Information Theoretic Foundations for Physics

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. Here we provide the first derivation of a general unattainability principle, which applies to arbitrary cooling processes, even those exploiting the laws of quantum mechanics or involving an infinite-dimensional reservoir. We quantify the resources needed to cool a system to any particular temperature, and translate these resources into a minimal time or number of steps by considering the notion of a Cooling Machine which obeys similar restrictions to universal computers. We generally find that the obtainable temperature scales as an inverse power of the cooling time, and place ultimate bounds on the speed at which information can be erased.

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. In view of this a new concept in communication theory is put forward whereby signal processing is carried out in the absence of a transmitter; hence mathematical machineries in communication theory serves as new powerful tools for describing a wide range of observed phenomena. In the quantum context, this leads to a tentative—and perhaps speculative—idea that the dynamical evolution of the state of a quantum particle is such that the particle itself acts as if it were a "signal processor", trying to identify the stable configuration that it should settle, and adjusts its own state accordingly. It will be shown that the mathematical scheme of such a hypothesis works well for a broad class of noise structures having stationary and independent increments. (The talk will be based on work carried out in collaboration with L. P. Hughston.)

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. While there is no doubt that this mathematical relationship has been heuristically useful in guiding developments in both statistical mechanics and QFT, this is a case in which the physical interpretation of the mathematics does not carry over from one theory to the other.