Motivation for the Thermal Interpretation

All quantitative observations in physics are based on readings from macroscopic objects - pointers, films, counters, etc. whose physics is described by thermodynamics. From these, with the help of physics theories and numerical calculations, we decode information about the microsystems that interest us.

Raw measurement variables are therefore *always* thermodynamic variables; according to the principles of statistical physics they are therefore functions of expectation values in a corresponding grand canonical ensemble. In a *single* raw measurement one also has an ensemble, not a real, statistical one, but a fictional, virtual one. This was introduced as a fiction by Gibbs to enable the formal apparatus of statistical mechanics to be used to derive the laws of thermodynamics.

In my thermal interpretation I use this fact to motivate the ascription of a virtual ensemble to *every* individual system, not only thermodynamic ones. Naturally one can no longer interpret this ensemble statistically, any more than one needs to measure the temperature of a cup of tea x times before being able to ascribe a value to it.

The reading of a pointer, or the evaluation of a picture, to determine a measurement result is normally performed only once, perhaps in practice a second time as a check. If, on checking, the reading proves reliable, the measurement is recorded, but it does not count as reproducible. Then (and only then) does one perform multiple observations and put these together statistically into a combined observation, for example in the form of an reported value (mean) and error (standard deviation).

Each individual observation is therefore a thermodynamic expectation value based on a virtual, non-statistical ensemble; the combined observation on the other hand is a statistical expectation value based on a real ensemble of many repeated measurements.

Both kinds of ensemble/expectation value satisfy the same mathematical laws, but have a completely different meaning.

Thus I prefer to define an abstract ensemble axiomatically, by means of the relevant properties (see EEEQ for details). Just as with the definition of a vector space (which was originally motivated by three dimensional vectors and is today used for all sorts of virtual vectors: sequences, matrices, functionsâ€¦) this enables a clean division between formal properties and their contextual significance.

The fundamental assumption of the thermal interpretation is therefore that the objective aspects of the universe are given by an ensemble in this abstract sense, and everything measurable by means of expectation values in this universal ensemble, or functions of such expectation values.

This assumption encompasses everything measurable in a unified fashion: natural constants, scattering amplitudes, random probabilities, reaction rates, transfer coefficients, thermodynamic variables - in short, everything that experimentalists can or wish to measure or control!

Proceeding from this fundamental assumption one finds objective foundations for a logical construction of quantum mechanics which:

1. is logically trouble free,

2. has a deterministic classical (symplectic or Poisson-)dynamics,

3. explains the appearance of classical and quantum mechanical randomness,

4. is complete in the sense that a knowledge of the state of the universe (that is the universal ensemble) implies a knowledge of all possible measurement values.

Thus the thermal interpretation postulates a deterministic dynamics for the universe as a whole, and deduces an approximate stochastic dynamics for each subsystem.

In traditional nomenclature it is therefore a theory of hidden variables, in which the classical variables are the complete set of quantum mechanical expectation values (or equivalently all correlation functions of the BBGKY hierarchy).

Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ