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We formalize a notion of discrete Lorentz transforms for quantum walks (QW) and quantum cellular automata (QCA), in -dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the clock QW and the clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.

In the current study, we conducted a theoretical study to derive the Lorentz transformation between inertial frames of reference moving in two‐dimensional spacetime continuum. The invariance of the space–time interval equation, with use of the derived two‐dimensional transformations, yields the notion of 2 + 2‐dimensional spacetime continuum which consists of two space and two time coordinates. The velocity addition formulas, Lorentz transformations of energy and momentum are then investigated in 2 + 2‐dimensional spacetime frame. Additionally, we investigated the concept of four‐vector in 2 + 2 dimensions and further discussed their transformation based on the matrix equation, which is fully consistent with the Lorentz invariant energy–momentum relation.

We present a theoretical framework for the dynamics of bosonic Bogoliubov quasiparticles. We call it Lorentz quantum mechanics because the dynamics is a continuous complex Lorentz transformation in complex Minkowski space. In contrast, in usual quantum mechanics, the dynamics is the unitary transformation in Hilbert space. In our Lorentz quantum mechanics, three types of state exist: space-like, light-like and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as a matrix form of a Lorentz transformation, and the construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in these mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where bosonic Bogoliubov quasi-particles and their Lorentz quantum dynamics can arise, are presented. They are a one-dimensional fermion gas, Bose-Einstein condensate (or superfluid), and one-dimensional antiferromagnet.

We study the mixing of the Gluino-Glue operator in N=1 Supersymmetric Yang–Mills theory (SYM), both in dimensional regularization and on the lattice. We calculate its renormalization, which is not merely multiplicative, due to the fact that this operator can mix with non-gauge invariant operators of equal or, on the lattice, lower dimension. These operators carry the same quantum numbers under Lorentz transformations and global gauge transformations, and they have the same ghost number. We compute the one-loop quantum correction for the relevant two-point and three-point Green’s functions of the Gluino-Glue operator. This allows us to determine renormalization factors of the operator in the MS¯ scheme, as well as the mixing coefficients for the other operators. To this end our computations are performed using dimensional and lattice regularizations. We employ a standard discretization where gluinos are defined on lattice sites and gluons reside on the links of the lattice; the discretization is based on Wilson’s formulation of non-supersymmetric gauge theories with clover improvement. The number of colors, Nc, the gauge parameter, β, and the clover coefficient, cSW, are left as free parameters.

We constructed gamma operators which are superalgebraic analogs of the Dirac gamma matrices as well as two additional gamma operators which have no analogs in the Dirac theory. We found a new mechanism of the left-right symmetry breaking which is absent in the usual theory of Dirac spinors. We constructed Lorentz invariant gamma operators from operators of creation and annihilation of spinors. These gamma operators are also analogs of the Dirac gamma matrices, however they are not related to Lorentz transformations but generate vector fields as affine spinor connections. We have shown that the theory is equivalent to an extended version of the Pati-Salam theory.

In the standard formulation, the f ( T ) field equations are not invariant under local Lorentz transformations, and thus the theory does not inherit the causal structure of special relativity. Actually, even locally violation of causality can occur in this formulation of f ( T ) gravity. A locally Lorentz covariant f ( T ) gravity theory has been devised recently, and this local causality problem seems to have been overcome. The non-locality question, however, is left open. If gravitation is to be described by this covariant f ( T ) gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its field equations allow homogeneous Gödel-type solutions, which necessarily leads to violation of causality on non-local scale. Here, to look into the potentialities and difficulties of the covariant f ( T ) theories, we examine whether they admit Gödel-type solutions. We take a combination of a perfect fluid with electromagnetic plus a scalar field as source, and determine a general Gödel-type solution, which contains special solutions in which the essential parameter of Gödel-type geometries, m 2 , defines any class of homogeneous Gödel-type geometries. We show that solutions of the trigonometric and linear classes ( m 2 < 0 and m = 0 ) are permitted only for the combined matter sources with an electromagnetic field matter component. We extended to the context of covariant f ( T ) gravity a theorem which ensures that any perfect-fluid homogeneous Gödel-type solution defines the same set of Gödel tetrads h A μ up to a Lorentz transformation. We also showed that the single massless scalar field generates Gödel-type solution with no closed time-like curves. Even though the covariant f ( T ) gravity restores Lorentz covariance of the field equations and the local validity of the causality principle, the bare existence of the Gödel-type solutions makes apparent that the covariant formulation of f ( T ) gravity does not preclude non-local violation of causality in the form of closed time-like curves.

In most approaches to fundamental physics, spacetime symmetries are postulated a priori and then explicitly implemented in the theory. This includes Lorentz covariance in quantum field theory and diffeomorphism invariance in quantum gravity, which are seen as fundamental principles to which the final theory has to be adjusted. In this paper, we suggest, within a much simpler setting, that this kind of reasoning can actually be reversed, by taking an operational approach inspired by quantum information theory. We consider observers in distinct laboratories, with local physics described by the laws of abstract quantum theory, and without presupposing a particular spacetime structure. We ask what information-theoretic effort the observers have to spend to synchronize their descriptions of local physics. If there are 'enough' observables that can be measured universally on several different quantum systems, we show that the observers' descriptions are related by an element of the orthochronous Lorentz group O + ( 3 , 1 ) , together with a global scaling factor. Not only does this operational approach predict the Lorentz transformations, but it also accurately describes the behavior of relativistic Stern-Gerlach devices in the WKB approximation, and it correctly predicts that quantum systems carry Lorentz group representations of different spin. This result thus hints at a novel information-theoretic perspective on spacetime.

This work explores a Standard Model extension possibility, that violates Lorentz invariance, preserving the space-time isotropy and homogeneity. In this sense HMSR represents an attempt to introduce an isotropic Lorentz Invariance Violation in the elementary particle SM. The theory is constructed starting from a modified kinematics, that takes into account supposed quantum effects due to interaction with the space-time background. The space-time structure itself is modified, resulting in a pseudo-Finsler manifold. The SM extension here provided is inspired by the effective fields theories, but it preserves covariance, with respect to newly introduced modified Lorentz transformations. Geometry perturbations are not considered as universal, but particle species dependent. Non universal character of the amended Lorentz transformations allows to obtain visible physical effects, detectable in experiments by comparing different perturbations related to different interacting particles species.

In classical electrodynamics, by motion, it always means a relative movement of two observers in inertia reference frames, so that the covariance of the Maxwell's equations is preserved respectively in two spaces under Lorentz transformation. The energy is thus conservative for the electromagnetic system. The theory for describing the electromagnetic behavior of the charged particles in vacuum space can be well described using the special relativity because of the invariance of the speed of light in vacuum. However, for engineering applications, the media have shapes and sizes and may move with acceleration, and a system may have multiple moving objects that may be correlated or independently under external mechanical triggering. This paper presents the theory for describing the electromagnetic phenomena in this electro-magnetic-mechano system. We mainly introduce the Maxwell's equations for a mechano-driven media system (MEs-f-MDMS) under low-speed approximation (v << c). We concluded that the MEs-f-MDMS are required for describing the electrodynamics inside a moving object that moves not only with accelerated translation motion but also has rotation motion. The classical Maxwell's equations are to describe the electrodynamics in the region where there is no local medium movement. The full solutions of the two regions satisfy the boundary conditions, so that the rotation of the object affects the electromagnetic field at vicinity. The theoretical approaches for solving the MEs-f-MDMS are also presented.

Using the analogy with the properties of plane electromagnetic waves in Minkowski space, a definition of an affine-metric space of the plane wave type is given, which is characterized by the null action of the Lie derivative on the 40 components of the nonmetricity 1-form in the 4-dimensional affine-metric space. This leads to the conclusion that the nonmetricity of a plane wave type is determined by five arbitrary functions of delayed time. A theorem on the structure of the nonmetricity of the plane wave type is proved, which states that parts of the nonmetricity 1-form irreducible with respect to the Lorentz transformations of the tangent space, such as the Weyl 1-form, the trace 1-form, and the symmetric 1-form, are defined by one arbitrary function each, and the antisymmetric 1-form is defined by two arbitrary functions. Presence of arbitrary functions in the description of nonmetricity plane waves allows transmitting information with the help of nonmetricity waves.