Sommerfeld Theory Colloquium (ASC)

Quantum field theory (QFT) is the universal language of theoretical physics, underlying the Standard Model of elementary particles, the physics of the early Universe and a host of condensed matter phenomena such as phase transitions and superconductivity. A great achievement of 20th-century physics was the understanding of weakly

coupled quantum field theories where interactions can be treated as small perturbations of otherwise freely moving particles. Critical challenges for the 21st century include solving the problem of strong coupling and mapping the whole space of consistent QFTs. In this lecture, I will overview the bootstrap approach, the idea that

theory space can be determined from the general principles of symmetry and quantum mechanics. This strategy provides a new unifying language for QFT and has allowed researchers to make predictions

for physical observables even in strongly coupled theories. I will illustrate the general framework in a few examples, ranging from the concrete (boiling water) to the abstract (supersymmetric theories in various spacetime dimensions).

A unified theory for space-time and matter might be based on finite

projective geometries instead of differentiable manifolds and gauge

groups. Each point is equipped with a quadratic form over a finite Galois field which define neighbors in the finite set of points.

Due to the projective equivalence of all quadratic forms this world

is necessarily a 4-dimensional Lorentz-invariant space-time with a

gauge symmetry G(3)xG(2)xG(1) for internal points which represent elementary particle degrees of freedom. Matter appears as a

geometric distortion by an inhomogeneous field of quadrics and all

physical properties (spins, charges) of the standard model seem to

follow from its geometric structure in a continuum limit. The finiteness inevitably induces a fermionic quantization of all matter fields

and a bosonic for gauge fields. This unity of space-time and matter

was already sought 1918 by Hermann Weyl in a gauge theory as an

extension of Einstein’s general theory of relativity, but not found -

probably because of the assumption of a continuous geometry.

I describe an algebraic scheme for quantizing the Ruijsenaars-Schneider

models in the R-matrix formalism. It is based on a special parametrization

of the cotangent bundle over GL(n,C). In new variables

the standard symplectic structure is described by a classical (Frobenius)

r-matrix and by a new dynamical r¯-matrix. Quantizing these

r-matrices, I will exhibit the quantum L-operator algebra and construct

its particular representation corresponding to the RuijsenaarsSchneider

system. I will also indicate a couple of open problems.

I describe an algebraic scheme for quantizing the Ruijsenaars-Schneider

models in the R-matrix formalism. It is based on a special parametrization

of the cotangent bundle over GL(n,C). In new variables

the standard symplectic structure is described by a classical (Frobenius)

r-matrix and by a new dynamical r¯-matrix. Quantizing these

r-matrices, I will exhibit the quantum L-operator algebra and construct

its particular representation corresponding to the RuijsenaarsSchneider

system. I will also indicate a couple of open problems.

The apparent crisis of black holes inconsistency with foundational physical principles provides a sharp focus for the conflict between quantum mechanics and classical spacetime. Various resolutions have

been proposed; a very plausible one is that small interactions can transfer sufficient information between the black hole and outgoing radiation, with a quantum enhancement from the enormous number of black hole states. Such interactions must however violate conventional notions of locality, perhaps as a symptom of the more basic subtlety of information localization in quantum gravity, and hinting at aspects of the fundamental structure of quantum gravity. An intriguing question is whether further clues can be found from new observational windows on black holes.

The study of black holes has revealed a deep connection between

quantum information and spacetime geometry. Its origin must lie

in the quantum theory of gravity, which offers a valuable hint in

our search for a unified theory. Precise formulations of this relation

recently led to new insights in Quantum Field Theory, some of which

have been rigorously proven. An important example is our discovery

of the first universal lower bound on the local energy density. The

energy near a point can be negative but it is bounded below by a

quantity related to the information flowing past the point.

Sommerfeld Theory Colloquium, The knowledge of all correlation functions of a system is equivalent to solving the corresponding quantum many-body

problem. If one can identify the relevant degrees of freedom, the knowledge of a finite set of correlation functions is in

many cases sufficient to determine a sufficiently accurate solution of the corresponding field theory. Complete

factorization is equivalent to identifying the relevant degrees of freedom where the Hamiltonian becomes diagonal. I

will give examples how one can apply this powerful theoretical concept in experiment.

A detailed study of non-translation invariant correlation functions reveals that the pre-thermalized state a system of

two 1-dimensional quantum gas relaxes to after a splitting quench [1], is described by a generalized Gibbs ensemble

[2]. This is verified through phase correlations up to 10th order.

Interference in a pair of tunnel-coupled one-dimensional atomic super-fluids, which realize the quantum Sine-Gordon /

massive Thirring models, allows us to study if, and under which conditions the higher correlation functions factorize

[3]. This allowed us to characterize the essential features of the model solely from our experimental measurements:

detecting the relevant quasi-particles, their interactions and the different topologically distinct vacuum-states the

quasi-particles live in. The experiment thus provides a comprehensive insight into the components needed to solve a

non-trivial quantum field theory.

Our examples establish a general method to analyse quantum systems through experiments. It thus represents a

crucial ingredient towards the implementation and verification of quantum simulators.

Work performed in collaboration with E.Demler (Harvard), Th. Gasenzer und J. Berges (Heidelberg).

Supported by the Wittgenstein Prize, the Austrian Science Foundation (FWF): SFB FoQuS: F40-P10 and

the EU: ERC-AdG QuantumRelax

[1] M. Gring et al., Science, 337, 1318 (2012);

[2] T. Langen et al., Science 348 207-211 (2015).

[3] T. Schweigler et al., Nature 545, 323 (2017), arXiv:1505.03126

Sommerfeld Theory Colloquium, Metagenomics has revealed hundreds to thousands of microbial species

coexisting in almost all microbiota. It is increasingly appreciated

that microbial communities condition their own environments.

To better understand the role of this environmental conditioning in

promoting diversity, we physically model the population dynamics of

microbes that compete for steadily supplied resources. In a model

where cells require multiple nutrients, we find that population dynamics

generally leads to the coexistence of different metabolic types,

which satisfy an extended competitive exclusion principle. Moreover,

we establish that these consortia of metabolic types act as cartels,

whereby population dynamics pins down resource concentrations at

values for which no other strategy can invade. Strikingly, these cartels

also yield maximum biomass, constituting a microbial example

of Adam Smith’s “invisible hand” leading to collective optimal usage

of resources. Curiously, in a model where only total resource acquisition

is considered, diversity can arbitrarily exceed that predicted by

the competitive exclusion principle.

Sommerfeld Theory Colloquium, The observed deviations from the laws of gravity of Newton and Einstein

in galaxies and clusters can logically speaking be either due to

the presence of unseen dark matter particles or due to a change in

the way gravity works in these situations. Until recently there was

little reason to doubt that general relativity correctly describes gravity

in all circumstances. In the past few year insights from black hole

physics and string theory have lead to a new theoretical framework in

which the gravitational laws are derived from the quantum entanglement

of the microscopic information that is underlying space-time.

An essential ingredient in the derivation is of the Einstein equations

is that the vacuum entanglement obeys an area law, a condition that

is known to hold in Anti-de Sitter space due to the work of Ryu

and Takayanagi. We will argue that in de Sitter space due to the

positive dark energy, that the microscopic entanglement entropy also

contains also a volume law contribution in addition to the area law.

This volume law contribution is related to the thermal properties of

de Sitter space and leads to a total entropy that precisely matches the

Bekenstein-Hawking formula for the cosmological horizon. We study

the effect of this extra contribution on the emergent laws of gravity,

and argue that it leads to a modification compared to Einstein gravity.

We provide evidence for the fact this modification explains the

observed phenomena in galaxies and clusters currently attributed to

dark matter.

Sommerfeld Theory Colloquium, In February 2016 the LIGO team announced the detection of gravitational

waves (GW) created by the merger of two black holes. In addition

to confirming a major prediction of general relativity, successful

GW detection would provide a powerful new tool for astrophysics.

Given their evident importance, the LIGO results and the methods

which led to them deserve independent critical analysis. This talk

will present the results of one such study in a manner suitable for

non-specialists.