Search Results (60)

Wigner’s little groups are the subgroups of the Poincaré group whose transformations leave the four-momentum of a relativistic particle invariant. The little group for a massive particle is SO(3)-like, whereas for a massless particle, it is E(2)-like. Multiple approaches to group contractions are discussed. It is shown that the Lie algebra of the E(2)-like little group for massless particles can be obtained from the SO(3) and from the SO(2, 1) group by boosting to the infinite-momentum limit. It is also shown that it is possible to obtain the generators of the E(2)-like and cylindrical groups from those of SO(3) as well as from those of SO(2, 1) by using the squeeze transformation. The contraction of the Lorentz group SO(3, 2) to the Poincaré group is revisited. As physical examples, two applications are chosen from classical optics. The first shows the contraction of a light ray from a spherical transparent surface to a straight line. The second shows that the focusing of the image in a camera can be formulated by the implementation of the focal condition to the [ABCD] matrix of paraxial optics, which can be regarded as a limiting procedure. Full article

Building on the first paper in this series (Paper I), a unified perspective on Poincaré and Galilei physics in a 5-dimensional spacetime setting is further pursued through a consideration of the kinematics of general relativity, with the gravitational dynamics to be addressed separately. The metric of the 5-dimensional affine spacetimes governed by the Bargmann groups considered in Paper I (central extensions of the Poincaré and Galilei groups) is generalized to curved spacetime by extending the usual 1 + 3 (traditionally ‘3 + 1’) formalism of general relativity on 4-dimensional spacetime to a 1 + 3 + 1 formalism, whose spacetime kinematics is shown to be consistent with that of the usual 1 + 3 formalism. Spacetime tensor laws governing the motion of an elementary classical material particle and the dynamics of a simple fluid are presented, along with their 1 + 3 + 1 decompositions; these reference the foliation of spacetime in a manner that partially reverts the Einstein perspective (accelerated fiducial observers, and geodesic material particles and fluid elements) to a Newton-like perspective (geodesic fiducial observers, and accelerated material particles and fluid elements subject to a gravitational force). These spacetime laws of motion for particles and fluids also suggest that a strong-field Galilei general relativity would involve a limit in which not only $c\to \infty$ but also $G\to \infty$, such that $G/c^2$ remains constant. Full article

This paper elucidates the Lorentz group, a fundamental subgroup of the Poincaré group. The orbits and little groups associated with the Lorentz group are described in detail, along with their corresponding properties. The Poincaré group is presented. Another fundamental aspect of the Poincaré group is Wigner’s little groups obtained from this group. An in-depth discussion on the cases of both massive and massless relativistic particles within the context of little groups is given. Our examination extends to the properties of various special groups associated with the Poincaré group. Applications of these groups are elaborated by physical examples taken from high-energy physics and optics from both classical and quantum domains. Specifically, covariant harmonic oscillators including entangled states, proton form factors, and the parton picture as proposed by Feynman are discussed. In this context, laser cavities and shear states are also addressed. We lay out the underlying mathematics that connects these apparently disparate realms of physics. Full article

The quantum description of relativistic orientable objects by a scalar field on the Poincaré group is considered. The position of the relativistic orientable object in Minkowski space is completely determined by the position of a body-fixed reference frame with respect to the position of the space-fixed reference frame, so that all the positions can be specified by elements q of the Poincaré group. Relativistic orientable objects are described by scalar wave functions $f\left(q\right)$, where the arguments $q=(x,z)$ consist of space–time points x and of orientation variables z from $SL(2,C)$ matrices. We introduce and study the double-sided representation $\mathit{T}\left(\mathit{g}\right)f\left(q\right)=f\left(g_l^{-1}q{g}_r\right)$, $\mathit{g}=(g_l,g_r)\in \mathit{M},$ of the group $\mathit{M}$. Here, the left multiplication by $g_l^{-1}$ corresponds to a change in a space-fixed reference frame, whereas the right multiplication by $g_r$ corresponds to a change in a body-fixed reference frame. On this basis, we develop a classification of orientable objects and draw attention to the possibility of connecting these results with particle phenomenology. In particular, we demonstrate how one may identify fields described by polynomials in z with known elementary particles of spins 0, ${\displaystyle \frac{1}{2}}$, and 1. The developed classification does not contradict the phenomenology of elementary particles and, in some cases, even provides a group-theoretic explanation for it. Full article

Background: Chronic pain has been reported as one of the leading causes of disability in the world, being associated with a potential impact on autonomic balance. Objective: The aim was to compare sympathetic and parasympathetic activity through heart rate variability (HRV) between adults with and without chronic low back pain (CLBP). Methods: An observational study was conducted in which HRV parameters were recorded using time-domain measures—root mean square of successive differences between consecutive RR intervals (rMSSD), minimum and maximum heart rate variability (Min HR and Max HR), and mean heart rate (Mean HR)—and nonlinear measures—Poincaré plot indices SD1 and SD2, Stress Score (SS), and sympathetic/parasympathetic ratio (S:PS). Results: The results showed statistically significant differences between groups (p < 0.05), with higher parasympathetic activity parameters in the group of healthy subjects (rMSSD: p < 0.001; SD1: p = 0.030) and higher sympathetic activity in the CLBP group (SD2, SS, and S:PS ratio: p < 0.001). All parameters showed large effect sizes. Conclusions: These findings show the association between autonomic balance mechanisms and pain regulation in adults with CLBP. Full article

A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector $n^{\mu }$ and then inducing a representation of $SL(2,C)$, based on $p^{\mu }$ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space $\{(x^{\mu },p^{\mu }),({\zeta }^{\mu },{\pi }^{\mu })\}$, inducing a representation on the momentum ${\pi }^{\mu }$, thus providing a novel dynamical interpretation of the timelike unit vector $n^{\mu }={\pi }^{\mu }/M$. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments. Full article

We geometrically derive the explicit form of the unitary representation of the Poincaré group for vector-valued wave functions and use it to apply speed-of-light boosts to a simple polarization basis to end up with a Hawton–Baylis photon position operator with commuting components. We give explicit formulas for other photon boost eigenmodes. We investigate the underlying affine connections on the light cone in momentum space and find that while the Pryce connection is metric semi-symmetric, the flat Hawton–Baylis connection is not semi-symmetric. Finally, we discuss the localizability of photon states on closed loops and show that photon states on the circle, both unnormalized improper states and finite-norm wave packet smeared-over washer-like regions are strictly localized not only with respect to Hawton–Baylis operators with commuting components but also with respect to the noncommutative Jauch–Piron–Amrein POV measure. Full article

A semantic adjustment to what physicists mean by the terms ‘special relativity’ and ‘general relativity’ is suggested, which prompts a conceptual shift to a more unified perspective on physics governed by the Poincaré group and physics governed by the Galilei group. After exploring the limits of a unified perspective available in the setting of 4-dimensional spacetime, a particular central extension of the Poincaré group—analogous to the Bargmann group that is a central extension of the Galilei group—is presented that deepens a unified perspective on Poincaré and Galilei physics in a 5-dimensional spacetime setting. The immediate focus of this paper is classical physics on affine 4-dimensional and 5-dimensional spacetimes (‘special relativity’ as redefined here), including the electrodynamics that gave rise to Poincaré physics in the first place, but the results here may suggest the existence of a ‘Galilei general relativity’ more extensive than generally known, to be pursued in the sequel. Full article

This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form $P/Q^k$ and $1/Q$, are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple $F_1^2$ hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model. Full article

We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (“ML spacetime”). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito’s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that—under the Robertson–Walker minimum length theory—rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents. Full article

We single out a class of Lagrangians on a group manifold, for which one can introduce non-canonical coordinates in the phase space, which simplify the construction of the Poisson structure without explicitly calculating the Dirac bracket. In the case of the $SO\left(3\right)$ manifold, the application of this formalism leads to the Poincaré–Chetaev equations. The general solution to these equations is written in terms of an exponential of the Hamiltonian vector field. Full article

The null conformal boundary $\mathcal{I}$ of Minkowski spacetime $\mathbb{M}$ plays a special role in scattering theory, as it is the locus where massless particle states are most naturally defined. We construct quantum fields on $\mathcal{I}$, which create these massless states from the vacuum and transform covariantly under Poincaré symmetries. Because the latter symmetries act as Carrollian conformal isometries of $\mathcal{I}$, these quantum fields are Carrollian conformal fields. This group theoretic construction is intrinsic to $\mathcal{I}$ by contrast to existing treatments in the literature. However, we also show that the standard relativistic massless quantum fields in $\mathbb{M}$, when pulled back to $\mathcal{I}$, provide a realisation of these Carrollian conformal fields. This correspondence between bulk and boundary fields should constitute a basic entry in the dictionary of flat holography. Finally, we show that $\mathcal{I}$ provides a natural parametrisation of the massless particles as described by irreducible representations of the Poincaré group and that in an appropriate conjugate basis, they indeed transform like Carrollian conformal fields. Full article

The dynamics of cardiac signals can be studied using methods for nonlinear analysis of heart rate variability (HRV). The methods that are used in the article to investigate the fractal, multifractal and informational characteristics of the intervals between heartbeats (RR time intervals) are: Rescaled Range, Detrended Fluctuation Analysis, Multifractal Detrended Fluctuation Analysis, Poincaré plot, Approximate Entropy and Sample Entropy. Two groups of people were studied: 25 healthy subjects (15 men, 10 women, mean age: 56.3 years) and 25 patients with arrhythmia (13 men, 12 women, mean age: 58.7 years). The results of the application of the methods for nonlinear analysis of HRV in the two groups of people studied are shown as mean ± std. The effectiveness of the methods was evaluated by t-test and the parameter Area Under the Curve (AUC) from the Receiver Operator Curve (ROC) characteristics. The studied 11 parameters have statistical significance (p < 0.05); therefore, they can be used to distinguish between healthy and unhealthy subjects. It was established by applying the ROC analysis that the parameters H_q=2(MFDFA), F(α)(MFDFA) and SD₂(Poincaré plot) have a good diagnostic value; H(R/S), α₁(DFA), SD₁/SD₂(Poincaré plot), ApEn and SampEn have a very good score; α₂(DFA), α_all(DFA) and SD₁(Poincaré plot) have an excellent diagnostic score. In conclusion, the methods used for nonlinear analysis of HRV have been evaluated as effective, and with their help, new perspectives are opened in the diagnosis of cardiovascular diseases. Full article

A statistical transformation model consists of a smooth data manifold, on which a Lie group smoothly acts, together with a family of probability density functions on the data manifold parametrized by elements in the Lie group. For such a statistical transformation model, the Fisher–Rao semi-definite metric and the Amari–Chentsov cubic tensor are defined in the Lie group. If the family of probability density functions is invariant with respect to the Lie group action, the Fisher–Rao semi-definite metric and the Amari–Chentsov tensor are left-invariant, and hence we have a left-invariant structure of a statistical manifold. In the present work, the general framework of statistical transformation models is explained. Then, the left-invariant geodesic flow associated with the Fisher–Rao metric is considered for two specific families of probability density functions on the Lie group. The corresponding Euler–Poincaré and the Lie–Poisson equations are explicitly found in view of geometric mechanics. Related dynamical systems over Lie groups are also mentioned. A generalization in relation to the invariance of the family of probability density functions is further studied. Full article

Exercise stress testing (EST) has limited power in diagnosing obstructive coronary artery disease (CAD). The heart rate variability (HRV) analysis might increase the sensitivity of CAD detection. This study aimed to evaluate the correlation between short-term HRV and myocardial ischemia during EST, including the acceleration, maximum, and recovery stages of heart rate (HR). The HRV during EST from 19 healthy (RHC) subjects and 35 patients with CAD (25 patients with insignificant CAD (iCAD), and 10 patients with significant CAD (sCAD)) were compared. As a result, all HRV indices decreased at the maximum stage and no significant differences between iCAD and sCAD were found. The low-frequency power of heart rate signal (LF) of the RHC group recovered relatively quickly from the third to the sixth minutes after maximum HR, compared with that of the sCAD group. The relative changes of most HRV indices between maximum HR and recovery stage were lower in the sCAD group than in the RHC group, especially in LF, the standard deviation of all normal to normal intervals (SDNN), and the standard deviation in the long axis direction of the Poincaré plot analysis (SD2) indices (p < 0.05). The recovery slope of LF was significantly smaller in the sCAD group than in the RHC group (p = 0.02). The result suggests that monitoring short-term HRV during EST provides helpful insight into the cardiovascular autonomic imbalance in patients with significant CAD. The relative change of autonomic tone, especially the delayed sympathetic recovery, could be an additional marker for diagnosing myocardial ischemia. Full article