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Incorporating higher curvature terms into gravity theories modifies the classical field equations, potentially leading to theoretical issues like Shapiro time advancements that violate the Camanho, Edelstein, Maldacena, and Zhiboedov (CEMZ) causality criterion. We explore this criterion within the context of Generalised Quadratic Gravity (GQG), a higher curvature theory with a distinct graviton three-point coupling from General Relativity (GR). By constructing an exact shock wave solution of GQG and calculating the Shapiro time shift for a massless probe graviton, we demonstrate that it can remain strictly positive within a classically allowed parameter space of couplings, ensuring the theory's adherence to the CEMZ criterion. This finding indicates that GQG can offer a causal extension beyond GR, paving the way for further exploration into the consistency of classical higher curvature gravity theories., Comment: 5 pages, 2 figures

The symmetries of the gravitational scattering are intimately tied to the symmetries which preserve asymptotic flatness at null infinity. In Penrose's definition of asymptotic flatness, a central role is played by the notion of asymptotic simplicity and the ensuing peeling behavior which dictates the decay rate of the Weyl tensor. However, there is now accumulating evidence that in a generic gravitational scattering the peeling property is broken, so that the spacetime is not asymptotically-flat in the usual sense. These obstructions to peeling can be traced back to the existence of universal radiative low frequency observables called ''tails to the displacement memory''. The universality of these tail modes is the statement of the classical logarithmic soft graviton theorem of Sahoo, Saha and Sen. Four-dimensional gravitation scattering therefore exhibits a rich infrared interplay between tail to the memory, loss of peeling, and universal logarithmic soft theorems. In this paper we study the solution space and the asymptotic symmetries for logarithmically-asymptotically-flat spacetimes. These are defined by a polyhomogeneous expansion of the Bondi metric which gives rise to a loss of peeling, and represent the classical arena which can accommodate a generic gravitational scattering containing tails to the memory. We show that while the codimension-two generalized BMS charges are sensitive to the loss of peeling at $\mathcal{I}^+$, the flux is insensitive to the fate of peeling. Due to the tail to the memory, the soft superrotation flux contains a logarithmic divergence whose coefficient is the quantity which is conserved in the scattering by virtue of the logarithmic soft theorem. In our analysis we also exhibit new logarithmic evolution equations and flux-balance laws, whose presence suggests the existence of an infinite tower of subleading logarithmic soft graviton theorems., Comment: 40 pages

Arkani-Hamed, Bai, He, and Yan (ABHY) discovered a convex realisation of the associahedron whose combinatorial and geometric structure generates tree-level amplitudes in bi-adjoint scalar theory. In this paper, we identify S-matrix of Yang-Mills theory with a scalar obtained by contracting the canonical form of ABHY associahedron with a multi-vector field (MVF) in the kinematic space. Components of this MVF are determined by the combinatorial structures that underlie the associahedron and Corolla polynomial that was introduced by Kreimer, Sars, and van Suijlekom (KSVS) in [2]. KSVS used the Corolla polynomial to obtain (at all orders in the loop expansion) the parametric representation of gauge theory Feynman integral from the corresponding Feynman integral in $\phi^{3}$ theory. Using the full power of Corolla polynomial, we then extend these results to obtain Yang-Mills one loop planar integrand by contracting the Corolla generated MVF with the canonical form defined by $\hat{D}_{n}$ polytope discovered by Arkani-Hamed, Frost, Plamondon, Salvatori, Thomas. We also demonstrate that KSVS representation of Corolla graph differential in the parametric space can be readily extended to "spin up" the curve integral formulae for $\textrm{Tr}\phi^{3}$ amplitude discovered in [3,4] and give an explicit construction of such formulae for tree-level and planar one loop gluon amplitudes., Comment: 68 pages, 12 figures

Gauge theories and perturbative gravity in four dimensions are governed by a tower of infinite-dimensional symmetries which arise from tree-level soft theorems. However, aside from the leading soft theorems which are all-loop exact, subleading ones receive loop corrections due to long-range infrared effects which result in new soft theorems with logarithmic dependence on the energy of the soft particle. The conjectured universality of these logarithmic soft theorems to all loop orders cries out for a symmetry interpretation. In this letter we initiate a program to compute long-range infrared corrections to the charges that generate the asymptotic symmetries in (scalar) QED and perturbative gravity. For late-time fall-offs of the electromagnetic and gravitational fields which give rise to infrared dressings for the matter fields, we derive finite charge conservation laws and show that in the quantum theory they correspond precisely to the first among the infinite tower of logarithmic soft theorems. This symmetry interpretation, by virtue of being universal and all-loop exact, is a key element for a holographic principle in spacetimes with flat asymptotics., Comment: 8 pages

In this note, we prove that the realization of associahedron discovered by Arkani-Hamed, Bai, He, and Yun (ABHY) is a positive geometry for tree-level S-matrix of scalars which have no color and which interact via cubic coupling. More in detail, we consider diffeomorphic images of the ABHY associahedron. The diffeomorphisms are linear maps parametrized by the right cosets of the Dihedral group on n elements. The set of all the boundaries associated with these copies of ABHY associahedron exhaust all the simple poles. We prove that the sum over the diffeomorphic copies of ABHY associahedron is a positive geometry and the total volume obtained by summing over all the dual associahedra is proportional to the tree-level S matrix of (massive or massless) scalar particles with cubic coupling. We then provide non-trivial evidence that the projection of the planar scattering forms parametrized by the Stokes polytope on these realizations of the associahedron leads to the tree-level amplitudes of scalar particles, which interact via quartic coupling. Our results build on ideas laid out in our previous works, leading to further evidence that a large class of positive geometries which are diffeomorphic to the ABHY associahedron defines an ``amplituhedron" for a tree-level S matrix of some local and unitary scalar theory. We also highlight a fundamental obstruction in applying these ideas to discover positive geometry for the one loop integrand when propagating states have no color., Comment: 33 Pages, 4 Figures

We study a novel asymptotic limit of massive scalar fields in nongravitational quantum field theories in four-dimensional flat space. We foliate the spacetime into a set of dS$_3$ slices that are spacelike to, and at a constant proper distance from, an arbitrarily chosen origin, and study the boundary dS$_3$ obtained in the infinite-distance limit. Massive bulk fields have an exponentially small tail in this limit, and by stripping off this tail we obtain observables that are intrinsic to the boundary dS$_3$. A single massive field in the bulk can be decomposed into an infinite set of dS$_3$ fields, and the Minkowski vacuum corresponds to the Euclidean vacuum for these fields. Our procedure for extrapolating bulk observables induces potential singularities in boundary correlators but we show how they can be cured in the free theory by smearing the boundary operators. We show that by integrating boundary operators with suitable smearing functions it is possible to reconstruct all local bulk operators in the free theory. We argue, using perturbation theory, that our extrapolation procedure continues to be well defined in the presence of interactions. We demonstrate a relationship between the width of the boundary smearing function and the localization of the bulk field. We study other interesting properties of the boundary algebra including the action of global translations and the manner in which local bulk interactions are encoded on the boundary., Comment: 54 pages; 4 figures

Building on the prior work in [1] we locate a family of positive geometries in the kinematic space which are a specific class of convex realisations of the associahedron. These realisations are obtained by scaling and translating the kinematic space associahedron discovered by by Arkani-Hamed, Bai, He and Yan (ABHY). We call the resulting polytopes, deformed realisations of the associahedron. The deformed realisations shed new light on the CHY formula. One of the striking discoveries in [2] was the fact that the CHY scattering equations generate diffeomorphism between the (compactified) CHY moduli space and the ABHY associahedron. As we argue, the deformed realisation of the associahedron can also be interpreted as an diffeomorphic image of the CHY moduli space under scattering equations that we call deformed scattering equations. The canonical form in the kinematic space is thus once again the push-forward of the Parke-Taylor form . A natural off-shoot of our analysis is the universality of the Parke-Taylor form as a CHY Integrand for a class of (tree-level and planar) multi-scalar field amplitudes. These ideas help us in proving the existence of positive geometries for certain specific multi-scalar interactions. We prove that in a field theory with a massless and a massive bi-adjoint scalar fields which interact via cubic interaction, the tree-level S-matrix with massless external states and at most one massive propagator is a weighted sum over the canonical forms defined by certain deformed realisations of the associahedron. Finally, we show that these ideas admit an extension to one-loop. In particular, the one loop S-matrix integrand with at most one massive propagator is a weighted sum over canonical forms of a family of deformed realisations of the type-D cluster polytope, discovered in [3,4]., Comment: 39 pages, 8 figures

In this note, we investigate the implications of classical soft theorems for the formalism developed by Kosower, Maybee and O'Connell (KMOC) to derive classical observables in gauge theory and gravity from scattering amplitudes. In particular, we show that the radiative electro-magnetic field at leading order in the soft expansion imposes an infinite hierarchy of constraints on the expectation value of the family of observables generated by \textit{monomials} of linear impulse. We perform an explicit check on these constraints at next to leading order (NLO) in the coupling and as a corollary show how up to NLO, soft radiation obtained from quantum amplitudes is consistent with the (leading) classical soft photon theorem. We also argue that in 4 dimensions the classical log soft theorem derived by Saha, Sahoo and Sen generates an infinite hierarchy of constraints on the expectation value of operators which are products of one angular momentum and an arbitrary number of linear momenta., Comment: v2: Appendix B included, references added and typos fixed v3: Journal version

Classifying consistent effective field theories for the gravitational interaction has recently been the subject of intense research. Demanding the absence of causality violation in high energy graviton scattering processes has led to a hierarchy of constraints on higher derivative terms in the Lagrangian. Most of these constraints have relied on analysis that is performed in general relativistic backgrounds, as opposed to a generic solution to the equations of motion which are perturbed by higher curvature operators. Hence, these constraints are necessary but may not be sufficient to ensure that the theory is consistent. In this context, we explore the so-called CEMZ causality constraints on Quadratic Gravity in a space of shock wave solutions beyond GR. We show that the Shapiro time delay experienced by a graviton is polarization-independent and positive, regardless of the strength of the gravitational couplings. Our analysis shows that as far as the causality constraints are concerned, albeit inequivalent to General Relativity due to additional propagating modes, Quadratic Gravity is causal as per as the diagnostic proposed by CEMZ., Comment: 18 pages, 1 figure, 1 table

The Lie algebra generated by supertranslation and superrotation vector fields at null infinity, known as the extended BMS (eBMS) algebra is expected to be a symmetry algebra of the quantum gravity S matrix. However, the algebra of commutators of the quantized eBMS charges has been a thorny issue in the literature. On the one hand, recent developments in celestial holography point towards a symmetry algebra which is a closed Lie algebra with no central extension or anomaly, and on the other hand, work of Distler, Flauger and Horn has shown that when these charges are quantized at null infinity, the commutator of a supertranslation and a superrotation charge does not close into a supertranslation but gets deformed by a 2 cocycle term, which is consistent with the original proposal of Barnich and Troessaert. In this paper, we revisit this issue in light of recent developments in the classical understanding of superrotation charges. We show that, for extended BMS symmetries, a phase space at null infinity is an extension of hitherto considered phase spaces which also includes a mode associated to the spin memory and its conjugate partner. We also show that for holomorphic vector fields on the celestial plane, quantization of the eBMS charges in the new phase space leads to an algebra which closes without a 2 cocycle. The degenerate vacua are labelled by the soft news and a Schwarzian mode which corresponds to deformations of the celestial metric by superrotations. The closed eBMS quantum algebra may also lead to a convergence between two manifestations of asymptotic symmetries, one via asymptotic quantization at null infinity and the other through celestial holography., Comment: 35 pages, No Figures

The geometric structure of S-matrix encapsulated by the "Amplituhedron program" has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan it is now understood that for a wide class of scalar quantum field theories, tree-level amplitudes are canonical forms associated to polytopes known as accordiohedra. Similarly the higher loop scalar integrands are canonical forms associated to so called type-D cluster polytopes for cubic interactions or recently discovered class of polytopes termed pseudo-accordiohedron for higher order scalar interactions. In this paper, we continue to probe the universality of these structures for a wider class of scalar quantum field theories. More in detail, we discover new realisations of the associahedron in planar kinematic space whose canonical forms generate (colour-ordered) tree-level S matrix of external massless particles with $n-4$ massless poles and one massive pole at $m^{2}$. The resulting amplitudes are associated to $\lambda_{1}\, \phi_{1}^{3}\, +\, \lambda_{2}\, \phi_{1}^{2}\phi_{2}$ potential where $\phi_{1}$ and $\phi_{2}$ are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the "decoupling limit" (where $m \rightarrow \infty, \lambda_{2} \rightarrow \infty$ such that $g := \frac{\lambda_{2}}{m} = \textrm{finite}$) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by $\lambda \phi_{1}^{3} + g \phi_{1}^{4}$., Comment: 43 pages, 11 Figures

Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST), we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions. We define a new class of combinatorial polytopes that we call pseudo-accordiohedra whose poset structures are associated to singularities of the one loop integrand associated to scalar quartic interactions. Pseudo-accordiohedra parametrize a family of projective forms on the abstract kinematic space defined by AHST and restriction of these forms to the type-D associahedra can be associated to one-loop integrands for quartic interactions. The restriction (of the projective form) can also be thought of as a canonical top form on certain geometric realisations of pseudo-accordiohedra. Our work explores a large class of geometric realisations of the type-D associahedra which include all the AHST realisations. These realisations are based on the pseudo-triangulation model for type-D cluster algebras discovered by Ceballos and Pilaud., Comment: 37 pages, 9 figures

We apply the recently developed formalism by Kosower, Maybee and O'Connell (KMO) to analyse the soft electromagnetic and soft gravitational radiation emitted by particles without spin in Four and higher dimensions. We use this formalism in conjunction with quantum soft theorems to derive radiative electro-magnetic and gravitational fields in low frequency expansion and to next to leading order in the coupling. We show that in all dimensions, the classical limit of sub-leading soft (photon and graviton) theorems is consistent with the classical soft theorems proved by Sen et al in a series of papers. In particular Saha, Sahoo and Sen proved classical soft theorems for electro-magnetic and gravitational radiation in Four dimensions. For the class of scattering processes that can be analyzed using KMO formalism, we show that the classical limit of quantum soft theorems is consistent with these classical soft theorems, paving the way for their proof from scattering amplitudes., Comment: 53 pages, Improved presentation, typos corrected

We argue that, in a theory of quantum gravity in a four dimensional asymptotically flat spacetime, all information about massless excitations can be obtained from an infinitesimal neighbourhood of the past boundary of future null infinity and does not require observations over all of future null infinity. Moreover, all information about the state that can be obtained through observations near a cut of future null infinity can also be obtained from observations near any earlier cut although the converse is not true. We provide independent arguments for these two assertions. Similar statements hold for past null infinity. These statements have immediate implications for the information paradox since they suggest that the fine-grained von Neumann entropy of the state defined on a segment $(-\infty,u)$ of future null infinity is independent of u. This is very different from the oft-discussed Page curve that this entropy is sometimes expected to obey. We contrast our results with recent discussions of the Page curve in the context of black hole evaporation, and also discuss the relation of our results to other proposals for holography in flat space., Comment: 44 pages (v2) refs added. typos fixed

In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3,4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any $n$-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2,5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra. We then use these kinematic space geometric constructions to write worldsheet forms for $\phi^{4}$ theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint $\phi^3$ scalar amplitudes, scattering equations can be used as diffeomorphisms between certain $\frac{n-4}{2}$ forms on the worldsheet and $\frac{n-4}{2}$ forms on ABHY associahedron that generate quartic amplitudes., Comment: 58 pages, 7 figures, Section 6 revised and a new Appendix (Appendix G) added

Classical soft graviton theorem gives an expression for the spectrum of low frequency gravitational radiation, emitted during a classical scattering process, in terms of the trajectories and spin angular momenta of ingoing and outgoing objects, including hard radiation. This has been proved to subleading order in the expansion in powers of frequency by taking the classical limit of the quantum soft graviton theorem. In this paper we give a direct proof of this result by analyzing the classical equations of motion of a generic theory of gravity coupled to interacting matter in space-time dimensions larger than four., Comment: LaTeX; 32 pages

Recently Sahoo and Sen obtained a series of remarkable results concerning sub-leading soft photon and graviton theorems in four dimensions. Even though the S- matrix is infrared divergent, they have shown that the sub-leading soft theorems are well defined and exact statements in QED and perturbative Quantum Gravity. However unlike the well studied Cachazo-Strominger soft theorems in tree-level amplitudes, the new sub-leading soft expansion is at the order ln {\omega} (where {\omega} is the soft frequency) and the corresponding soft factors structurally show completely different properties then their tree-level counterparts. Whence it is natural to ask if these theorems are associated to asymptotic symmetries of the S-matrix. We consider this question in the context of sub-leading soft photon theorem in scalar QED and show that there are indeed an infinity of conservation laws whose Ward identities are equivalent to the loop-corrected soft photon theorem. This shows that in the case of four dimensional QED, the leading and sub-leading soft photon theorems are equivalent to Ward identities of (asymptotic) charges., Comment: 33 pages, no figures

In a remarkable recent work [arXiv : 1711.09102] by Arkani-Hamed et al, the amplituhedron program was extended to the realm of non-supersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless $\phi^{3}$ theory (and its close cousin, bi-adjoint $\phi^{3}$ theory) a polytope known as the associahedron sits inside the kinematic space and is the amplituhedron for the theory. Precisely as in the case of amplituhedron, it was shown that scattering amplitude is nothing but residue of the canonical form associated to the associahedron. Combinatorial and geometric properties of associahedron naturally encode properties like locality and unitarity of (tree level) scattering amplitudes. In this paper we attempt to extend this program to planar amplitudes in massless $\phi^{4}$ theory. We show that tree-level planar amplitudes in this theory can be obtained from geometry of objects known as the Stokes polytope which sits naturally inside the kinematic space. As in the case of associahedron we show that residues of the canonical form on these Stokes polytopes can be used to compute scattering amplitudes for quartic interactions. However unlike associahedron, Stokes polytope of a given dimension is not unique and as we show, one must sum over all of them to obtain the complete scattering amplitude. Not all Stokes polytopes contribute equally and we argue that the corresponding weights depend on purely combinatorial properties of the Stokes polytopes. As in the case of $\phi^{3}$ theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes., Comment: typos corrected, improvement in explanation of Stokes polytopes (combinatorial vs. convex realisation) and the Q-compatibility rule; accepted for publication in JHEP

Hamada and Shiu have recently shown that tree level amplitudes in QED satisfy an infinite hierarchy of soft photon theorems, the first two of which are Weinberg and Low's theorems respectively. In this paper we propose that in tree level massless QED, this entire hierarchy is equivalent to a hierarchy of (asymptotic) conservation laws. We prove the equivalence explicitly for the case of sub-subleading soft photon theorem and give substantial evidence that the equivalence continues to hold for the entire hierarchy. Our work also brings out the (complimentary) relationship between the asymptotic charges associated to soft theorems and the well known Newman-Penrose charges., Comment: 37 pages

There has been a recent resurgence of interest in the structure of the gravitational field at null infinity, sparked by new results on soft charges and infrared issues related to the S matrix theory in perturbative quantum gravity. We summarize these developments and put them in the broader context of research in the relativity community that dates back to several decades. In keeping with intent of this series, this overview is addressed to gravitational scientists who are not experts in this specific area., Comment: 19 pages, no figures. Prepared for the "Introduction to Current Research Series." Clarifications added following correspondence with expert, typos corrected, version to appear in the journal

We show that the recently discovered logarithmic terms in the soft graviton theorem induce a late time component in the gravitational wave-form that falls off as inverse power of time, producing a tail term to the linear memory effect., Comment: 8 pages, v3 : Discussion on Non-relativistic limit changed, references added

It has been shown that in larger than four space-time dimensions, soft factors that relate the amplitudes with a soft photon or graviton to amplitudes without the soft particle also determine the low frequency radiative part of the electromagnetic and gravitational fields during classical scattering. In four dimensions the S-matrix becomes infrared divergent making the usual definition of the soft factor ambiguous beyond the leading order. However the radiative parts of the electromagnetic and gravitational fields provide an unambiguous definition of soft factor in the classical limit up to the usual gauge ambiguity. We show that the soft factor defined this way develops terms involving logarithm of the energy of the soft particle at the subleading order in the soft expansion., Comment: LaTeX file, 33 pages, v2 : Overall phase determined carefully, change in footnote 2

Classical limit of multiple soft graviton theorem can be used to compute the angular power spectrum of long wavelength gravitational radiation in classical scattering provided the total energy carried away by the radiation is small compared to the energies of the scatterers. We could ensure this either by taking the limit in which the impact parameter is large compared to the Schwarzschild radii of the scatterers, or by taking the probe limit where one object (the probe) has mass much smaller than the other object (the scatterer). We compute the results to subsubleading order in soft momentum and test them using explicit examples involving classical scattering. Our analysis also generalizes to the case where there are multiple objects involved in the scattering and the objects exchange mass, fragment or fuse into each other during the scattering. A similar analysis can be carried out for soft photons to subleading order, reproducing standard textbook results. We also discuss the modification of soft expansion in four dimensions beyond the leading order due to infrared divergences., Comment: LaTeX; 63 pages

In [1,2] it was shown that the subleading soft photon theorem in tree level amplitudes in massless QED is equivalent to a new class of symmetries of the theory parameterized by a vector field on the celestial sphere. In this paper, we extend these results to the subleading soft photon theorem in any Effective Field Theory containing photons and an arbitrary spectrum of massless particles. We show that the charges associated to the above class of symmetries are sensitive to certain three point functions of the theory and are corrected by irrelevant operators of specific dimensions. Our analysis shows that the subleading soft photon theorem in any tree level scattering amplitude is a statement about asymptotic symmetries of the ${\cal S}$-matrix., Comment: 26 pages, 3 figures

We analyze scattering amplitudes with one soft external graviton and arbitrary number of other finite energy external states carrying arbitrary mass and spin to sub-subleading order in the momentum of the soft graviton. Our result can be expressed as the sum of a universal part that depends only on the amplitude without the soft graviton and not the other details of the theory and a non-universal part that depends on the amplitude without the soft graviton, and the two and three point functions of the theory. For tree amplitudes our results are valid in all space-time dimensions while for loop amplitudes, infrared divergences force us to restrict our analysis to space time dimensions five or more. With this restriction the results are valid to all orders in perturbation theory. Our results agree with known results in quantum field theories and string theory., Comment: 52 pages, 7 figures v2: Proved that the non-universal sub-subleading corrections depend only on on-shell 3-point functions

We present strong evidence that the sub-subleading soft theorem in semi-classical (tree level) gravity discovered by Cachazo and Strominger is equivalent to the conservation of asymptotic charges associated to a new class of vector fields not contained within the previous extensions of BMS algebra. Our analysis crucially relies on analyzing the hitherto established equivalences between soft theorems and Ward identities from a new perspective. In this process we naturally (re)discover a class of `magnetic' charges at null infinity that are associated to the dual of the Weyl tensor., Comment: 41 pages

Lysov, Pasterski and Strominger have shown how Low's subleading soft photon theorem can be understood as Ward identities of new symmetries of massless QED. In this paper we offer a different perspective and show that there exists a class of large $U(1)$ gauge transformations such that (i) the associated (electric and magnetic) charges can be computed from first principles (ii) their Ward identities are equivalent to Low's theorem. Our framework paves the way to analyze the sub-subleading theorem in gravity in terms of Ward identities associated to large diffeomorphisms., Comment: 17 pages

Due to seminal works of Weinberg, Cachazo and Strominger we know that tree level quantum gravity amplitudes satisfy three factorization constraints. Building on previous works which relate two of these constraints to symmetries of quantum gravity at null infinity, we present rather strong evidence that the third constraint is also equivalent to a new set of symmetries of (perturbative) quantum gravity. Our analysis implies that the symmetry group of quantum gravity may be even richer than the BMS group (or infinite dimensional extension thereof) previously considered., Comment: 5 pages

The existing equivalence between (generalized) BMS Ward identities with leading and subleading soft graviton theorems is extended to the case where the scattering particles are massive scalars. By extending the action of generalized BMS group off null infinity at late times, we show that there is a natural action of such group not only on the radiative data at null infinity but also on the scattering data of the massive scalar field. This leads to a formulation of Ward identities associated to the generalized BMS group when the scattering states are massive scalars or massless gravitons and we show that these Ward identities are equivalent to the leading and subleading soft graviton theorems., Comment: 30 pages

Various equivalences between so-called soft theorems which constrain scattering amplitudes and Ward identities related to asymptotic symmetries have recently been established in gauge theories and gravity. So far these equivalences have been restricted to the case of massless matter fields, the reason being that the asymptotic symmetries are defined at null infinity. The restriction is however unnatural from the perspective of soft theorems which are insensitive to the masses of the external particles. In this work we remove the aforementioned restriction in the context of scalar QED. Inspired by the radiative phase space description of massless fields at null infinity, we introduce a manifold description of time-like infinity on which the asymptotic phase space for massive fields can be defined. The "angle dependent" large gauge transformations are shown to have a well defined action on this phase space, and the resulting Ward identities are found to be equivalent to Weinberg's soft photon theorem., Comment: 19 pages, no figures

In [15] we proposed a generalization of the BMS group G which is a semidirect product of supertranslations and smooth diffeomorphisms of the conformal sphere. Although an extension of BMS, G is a symmetry group of asymptotically flat space times. By taking G as a candidate symmetry group of the quantum gravity S-matrix, we argued that the Ward identities associated to the generators of Diff(S^2) were equivalent to the Cachazo-Strominger subleading soft graviton theorem. Our argument however was based on a proposed definition of the Diff(S^2) charges which we could not derive from first principles as G does not have a well defined action on the radiative phase space of gravity. Here we fill this gap and provide a first principles derivation of the Diff(S^2) charges. The result of this paper, in conjunction with the results of [4, 15] prove that the leading and subleading soft theorems are equivalent to the Ward identities associated to G., Comment: 19 pages

Motivated by the equivalence between soft graviton theorem and Ward identities for the supertranslation symmetries belonging to the BMS group, we propose a new extension (different from the so-called extended BMS) of the BMS group which is a semi-direct product of supertranslations and Diff(S^2). We propose a definition for the canonical generators associated to the smooth diffeomorphisms and show that the resulting Ward identities are equivalent to the subleading soft graviton theorem of Cachazo and Strominger., Comment: 25 pages, published version

We initiate the hunt for a definition of Hamiltonian constraint in Euclidean Loop Quantum Gravity (LQG) which faithfully represents quantum Dirac algebra. Borrowing key ideas from previous works on Hamiltonian constraint in LQG and several toy models, we present some evidence that there exists such a continuum Hamiltonian constraint operator which is well defined on a suitable generalization of the Lewandowski-Marolf Habitat and is anomaly free off-shell., Comment: 68 pages, 6 figures

In [1] we initiated an approach towards quantizing the Hamiltonian constraint in Loop Quantum Gravity (LQG) by requiring that it generates an anomaly-free representation of constraint algebra off-shell. We investigated this issue in the case of a toy model of a 2+1-dimensional $U(1)^{3}$ gauge theory, which can be thought of as a weak coupling limit of Euclidean three dimensional gravity. However in [1] we only focused on the most non-trivial part of the constraint algebra that involves commutator of two Hamiltonian constraints. In this paper we continue with our analysis and obtain a representation of full constraint algebra in loop quantized framework. We show that there is a representation of the Diffeomorphism group with respect to which the Hamiltonian constraint quantized in [1] is diffeomorphism covariant. Our work can be thought of as a potential first step towards resolving some long standing issues with the Hamiltonian constraint in canonical LQG., Comment: 30 pages, 1 figure

We analyze the issue of anomaly-free representations of the constraint algebra in Loop Quantum Gravity (LQG) in the context of a diffeomorphism-invariant gauge theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the "mu-bar"-scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework., Comment: 60 pages, 6 figures

We review some approaches to the Hamiltonian dynamics of (loop) quantum gravity, the main issues being the regularization of the Hamiltonian and the continuum limit. First, Thiemann's definition of the quantum Hamiltonian is presented, and then more recent approaches. They are based on toy models which provide new insights into the difficulties and ambiguities faced in Thiemann's construction. The models we use are parametrized field theories, the topological BF model of which a special case is three-dimensional gravity which describes quantum flat space, and Regge lattice gravity.

We construct the smeared diffeomorphism constraint operator at finite triangulation from the basic holonomy- flux operators of Loop Quantum Gravity, evaluate its continuum limit on the Lewandowski- Marolf habitat and show that the action of the continuum operator provides an anomaly free representation of the Lie algebra of diffeomorphisms of the 3- manifold. Key features of our analysis include: (i) finite triangulation approximants to the curvature, $F_{ab}^i$ of the Ashtekar- Barbero connection which involve not only small loop holonomies but also small surface fluxes as well as an explicit dependence on the edge labels of the spin network being acted on (ii) the dependence of the small loop underlying the holonomy on both the direction and magnitude of the shift vector field (iii) continuum constraint operators which do {\em not} have finite action on the kinematic Hilbert space, thus implementing a key lesson from recent studies of parameterised field theory by the authors. Features (i) and (ii) provide the first hints in LQG of a conceptual similarity with the so called "mu- bar" scheme of Loop Quantum Cosmology. We expect our work to be of use in the construction of an anomaly free quantum dynamics for LQG., Comment: 37 pages, 6 figures

Recently, a generally covariant reformulation of 2 dimensional flat spacetime free scalar field theory known as Parameterised Field Theory was quantized using Loop Quantum Gravity (LQG) type `polymer' representations. Physical states were constructed, without intermediate regularization structures, by averaging over the group of gauge transformations generated by the constraints, the constraint algebra being a Lie algebra. We consider classically equivalent combinations of these constraints corresponding to a diffeomorphism and a Hamiltonian constraint, which, as in gravity, define a Dirac algebra. Our treatment of the quantum constraints parallels that of LQG and obtains the following results, expected to be of use in the construction of the quantum dynamics of LQG:(i) the (triangulated) Hamiltonian constraint acts only on vertices, its construction involves some of the same ambiguities as in LQG and its action on diffeomorphism invariant states admits a continuum limit (ii)if the regulating holonomies are in representations tailored to the edge labels of the state, all previously obtained physical states lie in the kernel of the Hamiltonian constraint, (iii) the commutator of two (density weight 1) Hamiltonian constraints as well as the operator correspondent of their classical Poisson bracket converge to zero in the continuum limit defined by diffeomorphism invariant states, and vanish on the Lewandowski- Marolf (LM) habitat (iv) the rescaled density 2 Hamiltonian constraints and their commutator are ill defined on the LM habitat despite the well defined- ness of the operator correspondent of their classical Poisson bracket there (v) there is a new habitat which supports a non-trivial representation of the Poisson- Lie algebra of density 2 constraints, Comment: 53 pages

Building on prior work, a generally covariant reformulation of free scalar field theory on the flat Lorentzian cylinder is quantized using Loop Quantum Gravity (LQG) type `polymer' representations. This quantization of the {\em continuum} classical theory yields a quantum theory which lives on a discrete spacetime lattice. We explicitly construct a state in the polymer Hilbert space which reproduces the standard Fock vacuum- two point functions for long wavelength modes of the scalar field. Our construction indicates that the continuum classical theory emerges under coarse graining. All our considerations are free of the "triangulation" ambiguities which plague attempts to define quantum dynamics in LQG. Our work constitutes the first complete LQG type quantization of a generally covariant field theory together with a semi-classical analysis of the true degrees of freedom and thus provides a perfect infinite dimensional toy model to study open issues in LQG, particularly those pertaining to the definition of quantum dynamics., Comment: 58 pages

Free scalar field theory on 2 dimensional flat spacetime, cast in diffeomorphism invariant guise by treating the inertial coordinates of the spacetime as dynamical variables, is quantized using LQG type `polymer' representations for the matter field and the inertial variables. The quantum constraints are solved via group averaging techniques and, analogous to the case of spatial geometry in LQG, the smooth (flat) spacetime geometry is replaced by a discrete quantum structure. An overcomplete set of Dirac observables, consisting of (a) (exponentials of) the standard free scalar field creation- annihilation modes and (b) canonical transformations corresponding to conformal isometries, are represented as operators on the physical Hilbert space. None of these constructions suffer from any of the `triangulation' dependent choices which arise in treatments of LQG. In contrast to the standard Fock quantization, the non- Fock nature of the representation ensures that the algebra of conformal isometries as well as that of spacetime diffeomorphisms are represented in an anomaly free manner. Semiclassical states can be analysed at the gauge invariant level. It is shown that `physical weaves' necessarily underly such states and that such states display semiclassicality with respect to, at most, a countable subset of the (uncountably large) set of observables of type (a). The model thus offers a fertile testing ground for proposed definitions of quantum dynamics as well as semiclassical states in LQG., Comment: 44 pages, no figures

We present a polymer(loop) quantization of a two dimensional theory of dilatonic gravity known as the CGHS model. We recast the theory as a parametrized free field theory on a flat 2-dimensional spacetime and quantize the resulting phase space using techniques of loop quantization. The resulting (kinematical) Hilbert space admits a unitary representation of the spacetime diffeomorphism group. We obtain the complete spectrum of the theory using a technique known as group averaging and perform quantization of Dirac observables on the resulting Hilbert space. We argue that the algebra of Dirac observables gets deformed in the quantum theory. Combining the ideas from parametrized field theory with certain relational observables, evolution is defined in the quantum theory in the Heisenberg picture. Finally the dilaton field is quantized on the physical Hilbert space which carries information about quantum geometry., Comment: 50 pages, 2 figures