Colloquium

The standard method to study nonperturbative properties of quantum field theories is to Wick rotate the theory to Euclidean space and regulate it on a Euclidean Lattice. An alternative is "fuzzy field theory". This involves replacing the lattice field theory by a matrix model that approximates the field theory of interest, with the approximation becoming better as the matrix size is increased. The regulated field theory is one on a background noncommutative space. I will describe how this method works and present recent progress and surprises.

Several current experiments probe physics in the approximation in which Planck's constant and Newton's constant may be neglected, but, the Planck mass, is relevant. These include tests of the symmetry of the ground state of quantum gravity such as time delays in photons of different energies from gamma ray bursts. I will describe a new approach to quantum gravity phenomenology in this regime, developed with Giovanni Amelino-Camelia, Jersy Kowalski-Glikman and Laurent Freidel. This approach is based on a deepening of the relativity principle, according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry, such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments.

Traditional condensed matter physics is based on two theories: symmetry breaking theory for phases and phase transitions, and Fermi liquid theory for metals. Mean-field theory is a powerful method to describe symmetry breaking phases and phase transitions by assuming the ground state wavefunctions for many-body systems can be approximately described by direct product states. The Fermi liquid theory is another powerful method to study electron systems by assuming that the ground state wavefunctions for the electrons can be approximately described by Slater determinants. From the encoding point of view, both methods only use a polynomial amount of information to approximately encode many-body ground state wavefunctions which contain an exponentially large amount of information. Moreover, another nice property of both approaches is that all the physical quantities (energy, correlation functions, etc.) can be efficiently calculated (polynomially hard). In this talk, I'll introduce a new class of states: (Grassmann-number) tensor-net states. These states only need polynomial amount of information to approximately encode many-body ground states. Many classes of states, such as Slater determinant states, projective states, string-net states and their generalizations, etc., are subclasses of (Grassmann-number) tensor-net states. However, calculating the physical quantities for these state can be exponentially hard in general. To solve this difficulty, we develop the Tensor-Entanglement Renormalization Group (TERG) method to efficiently calculate the physical quantities. We demonstrate our algorithm by studying several interesting boson/fermion models, including t-J model on a honeycomb lattice.

How many interacting quantum (field) theories of four-dimensional geometry are there which have General Relativity as their classical limit? Some of us still harbour hopes that a quantum theory of gravity is "reasonably unique", i.e. characterized by a finite number of free parameters. One framework in which such universality may manifest itself is that of "Quantum Gravity from Causal Dynamical Triangulations (CDT)". I will summarize the rationale behind this nonperturbative formulation and CDT's main achievements in trying to explain the micro- and macro-structure of spacetime from first principles. This includes the remarkable property of "dynamical reduction" of the spacetime dimension from four to two at the Planck scale.

Cosmic strings, generic in brane inflationary models, may be detected by the current generation of gravitational wave detectors. An important source of gravitational wave emission is from isolated events on the string called cusps and kinks. I first review cosmic strings, discussing their effective action and motion, and showing how cusps and kinks arise dynamically. I then show how allowing for the motion of the strings in extra dimensions gives a potentially significant reduction in signal strength, and comment on current LIGO bounds.

Scattering amplitudes in gauge theories and gravity have extraordinary properties that are completely invisible in the textbook formulation of quantum field theory using Feynman diagrams. In the standard approach--going back to the birth of quantum field theory--space-time locality and quantum-mechanical unitarity are made manifest at the cost of introducing huge gauge redundancies in our description of physics. As a consequence, apart from the very simplest processes, Feynman diagram calculations are enormously complicated, while the final results turn out to be amazingly simple, exhibiting hidden infinite-dimensional symmetries. This strongly suggests the existence of a new formulation of quantum field theory where locality and unitarity are derived concepts, while other physical principles are made more manifest. Rapid advances have been made towards uncovering this new picture, especially for the maximally supersymmetric gauge theory in four dimensions. These developments have interwoven and exposed connections between a remarkable collection of ideas from string theory, twistor theory and integrable systems, as well as a number of new mathematical structures in algebraic geometry. In this talk I will review the current state of this subject and and describe a number of ongoing directions of research.

To a first approximation, everything that happens at the Large Hadron Collider at CERN is a strong interaction process. If signals of supersymmetric particles or other new states are found at the LHC, the events that produce those signals will represent parts per trillion of the total sample of proton-proton scattering events and parts per billion of the sample of events with hard scattering of quarks and gluons. Can we predict the rates of QCD processes well enough to control their contribution to a tantalizing signal? What physics insights can assist this process? Can string theory help? In this lecture, I will describe the current status of Quantum Chromodynamics and its application to the predictions for hadron colliders. Note: Those who would like more background on the LHC should attend my talk at the Physics Department Colloquium of the University of Waterloo, Thursday, Jan. 13, at 4:00 pm: The Search for New Elementary Particles at the CERN Large Hadron Collider

Graphene-like materials provide a unique opportunity to explore quantum-relativistic phenomena in a condensed matter laboratory. Interesting phenomena associated with the internal degrees of freedom, spin and valley, including quantum spin-Hall effect, have been theoretically proposed, but could not be observed so far largely due to disorder and density inhonogeneity. We show that weak magnetic field breaks the symmetries that protect flavor (spin, valley) degeneracy, and induces large bulk non-quantized flavor-Hall effect in graphene. The effect occurs due to flavor splitting which generates the imbalance of the Hall resistivities of the two flavor species. At the Dirac point, flavor-Hall effect is greatly enhanced due to anomalous magnetotransport of Dirac-like carriers. The flavor-Hall effect is robust in the presence of disorder and interactions, and can serve as a hallmark of the quasi-relativistic carriers in the system. It manifests itself in large nonlocal transport mediated by long-lived flavor currents, as well as in flavor injection and accumulation experiments. Recent experimental observation of the giant nonlocal transport in graphene following our prediction opens up new opportunities for creating and manipulating flavor currents. Our work also shows that nonlocal electric transport provides a tool to study neutral modes in solid-states systems.

Quantum error correcting codes and topological quantum order (TQO) are inter-connected fields that study non-local correlations in highly entangled many-body quantum states. In this talk I will argue that each of these fields offers valuable techniques for solving problems posed in the other one. First, we will discuss the zero-temperature stability of TQO and derive simple conditions that guarantee stability of the spectral gap and the ground state degeneracy under generic local perturbations. These conditions thus can be regarded as a rigorous definition of TQO. Our results apply to any quantum spin Hamiltonian that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice. This large class of Hamiltonians includes Levin-Wen string-net models and Kitaev's quantum double models. Secondly, we derive upper bounds on the parameters of quantum codes with local check operators and discuss the implications for feasibility of a quantum self-correcting memory.

The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, which is obviously not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While $H=p^2+ix^3$ is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined space reflection P and time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. Some of these properties have recently been verified in laboratory experiments. If one generalizes classical mechanics into the complex domain, the resulting theories have equally remarkable properties.

This talk will describe the best current understanding of the interior structure of astronomically realistic black holes. A common misconception is that matter falling into a black hole simply falls to a central singularity, and that's that. Reality is much more interesting. Rotating black holes have not only outer horizons, but also inner horizons. Penrose (1968) first pointed out that an infaller falling through the inner horizon would see the outside Universe infinitely blueshifted, and he speculated that this would destabilize the inner horizon. The expectation was supported by linear perturbation theory, but it was not until 1990 that Poisson & Israel were able to clarify the nonlinear evolution of the instability at the inner horizon, which they called mass inflation. Inflation accelerates ingoing (positive energy) and outgoing (negative energy) streams to exponentially huge energies. The black hole thus behaves like a particle accelerator of extraordinary power, accelerating ingoing and outgoing particles to collide with each other at super-Planckian energies. The talk raises the fundamental question: What does Nature do with this remarkable accelerator?

The gravitational observatory LISA will detect radiation from massive black hole sources at cosmological distances, accurately measure their luminosity distance and help identify the electromagnetic counterparts that such sources may generate. I will describe various astrophysical scenarios for the generation of electromagnetic counterparts and discuss observational strategies aimed at identifying them. Successful identifications will enable novel studies of black hole astrophysics and cosmological physics.