Representations Of Lorentz Group Research Articles

An operational approach to spacetime symmetries: Lorentz transformations from quantum communication

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 , 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.

GR uniqueness and deformations

In the metric formulation gravitons are described with the parity symmetric $S_+^2\otimes S_-^2$ representation of Lorentz group. General Relativity is then the unique theory of interacting gravitons with second order field equations. We show that if a chiral $S_+^3\otimes S_-$ representation is used instead, the uniqueness is lost, and there is an infinite-parametric family of theories of interacting gravitons with second order field equations. We use the language of graviton scattering amplitudes, and show how the uniqueness of GR is avoided using simple dimensional analysis. The resulting distinct from GR gravity theories are all parity asymmetric, but share the GR MHV amplitudes. They have new all same helicity graviton scattering amplitudes at every graviton order. The amplitudes with at least one graviton of opposite helicity continue to be determinable by the BCFW recursion.

Canonical quantization of spin 1 matter fields. Preliminary results

In this work we present preliminary results on the canonical quantization of spin 1 matter fields, i.e. massive fields transforming in the (1, 0) ⊕ (0,1) representation of the Homogeneous Lorentz Group. The classical field theory is based on the projection onto subspaces of well defined parity in this representation.

Deformed twistors and higher spin conformal (super-)algebras in four dimensions

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of deformations labelled by the spin t of an SU(2)_T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS_7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS_7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8*|2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS_7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

Isgur-Wise functions and unitary representations of the Lorentz group: The meson case withj=12light cloud

We pursue the group theoretical method to study Isgur-Wise functions. We apply the general formalism, formerly applied to the baryon case j^P = 0^+ (for \Lambda_b -> \Lambda_c \ell \nu), to mesons with j^P = 1/2^-, i.e. $\overline{B} -> D(D^{(*)})\ell\nu. In this case, more involved from the angular momentum point of view, only the principal series of unitary representations of the Lorentz group contribute. We obtain an integral representation for the IW function xi(w) with a positive measure, recover the bounds for the slope and the curvature of xi(w) obtained from the Bjorken-Uraltsev sum rule method, and get new bounds for higher derivatives. We demonstrate also that if the lower bound for the slope is saturated, the measure is a delta-function, and xi(w) is given by an explicit elementary function. Inverting the integral formula, we obtain the measure in terms of the IW function, allowing to formulate criteria to decide if a given ansatz for the Isgur-Wise function is compatible or not with the sum rule constraints. Moreover, we have obtained an upper bound on the IW function valid for any value of w. We compare these theoretical constraints to a number of forms for \xi(w) proposed in the literature. The "dipole" function \xi(w) = (2/(w+1))^(2c) satisfies all constraints for c \geq 3/4, while the QCD Sum Rule result including condensates does not satisfy them. Special care is devoted to the Bakamjian-Thomas relativistic quark model in the heavy quark limit and to the description of the Lorentz group representation that underlies this model. Consistently, the IW function satisfies all Lorentz group criteria for any explicit form of the meson Hamiltonian at rest.

Deformed twistors and higher spin conformal (super-)algebras in six dimensions

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of labelled by the spin t of an SU(2)T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8 ∗ |2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

Wigner and the groups in classifying elementary particles and nuclear states

The application of group-representations in nuclear and particle physics, as ini- tiated by Wigner (1), is considered. The general classification of the elementary particles according to the representations of the inhomogenous Lorentz-group (2) is recalled. The role of the U(4) group is discussed, first in relation with the spin-isospin supermultiplets (3) both in nuclear and in particle physics, then as a dynamical group of the two-body systems in both disciplines. Finally the classification of the nuclear states is mentioned both for the single shell, and for the multi-shell problems.

Covariant basis induced by parity for the(j,0)⊕(0,j)representation

In this work, we build a covariant basis for operators acting on the $(j,0)\oplus(0,j)$ Lorentz group representations. The construction is based on an analysis of the covariant properties of the parity operator, which for these representations transforms as the completely temporal component of a symmetrical tensor of rank $2j$. The covariant properties of parity involve the Jordan algebra of anti commutators of the Lorentz group generators which unlike the Lie algebra is not universal. We make the construction explicit for $j=1/2,1$ and $3/2$, reproducing well-known results for the $j=1/2$ case. We provide an algorithm for the corresponding calculations for arbitrary $j$. This covariant basis provides an inventory of all the possible interaction terms for gauge and non-gauge theories of fields for these representations. In particular, it supplies a single second rank antisymmetric structure, which in the Poincar\'e projector formalism implies a single Pauli term arising from gauge interactions and a single (free) parameter $g$, the gyromagnetic factor. This simple structure predicts that for an elementary particle in this formalism all multipole moments, $Q^{l}_{E}$ and $Q^{l}_{M}$, are dictated by the complete algebraic structure of the Lorentz generators and the value of $g$. We explicitly calculate the multipole moments, for arbitrary $j$ up to $l=8$. Comparing with results in the literature we find that only the electric charge and magnetic moment of a spin $j$ particle are independent of the Lorentz representation under which it transforms, all higher multipoles being representation dependent. Finally we show that the propagation of the corresponding spin $j$ waves in a electromagnetic background is causal.

Compton scattering off massive fundamental bosons of pure spin 1

Relativistic particles with spins $J>0$ are described by means of multicomponent wave functions which transform covariantly according to Lorentz-group representations that contain at rest the spin of interest. The symmetry group of space-time provides not one but an infinity of such representations which are equivalent for free particles but yield different electromagnetic couplings upon gauging; thus the challenge is to develop criteria which allow us to select those of them which relate to physically detectable particles. We here take the position that the unitarity of the Compton scattering cross sections in the ultrarelativistic limit, when predicted by a consistent method for a spin-$1$ description, could provide such a criterion. We analyze the properties of massive fundamental spin-$1$ bosons transforming as antisymmetric tensors of second rank, $(1,0)\oplus(0,1)$. For this purpose, we employ the Poincar\'e covariant projector method, which provides consistent, causal, and representation specific Lagrangians. This formalism yields a twofold extension of the Proca Lagrangian for the description of spin-$1$ bosons, first from an in-built $g=1$ value of the gyromagnetic ratio to an unspecified $g\not=1$, and from, single-parity, to parity-doublet degrees of freedom. We find different results for Compton scattering in these theories and track the differences to the lack of universality of the vector-antisymmetric-tensor equivalence theorem which is specific only to Proca's framework, and valid for $g=1$, while it is violated within the covariant projector formalism. Our main result is that a finite Compton scattering differential cross section in the ultrarelativistic limit requires us to consider the contributions of both parities in $(1,0)\oplus(0,1)$. On that basis, we conclude that massive spin-$1$ bosons transforming as antisymmetric tensors are physical parity doublets.

Six-dimensional methods for four-dimensional conformal field theories. II. Irreducible fields

This note supplements an earlier paper on conformal field theories. There it was shown how to construct tensor, spinor, and spinor-tensor primary fields in four dimensions from their counterparts in six dimensions, where conformal transformations act simply as $SO(4,2)$ Lorentz transformations. Here we show how to constrain fields in six dimensions so that the corresponding primary fields in four dimensions transform according to irreducible representations of the four-dimensional Lorentz group, even when the irreducibility conditions on these representations involve the four-component Levi-Civita tensor ${ϵ}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\sigma}}$.

Baryon tri-local interpolating fields

We systematically investigate tri-local (non-local) three-quark baryon fields with U_L(2)*U_R(2) chiral symmetry, according to their Lorentz and isospin (flavor) group representations. We note that they can also be called as "nucleon wave functions" due to this full non-locality. We study their chiral transformation properties and find all the possible chiral multiplets consisting J=1/2 and J=3/2 baryon fields. We find that the axial coupling constant |g_A| = 5/3 is only for nucleon fields belonging to the chiral representation (1/2,1)+(1,1/2) which contains both nucleon fields and Delta fields. Moreover, all the nucleon fields belonging to this representation have |g_A| = 5/3.

Some formulas for Legendre functions related to the Poisson transform and Lorentz group representation

We consider the representation of Lorentz group in the spaces of homogeneous functions on cone and hyperboloid. Using the Poisson transform, which maps a function from the first space into the corresponding function from the second space, we derive some formulas containing Legendre functions. Using the restriction of the representation in the first space on a one-parameter subgroup of hyperbolic rotations, we obtain another formula for Legendre functions. All obtained formulas are new.

Relativistic implications of the quantum phase

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics.We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations.We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.

CPT Groups of Higher Spin Fields

$CPT$ groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the fields on the Poincar\'{e} group which describe orientable (extended) objects. A general method for construction of $CPT$ groups of the fields of any spin is given. $CPT$ groups of the fields of spin-1/2, spin-1 and spin-3/2 are considered in detail. $CPT$ groups of the fields of tensor type are discussed. It is shown that tensor fields correspond to particles of the same spin with different masses.

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.

Second order formalism for spin12fermions and Compton scattering

We develop a second order formalism for spin 1/2 fermions based on the projection over Poincar\'{e} invariant subspaces in the $(1/2,0)\oplus(0,1/2)$ representation of the homogeneous Lorentz group. Using $U(1)_{em}$ gauge principle we obtain second order description for the electromagnetic interactions of a spin 1/2 fermion with two free parameters, the gyromagnetic factor $g$ and a parameter $\xi$ related to odd-parity Lorentz structures. We calculate Compton scattering in this formalism. In the particular case $g=2, \xi=0$ and for states with well defined parity we recover Dirac results. In general, we find the correct classical limit and a finite value $r_{c}^{2}$ for the forward differential cross section, independent of the photon energy and of the value of the parameters $g$ and $\xi$. The differential cross section vanishes at high energies for all $g, \xi$ except in the forward direction. The total cross section at high energies vanishes only for $g=2, \xi=0$. We argue that this formalism is more convenient than Dirac theory in the description of low energy electromagnetic properties of baryons and illustrate the point with the proton case.

Bi-local baryon interpolating fields with two flavors

We construct bi-local interpolating field operators for baryons consisting of three quarks with two flavors, assuming good isospin symmetry. We use the restrictions following from the Pauli principle to derive relations/identities among the baryon operators with identical quantum numbers. Such relations that follow from the combined spatial, Dirac, color, and isospin Fierz transformations may be called the (total/complete) Fierz identities. These relations reduce the number of independent baryon operators with any given spin and isospin. We also study the Abelian and non-Abelian chiral transformation properties of these fields and place them into baryon chiral multiplets. Thus we derive the independent baryon interpolating fields with given values of spin (Lorentz group representation), chiral symmetry (U L (2)×U R (2) group representation) and isospin appropriate for the first angular excited states of the nucleon.

A general theory of matrix transformation and its application to quantum mechanical problems

A general formalism is given to construct a transformation matrix TAB which connects two matrices A and B of order n × n satisfying any given polynomial equations of degree r(≤n). The matrix TAB is given explicitly by a polynomial of degree (r − 1) in A and B. A special case where B is a diagonal matrix equivalent to A leads to a general theory of matrix diagonalization. The formalism contains the projection operator method as a special case. Illustrative examples are given on the Dirac theory of the electron and the basic 2 × 2 spinor representation of the proper orthochronous Lorentz group; the latter is completely prameterized by the spatial rotation and the velocity of the parallel translation.

Electromagnetic form factors and polarizations of non-Dirac particles with rest spin 1/2

We consider one aspect of the theoretical foundations of polarization experiments on elastic scattering of electrons on protons yielding form factor ratios incompatible with those that are extracted from nonpolarization experiments. We analyze the consequences of abandoning the assumption that the nucleon is a Dirac particle. We show that the process of elastic scattering of electrons on nucleons is described by the same formulas, irrespective of the proper Lorentz group representation associated with the nucleon as a particle with the rest spin 1/2.

Note on quantum Minkowski space

In this work, some interesting details about quantum Minkowski space and quantum Lorentz group structures are revealed. The task is accomplished by generalizing an approach adopted in a previous work where quantum rotation group and quantum Euclidean space structures have been investigated. The generalized method is based on a mapping relating the $q$-spinors (precisely the tensor product of dotted and undotted fondamental $q$-spinors) to Minkowski $q$-vectors. As a result of this mapping, the quantum analog of Minkowski space is constructed (with a definite metric). Also, the matrix representation of the quantum Lorentz group is determined together with its corresponding $q$-deformed orthogonality relation.