# Past Quantum Field Theory Seminar

[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.