The nonrelativistic limit of the centrally extended Poincaré group is considered and their consequences in the modal Hamiltonian interpretation of quantum mechanics are discussed [ O. Lombardi and M. Castagnino, Stud. Hist. Philos. Mod. Phys 39, 380 (2008) ; J. Phys, Conf. Ser. 128, 012014 (2008) ]. Through the assumption that in quantum field theory the Casimir operators of the Poincaré group actualize, the nonrelativistic limit of the latter group yields to the actualization of the Casimir operators of the Galilei group, which is in agreement with the actualization rule of previous versions of modal Hamiltonian interpretation [ Ardenghi et al., Found. Phys. (submitted) ]
The velocity basis of the Poincare group is used in the direct product space of two irreducible unitary representations of the Poincare group. The velocity basis with total angular momentum j will be used for the definition of relativistic Gamow vectors.
Though the irreducible representations of the Poincare' group form the groundwork for the formulation of relativistic quantum theories of a particle, robust classes of such representations are missed in current formulations of these theories. In this work the extended class of irreducible representations with positive `mass' parameter is explicitly determined. Several new representations in such extension, so far excluded, give rise to consistent theories for Klein-Gordon particles and also to new species of particle theories.
The Lorentz covariance of a non-linear, time-dependent relativistic wave equation is demonstrated; the equation has recently been shown to have highly interesting and significant empirical consequences. It is established here that an operator already exists which ensures the relativistic properties of the equation. Furthermore, we show that the time-dependent equation is gauge invariant. The equation however, breaks Poincare symmetry via time translation in a way consistent with its physical interpretation. It is also shown herein that the Casimir invariant PmuPmu of the Poincare group, which …
Gamow vectors are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues $(E_{R}\mp iΓ/2)$ describing quasistable states. In the relativistic domain this leads to Poincaré semigroup representations which are characterized by spin $j$ and by complex invariant mass square ${\mathsf{s}}={\mathsf{s}}_{R}=(M_{R}-\frac{i}{2}Γ_{R})^{2}$. Relativistic Gamow kets have all the properties required to describe relativistic resonances and quasistable particles with resonance mass $M_{R}$ and lifetime $\hbar/Γ_{R}$.
Based on the principle of relativity with two universal constants (c, l) and in the inertial motion group IM(1,3)\sim PGL(5,R), with Lorentz isotropy, in addition to Poincaré group of Einstein's SR the dual Poincaré group preserves the origin lightcone and its space/time-like region R_\pm appeared at common origin of intersected Minkowski/dS/AdS space. The dual Poincare kinematics is on a pair of degenerate Einstein manifolds with Λ_\pm=\pm3l^{-2} for R_\pm, respectively. Thus, there is a Poincaré double and the dS double for dS/AdS SR. Further, with other four doubles they form a relativistic…
The Biedenharn type relativistic wavefunctions are considered on the group manifold of the Poincaré group. It is shown that the wavefunctions can be factorized on the group manifold into translation group and Lorentz group parts. A Lagrangian formalism and field equations for such factorizations are given. Parametrizations of the functions obtained are studied in terms of a ten-parameter set of the Poincaré group. An explicit construction of the wavefunction for the spin 1/2 is given. A relation of the proposed description with the quantum field theory and harmonic analysis on the Poincaré gro…
A relativistic resonance which was defined by a pole of the $S$-matrix, or by a relativistic Breit-Wigner line shape, is represented by a generalized state vector (ket) which can be obtained by analytic extension of the relativistic Lippmann-Schwinger kets. These Gamow kets span an irreducible representation space for Poincaré transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincaré transformations of the Gamow kets, as well as of …
We introduce a complex relativistic phase space as the space $\mathbb{C}^4$ equipped with the Minkowski metric and with a geometric tri-product on it. The geometric tri-product is similar to the triple product of the bounded symmetric domain of type IV in Cartan's classification, called the spin domain. We define a spin 1 representations of the Lie algebra of the Poincaré group by natural operators of this tri-product on the complex relativistic phase space. This representation is connected with the electromagnetic tensor. A spin 1/2 representation on the complex relativistic phase space is co…
We show that the relativistic analogue of the two types of time translation in a non-relativistic history theory is the existence of two distinct Poincaré groups. The `internal' Poincaré group is analogous to the one that arises in the standard canonical quantisation scheme; the `external' Poincaré group is similar to the group that arises in a Lagrangian description of the standard theory. In particular, it performs explicit changes of the spacetime foliation that is implicitly assumed in standard canonical field theory.
While internal space-time symmetries of relativistic particles are dictated by the little groups of the Poincaré group, it is possible to construct representations of the little group for massive particles starting from harmonic oscillator wave functions for the quark model. The resulting oscillator wave functions are covariant and can be Lorentz-boosted. It is thus possible to obtain the parton model by boosting the quark model. A review of Wigner's theory of the little groups is given. It is shown that the covariant oscillator wave functions become squeezed as as the system becomes boosted. …
Relativistic invariance in Euclidean formulations of quantum mechanics is discussed. Relativistic treatments of quantum theory are needed to study hadronic systems at sub-hadronic distance scales. Euclidean formulations of relativistic quantum mechanics have some computational advantages. In the Euclidean representation the physical Hilbert space inner product is expressed in terms of Euclidean space-time variables with no need for any analytic continuation. The identification of the complex Euclidean group with the complex Poincaré group relates the infinitesimal generators of both groups. In…
A general technique is presented for constructing a quantum theory of a finite number of interacting particles satisfying Poincaré invariance, cluster separability, and the spectral condition. Irreducible representations and Clebsch-Gordan coefficients of the Poincaré group are the central elements of the construction. A different realization of the dynamics is obtained for each basis of an irreducible representation of the Poincaré group. Unitary operators that relate the different realizations of the dynamis are constructed. This technique is distinguished from other solutions of this proble…
The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous $C^\ast$-algebraic formulation of the theory's axioms. Conversely, the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a notion of functional integration based on topological groups that allows to construct functional integral …
Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the $S$-matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in- and out-scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several ``relativistic Breit-Wigner'' amplitudes, which represent the resonances per se. Thes…
In this paper a generalized form of relativistic dynamics is presented: A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in $\mathbb{R}^4$. The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincaré algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the ``eleventh generator'' formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many propos…
Based on a recent classification of coadjoint orbits of the full Poincaré group, we give a new group theoretic interpretation for the mass of a classical relativistic particle.
A construction of relativistic wave equations on the homogeneous spaces of the Poincaré group is given for arbitrary spin chains. Parametrizations of the field functions and harmonic analysis on the homogeneous spaces are studied. It is shown that a direct product of Minkowski spacetime and two-dimensional complex sphere is the most suitable homogeneous space for the physical applications. The Lagrangian formalism and field equations on the Poincaré and Lorentz groups are considered. A boundary value problem for the relativistically invariant system is defined. General solutions of this proble…
We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
The unification of electricity and magnetism achieved by special relativity has remained for decades a model of unification in theoretical physics. We discuss the relationship between electric and magnetic fields from a classical point of view, and then examine how the four main relevant authors (Lorentz, Poincaré, Einstein, Minkowski) dealt with the problem of establishing the transformation laws of the fields in different inertial systems. We argue that Poincaré's derivation of the transformation laws for the potentials and the fields was definitely less arbitrary than those of the other cit…
When utilizing a cluster decomposible relativistic scattering formalism, it is most convenient that the covariant field equations take on a linear form with respect to the energy and momentum dispersion on the fields in the manner given by the Dirac form for spin ${1 \over 2}$ systems. The general spinor formulation for arbitrary spins given in a previous paper is extended to include momentum operators. Unitary quantum mechanical representations are developed for these operators, and physical interpretations are suggested.
We present the consistent approach to finding the discrete transformations in the representation spaces of the proper Poincaré group. To this end we use the possibility to establish a correspondence between involutory automorphisms of the proper Poincaré group and the discrete transformations. As a result, we derive rules of the discrete transformations for arbitrary spin-tensor fields without the use of relativistic wave equations. Besides, we construct explicitly fields carrying representations of the extended Poincaré group, which includes the discrete transformations as well.
Several issues concerning quantum kappa-Poincare algebra are discussed and reconsidered here. We propose two different formulations of kappa-Poincare quantum algebra. Firstly we present a complete Hopf algebra formulae of kappa-Poincare in classical Poincare basis. Further by adding one extra generator, which modifies the classical structure of Poincare algebra, we eliminate non polynomial functions in the kappa-parameter. Hilbert space representations of such algebras make Doubly Special Relativity (DSR) similar to the Stueckelberg's version of (proper-time) relativistic Quantum Mechanics.
We prove that the extended Poincare group in (1+1) dimensions is non-nilpotent solvable exponential, and therefore that it belongs to type I. We determine its first and second cohomology groups in order to work out a classification of the two-dimensional relativistic elementary systems. Moreover, all irreducible unitary representations of the extended Poincare group are constructed by the orbit method. The most physically interesting class of irreducible representations corresponds to the anomaly-free relativistic particle in (1+1) dimensions, which cannot be fully quantized. However, we show …
Born proposed a unification of special relativity and quantum mechanics that placed position, time, energy and momentum on equal footing through a reciprocity principle and extended the usual position-time and energy-momentum line elements to this space by combining them through a new fundamental constant. Requiring also invariance of the symplectic metric yields U(1,3) as the invariance group, the inhomogeneous counterpart of which is the canonically relativistic group CR(1,3) = U(1,3) *s H(1,3) where H(1,3) is the Heisenberg Group in 4 dimensions and "*s" is the semidirect product. This is t…
Viewing gravitational energy-momentum $p_G^μ$ as equal by observation, but different in essence from inertial energy-momentum $p_I^μ$ naturally leads to the gauge theory of volume-preserving diffeormorphisms of an inner Minkowski space ${\bf M}^{\sl 4}$. To extract its physical content the full gauge group is reduced to its Poincaré subgroup. The respective Poincaré gauge fields, field strengths and Poincaré-covariant field equations are obtained and point-particle source currents are derived. The resulting set of non-linear field equations coupled to point matter is solved in first order resu…
Based on a linear realization formulation of a quantum relativity -- the proposed relativity for quantum `space-time', we introduce the Poincaré-Snyder relativity and Snyder relativity as relativities in between the latter and the well known Galilean and Einstein cases. We discuss how the Poincaré-Snyder relativity may provide a stronger framework for the description of the usual (Einstein) relativistic quantum mechanics and beyond. In particular, we discuss a geometric quantization picture through the U(1) central extension of the relativity group, which had been establish to work well for th…
Recent work showed that $κ$-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries, one can toggle the curvature parameter $Λ$, the Planck scale quantum deformation parameter $κ$ and the speed of light parameter $c$ to move to the well-studied $κ$-Poincaré algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these al…
The capabilities of some approaches to the relativistic description of hadronic states with any rest spin are analysed. The key feature in the Wigner's construction of irreducible representations of the Poincare group which makes this construction fruitless in the particle physics is picked out. A realization of unitary irreducible representations of the Poincare group of the standard type, which yet have never been considered, is discussed. The viability of the description of hadrons by the Poincare group representations of the standard type in the space of the infinite-component ISFIR-class …
The derivation of the non-relativistic Cosserat equations that was described in Part I of this series of papers is extended from the group of rigid motions in three-dimensional Euclidian space to the Poincaré group of four-dimensional Minkowski space. Examples of relativistic Cosserat media are then given in the form of the free Dirac electron and the Weyssenhoff fluid.
A novel approach to study the properties of models with quantum-deformed relativistic symmetries relies on a noncommutative space of worldlines rather than the usual noncommutative spacetime. In this setting, spacetime can be reconstructed as the set of events, that are identified as the crossing of different worldlines. We lay down the basis for this construction for the $κ$-Poincaré model, analyzing the fuzzy properties of $κ$-deformed time-like worldlines and the resulting fuzziness of the reconstructed events.
The way to obtain massive non-relativistic states from the Poincare algebra is twofold. First, following Inonu and Wigner the Poincare algebra has to be contracted to the Galilean one. Second, the Galilean algebra is to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any Inonu-Wigner contraction of one algebra to another, corresponds to an infinite tower of abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apa…
We extend some results of group representation theory and von Neumann algebras to the quaternionic Hilbert space case, proving the double commutant theorem (whose quaternionic proof requires a different procedure) and extend to the quaternionic case a result concerning the classification of irreducible von Neumann algebras. Secondly, we consider elementary relativistic systems in Wigner's view defined as a locally-faithful irreducible strongly-continuous unitary representation of Poincaré group in quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the syst…
As established by Solèr, Quantum Theories may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided physical reasons for ruling out real Hilbert spaces relying on Heisenberg principle. Focusing on this issue from another viewpoint, we argue that there is a fundamental reason why elementary quantum systems are not described in real Hilbert spaces: their symmetry group. We consider an elementary relativistic system within Wigner's approach defined as a locally-faithful irreducible continuous unitary representation of the Poincaré group in a real Hilbert space. …
In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincaré algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the Poincaré algebra or Supersymmetry. Some general features on the so-called Wess-Zumino model (the simplest field theory invariant under Supersymmetry) are then given. We further introduce an additional algebraic structure called Lie algebras of order F, which naturally comprise the concepts of ordinary Lie algebras and superalgebras. This structure enables us to …
The stabilized Poincare-Heisenberg algebra (SPHA) is the Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after attempting to combine the Lie algebras of quantum mechanics and relativity which by themselves are stable, however not when combined. In this paper we show how the sixteen dimensional Clifford algebra CL(1,3) can be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional stability considerations, relying instead on the fact that CL(1,3) is a semi-simple algebra and therefo…
We discuss a formulation of exactly Poincaré invariant quantum mechanics where the input is model Euclidean Green functions or their generating functional. We discuss the structure of the models, the construction of the Hilbert space, the construction and transformation properties of single-particle states, and the construction of GeV scale transition matrix elements. A simple model is utilized to demonstrate the feasibility of this approach.
It is possible to construct representations of the Lorentz group using four-dimensional harmonic oscillators. This allows us to construct three-dimensional wave functions with the usual rotational symmetry for space-like coordinates and one-dimensional wave function for time-like coordinate. It is then possible to construct a representation of the Poincaré group for a massive particles having the O(3) internal space-time symmetry in its rest frame. This oscillator can also be separated into two transverse components and the two-dimensional world of the longitudinal and time-like coordinates. T…
We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the …
A new formulation of relativistic classical mechanics allows a revisiting of old unsolved problems in relativistic kinetic theory and in relativistic statistical mechanics. In particular a definition of the relativistic micro-canonical partition function is given strictly in terms of the Poincaré generators of an interacting N-particle system both in the inertial and non-inertial rest frames. The non-relativistic limit allows a definition of both the inertial and non-inertial micro-canonical ensemble strictly in terms of the Galilei generators. Also the one-particle relativistic distribution f…
