), is reflected in the conservation of energy and momentum in NC theory. The constant tensor θ µν (the noncommutativity parameter) destroys invariance under the latter.

Physical Review D, 2020

We generalise our previous formulation of gauge-invariant PT-symmetric field theories to include models with non-Abelian symmetries and discuss the extension to such models of the Englert-Brout-Higgs-Kibble mechanism for generating masses for vector bosons. As in the Abelian case, the non-Abelian gauge fields are coupled to non-conserved currents. We present a consistent scheme for gauge fixing, demonstrating Becchi-Rouet-Stora-Tyutin invariance, and show that the particle spectrum and interactions are gauge invariant. We exhibit the masses that gauge bosons in the simplest two-doublet SU(2)×U(1) model acquire when certain scalar fields develop vacuum expectation values: they and scalar masses depend quartically on the non-Hermitian mass parameter µ. The bosonic mass spectrum differs substantially from that in a Hermitian two-doublet model. This non-Hermitian extension of the Standard Model opens a new direction for particle model building, with distinctive predictions to be explored further.

2016

A didactic and systematic derivation of Noether point symmetries and conserved currents is put forward in special relativistic field theories, without a priori assumptions about the transformation laws. Given the Lagrangian density, the invariance condition develops as a set of partial differential equations determining the symmetry transformation. The solution is provided in the case of real scalar, complex scalar, free electromagnetic, and charged electromagnetic fields. Besides the usual conservation laws, a less popular symmetry is analyzed: the symmetry associated with the linear superposition of solutions, whenever applicable. The role of gauge invariance is emphasized. The case of the charged scalar particle under external electromagnetic fields is considered, and the accompanying Noether point symmetries determined.