The only observables of physics are expectation values

The thermal interpretation states that the only observables of physics are expectation values that vary sufficiently slowly in time and space.

This includes:

1. At the global level: the properties of matter, including masses of atoms, protons, electrons as well as the masses and decay rates of unstable particles, the spectra of atoms and molecules, etc.

2. At the level of local equilibrium: currents, the densities of materials, pressure, temperature, mechanical and electrodynamic stresses, etc. which all obey the hydrodynamic equations (Navier-Stokes) approximately, and are all expectation values of microscopic operators or variables that can be calculated from these by means of thermodynamics.

3. At the level of the kinetic description (micro-local equilibrium): phase space densities (Wigner functions) that satisfy the Boltzmann or Vlasov equations approximately and are all expectation values of single particle operators.

4. At the molecular level: density functions that obey the Hartree-Fock equations or CI equations approximately and are likewise expectation values of single particle operators.

5. At still more fundamental levels: effective field equations for atomic nuclei, that satisfy the Hartree-Fock-Bogoliubov equations approximately and are likewise expectation values of single particle operators.

6. In addition: spatial and temporal correlation functions that describe a linear response to excitation; these are also expectation values, but this time of two-particle operators.

Nowhere does one observe anything but expectation values.

Interference patterns on photographic plates are also expectation values of particle density fields, and clicks in a Geiger counter are expectation values of pressure fields, which are perceived by our ear (or a corresponding detector). It is admittedly an old tradition to interpret the clicks as an irreducible measurement of a discrete state of a single microsystem, but this is justified very indirectly (by theories themselves requiring interpretation) and is simply dropped in the thermal interpretation. As soon as one does so, the interpretational problems of quantum mechanics disappear.

The statistical interpretation of expectation values in statistical mechanics is a pure historical accident, as no formal probability theory based on measure theory existed at the time of Gibbs. But Gibbs, the founder of statistical thermodynamics, was more far-seeing than most of his contemporaries. In his book

W. Gibbs,

Elementary Principles in Statistical Mechanics,

Yale Univ. Press, 1902

he introduces ensembles as fictional, identical copies of the observed microsystem which are in thermal equilibrium, in order to justify his 'misuse' of the probability calculus for a single system:

''Let us imagine a great number of independent systems, identical
in nature, but differing in phase, that is, in their condition with
respect to configuration and velocity.'' (P.5)

''The application of this principle is not limited to cases in
which there is a formal and explicit reference to an ensemble of
systems. Yet the conception of such an ensemble may serve to give
precision to notions of probability.'' (P.17)

It is clear that Gibbs ultimately assumes that his theory is valid for each individual measurement of a substance in equilibrium. And experience confirms his assumption.

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