Information Theoretic Foundations for Physics
Perhaps the first use of the mathematical theory of heat to develop another theory was Thomson’s use of Fourier’s equations to formulate equations for electrostatics in the 1840s. After extracting a lesson from this historical case, I will fast forward more than a century to examine the relationship between classical statistical mechanics and QFT that is induced by analytic continuation.
It is sometimes envisaged that the behaviour of elementary particles can be characterised by the information content it carries, and that exchange of energy and momentum, or more generally the change of state through interactions, can likewise be characterised in terms of its information content. But exchange of information occurs only in the context of a (typically noisy) communication channel, which traditionally requires a transmitter and a receiver; whereas particles evidently are not equipped with such devices.
The third law of thermodynamics has a controversial past and a number of formulations due to Planck, Einstein, and Nernst. It's most accepted version, the unattainability principle, states that "any thermodynamic process cannot reach the temperature of absolute zero by a finite number of steps and within a finite time". Although formulated in 1912, there has been no general proof of the principle, and the only evidence we have for it is that particular cooling methods become less efficient as a the temperature lowers.
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.
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.
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.
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.
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.