Past Quantum Field Theory Seminar

I'll explain the formalism of extended QFT, while
focusing on the cases of two dimensional conformal field theories,
and three dimensional topological field theories.

In this talk, I will review an inverse scattering construction of interacting integrable quantum field theories on two-dimensional Minkowski space and its ramifications. The construction starts from a given two-body S-matrix instead of a classical Lagrangean, and defines corresponding quantum field theories in a non-perturbative manner in two steps: First certain semi-local fields are constructed explicitly, and then the analysis of the local observable content is carried out with operator-algebraic methods (Tomita-Takesaki modular theory, split subfactor inclusions). I will explain how this construction solves the inverse scattering problem for a large family of interactions, and also discuss perspectives on extensions of this program to higher dimensions and/or non-integrable theories.

Amplitudes in quantum field theory have discontinuities when regarded as
functions of
the scattering kinematics. Such discontinuities can be determined from
Cutkosky rules.
We present a structural analysis of such rules for massive quantum field
theory which combines
algebraic geometry with the combinatorics of Karen Vogtmann's Outer Space.
This is joint work with Spencer Bloch (arXiv:1512.01705).

Hawking radiation and particle creation by an expanding Universe
are paradigmatic predictions of quantum field theory in curved spacetime.
Although the theory is a few decades old, it still awaits experimental
demonstration. At first sight, the effects predicted by the theory are too
small to be measured in the laboratory. Therefore, current experimental
efforts have been directed towards siumlating Hawking radiation and
studying quantum particle creation in analogue spacetimes.
In this talk, I will present a proposal to test directly effects of
quantum field theory in the Earth's spacetime using quantum technologies.
Under certain circumstances, real spacetime distortions (such as
gravitational waves) can produce observable effects in the state of
phonons of a Bose-Einstein condensate. The sensitivity of the phononic
field to the underlying spacetime can also be used to measure spacetime
parameters such as the Schwarzschild radius of the Earth.

[based on joint work with Li Guo and  Bin Zhang]

 We apply to  the study of exponential sums on lattice points in
convex rational polyhedral cones, the generalised algebraic approach of
Connes and Kreimer to  perturbative quantum field theory.  For this purpose
we equip the space of    cones   with a connected coalgebra structure.
The  algebraic Birkhoff factorisation of Connes and Kreimer   adapted  and
generalised to this context then gives rise to a convolution factorisation
of exponential sums on lattice points in cones. We show that this
factorisation coincides with the classical Euler-Maclaurin formula
generalised to convex rational polyhedral cones by Berline and Vergne by
means of  an interpolating holomorphic function.
We define  renormalised conical zeta values at non-positive integers as the
Taylor coefficients at zero of the interpolating holomorphic function.  When
restricted to Chen cones, this  yields yet another way to renormalise
multiple zeta values  at non-positive integers.

In a quantum quench, a system is prepared in some state
$|\psi_0\rangle$, usually the ground state of a hamiltonian $H_0$, and then
evolved unitarily with a different hamiltonian $H$. I study this problem
when $H$ is a 1+1-dimensional conformal field theory on a large circle of
length $L$, and the initial state has short-range correlations and
entanglement. I argue that (a) for times $\ell/2<t<(L-\ell)/2$  the
reduced density matrix of a subinterval of length $\ell$ is exponentially
close to that of a thermal ensemble; (b) despite this, for a rational CFT
the return amplitude $\langle\psi_0|e^{-iHt}|\psi_0\rangle$ is $O(1)$ at
integer multiples of $2t/\ell$ and has interesting structure at all rational
values of this ratio. This last result is related to the modular properties
of Virasoro characters.

Axions are ubiquitous in string theory compactifications. They are
pseudo goldstone bosons and can be extremely light, contributing to
the dark sector energy density in the present-day universe. The
mass defines a characteristic length scale. For 1e-33 eV<m< 1e-20
eV this length scale is cosmological and axions display novel
effects in observables. The magnitude of these effects is set by
the axion relic density. The axion relic density and initial
perturbations are established in the early universe before, during,
or after inflation (or indeed independently from it). Constraints
on these phenomena can probe physics at or beyond the GUT scale. I
will present multiple probes as constraints of axions: the Planck
temperature power spectrum, the WiggleZ galaxy redshift survey,
Hubble ultra deep field, the epoch of reionisation as measured by
cmb polarisation, cmb b-modes and primordial gravitational waves,
and the density profiles of dwarf spheroidal galaxies. Together

these probe the entire 13 orders of magnitude in axion mass where
axions are distinct from CDM in cosmology, and make non-trivial
statements about inflation and axions in the string landscape. The
observations hint that axions in the range 1e-22 eV<m<1e-20 eV may
play an interesting role in structure formation, and evidence for
this could be found in the future surveys AdvACT (2015), JWST, and
Euclid (>2020). If inflationary B-modes are observed, a wide range
of axion models including the anthropic window QCD axion are
excluded unless the theory of inflation is modified. I will also
comment briefly on direct detection of QCD axions.

We treat the problem of geometric interpretation of the formalism
of algebraic quantum mechanics as a special case of the general problem of
extending classical 'algebra - geometry' dualities (such as the
Gel'fand-Naimark theorem) to non-commutative setting.  
I will report on some progress in establishing such dualities. In
particular, it leads to a theory of approximate representations of Weyl
algebras
in finite dimensional  "Hilbert spaces". Some calculations based on this
theory will be discussed.

Topological phases of matter exhibit Bott-like periodicity with respect to
time-reversal, charge conjugation, and spatial dimension. I will explain how
the non-commutative topology in topological phases originates very generally
from symmetry data, and how operator K-theory provides a powerful and
natural framework for studying them.

We will discuss the relation between perturbative gauge theory and
perturbative gravity, and look at how this relation extends to some exact
classical solutions. First, we will review the double copy prescription that
takes gauge theory amplitudes into gravity amplitudes, which has been
crucial to progress in perturbative studies of supergravity. Then, we will
see how the relation between the two theories can be made manifest when we
restrict to the self-dual sector, in four dimensions. A key role is played
by a kinematic algebraic structure mirroring the colour structure, which can
be extended from the self-dual sector to the full theory, in any number of
dimensions. Finally, we will see how these ideas can be applied also to some
exact classical solutions, namely black holes and plane waves. This leads to
a relation of the type Schwarzschild as (Coulomb charge)^2.

The talk will give a definition of matrix geometries, which are

particular types of finite real spectral triple that are useful for

approximating manifolds. Examples include fuzzy spheres and also the

internal space of the standard model. If time permits, the relation of

matrix geometries with 2d state sum models will also be sketched.

Quantum field theory (QFT) originated in physics in the context of

elementary particles. Although, over the years, surprising and profound

connections to very diverse branches of mathematics have been discovered,

QFT does not have, as yet, found a universally accepted "standard"

mathematical formulation. In this talk, I shall outline an approach to QFT

that emphasizes its underlying algebraic structure. Concretely, this is

represented by a concept called "Operator Product Expansion". I explain the

properties of such expansions, how they can be constructed in concrete QFT

models, and the emergent relationship between "perturbation theory" on the

physics side and

"Hochschild cohomology" on the physics side. This talk is based on joint

work

with Ch. Kopper and J. Holland from Ecole Polytechnique, Paris.

The observations of the first traces of cosmic structure in the

Cosmic Microwave Background are in excellent agreement with the

predictions of Inflation. However as we shall see, that account

is not fully satisfactory, as it does not address the transition

from an homogeneous and isotropic early stage to a latter one

lacking those symmetries. We will argue that new physics along the

lines of the dynamical quantum state reduction theories is needed

to account for such transition and, motivated by Penrose's ideas

suggest that quantum gravity might be the place from where

this new physics emerges. Moreover we will show that observations

can be used to constrain the various phenomenological proposals

made in this regard.

The talk is based on my paper with E. Beggs appearing in Class. Quantum

Gravity.

Working within a bimodule approach to noncommutative geometry, we show that

even a small amount of noncommutativity drastically constrains the moduli

space of

noncommutative metrics. In particular, the algebra [x,t]=x is forced to have

a geometry

corresponding to a gravitational source at x=0 so strong that even light

cannot

escape. This provides a non-trivial example of noncommutative Riemannian

geometry

and also serves as an introduction to some general results.

The modular invariant partition functions for SU(2) and SU(3)

conformal field theories have been classified. The SU(2) theory is closely

related to the preprojective algebras of Coxeter-Dynkin quivers. The

analogous finite dimensional superpotential algebras, which we call almost

Calabi-Yau algebras, associated to the SU(3) invariants will be discussed.

Cutting-edge experiments in quantum communications are reaching regimes

where relativistic effects can no longer be neglected. For example, there

are advanced plans to use satellites to implement teleportation and quantum

cryptographic protocols. Relativistic effects can be expected at these

regimes: the Global Positioning System (GPS), which is a system of

satellites that is used for time dissemination and navigation, requires

relativistic corrections to determine time and positions accurately.

Therefore, it is timely to understand what are the effects of gravity and

motion on entanglement and other quantum properties exploited in quantum

information.

In this talk I will show that entanglement can be created or degraded by

gravity and non-uniform motion. While relativistic effects can degrade the

efficiency of teleportation between moving observers, the effects can also

be exploited in quantum information. I will show that the relativistic

motion of a quantum system can be used to perform quantum gates. Our

results, which will inform future space-based experiments, can be

demonstrated in table-top experiments using superconducting circuits.

              Conventional decoherence (usually called 'Environmental

Decoherence') is supposed to be a result of correlations

established between some quantum system and the environment.

'Intrinsic decoherence' is hypothesized as being an essential

feature of Nature - its existence would entail a breakdown of

quantum mechanics. A specific mechanism of some interest is

'gravitational decoherence', whereby gravity causes intrinsic

decoherence.

I will begin by discussing what is now known about the mechanisms of

environmental decoherence, noting in particular that they can and do

involve decoherence without dissipation (ie., pure phase decoherence).

I will then briefly review the fundamental conflict between Quantum

Mechanics and General Relativity, and several arguments that suggest

how this might be resolved by the existence of some sort of 'gravitational

decoherence'.  I then outline a theory of gravitational decoherence

(the 'GR-Psi' theory) which attempts to give a quantitative discussion of

gravitational decoherence, and which makes predictions for

experiments.

The weak field regime of this theory (relevant to experimental

predictions) is discussed in detail, along with a more speculative

discussion of the strong field regime.

We will first review the construction of N =1

supersymmetric Yang-Mills theory in three dimensions. Then we will

construct a superloop space formulation for this super-Yang-Mills

theory in three dimensions.Thus, we will obtain expressions for loop

connection and loop curvature in this superloop space. We will also

show that curvature will vanish, unless there is a monopole in the

spacetime. We will also construct a quantity which will give the

monopole charge in this formalism. Finally, we will show how these

results hold even in case of deformed superspace.

Topological quantum error correcting codes, such as the Toric code, are
ideal candidates for protecting a logical quantum bit against local noise.
How are we to get the best performance from these codes when an unknown
local perturbation is applied? This talk will discuss how knowledge, or lack
thereof, about the error affects the error correcting threshold, and how
thresholds can be improved by introducing randomness to the system. These
studies are directed at trying to understand how quantum information can be
encoded and passively protected in order to maximise the span of time between successive rounds of error correction, and what properties are
required of a topological system to induce a survival time that grows
sufficiently rapidly with system size. The talk is based on the following
papers: arXiv:1208.4924 and Phys. Rev. Lett. 107, 270502 (2011).

This is a report of joint work with T. Koppe, P. Majumdar, and K.

 Ray.

I will define new partition functions for theories with targets on toric

singularities via

products of old partition functions on  crepant resolutions. I will

present explicit examples 

and show that the  new partition functions turn out to be homogeneous on

MacMahon factors.