Poincare Algebra Research Articles

Anatomy of null contractions

We introduce null contractions of the Poincare and relativistic conformal algebras. The longitudinal null contraction involves writing the algebra in lightcone coordinates and contracting one of the null directions. For the Poincare algebra, this yields two non-overlapping co-dimension one Carroll algebras. The transverse contraction is a limit on the spatial dimensions and yields two non-overlapping co-dimension one Galilean algebras. We find, similar to Susskind’s original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension. The effect of the longitudinal contraction, which essentially focusses on the null direction, is to disentangle the two Carroll algebras that now correspond to the symmetries of the two null boundaries. The transverse contraction similarly separates the overlapping Galilean sub-algebras of the original Poincare algebra. We discuss aspects of the conformal case, where we get lower dimensional Carroll Conformal algebras and Schrödinger algebras.

A Very Fast Reasoning in Favor of the Poincare Group, or of Minkowskian Rather than Euclidean Line Elements

1. We present a very fast derivation of Minkowski’s line element which is the geometric foundation not only of the special Lorentz transformation but of the whole Poincar’e group aka inhomogeneous Lorentz group [1]. The Euclidean line element is excluded using the wave equation and Einstein’s special principle of relativity (Einstein’s first postulate) [2]. For doing so, it is not necessary, 1. to assume the speed of light to be independent of the motion of the observer (Einstein’s second postulate [2], 2. to synchronize watches, 3. to interrelate the coordinate systems of the inertial frames involved; their relative velocity (and hence reciprocity [3]) even plays not any role in the derivation. 4. to make any assumption about the properties of the underlying coordinate transformation such as linearity, uniqueness etc

Beyond Wilson? Carroll from current deformations

At extreme energies, both low and high, the spacetime symmetries of relativistic quantum field theories (QFTs) are expected to change with Galilean symmetries emerging in the very low energy domain and, as we will argue, Carrollian symmetries appearing at very high energies. The formulation of Wilsonian renormalisation group seems inadequate for handling these changes of the underlying Poincare symmetry of QFTs and it seems unlikely that these drastic changes can be seen within the realms of relativistic QFT. We show that contrary to this expectation, changes in the spacetime algebra occurs at the very edges of parameter space. In particular, we focus on the very high energy sector and show how bilinears of U(1) currents added to a two dimensional (massless) scalar field theory deform the relativistic spacetime conformal algebra to conformal Carroll as the effective coupling of the deformation is dialed to infinity. We demonstrate this using both a symmetric and an antisymmetric current-current deformation for theories with multiple scalar fields. These two operators generate distinct kinds of quantum flows in the coupling space, the symmetric driven by Bogoliubov transformations and the antisymmetric by spectral flows, both leading to Carrollian CFTs at the end of the flow.

Testing Poincare Invariance with QED

Testing Poincare Invariance with QED

Three point amplitudes in matrix theory

We compute the three graviton amplitude in the Banks-Fischler-Shenker-Susskind matrix model for M-theory. Even though the three point amplitude is determined by super Poincare invariance in eleven dimensional M-theory, it requires a non-trivial computation in the matrix model. We consider a configuration where all three gravitons carry non-zero longitudinal momentum. To simplify the problem, we compactify one additional dimension and relate the amplitude to a supersymmetric index computation. We find agreement with the expected answer even at finite values of N.

The hidden Lorentz covariance of quantum mechanics

This paper introduces a systematic algorithm for deriving a new unitary representation of the Lorentz algebra (so(1,3)) and an irreducible unitary representation of the extended (anti) de-Sitter algebra (so(2,4)) on L2(R3,1r). This representation is equivalent to a representation on L2(R3), and the corresponding similarity transformation is identified. An explicit representation in terms of differential operators is given, and it is shown that the inner product is Lorentz invariant. Ensuring Lorentz covariance demands a modification of the Heisenberg algebra, recognized as a phase space algebra at the interface of gravitational and quantum realms (IGQR), which we consider subordinate to Lorentz covariance. It is also demonstrated that time evolution can be cast in a manifestly covariant form. Each mass sector of the Hilbert space carries a representation of the Lorentz algebra, and the (anti) de-Sitter algebra on each mass sector contracts to the Poincare algebra in the flat configuration and momentum space limits. Finally, we show that three-dimensional fuzzy space also carries a unitary representation of these algebras, algebraically equivalent to the L2(R3,1r) representation but not necessarily equivalent as representations. Several outstanding issues are identified for future exploration.

Solving Particle–Antiparticle and Cosmological Constant Problems

We solve the particle-antiparticle and cosmological constant problems proceeding from quantum theory, which postulates that: various states of the system under consideration are elements of a Hilbert space H with a positive definite metric; each physical quantity is defined by a self-adjoint operator in H; symmetry at the quantum level is defined by a representation of a real Lie algebra A in H such that the representation operator of any basis element of A is self-adjoint. These conditions guarantee the probabilistic interpretation of quantum theory. We explain that in the approaches to solving these problems that are described in the literature, not all of these conditions have been met. We argue that fundamental objects in particle theory are not elementary particles and antiparticles but objects described by irreducible representations (IRs) of the de Sitter (dS) algebra. One might ask why, then, experimental data give the impression that particles and antiparticles are fundamental and there are conserved additive quantum numbers (electric charge, baryon quantum number and others). The reason is that, at the present stage of the universe, the contraction parameter R from the dS to the Poincare algebra is very large and, in the formal limit R→∞, one IR of the dS algebra splits into two IRs of the Poincare algebra corresponding to a particle and its antiparticle with the same masses. The problem of why the quantities (c,ℏ,R) are as are does not arise because they are contraction parameters for transitions from more general Lie algebras to less general ones. Then the baryon asymmetry of the universe problem does not arise. At the present stage of the universe, the phenomenon of cosmological acceleration (PCA) is described without uncertainties as an inevitable kinematical consequence of quantum theory in semiclassical approximation. In particular, it is not necessary to involve dark energy the physical meaning of which is a mystery. In our approach, background space and its geometry are not used and R has nothing to do with the radius of dS space. In semiclassical approximation, the results for the PCA are the same as in General Relativity if Λ=3/R2, i.e., Λ>0 and there is no freedom for choosing the value of Λ.

Symmetries of κ-Minkowski space-time: a possibility of exotic momentum space geometry?

The quest for a quantum gravity phenomenology has inspired a quantum notion of space-time, which motivates us to study the fate of the relativistic symmetries of a particular model of quantum space-time, as well as its intimate connection with the plausible emergent curved “physical momentum space”. We here focus on the problem of Poincare symmetry of κ-Minkowski type non-commutative (quantum) space-time, where the Poincare algebra, on its own, remains undeformed, but in order to preserve the structure of the space-time non-commutative (NC) algebra, the actions of the algebra generators on the operator-valued space-time manifold must be enveloping algebra valued that lives in entire phase space i.e. the cotangent bundle on the space-time manifold (at classical level). Further, we constructed a model for a spin-less relativistic massive particle enjoying the deformed Poincare symmetry, using the first order form of geometric Lagrangian, that satisfies a new deformed dispersion relation and explored a feasible regime of a future Quantum Gravity theory in which the momentum space becomes curved. In this scenario there is only a mass scale (Planck mass mp), but no length scale. Finally, we relate the deformed mass shell to the geodesic distance in this curved momentum space, where the mass of the particle gets renormalized as a result of noncommutativity. We show, that under some circumstances, the Planck mass provides an upper bound for the observed renormalized mass.

Scalar-tensor theory with EGB term from Einstein Chern-Simons gravity

It is shown that the compactification á la Randall-Sundrum of the so-called, five-dimensional Einstein-Chern-Simons action gravity leads to an action for a four-dimensional scalar-tensor gravity that includes a Gauss-Bonnet term, which belongs to a particular case of the action of the Horndeski theory. The five-dimensional action includes new gravitational degrees of freedom that were introduced requiring that the action be invariant under symmetries greater than the usual Poincaré or (A)dS symmetries, namely the so-called generalized Poincare algebras B5.

The Maslov index and some applications to dispersion relations in curved space times

The aim of the present work is to generalize the results given in Osorio Morales and Santillán [Eur. Phys. J. C 82, 353 (2022)] to a generic situation for causal geodesics. It is argued that these results may be of interest for causality issues. Recall that the presence of superluminal signals in a generic space time (M, gμν) does not necessarily imply violations of the principle of causality {[G. M. Shore, Nucl. Phys. B 778, 219 (2007)] and [T. J. Hollowood and G. M. Shore, Phys. Lett. B 655, 67 (2007)]}. In flat spaces, global Lorenz invariance leads to the conclusion that closed time-like curves appear if these signals are present. In a curved space instead, there is only local Poincare invariance, and the presence of closed causal curves may be avoided even in the presence of a superluminal mode, especially when terms violating the strong equivalence principle appear in the action. This implies that the standard analytic properties of the spectral components of these functions are therefore modified, and in particular, the refraction index n(ω) is not analytic in the upper complex ω plane. The emergence of these singularities may also take place for non-superluminal signals due to the breaking of global Lorenz invariance in a generic space time. In the present work, it is argued that the homotopy properties of the Maslov index are useful for studying how the singularities of n(ω) vary when moving along a geodesic congruence. In addition, several conclusions obtained in Shore [Nucl. Phys. B 778, 219 (2007)] and Hollowood and Shore [Phys. Lett. B 655, 67 (2007)] are based on the Penrose limit along a null geodesic, and they are restricted to GR with matter satisfying strong energy conditions. The use of the Maslov index may allow a more intrinsic description of singularities, not relying on that limit, and a generalization of these results about non-analyticity to generic gravity models with general matter content.

Remarks on the two-point string amplitudes

The two-point string amplitude at tree level in flat spacetime reproduces the expected expression for free particles. This has been proven by Erbin, Maldacena and Skliros in [JHEP 07 (2019) 139] by two methods. Here, we provide an alternative proof of this result. Our method consists in considering a timelike Liouville direction as a regularization parameter, which suffices to break time translation invariance and, at the same time, to stabilize the residual conformal symmetry in the worldsheet. This allows taking a limit and restoring Poincare symmetry in a controlled way, thus reproducing the correct expression of the two-point amplitude.

Gauge and Poincaré properties of the UV cutoff and UV completion in quantum field theory

The ultraviolet (UV) cutoff on a quantum field theory (QFT) can explicitly break or conserve the Poincare (translation) symmetry. And the very same cutoff can explicitly break or conserve the gauge symmetry. In the present work, we perform a systematic study of the UV cutoff in regard to its gauge and Poincare properties, and construct UV completions restoring the broken gauge symmetry. In the case of Poincare-conserving UV cutoff, we find that the gauge symmetry gets restored via the Higgs mechanism. In the case of Poincare-breaking UV cutoff, however, we find that the flat spacetime affine curvature takes the place of the Higgs field and, when taken to curved spacetime, gauge symmetry gets restored at the extremum of the metric-affine action. We also find that gravity emerges at the extremum if the QFT under concern consists of new particles beyond the known ones. The resulting emergent gravity plus renormalized QFT setup has the potential to reveal itself in various astrophysical, cosmological and collider phenomena.

Magic fermions: Carroll and flat bands

The Carroll algebra is constructed as the c → 0 limit of the Poincare algebra and is associated to symmetries on generic null surfaces. In this paper, we begin investigations of Carrollian fermions or fermions defined on generic null surfaces. Due to the availability of two different (degenerate) metrics on Carroll spacetimes, there is the possibility of two different versions of Carroll Clifford algebras. We consider both possibilities and construct explicit representations of Carrollian gamma matrices and show how to build higher spacetime dimensional representations out of lower ones. Actions for Carroll fermions are constructed with these gamma matrices and the properties of these actions are investigated.We show that in condensed matter systems where the dispersion relation becomes trivial i.e. the energy is not dependent on momentum and bands flatten out, Carroll symmetry generically appears. We give explicit examples of this including that of twisted bi-layer graphene, where superconductivity appears at so called magic angles and connect this to Carroll fermions.

Maxwell extension of f(R) gravity

Inspired by the Maxwell symmetry generalization of general relativity (Maxwell gravity), we have constructed the Maxwell extension of f(R) gravity. We found that the semi-simple extension of the Poincare symmetry allows us to introduce geometrically a cosmological constant term in four-dimensional f(R) gravity. This symmetry also allows the introduction of a non-vanishing torsion to the Maxwell f(R) theory. It is found that the antisymmetric gauge field B^{ab} associated with Maxwell extension is considered as a source of the torsion. It is also found that the gravitational equation of motion acquires a new term in the form of an energy–momentum tensor for the background field. The importance of these new equations is briefly discussed.

Mass dimension one fields with Wigner degeneracy: A theory of dark matter

Whatever dark matter is, it must be one irreducible unitary representation of the inhomogeneous Lorentz group or another. We here develop a formalism of mass dimension one fermions and bosons of spin one half, and show that they provide natural dark matter candidates. By construction, they are covariant under space-time translations and boosts. However, incorporating the rotational symmetry is non-trivial and requires introducing a two-fold Wigner degeneracy thus doubling the degrees of freedom for particles and anti particles from two to four. With Wigner degeneracy, we have a well-defined theory of mass dimension one fields of spin one half that are physically distinct from the Dirac field. They are local, Lorentz covariant and have positive definite Hamiltonians. The developed framework also has the potential to resolve the cosmological constant problem, and supply dark energy.

Observational Constraints on F(T,TG) Gravity with Hubble’s Parametrization

Any new gravitational theories can be built with the help of a gauge theory with local Poincare symmetry. This local Poincare symmetry can set up a space-time with torsion. In the present study, the authors working on the parametrization approach towards Hubble’s parameter in the frame of modified teleparallel Gauss-Bonnet gravity which is established on the torsion invariant T and the teleparallel equivalent of the Gauss-Bonnet term TG, say F(T,TG) gravity. In particular, gravity is responsible for an integrated explanation of the cosmological history from early-time inflation to late-time acceleration expansion, by lacking the addition of a cosmological constant. The domino effect acquired is reliable with recent cosmological outcomes. A transition scenario from a decelerating phase to an accelerating phase of cosmic evolution has been detected. Using the combined datasets (SNe-Ia+BAO+CMB+H(z)), we have constrained the transition redshift zt (at which the universe transit from a decelerating phase to an accelerating) and established the best fit value of zt. Next, we paralleled the renovated results of q(z) and ω(z) and found that the outcomes are well-suited with a ΛCDM universe.

The bottom-up EFT: complete UV resonances of the SMEFT operators

The standard model effective field theory (SMEFT) provides systematic parameterization of all possible new physics above the electroweak scale. According to the amplitude-operator correspondence, an effective operator can be decomposed into a linear combination of several j-basis operators, which correspond to local amplitudes carrying certain spin and gauge quantum numbers in a particular scattering channel. Based on the Poincare and gauge symmetries of scattering amplitude, we construct the j-basis using the Casimir method for both the Lorentz and gauge sectors. The quantum numbers of the j-basis operators fix the quantum numbers of any intermediate state in the corresponding amplitudes, such as a UV resonance. This can be re-interpreted as the j-basis/UV correspondence, thus obtaining the j-bases in all partitions of fields for an operator amounts to finding all of its UV origins at tree level, constituting the central part of the bottom-up EFT framework. Applying the j-basis analysis to SMEFT, we obtain a complete list of possible tree-level UV origins of the effective operators at the dimension 5, 6, 7, and all the bosonic operators at dimension 8.

Massless fermions on a half-space: the curious case of 2+1-dimensions

Boundary conditions for a massless Dirac fermion in 2+1 dimensions where the space is a half-plane are discussed in detail. It is argued that linear boundary conditions that leave the Hamiltonian Hermitian generically break C P and T symmetries as well as Lorentz and conformal symmetry. We show that there is essentially one special case where a single species of fermion has C PT and the full Poincare and conformal symmetry of the boundary. We show that, with doubled fermions, there is a second special case which respects C PT but still violates Lorentz and conformal symmetry. This second special case is essentially the unique boundary condition where the Dirac operator has fermion zero mode edge states. We discuss how the edge states lead to exotic representations of scale, phase and translation symmetries and how imposing a symmetry requirement leads to edge ferromagnetism of the system. We prove that the exotic ferromagnetic representations are indeed carried by the ground states of the system perturbed by a class of interaction Hamiltonians which includes the non-relativistic Coulomb interaction.

Radiated Angular Momentum and Dissipative Effects in Classical Scattering.

We present a new formula for the angular momentum J^{μν} carried away by gravitational radiation in classical scattering. This formula, combined with the known expression for the radiated linear momentum P^{μ}, completes the set of radiated Poincaré charges due to scattering. We parametrize P^{μ} and J^{μν} by nonperturbative form factors and derive exact relations using the Poincaré algebra. There is a contribution to J^{μν} due to static (zero-frequency) modes, which can be derived from Weinberg's soft theorem. Using tools from scattering amplitudes and effective field theory, we calculate the radiated J^{μν} due to the scattering of two spinless particles to third order in Newton's constant G, but to all orders in velocity. Our form-factor analysis elucidates a novel relation found by Bini, Damour, and Geralico between energy and angular momentum loss at O(G^{3}). Our new results have several nontrivial implications for binary scattering at O(G^{4}). We give a procedure to bootstrap an effective radiation reaction force from the loss of Poincaré charges due to scattering.

Bounds on quantum information storage and retrieval.

We present certain universal bounds on the capacity of quantum information storage and on the time scale of its retrieval for a generic quantum field theoretic system. The capacity, quantified by the microstate entropy, is bounded from above by the surface area of the object measured in units of a Goldstone decay constant. The Goldstone bosons are universally present due to the spontaneous breaking of Poincare and internal symmetries by the information-storing object. Applied to a black hole, the bound reproduces the Bekenstein-Hawking entropy. However, the relation goes beyond gravity. The minimal time-scale required for retrieving the quantum information from a system is equal to its volume measured in units of the same Goldstone scale. For a black hole, this reproduces the Page time as well as the quantum break-time. Again, the expression for the information retrieval time is very general and is shared by non-gravitational saturated states in gauge theories including QCD. All such objects exhibit universal signatures such as the emission of ultra-soft radiation. Similar bounds apply to non-relativistic many-body systems. This article is part of the theme issue 'Quantum technologies in particle physics'.