We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED `no-triangle' property is true for all helicity configurations and has similarities to the `no-triangle' property found in the case of maximal N = 8 supergravity. Results are derived both via a world-line formalism as well as using on-shell unitarity methods. We show that the simple structure of the loop amplitude originates from the extremely good BCFW scaling behaviour of the QED tree-amplitude.
We present a general study of 3-point functions of conformal field theory (CFT) in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWIs), introduced in recent works. We investigate and detail the structure of the CWIs and their non-perturbative solutions, and compare them to perturbation theory, taking QED and QCD as examples. Exact solutions of CFT's in the flat background limit in momentum space are matched by the perturbative realizations in free field theories, showing that the origin the conformal anomaly is…
At leading order, the $S$-matrices in QED and gravity are known to factorise, providing unambiguous determinations of the parts divergent due to infrared contributions. The soft $S$-matrices defined in this fashion are shown to be defined entirely in terms of $2$ dimensional models on the celestial sphere, involving two real scalar fields, allowing us to express the soft $S$-matrices for real as well as virtual divergences as two dimensional correlation functions. We discuss what this means for finding holographic representations of scattering amplitudes in QED and gravity and comment on simpl…
Infrared physics of QED and gravity from representation theory
SISSA（的里雅斯特国际高等研究院）；INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Institut Denis Poisson, Université de Tours, Université d’Orléans, CNRS, IDP, UMR 7013（都灵大学、奥尔良大学、法国国家科学研究中心、德尼·普瓦松研究所）
Strominger and his collaborators pioneered the study of equivalence between soft theorems and asymptotic conservation laws. We study this equivalence in the context of loop level subleading soft photon theorem for massless scalar QED in presence of dynamical gravity. Motivated by Campiglia and Laddha \cite{1903.09133}, we show that the Sahoo-Sen soft photon theorem \cite{1808.03288} for loop amplitudes is equivalent to an asymptotic conservation law. This asymptotic charge is directly related to the dressing of fields due to long range forces exclusively present in four spacetime dimensions. I…
We reconsider the Lee-Wick (LW) models and compare their properties to the properties of the models that contain purely virtual particles. We argue against the LW premise that unstable particles can be removed from the sets of incoming and outgoing states in scattering processes. The removal leads to a non-Hermitian classical limit, besides clashing with the observation of the muon. If, on the other hand, all the states are included, the LW models have a Hamiltonian unbounded from below or negative norms. Purely virtual particles, on the contrary, lead to a Hermitian classical limit and are ab…
We recast the soft $S$-matrices on the celestial sphere as correlation functions of certain $2$-dimensional models of topological defects. In pointing out the double copy structure between the soft photon and soft graviton cases, we arrive at a putative classical double copy between the corresponding topological models and a rederivation of gauge invariance and the equivalence principle as Ward identities of the $2$-dimensional theories.
We derive new representations of the Einstein-Hilbert action in which graviton perturbation theory is immensely simplified. To accomplish this, we recast the Einstein-Hilbert action as a theory of purely cubic interactions among gravitons and a single auxiliary field. The corresponding equations of motion are the Einstein field equations rewritten as two coupled first-order differential equations. Since all Feynman diagrams are cubic, we are able to derive new off-shell recursion relations for tree-level graviton scattering amplitudes. With a judicious choice of gauge fixing, we then construct…
The simplicity constraint is studied in the context of 4d spinfoam models with cosmological constant. We find that the quantum simplicity constraint is realized as the 2d surface defect in SL(2,$\mathbb{C}$) Chern-Simons theory in the construction of spinfoam amplitudes. By this realization of simplicity constraint in Chern-Simons theory, we are able to construct the new spinfoam amplitude with cosmological constant for arbitrary simplicial complex (with many 4-simplices). The semiclassical asymptotics of the amplitude is shown to reproduce correctly the 4-dimensional Einstein-Regge action wit…
We construct Faddeev-Kulish states in QED and perturbative quantum gravity to subleading order in the soft momentum expansion and to first order in the coupling constant, using the charge conservation formula of asymptotic symmetries associated with the tree-level subleading soft theorems. We demonstrate that the emission and absorption of soft photons/gravitons in dressed amplitudes vanish. The fact that no additional soft radiation may be added to a dressed amplitude supports the claim that, in the dressed state formalism, the soft and hard sectors of scattering processes are correlated. We …
It is shown that the ground state or vacuum to the lattice Euclidean quantum gravity is significantly different from the ground states to the well-known vacua in QED, QCD, et cetera. In the case of the lattice Euclidean quantum gravity, the long-wavelength scale vacuum structure is similar to that in QED, moreover the quantum fluctuations to gravity are very reduced in comparison with the situation in QED. But the small scale (of the order of the lattice scale) vacuum structure to gravity is significantly different from that to the long-wavelength scales: the fluctuation values of geometrical …
The in-out formalism is a systematic and powerful method for finding the effective actions in an electromagnetic field and a curved spacetime provided that the field equation has explicitly known solutions. The effective action becomes complex when pairs of charged particles are produced due to an electric field and curved spacetime. This may lead to a conjecture of one-to-one correspondence between the vacuum polarization (real part) and the vacuum persistence (imaginary part). We illustrate the one-loop effective action in a constant electric field in a Minkowski spacetime and in a uniform e…
In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincaré and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints…
We apply positivity bounds directly to a $U(1)$ gauge theory with charged scalars and charged fermions, i.e. QED, minimally coupled to gravity. Assuming that the massless $t$-channel pole may be discarded, we show that the improved positivity bounds are violated unless new physics is introduced at the parametrically low scale $Λ_{\rm new} \sim (e m M_{\rm pl})^{1/2}$, consistent with similar results for scalar field theories, far lower than the scale implied by the weak gravity conjecture. This is sharply contrasted with previous treatments which focus on the application of positivity bounds t…
We revise the classical continuum formulation behind the Spin Foam approach to the quantization of gravity. Based on the recent applications of the current EPRL-FK model beyond triangulations, we identify the tension with the implementation of the 'volume' part of simplicity constraints, required for the passage from the topological BF theory to gravity. The crucial role, played by 4d normals in the linear version of constraints, necessitates the extension of the configuration space, and we argue to switch from normal 3-forms directly to tetrads. The requirement of vanishing torsion leads to c…
The self-consistent matter coupling is found in a broad class of minimally modified gravity theories which was discovered recently. All constraints in the theories remain first class and thus a graviton has only 2 local degrees of freedom. The cosmological solution of one of the examples in this class, the so-called square root gravity, exhibits a singularity freeness at high energy limit. At low energy limit, the theory smoothly connects to GR. A general feature of the theories in this class, with the self-consistent matter coupling discovered in our current work, is the non-trivial interacti…
We study the implications of the simplicity constraint in the spincube model of quantum gravity. By relating the edge-lengths to the integer areas of triangles, the simplicity constraint imposes very strong restrictions between them, ultimately leading to a requirement that all 4-simplices in the triangulation must be almost mutually identical. As a surprising and unexpected consequence of this property, one can obtain the CDT state sum as a special case of the spincube state sum. This relationship brings new insight into the long-standing problem of the relationship between the spinfoam appro…
We extend a previously developed approach to relate thermal currents in the high temperature regime and classical limits of amplitudes. We consider the bi-adjoint scalar theory, which has the basic structure of a cubic theory and which is related to QCD and gravity through the double copy. In addition, we consider a generalization of scalar QED to model classical spin, where massive scalars are complex higher-spin fields. We derive Vlasov-type kinetic equations for bi-adjoint scalars and study their iterative solutions, while for QED we use well-known kinetic equations. In both cases we find c…
Gluon amplitudes at most-subleading order in the $1/N$ expansion share a remarkable simplicity with graviton amplitudes: collinear divergences are completely absent in both and, as a consequence, their full IR behavior arises from soft gluon/graviton exchange among the external states. In this paper we study the effect of all-loop IR divergences of celestial most-subleading color gluon amplitudes and their similarities with the celestial gravity case. In particular, a simple celestial exponentiation formula for the dipole part can be written. We also analize how this exponentiation is modified…
This is a redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. The lectures may be viewed online at https://goo.gl/3DJdOr. Please send typos or corrections to strominger@physics.harvard.edu.
Gravity and electroweak sector from symmetry breaking of an $SO(3,3)$ BF theory
University of Madras（马德拉斯大学）；Tata Institute of Fundamental Research（塔塔基础研究所）；Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia（瓦伦西亚理工大学纯数学与应用数学大学研究所）
Spin Foam and Loop approaches to Quantum Gravity reformulate Einstein's theory of relativity in terms of connection variables. The metric properties are encoded in face bivectors/conjugate fluxes that are required to satisfy certain conditions, in order to allow for their geometric interpretation. We show that the (sub-)set of the so-called `volume simplicity constraints' is not implemented properly in the current EPRL-FK spinfoam vertex amplitude, if extended to arbitrary polyhedra. We then propose that a certain knot-invariant of the bivector geometry, induced on the boundary graph, encodes …
We present an improved formulation of 4-dimensional Lorentzian spinfoam quantum gravity with cosmological constant. The construction of spinfoam amplitudes uses the state-integral model of PSL(2,$\mathbb{C}$) Chern-Simons theory and the implementation of simplicity constraint. The formulation has 2 key features: (1) spinfoam amplitudes are all finite, and (2) With suitable boundary data, the semiclassical asymptotics of the vertex amplitude has two oscillatory terms, with phase plus or minus the 4-dimensional Lorentzian Regge action with cosmological constant for the constant curvature 4-simpl…
Reflective equilibrium between physics and philosophy, and between GR and particle physics, is fruitful and rational. I consider the virtues of simplicity, conservatism, and conceptual coherence, along with perturbative expansions. There are too many theories to consider. Simplicity supplies initial guidance, after which evidence increasingly dominates. One should start with scalar gravity; evidence required spin 2. Good beliefs are scarce, so don't change without reason. But does conservatism prevent conceptual innovation? No: considering all serious possibilities (Feynman, Weinberg, etc.) co…
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 ω (where ω is the soft frequency) and the corresponding soft factors structurally show completely different properties then their tree-l…
Within the low-energy effective field theories of QED and gravity, the low-energy speed of light or that of gravitational waves can typically be mildly superluminal in curved spacetimes. Related to this, small scattering time advances relative to the curved background can emerge from known effective field theory coefficients for photons or gravitons. We clarify why these results are not in contradiction with causality, analyticity or Lorentz invariance, and highlight various subtleties that arise when dealing with superluminalities and time advances in the gravitational context. Consistent low…
