After outlining the principles of Algebraic Quantum Field Theory (AQFT) I will describe the generalization of Hochschild cohomology that is relevant to describing deformations in AQFT. An interaction is described by a cohomology class.

# Past Quantum Field Theory Seminar

I will discuss a classical six-dimensional superconformal field theory containing a non-abelian tensor multiplet which we recently constructed in arXiv:1712.06623.

This theory satisfies many of the properties of the mysterious (2,0)-theory: non-abelian 2-form potentials, ADE-type gauge structure, reduction to Yang-Mills theory and reduction to M2-brane models. There are still some crucial differences to the (2,0)-theory, but our action seems to be a key stepping stone towards a potential classical formulation of the (2,0)-theory.

I will review in detail the underlying mathematics of categorified gauge algebras and categorified connections, which make our constructions possible.

Algebraic quantum field theories (AQFTs) are traditionally described as functors that assign algebras (of observables) to spacetime regions. These functors are required to satisfy a list of physically motivated axioms such as commutativity of the multiplication for spacelike separated regions. In this talk we will show that AQFTs can be described as algebras over a colored operad. This operad turns out to be interesting as it describes an interpolation between non-commutative and commutative algebraic structures. We analyze our operad from a homotopy theoretical perspective and determine a suitable resolution that describes the commutative behavior up to coherent homotopies. We present two concrete constructions of toy-models of algebras over the resolved operad in terms of (i) forming cochains on diagrams of simplicial sets (or stacks) and (ii) orbifoldization of equivariant AQFTs.

In this talk, I will present the construction of a family of solutions to

vacuum Einstein equations which consist of an arbitrary number of high

frequency waves travelling in different directions. In the high frequency

limit, our family of solutions converges to a solution of Einstein equations

coupled to null dusts. This construction is an illustration of the so called

backreaction, studied by physicists (Isaacson, Burnet, Green, Wald...) : the

small scale inhomogeneities have an effect on the large scale dynamics in

the form of an energy impulsion tensor in the right-hand side of Einstein

equations. This is a joint work with Jonathan Luk (Stanford).

Some of the most studied examples of conformal field theories

include

the Wess-Zumino-Witten models. These are conformal field theories exhibiting

affine Lie algebra symmetry at non-negative integers levels. In this talk I

will

discuss conformal field theories exhibiting affine Lie algebra symmetry at

certain rational (hence fractional) levels whose structure is arguably even

more intricate than the structure of the non-negative integer levels,

provided

one is prepared to look beyond highest weight modules.

Hardy's axiomatic approach to quantum theory revealed that just one axiom

distinguishes quantum theory from classical probability theory: there should

be continuous reversible transformations between any pair of pure states. It

is the single word `continuous' that gives rise to quantum theory. This

raises the question: Does there exist a finite theory of quantum physics

(FTQP) which can replicate the tested predictions of quantum theory to

experimental accuracy? Here we show that an FTQP based on complex Hilbert

vectors with rational squared amplitudes and rational phase angles is

possible providing the metric of state space is based on p-adic rather than

Euclidean distance. A key number-theoretic result that accounts for the

Uncertainty Principle in this FTQP is the general incommensurateness between

rational $\phi$ and rational $\cos \phi$. As such, what is often referred to

as quantum `weirdness' is simply a manifestation of such number-theoretic

incommensurateness. By contrast, we mostly perceive the world as classical

because such incommensurateness plays no role in day-to-day physics, and

hence we can treat $\phi$ (and hence $\cos \phi$) as if it were a continuum

variable. As such, in this FTQP there are two incommensurate Schr\"{o}dinger

equations based on the rational differential calculus: one for rational

$\phi$ and one for rational $\cos \phi$. Each of these individually has a

simple probabilistic interpretation - it is their merger into one equation

on the complex continuum that has led to such problems over the years. Based

on this splitting of the Schr\"{o}dinger equation, the measurement problem

is trivially solved in terms of a nonlinear clustering of states on $I_U$.

Overall these results suggest we should consider the universe as a causal

deterministic system evolving on a finite fractal-like invariant set $I_U$

in state space, and that the laws of physics in space-time derive from the

geometry of $I_U$. It is claimed that such a deterministic causal FTQP will

be much easier to synthesise with general relativity theory than is quantum

theory.

I will review some classical results on geometric scattering

theory for linear hyperbolic evolution equations

on globally hyperbolic spacetimes and its relation to particle and charge

creation in QFT. I will then show that some index formulae for the

scattering matrix can be interpreted as a special case of the Lorentzian

analog of the Atyiah-Patodi-Singer index theorem. I will also discuss a

local version of this theorem and its relation to anomalies in QFT.

(Joint work with C. Baer)

I will start with a quick reminder of what we have learned so far about

transplanckian-energy collisions of particles, strings and branes.

I will then address the (so-far unsolved) problem of gravitational

bremsstrahlung from massless particle collisions at leading order in the

gravitational deflection angle.

Two completely different calculations, one classical and one quantum, lead

to the same final, though somewhat puzzling, result.