Mathematical physics

As is well known, time plays a special role in the standard formulation of quantum theory, bringing the latter into severe conflict with the principles of general relativity. This suggests the existence of a more fundamental and (as it turns out) covariant and timeless formulation of quantum theory. A conservative way to look for such a formulation would be to start from quantum theory as we know it, taken in its experimentally most successful form of quantum field theory, and try to uncover structure in the formalism made for actual physical predictions. A radical way to look for such a formulation would be to forget the standard formulation, take only a few first principles (locality and operationalism turn out to be good ones) and try to construct things from there. Remarkably, approaches following these apparently opposite paths have recently been shown to converge in a single framework. In this talk I want to provide an overview of the current understanding of the resulting "positive formalism", its implications, and the paths that led to it. This includes relations to works of Witten and Segal in mathematical physics and of Aharonov, Hardy and others in quantum foundations.

The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting space-time manifold, the general concept is to derive space-time as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. The dynamics of the system is described by the causal action principle. I will give a non-technical introduction, with an emphasis on conceptual issues related to information theory.

The CA interpretation presents a view on the origin of quantum mechanical behavior of physical degrees of freedom, suggesting that, at the Planck scale, bits and bytes are processed, rather than qubits or qubites, so that we are dealing with an ordinary classical cellular automaton. We demonstrate how this approach naturally leads to Born's expression for probabilities, shows how wave functions collapse at a measurement, and provides a natural resolution to Schroedinger's cat paradox without the need to involve vague decoherence arguments. We then continue to discuss the implications of Bell's inequalities, and other issues.

Renormalization to low energies is widely used in condensed matter theory to reveal the low energy degrees of freedom of a system, or in high energy physics to cure divergence problems. Here we ask which states can be seen as the result of such a renormalization procedure, that is, which states can “renormalized to high energies". Intuitively, the continuum limit is the limit of this "renormalization" procedure. We consider three definitions of continuum limit and characterise which states satisfy either one in the context of Matrix Product States. Joint work with N. Schuch, D. Perez-Garcia and I. Cirac.

The modern understanding of quantum field theory underlines its effective nature: it describes only those properties of a system relevant above a certain scale. A detailed understanding of the nature of the neglected information is essential for a full application of quantum information-theoretic tools to continuum theories. I will present an operationally motivated method for deriving an effective field theory from any microscopic description of a state. The approach is based on dimensional reduction relative to a quantum distinguishability metric. It relies on a microscopic description of experimental limitations, such as a finite spatial resolution. In this picture, the emergent field observables represent cotangent vectors on the manifold of states, and are not necessarily endowed with the full semantic of standard quantum observables.

Our subject is Entropic Dynamics, a framework that emphasizes the deep connections between the laws of physics and information. In attempting to understand quantum theory it is quite natural to assume that it reflects laws of physics that operate at some deeper level and the goal is to discover what these underlying laws might be. In contrast, in the entropic view no fundamental underlying dynamics is invoked. Quantum theory is an application of entropic methods of inference and the goal is to make the best possible predictions on the basis of some limited information represented by appropriate constraints. It is through the choice of microstates and of these constraints that the “physics” is introduced. In Entropic Dynamics a relational notion of entropic time is introduced as a book-keeping device to keep track of changes. We show that a non-dissipative entropic dynamics naturally leads to generic forms of Hamiltonian dynamics, and notions of information geometry naturally lead to those specific Hamiltonians (that is, those that include the correct quantum potential) that describes quantum mechanics.

In the last decade there were proposed several new information theoretic frameworks (in particular, symmetric monoidal categories and "operational" convex sets), allowing for an axiomatic derivation of finite dimensional quantum mechanics as a specific case of a larger universe of information processing theories. Parallel to this, there was an influential development of quantum versions of bayesianism and causality, and relationships between quantum information and space-time structure. In the face of structural problems encountered when moving beyond finite dimensional quantum mechanics, as well as the lack of a mathematically and predictively sound nonperturbative framework for quantum field theories, a question appears: which of the existing structural assumptions of quantum information theory should be relaxed, and how? In this talk I will present a new approach to the information theoretic foundations of a "general" quantum theory (i.e., beyond quantum mechanics), that is a specific answer to the above question, with a hope to reconstruct both emergent space-times and emergent QFTs. Its mathematical setting is based on using quantum information geometry and integration over noncomutative algebras as structural and conceptual replacements of spectral theory and probability theory, respectively. This corresponds to a paradigmatic change: considering expectation values as more fundamental than eigenvalues. We construct a nonlinear generalisation of quantum kinematics using quantum relative entropies and spaces of states over W*-algebras. Unitary evolution is generalised to nonlinear hamiltonian flows, while Bayes' and Lueders' rules are generalised to constrained relative entropy maximisations. Combined together, they provide a framework for nonlinear causal inference (information dynamics), that is a generalisation and replacement of completely positive maps. As a result, we construct a large class of information processing theories, containing Hilbert space based QM and probability theory as two special cases. On the conceptual level, we propose a new approach to quantum bayesianism, that is ontically agnostic, intersubjective, and concerned with the relationships between experimental design, model construction, and their mutual predictive verifiability. Finally, we propose a procedure for the emergence of space-times from the geometry of quantum correlations and quantum causality structure, and discuss (briefly) the possibility of reconstructing emergent QFTs.

In science, we often see new advances and insights emerging from the intersection of different ideas coming from what appeared to be disconnected research areas. The theme of my seminar will be an ongoing collision between the three topics listed in my title which has been generating interesting new insights into a variety of fields, eg, condensed matter physics, quantum field theory and quantum gravity.

The second law of thermodynamics appears to be a universal law of physics. This universality suggests that entropy and with it information theory is part of the foundations of physics. In this talk I take the opposite (probably more conservative) approach: I assume that the dynamics of relational degrees of freedom form the foundation of physics. Physical information is an emergent phenomenon. This approach has an interesting consequence: Typical (initial) data for a gravitationally dominated universe leads to the spontaneous emergence of a gravitational arrow of time for the universe as a whole. This primary gravitational arrow of time generates secondary thermodynamic arrows of time in sufficiently isolated subsystems of the universe, which coincide with the gravitational arrow of time. This coincidence explains the universality of the second law of thermodynamics. I conclude the talk with a speculation about the emergence of quantum information: I assume (1) that the purpose of a physical law is the prediction of future properties of the universe based on the knowledge of records and (2) that gravity generates physical records (as in the classical part of the talk). This suggests a scenario in which stable quantum information is spontaneously generated. Collaborators for the classical part of the talk where Julian Barbour and Flavio Mercati

Classical and quantum theories are very different, but the gap between them may look narrow particularly if the notion of classicality is broadened. For example, if we do not impose all the classical assumptions at the same time, hidden variable theories reproduce the results of quantum mechanics. If a quantum system is restricted to Gaussian states, evolution and measurements, then classical phase space mechanics with a finite resolution fully reproduces its behavior. We discuss two examples of such extensions. In a version of the delayed-choice experiment we allow an otherwise classical system to exhibit two types of behavior ("P" or "W"), requiring, however, objectivity: the system is at any moment either "P" or "W", but not both. It turns out that the three conditions of objectivity, determinism, and independence of hidden variables are incompatible with any theory, not only with quantum mechanics. We then consider two harmonic oscillators with a Gaussian interaction between them. If one is treated as quantum and one is described by a classical theory with a finite phase space resolution, no consistent description of this interaction is possible. The lesson is that it is hard to be a little bit quantum: it is either pointless or quantumness takes over altogether.