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The Lorentz transformation equations, which are considered the cornerstone of the special theory of relativity, are fundamental theoretical tools to describe the transformation of spacetime coordinates between frames of reference moving at a constant velocity relative to each other. Here we obtain the Lorentz transformation equations between frames of reference moving in the three dimensions of space and further we present the matrix form of the obtained transformation equations with the help of the position six-vector. Moreover, we discuss the notion of six-velocity and six-momentum, which appear to be unknown in the literature, and we also investigate its main consequences, including demonstrating that it is completely consistent with many of the familiar outcomes of four-vector and deriving the transformation of six-momentum and energy that are completely compatible with the Lorentz invariant energy-momentum relation.

It is shown that Lorentz invariance implies that in general flavor neutrinos in oscillation experiments are superpositions of massive neutrinos with different energies and different momenta. It is also shown that for each process in which neutrinos are produced, there is either a Lorentz frame in which all massive neutrinos have the same energy or a Lorentz frame in which all massive neutrinos have the same momentum. In the case of neutrinos produced in two-body decay processes, there is a Lorentz frame in which all massive neutrinos have the same energy.

A review of experimental tests of time dilation in special relativity is given, with an emphasis on the recent heavy-ion storage-ring work, currently the most sensitive method. The experimental results are evaluated in the context of the kinematic test theories of Robertson and Mansouri and Sexl, which assume the existence of a preferred inertial reference frame. Prospects for future improvements and the sensitivity of time dilation experiments to Lorentz and CPT violating extensions of the standard model are discussed.

Lorentz invariant supersymmetric deformations of superspaces based on Moyal star product parametrized by Majorana spinor λ_a and Ramond Grassmannian vector in the spinor realization⁴² are proposed. The map of supergravity background into composite supercoordinates: valid up to the second order corrections in deformation parameter h and transforming the background dependent Lorentz noninvariant (anti)commutators of supercoordinates into their invariant Moyal brackets is revealed. We found one of the deformations to depend on the axial vector and to vanish for the θ components with the same chiralities. The deformations in the (super)twistor picture are discussed.

We show that a general principle of physical independence or physical invariance of mathematical background manifold leads to a replacement of the common derivative operators by the covariant co-derivative ones. This replacement naturally induces a background matrix, by means of which we obtain an effective Lagrangian for the minimal standard model with supplement terms characterizing Lorentz invariance violation or anisotropy of spacetime. We construct a simple model of the background matrix and find that the strength of Lorentz violation of proton in the photopion production is of the order 10^-23.

One of the disadvantages of the Hamiltonian formulation is that Lorentz invariance is not manifest in the former. Given a Hamiltonian, there is no simple way to check whether it is relativistic or not. One would either have to solve for the equations of motion or calculate the Poisson brackets of the Noether charges to perform such a check. In this paper we show that, for a class of Hamiltonians, it is possible to check Lorentz invariance directly from the Hamiltonian. Our work is particularly useful for theories where the other methods may not be readily available.

We report on a measurement of the constancy and anisotropy of the speed of light relative to the electrons in photon–electron scattering. We used the Compton scattering asymmetry measured by the new Compton polarimeter in Hall C at Jefferson Lab (JLab) to test for deviations from unity of the vacuum refractive index (n). For photon energies in the range of 9–46 MeV, we obtain a new limit of 1 − n < 1.4 × 10${}^{-8}$. In addition, the absence of sidereal variation over the six-month period of the measurement constrains any anisotropies in the speed of light. These constitute the first study of Lorentz invariance (LI) using Compton asymmetry. Within the minimal Standard Model extension (MSME) framework, our result yield limits on the photon and electron coefficients ${\tilde{\kappa }}_{0^{+}}^{YZ}$, $c_{TX}$, ${\tilde{\kappa }}_{0^{+}}^{ZX}$ and $c_{TY}$. Although these limits are several orders of magnitude larger than the current best limits, they demonstrate the feasibility of using Compton asymmetry for tests of LI. Future parity-violating electron-scattering experiments at JLab will use higher energy electrons enabling better constraints.

One of the most powerful probes of new physics is the polarized cosmic microwave background (CMB). The detection of a nonzero polarization angle rotation between the CMB surface of last scattering and today could provide evidence of Lorentz-violating physics. The purpose of this paper is two-fold. First, we review one popular mechanism for polarization rotation of CMB photons: the pseudo-Nambu–Goldstone boson (PNGB). Second, we propose a method to use the Polarbear experiment to constrain Lorentz-violating physics in the context of the Standard Model Extension (SME), a framework to standardize a large class of potential Lorentz-violating terms in particle physics.

Observations of high energy neutrinos, both in the laboratory and from cosmic sources, can be a useful probe in searching for new physics. Such observations can provide sensitive tests of Lorentz invariance violation (LIV), which may be the result of quantum gravity physics (QG). We review some observationally testable consequences of LIV using effective field theory (EFT) formalism. To do this, one can postulate the existence of additional small LIV terms in free particle Lagrangians, suppressed by powers of the Planck mass. The observational consequences of such terms are then examined. In particular, one can place limits on a class of non-renormalizable, mass dimension five and six Lorentz invariance violating operators that may be the result of QG.

The interactions of different particle species with the foamy space–time fluctuations expected in quantum gravity theories may not be universal, in which case different types of energetic particles may violate Lorentz invariance by varying amounts, violating the equivalence principle. We illustrate this possibility in two different models of space–time foam based on D-particle fluctuations in either flat Minkowski space or a stack of intersecting D-branes. Both models suggest that Lorentz invariance could be violated for energetic particles that do not carry conserved charges, such as photons, whereas charged particles such electrons would propagate in a Lorentz-inavariant way. The D-brane model further suggests that gluon propagation might violate Lorentz invariance, but not neutrinos. We argue that these conclusions hold at both the tree (lowest-genus) and loop (higher-genus) levels, and discuss their implications for the phenomenology of quantum gravity.

The reason why the elementary-particle theory must be Lorentz invariant is investigated under the principle of quantum priority, which implies that the existence of any space–time manifold such as the Minkowski space should not be assumed logically prior to the formulation of quantum-theoretical framework. The consideration is based on the canonical operator formalism of quantum Einstein gravity, which is formulated without introducing any classical background space–time nor any particular metric signature. The Lorentz invariance of the elementary-particle theory is derived as a consequence of the spontaneous breakdown of general linear invariance on the basis of quantum Einstein gravity. The pseudo-Riemannian space–time can appear not as a consequence of the presence of gravitation, but as that of the spontaneous breakdown of translational invariance.

These are intended to be review notes on emergent symmetries, i.e. symmetries which manifest themselves in specific sectors of energy in many systems. The emphasis is on the physical aspects rather than computation methods. We include some background material and go through more recent problems in field theory, statistical mechanics and condensed matter. These problems illustrate how some important symmetries, such as Lorentz invariance and supersymmetry, usually believed to be fundamental, can arise naturally in low-energy regimes of systems involving a large number of degrees of freedom. The aim is to discuss how these examples could help us to face other complex and fundamental problems.

A polynomial basis for parity-even three-point amplitudes of higher-spin massless fermions and bosons are derived in four-dimensional space–time from first principles. This basis can be used to construct three-point amplitudes of polarizations of any rank. The results are presented using polarization tensors and tensor-spinors, which is convenient when they are applied to Lagrangian construction.

Basis tensor gauge theory (BTGT) is a vierbein analog reformulation of ordinary gauge theories in which the vierbein field describes the Wilson line. After a brief review of the BTGT, we clarify the Lorentz group representation properties associated with the variables used for its quantization. In particular, we show that starting from an SO(1,3) representation satisfying the Lorentz-invariant U(1,3) matrix constraints, BTGT introduces a Lorentz frame choice to pick the Abelian group manifold generated by the Cartan subalgebra of U(1,3) for the convenience of quantization even though the theory is frame independent. This freedom to choose a frame can be viewed as an additional symmetry of BTGT that was not emphasized before. We then show how an $S_4$ permutation symmetry and a parity symmetry of frame fields natural in BTGT can be used to construct renormalizable gauge theories that introduce frame-dependent fields but remain frame independent perturbatively without any explicit reference to the usual gauge field.

Comparisons of the positive and negative halves of the distributions of parity-odd event variables in particle-physics experimental data can provide sensitivity to sources of nonstandard parity violation. Such techniques benefit from lacking first-order dependence on simulations or theoretical models, but have hitherto lacked systematic means of enumerating all discoverable signals. To address that issue this paper seeks to construct sets of parity-odd event variables which may be proved to be able to reveal the existence of any Lorentz-invariant source of nonstandard parity violation which could be visible in data consisting of groups of real nonspace-like four-momenta exhibiting certain permutation symmetries.

The Lorentz transformation describes differential simultaneity, which reflects the offsetting of time with distance between reference frames. Differential simultaneity is essential for Lorentz invariance. Here, the current experimental support for differential simultaneity is assessed. Differential simultaneity can be mathematically removed from the Lorentz transformation to produce an alternate transformation that describes relativistic effects in the absence of differential simultaneity. Differential simultaneity is shown to be required for Lorentz symmetry. The alternate transformation is used as a contrast for the Lorentz transformation to assess whether relativistic experiments provide definitive evidence for differential simultaneity. If the alternate transformation is compatible with an experiment, then the experiment does not provide definitive evidence for differential simultaneity because it can be explained in the absence of differential simultaneity. Here, the alternate transformation is shown to be compatible with the broad array of current relativistic experiments. This implies that current experiments do not provide definitive evidence for differential simultaneity. Thus, an essential aspect of Lorentz symmetry currently lacks definitive experimental support.

A Ramond open superstring quantization is studied in a flat D-dimensional Minkowski spacetime in a static gauge. This gauge degenerates the boost generators. A deformation in the structure of the boosts is proposed at the classical level with some energy-dependent coefficients. We quantize the theory and propose the quantum version of the deformation of the boost operators. The energy-dependent coefficients are determined in the first and second excited levels in order to keep the physical Hilbert space Lorentz invariant. The critical space–time dimension $D=10$ and ordering ambiguity constant $a=0$ are recovered. We give the connection between the covariant and static gauge quantizations.

We review some optical experiments, carried out in the last decade, which evidence an anomalous behavior of photon systems. Their results are apparently at variance with both standard quantum mechanics (in the Copenhagen interpretation) and usual (classical and quantum) electrodynamics. In particular, they can be interpreted as a virtual interference involving the pilot waves associated to photons (according to Bohmian quantum mechanics). The anomalous effects exhibit a threshold behavior in energy and space, which agrees with results obtained on the electromagnetic breakdown of local Lorentz invariance. A possible connection between these seemingly unrelated implications of the observed phenomenon can be set by assuming that the pilot wave of a photon is a deformation of spacetime ("shadow of light").

The question of the superluminal speed of information was stopped at crossroads in the last few years. According to one point of view, this speed must be limited to the light velocity in vacuum, whereas a different point of view is more open in this respect and, under specific conditions, this limit is considered surmountable. Very recently, a third approach (based on the hypothesis of a local broken Lorentz-invariance) was proposed and, if confirmed, would go beyond the controversy of the two points of view mentioned above. It is therefore worthwhile to recall attention to this problem, which is far from having a definite solution. The present paper reports some experimental results (similar to those of Ref. 1) which can contribute to these discussions, and also considers the fact that they seem to give some support to the aforesaid third approach, although revised in terms of decaying waves.

We discuss the possible violation of local Lorentz invariance (LLI) arising from a faster-than-light neutrino speed. A toy calculation of the LLI violation parameter $\delta$, based on the (disclaimed) OPERA data, suggests that the values of $\delta$ are determined by the interaction involved, and not by the energy range. This hypothesis is further corroborated by the analysis of the more recent results of the BOREXINO, LVD and ICARUS experiments.