Space-time Translations Research Articles

Tachyons in “momentum-space” representation

Obtaining the momentum space associated with tachyonic “particles” from the Poincaré group manifold proves to be rather intricate, departing very much from the ordinary dual to Minkowski space directly parametrized by space-time translations of the Poincaré group. In fact, although described by the constants of motion (Noether invariants) associated with space-time translations, they depend non-trivially on the parameters of the rotation subgroup. However, once the momentum space is parametrized by the Noether invariants, it behaves as that of ordinary particles. On the other hand, the evolution parameter is no longer the one associated with time translation, whose Noether invariant, Po, is now a basic one. Evolution takes place in a spatial direction. These facts not only make difficult the computation of the corresponding representation, but also force us to a sound revision of several traditional ingredients related to Cauchy hypersurface, scalar product and, of course, causality. After that, the theory becomes consistent and could shed new light on some special physical situations like inflation or traveling inside a black hole.

Space-time symmetry and nonreciprocal parametric resonance in mechanical systems.

Linear mechanical systems with time-modulated parameters can harbor oscillations with amplitudes that grow or decay exponentially with time due to the phenomenon of parametric resonance. While the resonance properties of individual oscillators are well understood, those of systems of coupled oscillators remain challenging to characterize. Here we determine the parametric resonance conditions for time-modulated mechanical systems by exploiting the internal symmetries arising from the real-valued and symplectic nature of classical mechanics. We also determine how these conditions are further constrained when the system exhibits external symmetries. In particular, we analyze systems with space-time symmetry where the system remains invariant after a combination of discrete translation in both space and time. For such systems, we identify a combined space-time translation operator that provides more information about the dynamics of the system than the Floquet operator does and use it to derive conditions for one-way amplification of traveling waves. Our exact theoretical framework based on symmetries enables the design of exotic responses such as nonreciprocal transport and one-way amplification in dynamic mechanical metamaterials and is generalizable to all physical systems that obey space-time symmetry.

Celestial leaf amplitudes

Celestial amplitudes may be decomposed as weighted integrals of AdS3-Witten diagrams associated to each leaf of a hyperbolic foliation of spacetime. We show, for the Kleinian three-point MHV amplitude, that each leaf subamplitude is smooth except for the expected light-cone singularities. Moreover, we find that the full translationally-invariant celestial amplitude is simply the residue of the pole in the leaf amplitude at the point where the total conformal weights of the gluons equals three. This full celestial amplitude vanishes up to light-cone contact terms, as required by spacetime translation invariance, and reduces to the expression previously derived by Mellin transformation of the Parke-Taylor formula.

Soft theorems for boostless amplitudes

We consider effective field theories (EFTs) of scalar fields with broken Lorentz boosts, which arise by taking the decoupling and flat-space limits of the EFT of inflation, and derive constraints that must be satisfied by the corresponding scattering amplitudes if there is an underlying non-linearly realised symmetry. We primarily concentrate on extended shift symmetries which depend on the space-time coordinates, and find that combinations of scattering amplitudes obey enhanced Adler zeros. That is, such combinations vanish as one external momentum is taken soft, with the rate at which they vanish dictated by the corresponding symmetry. In our soft theorem derivation, we pay particular care to the energy and momentum-conserving delta functions that arise due to space-time translations, and show that when acted upon by derivatives with respect to spatial momenta, they yield a tower of soft theorems which are ultimately required for closure of the underlying symmetry algebra. All of our soft theorems correspond to constraints that must be satisfied by on-shell amplitudes and, even for symmetries that depend on the time coordinate, our soft theorems only require derivatives to be taken with respect to spatial momenta. We perform a soft bootstrap procedure to find solutions to our soft theorems, and compare these solutions to what we find from an off-shell analysis using the coset construction.

Three Puzzles with Covariance and Supertranslation Invariance of Angular Momentum Flux and Their Solutions.

We describe and solve three puzzles arising in covariant and supertranslation-invariant formulas for the flux of angular momentum and other Lorentz charges in asymptotically flat spacetimes: (i)Supertranslation invariance and covariance imply invariance under spacetime translations; (ii)the flux depends on redundant auxiliary degrees of freedom that cannot be set to zero in all Lorentz frames without breaking Lorentz covariance; (iii)supertranslation-invariant Lorentz charges do not generate the transformations of the Bondi mass aspect implied by the isometries of the asymptotic metric. In this Letter, we solve the first two puzzles by presenting covariant formulas that unambiguously determine the auxiliary degrees of freedom and clarify the last puzzle by explaining the different roles played by covariant and canonical charges. Our construction makes explicit the choice of reference frame underpinning seemingly unambiguous results presented in the current literature.

Modelling ductile damage in metals and alloys through Weyl condition exploiting local gauge symmetries

Local translational and scaling symmetries in space–time is exploited for modelling ductile damage in metals and alloys over wide ranges of strain rate and temperature. The invariant energy density corresponding to the ductile deformation is constructed through the gauge invariant curvature tensor by imposing the Weyl like condition. In contrast, the energetics of the plastic deformation is brought in through the gauge compensating field emerged due to local translation and attempted to explore the geometric interpretation of certain internal variables often used in classical viscoplasticity models. Invariance of the energy density under the local action of translation and scaling is preserved through minimally replaced space–time gauge covariant operators. Minimal replacement introduces two non-trivial gauge compensating fields pertaining to local translation and scaling. These are used to describe ductile damage, including plastic flow and micro-crack evolution in the material. A space–time pseudo-Riemannian metric is used to lay out the kinematics in a finite-deformation setting. Recognizing the available insights in classical theories of viscoplasticity, we also establish a correspondence of the gauge compensating field due to spatial translation with Kröner’s multiplicative decomposition of the deformation gradient. Thermodynamically consistent coupling between viscoplasticity and ductile damage is ensured through an appropriate degradation function. Non-ordinary state-based (NOSB) peridynamics (PD) discretization of the model is used for numerical implementation. We conduct simulations of uniaxial deformation to validate the model against available experimental evidence and to assess its predictive features. The model’s viability is tested in reproducing a few experimentally known facts, viz., strain rate locking in the stress–strain response, whose origin is traced to a nonlinear microscopic inertia term arising out of the space–time translation symmetry. Finally, we solved 2D and axisymmetric deformation problems for qualitatively validating the model’s viability. NOSB peridynamics axisymmetric formulation in finite deformation setup is also presented.

Integrable coupled massive Thirring model with field values in a Grassmann algebra

A coupled massive Thirring model of two interacting Dirac spinors in 1 + 1 dimensions with fields taking values in a Grassmann algebra is introduced, which is closely related to a SU(1) version of the Grassmannian Thirring model also introduced in this work. The Lax pair for the system is constructed, and its equations of motion are obtained from a zero curvature condition. It is shown that the system possesses several infinite hierarchies of conserved quantities, which strongly confirms its integrability. The model admits a canonical formulation and is invariant under space-time translations, Lorentz boosts and global U(1) gauge transformations, as well as discrete symmetries like parity and time reversal. The conserved quantities associated to the continuous symmetries are derived using Noether’s theorem, and their relation to the lower-order integrals of motion is spelled out. New nonlocal integrable models are constructed through consistent nonlocal reductions between the field components of the general model. The Lagrangian, the Hamiltonian, the Lax pair and several infinite hierarchies of conserved quantities for each of these nonlocal models are obtained substituting its reduction in the expressions of the analogous quantities for the general model. It is shown that, although the Lorentz symmetry of the general model breaks down for its nonlocal reductions, these reductions remain invariant under parity, time reversal, global U(1) gauge transformations and space-time translations.

Almost Everywhere Ergodicity in Quantum Lattice Models

We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the C^* algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C^* algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.

Arrow of Time and Quantum Physics

Based on the hypothesis that the (non-reversible) arrow of time is intrinsic in any system, no matter how small, the consequences are discussed. Within the framework of local quantum physics it is shown how such a semi-group action of time can consistently be extended to that of the group of spacetime translations in Minkowski space. In presence of massless excitations, however, there arise ambiguities in the theoretical extensions of the time translations to the past. The corresponding loss of quantum information on states upon time is determined. Finally, it is explained how the description of operations in classical terms combined with constraints imposed by the arrow of time leads to a quantum theoretical framework. These results suggest that the arrow of time is fundamental in nature and not merely a consequence of statistical effects on which the Second Law is based.

The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss–Bonnet Gravity

We sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangians to select symmetries. Furthermore, we extend the prescription to the first prolongation of the Noether vector. It is possible to show that the latter application provides a general constraint on the infinitesimal generator ξ, related to the spacetime translations. This approach can be used for several applications. In the second part of the work, we consider a gravity theory, including the coupling between a scalar field ϕ and the Gauss–Bonnet topological term G. In particular, we study a gravitational action containing the function F(G,ϕ) and select viable models by the existence of symmetries. Finally, we evaluate the selected models in a spatially flat cosmological background and use symmetries to find exact solutions.

Soft phonon theorems

A variety of condensed matter systems describe gapless modes that can be interpreted as Nambu-Goldstone bosons of spontaneously broken Poincaré symmetry. In this paper we derive new soft theorems constraining the tree-level scattering of these degrees of freedom, as exhibited in solids, fluids, superfluids, and framids. These soft theorems are in one-to-one correspondence with various broken symmetries, including spacetime translations, Lorentz boosts, and, for the case of fluids, volume-preserving diffeomorphisms. We also implement a bootstrap in which the enhanced vanishing of amplitudes in the soft limit is taken as an input, thus sculpting out a subclass of exceptional solid, fluid, and framid theories.

A symmetry group based supervised learning method for solving partial differential equations

Deep neural network provides an effective way to solve partial differential equations (PDEs) where physics-informed neural network (PINN) is a typical representative of which the outputs are constrained to approximate the solution of PDEs. However, the PINN method still fails to learn some solutions of particular properties and the reasons of failure are still unclear. In this paper, we employ both the loss landscape of the final optimization parameters of network and the imbalance of back-propagation gradient during the training to analyze why the PINN method fails to learn large amplitude solution and high-frequency solution of the Klein–Gordon equation. Consequently, we find that the PDE-based soft constraints in the loss function take a major responsibility for the failures of the PINN method in predicting such types of solutions. To remedy the pathologies we remove the PDE-based soft constraints in the loss function and calibrate a supervised learning model for predicting such types of particular solutions for a class of PDEs which cannot be learned by the PINN method and the two weight-related improved PINN methods. Specifically, if the governing PDEs under study admit a Lie symmetry group and the unique solution of the corresponding initial boundary problem is obtained via the symmetry group, we utilize the symmetry group to generate labeled data in the interior domain of PDEs from the discrete points on the known initial and boundary conditions. Then, with the labeled data we leverage the supervised learning model to predict the large amplitude solution and the high-frequency solution of the Klein–Gordon equation on a rectangular domain by means of a space–time translation group and a singular solution of the heat equation on a moving boundary domain with a physical symmetry group. Numerical experiments confirm the good performances of the proposed supervised learning method.

Addendum to: 4D supersymmetric gauge theories of spacetime translations

Addendum to: 4D supersymmetric gauge theories of spacetime translations

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.

Control Strategy for Energy-Storage Systems to Smooth Wind Power Fluctuation Based on Interval and Fuzzy Control

The anti-peak shaving characteristics of wind power is an important factor that limits the consumption of wind power. The use of the space-time translation capability of a battery energy storage system is one of the effective means for promoting wind power consumption. Thus, this study proposes an energy storage system smoothing wind power fluctuation control strategy considering wind power consumption to improve the utilization level and economy of an energy storage system. First, the statistical characteristics of wind power are analyzed, and the scene division principle of a multiscenario regulation of an energy storage system is formulated using a correlation coefficient. On this basis, the charging and discharging intervals of the energy storage system in each scenario are determined by comprehensively considering the wind power and the time domain distribution law of load. Then, the control strategy of energy storage system based on inter-zone control is proposed based on the division of charging and discharging intervals. During the charging interval, smoothing the upward fluctuation of wind power or reducing the amount of abandoned wind power is aimed at storing electricity. During the discharge interval, electricity is released to smooth the downward fluctuation of wind power. Finally, considering the real-time state of charge and wind power consumption of the energy storage system, a method of charging and discharging power correction based on fuzzy control is proposed. Taking the actual data of a wind farm as an example, a test is performed on a simulation platform of a combined wind storage power generation system to verify the feasibility and superiority of the proposed control strategy.

Noether’s theorems and the energy-momentum tensor in quantum gauge theories

Noether's first and second theorems both imply conserved currents that can be identified as an energy-momentum tensor (EMT). The first theorem identifies the EMT as the conserved current associated with global spacetime translations, while the second theorem identifies it as a conserved current associated with local spacetime translations. This work obtains an EMT for quantum electrodynamics and quantum chromodynamics through the second theorem, which is automatically symmetric in its indices and invariant under the expected symmetries (e.g., BRST invariance) without the need for introducing an ad hoc improvement procedure.

Topological classification of gapped/gap-preserving rational space-time crystal systems

The traditional systems researched in condensed matter physics always have spatial translation symmetry. However for space-time crystal systems, the spatial translation symmetry is no longer preserved and the lattice potential have space-time translation symmetry instead. We show that a rational space-time crystal system is equal to a traditional floquet system. Then, we find a way to solve the floquet equation analytically and construct an effective Hamiltonian of the rational space-time crystal system. By this effective Hamiltonian, we obtain the topological classification of gapped and gap-preserving rational space-time crystal systems. Our works reveal the correlation between space-time crystal systems and floquet systems and give a systematic method to explore the properties of rational space-time crystal systems.

Celestial amplitudes as AdS-Witten diagrams

Both celestial and momentum space amplitudes in four dimensions are beset by divergences resulting from spacetime translation and sometimes scale invariance. In this paper we consider a (linearized) marginal deformation of the celestial CFT for Yang-Mills theory which preserves 2D conformal invariance but breaks both spacetime translation and scale invariance and involves a chirally coupled massive scalar. The resulting MHV celestial amplitudes are completely finite (apart from the usual soft and collinear divergences and isolated poles in the sum of the weights) and take the canonical CFT form. Moreover, we show they can be simply rewritten in terms of AdS3-Witten contact diagrams which evaluate to the well-known D-functions, thereby forging a direct connection between flat and AdS holography.

4D supersymmetric gauge theories of spacetime translations

The paper addresses the question whether in four spacetime dimensions, besides standard supergravity theories, field theories exist whose symmetries include local spacetime translations and supersymmetries generated by transformations whose commutators contain infinitesimal local spacetime translations. Several new supersymmetric free field theories are found which may provide theories of that type. It is outlined how these free field theories may be extended to globally supersymmetric theories with the desired properties. One class of such theories is constructed explicitly. These theories are similar to globally supersymmetric Yang-Mills theories in flat spacetime but have supersymmetries generated by transformations whose commutators generate infinitesimal general coordinate transformations with field dependent gauge parameters.

Second-Order Approximate Equations of the Large-Scale Atmospheric Motion Equations and Symmetry Analysis for the Basic Equations of Atmospheric Motion

In this paper, symmetry properties of the basic equations of atmospheric motion are proposed. The results on symmetries show that the basic equations of atmospheric motion are invariant under space-time translation transformation, Galilean translation transformations and scaling transformations. Eight one-parameter invariant subgroups and eight one-parameter group invariant solutions are demonstrated. Three types of nontrivial similarity solutions and group invariants are proposed. With the help of perturbation method, we derive the second-order approximate equations for the large-scale atmospheric motion equations, including the non-dimensional equations and the dimensional equations. The second-order approximate equations of the large-scale atmospheric motion equations not only show the characteristics of physical quantities changing with time, but also describe the characteristics of large-scale atmospheric vertical motion.