Symmetries from Locality. II. Gravitation and Lorentz Boosts
Mark P. Hertzberg, Jacob A. Litterer, McCullen Sandora
备注：
13 pages in double column format. V2 further clarification and fixed some typos. V3: Further clarifications, including mass terms. Updated towards version accepted for publication in Phys. Rev. D, including journal title format
It is known that local, Lorentz invariant, unitary theories involving particles with spin 1 demand that the matter sector they couple to are organized by internal physical symmetries and the associated charge conservation, while spin 3/2 demands supersymmetry. However, the introduction of a spin 2 graviton does not obviously demand new symmetries of the matter sector (although it does demand a universal coupling). In this work we relax the assumption of Lorentz boost symmetry, while maintaining a basic notion of locality that there is no instantaneous signaling at a distance. This extends and complements our accompanying work in Part 1 on related issues for spin 1 particles in electromagnetism. In order to avoid potential problems with longitudinal modes of the graviton, we choose to project them out, leaving only two degrees of freedom. We study large classes of theories that a priori may violate Lorentz boost invariance. By requiring the tree-level exchange action be local, we find that consistency demands that the Lorentz boost symmetry must be satisfied by the graviton and the matter sector, and in turn we recover general relativity at this order of analysis.
We relax the assumption of time translation and Lorentz boost symmetry in theories involving massless spin 2 gravitons, while maintaining a basic notion of locality that there is no instantaneous signaling at a distance. We project out longitudinal modes, leaving only two degrees of freedom of the graviton. Our previous work, which assumed time translation symmetry, found that the Lorentz boost symmetry is required to ensure locality at leading order. In this work, without assuming time translations or Lorentz boosts, we show that locality of the exchange action between matter sources demands …
We investigate the relativistic properties under boost transformations of the $κ$-Poincaré model with multiple causally connected interactions, both at the level of its formulation in momentum space only and when it is endowed with a full phase space construction, provided by the relative locality framework. Previous studies focussing on the momentum space picture showed that in presence of just one interaction vertex the model is relativistic, provided that the boost parameter acting on each given particle receives a "backreaction" from the momenta of the other particles that participate in t…
We show that the $κ$-Poincaré Hopf algebra can be interpreted in the framework of curved momentum space leading to the relativity of locality \cite{AFKS}. We study the geometric properties of the momentum space described by $κ$-Poincaré, and derive the consequences for particles propagation and energy-momentum conservation laws in interaction vertices, obtaining for the first time a coherent and fully workable model of the deformed relativistic kinematics implied by $κ$-Poincaré. We describe the action of boost transformations on multi-particles systems, showing that in order to keep covariant…
This essay argues that when measurement processes involve energies of the order of the Planck scale, the fundamental assumption of locality may no longer be a good approximation. Idealized position measurements of two distinguishable spin-$0$ particles are considered. The measurements alter the space-time metric in a fundamental manner governed by the commutation relations $[x_i\,\,p_j]= i\hbar\,δ_{ij}$ and the classical field equations of gravitation. This {\it in-principle} unavoidable change in the space-time metric destroys the commutativity (and hence locality) of position measurement ope…
It is well known that a theory of the (i) Lorentz invariant and (ii) locally interacting (iii) two degrees of freedom of a massless spin 1 particle, the photon, leads uniquely to electromagnetism at large distances. In this work, we remove the assumption of (i) Lorentz boost invariance, but we still demand (ii) and (iii). We consider several broad classes of theories of spin 1, which in general explicitly violate Lorentz symmetry. We restrict to the familiar two degrees of freedom of the photon. We find that most theories lead to non-locality and instantaneous signaling at a distance. By deman…
We revisit the construction of local bulk operators in AdS/CFT with special focus on gravitational dressing and its consequences for bulk locality. Specializing to 2+1-dimensions, we investigate these issues via the proposed identification between bulk operators and cross-cap boundary states. We obtain explicit expressions for correlation functions of bulk fields with boundary stress tensor insertions, and find that they are free of non-local branch cuts but do have non-local poles. We recover the HKLL recipe for restoring bulk locality for interacting fields as the outcome of a natural CFT cr…
Breakdown of local physics in string theory at distances longer than the string scale is investigated. Such nonlocality would be expected to be visible in ultrahigh-energy scattering. The results of various approaches to such scattering are collected and examined. No evidence is found for non-locality from strings whose length grows linearly with the energy. However, local quantum field theory does apparently fail at scales determined by gravitational physics, particularly strong gravitational dynamics. This amplifies locality bound arguments that such failure of locality is a fundamental aspe…
In this paper, we study the implications of bulk locality on the celestial amplitude. In the context of the four-point amplitude, the fact that the bulk S-matrix factorizes locally in poles of Mandelstam variables is reflected in the imaginary part of the celestial amplitude. In particular, on the real axis in the complex plane of the boost weight, the imaginary part of the celestial amplitude can be given as a positive expansion on the Poincaré partial waves, which are nothing but the projection of flat-space spinning polynomials onto the celestial sphere. Furthermore, we derive the celestial…
The observation of primordial correlators by cosmological surveys is a very promising avenue to probe high energies and the perturbative regime of quantum gravity. Hence, it is imperative that we understand how these observables are shaped by the pillars of fundamental physics, namely unitarity, locality and symmetries. To this end, we study the three-point correlators of gravitons and scalar curvature perturbations around a quasi de Sitter spacetime. We identify a set of Bootstrap Rules that fully fix the form of these correlators in the asymptotic future, i.e. at the boundary, and make no re…
We derive the first positivity bounds for low-energy Effective Field Theories (EFTs) that are not invariant under Lorentz boosts. "Positivity bounds" are the low-energy manifestation of certain fundamental properties in the UV -- to date they have been used to constrain a wide variety of EFTs, however since all of the existing bounds require Lorentz invariance they are not directly applicable when this symmetry is broken, such as for most cosmological and condensed matter systems. From the UV axioms of unitarity, causality and locality, we derive an infinite family of bounds which (derivatives…
We describe a completely general and fully non-perturbative framework for constructing dynamical reference frames in generally covariant theories, and for understanding the gauge-invariant observables that they yield. Our approach makes use of a 'universal dressing space', which contains as a subset every possible dynamical frame. We describe examples of such frames, including matter frames, a popular construction based on boundary-anchored geodesics and one using minimal surfaces -- but our formalism does not depend on the existence of a boundary. The class of observables we construct general…
Spacetime---the union of space and time---is both the actor and the stage during physical processes in our fascinating Universe. In Lorentz invariant local theories, the existence of a maximum signalling speed (the "speed of light") determines a notion of causality in spacetime, distinguishing the past from the future, and the cause from the effect. This thesis is dedicated to the study of \emph{deviations} from locality. Focussing on a particular class of \emph{non-local} theories that is both Lorentz invariant and free of ghosts, we aim to understand the effects of such non-local physics in …
This work refines arguments forbidding non-linear dynamical gravity from appearing in the low energy effective description of field theories with local kinematics, even for those with instantaneous long-range interactions. Specifically, we note that gravitational theories with universal coupling to energy -- an intrinsically non-linear phenomenon -- are characterized by Hamiltonians that are pure boundary terms on shell. In order for this to be the low energy effective description of a field theory with local kinematics, all bulk dynamics must be frozen and thus irrelevant to the construction.…
Poincaré invariance is a well-tested symmetry of nature and sits at the core of our description of relativistic particles and gravity. At the same time, in most systems Poincaré invariance is not a symmetry of the ground state and is hence broken spontaneously. This phenomenon is ubiquitous in cosmology where Lorentz boosts are spontaneously broken by the existence of a preferred reference frame in which the universe is homogeneous and isotropic. This motivates us to study scattering amplitudes without requiring invariance of the interactions under Lorentz boosts. In particular, using on-shell…
A local phenomenological model that reduces to a non-local gravitational theory giving dark energy is proposed. The non-local gravity action is known to fit the data as well as $Λ$-CDM thereby demanding a more fundamental local treatment. It is seen that the scale-invariant higher-derivative scalar-tensor theory of gravity, which is known to be ultraviolet perturbative renormalizable to all loops and where ghosts become innocuous, generates non-locality at low energies. The local action comprises of two real scalar fields coupled non-minimally with the higher-derivative gravity action. When on…
We study all translationally and rotationally invariant local theories involving massless spin 2 and spin 1 particles that mediate long range forces, allowing for general energy relations and violation of boost invariance. Although gauge invariance is not a priori required to describe non Lorentz invariant theories, we first establish that locality requires `soft gauge invariance'. Then by taking the soft graviton limit in scattering amplitudes, we prove that in addition to the usual requirement of universal graviton couplings, the special relativistic energy-momentum relation is also required…
We discuss some main aspects of theories of gravity containing non-local terms in view of cosmological applications. In particular, we consider various extensions of General Relativity based on geometrical invariants as $f(R, \Box^{-1} R)$, $f({\cal G}, \Box^{-1} {\cal G})$ and $f(T, \Box^{-1} T)$ gravity where $R$ is the Ricci curvature scalar, $\cal G$ is the Gauss-Bonnet topological invariant, $T$ the torsion scalar and the operator $\Box^{-1}$ gives rise to non-locality. After selecting their functional form by using Noether Symmetries, we find out exact solutions in a cosmological backgro…
The Special Theory of Relativity and Quantum Mechanics merge in the key principle of Quantum Field Theory, the Principle of Locality. We review some examples of its ``unreasonable effectiveness'' (which shows up best in the formulation of Quantum Field Theory in terms of operator algebras of local observables) in digging out the roots of Global Gauge Invariance in the structure of the local observable quantities alone, at least for purely massive theories; but to deal with the Principle of Local Gauge Invariance is still a problem in this frame. This problem emerges also if one attempts to fig…
Higher-spin vertices containing up to quintic interactions at the Lagrangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a $β\to-\infty$--shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter $η$ in the HS equations. All obtained vertices are shown to be spin-local containing a finite number of derivatives in the spinor space for any given set of spins. The vertices proportional to $η^2$ and $\bar η^2$ are in addition ultra-local, i.…
In this paper we will show all the linearized curvature tensors in the infinite derivative ghost and singularity free theory of gravity in the static limit. We have found that in the region of non-locality, in the ultraviolet regime (at short distance from the source), the Ricci tensor and the Ricci scalar are not vanishing, meaning that we do not have a vacuum solution anymore due to the smearing of the source induced by the presence of non-local gravitational interactions. It also follows that, unlike in Einstein's gravity, the Riemann tensor is not traceless and it does not coincide with th…
We discuss hidden symmetries of M-theory, its feedback on the construction of the M-theory effective action, and a response of the effective action when locality is preserved. In particular, the locality of special symmetries of the duality-symmetric linearized gravity constraints the index structure of the dual to graviton field in the same manner as it is required to separate the levels 0 and 1 generators subalgebra from the infinite-dimensional hidden symmetry algebra of gravitational theory. This conclusion fails once matter fields are taken into account and we give arguments for that. We …
