The thermal interpretation of quantum mechanics

Arnold Neumaier (Vienna)


The thermal interpretation of quantum mechanics and quantum field theory allows a consistent quantum description of the universe from the smallest to the largest levels of modeling, including its classical aspects.
These foundations are easily stated and to motivate since they are essentially the foundations used everywhere for uncertainty quantification, just slightly extended to accommodate quantum effects by not requiring that observables commute.


The thermal interpretation of quantum mechanics says that, consistent with statistical thermodynamics, an expectation (ensemble mean) should not be interpreted as a statistical average over many realizations (except when the statistical context is immediate). Instead, it should be interpreted as an in principle approximately measurable quantity. Therefore, the notion of ensemble is to be understood not necessarily as an actual repetition by repeated preparation. It should be understood instead in the original sense used by Gibbs - who coined the notion of an ensemble as a collection of imagined copies of which only one is actually realized -, giving him an intuitive excuse to be able to use the statistical formalism for a single system. What is conventionally called expectation becomes in the thermal interpretation simply the uncertain value.


According to the thermal interpretation of quantum mechanics, we need for a description of the universe a mathematical framework consisting of a Hilbert space carrying a unitary representation of the Poincare group to account for conservative dynamics and relativity, a representation of the standard model plus some form of gravity (not yet fully known) to describe the fundamental field content, density operators ρ encoding Heisenberg states, the formula Ã=< A>:=tr ρA defining the uncertain value (generally called expectation value) of the operator A, and for its interpretation the following simple rule generalizing statistical intuition:
Measurement rule: Upon measuring a Hermitian operator A, the measured result will be approximately Ã, with an uncertainty at least of the order of σA, the square root of <(A-Ã)2>. If the measurement can be sufficiently often repeated (on an object with the same or a sufficiently similar state) then σA will be a lower bound on the standard deviation of the measurement results.
Physicists doing quantum mechanics (even those adhering to the shut-up-and-calculate mode of working) use this rule routinely, and the rule applies universally. No probabilistic interpretation is needed, so it applies also to single systems.


All descriptions in physics are either very coarse-grained or of very small objects. The detailed state can be found with a good approximation only for fairly stationary sources of very small objects, that prepare sufficiently many of these in essentially the same quantum state. In this case, one can calculate sufficiently many expectations by averaging over the results of multiple experiments on these objects, and use these to determine the state. Except in very simple situations, the result is a mixed state described by a density operator. Thus in the thermal interpretation, any realistic state must be correctly described by a density operator, not by a state vector as in conventional interpretations.
For macroscopic systems, one must necessarily use a coarse-grained description in terms of a limited number of parameters. In the quantum field theory of macroscopic objects, the averaging is always done inside the definition of the macroscopic operator to be measured; this is sufficient to guarantee very small uncertainties of macroscopic observables. Thus one does not need an additional averaging in terms of multiple experiments on similarly prepared copies of the system. This is the deeper reason why quantum field theory can make accurate predictions for single macroscopic systems.
Everything deduced in quantum field theory about macroscopic properties follows, and one has a completely self-consistent setting. The transition to classicality is automatic and needs no deep investigations - the classical situation is simply the limit of a huge number of particles. Whereas on the microscopic level, uncertainties of single events are large, so that state determination must be based by the statistics of multiple events with a similar preparation. (In this case, one can derive Born's traditional rule for perfect binary measurements in pure states.)
Although only a coarse-grained description of a macroscopic system is possible, this doesn't mean that the detailed state doesn't exist. Even in classical mechanics, it is impossible to know a highly accurate state of a many-particle system (not even of the solar system with sun, planets, planetoids, and comets treated as rigid bodies). But its existence is never questioned. The existence of an exact state for large objects has always been a metaphysical but unquestioned assumptions.


Unlike in conventional single-world interpretations of quantum mechanics, nothing in the thermal interpretation depends on the existence of measurement devices (which were not available in the very far past of the universe). Thus the thermal interpretation allows one to consider the single universe we live in as a quantum system, the smallest closed physical system containing us, hence strictly speaking the only system to which unitary quantum mechanics applies rigorously.
There is no longer a physical reason to question the existence of the state of the whole universe, even though all its details may be unknown for ever. Measuring all observables or finding its exact state is already out of the question for a small macroscopic quantum system such as a piece of metal. Thus, as for a metal, one must be content with describing the state of the universe approximately.
What matters for a successful physics of the universe is only that we can model (and then predict) the observables that are accessible to measurement. Since all quantities of interest in a study of the universe as a whole are macroscopic, they have a tiny uncertainty and are well-determined even by an approximate state. For example, one could compute from a proposed model of the universe the (expectation) values of the electromagnetic field at points where we can measure it, and (if the computations could be done) should get excellent agreement with the measurements.
Since every observable of a subsystem is also an observable of the whole system, the state of the universe must be compatible with everything we have ever empirically observed in the universe! This is a very stringent test of adequacy - the state of the universe is highly constrained since knowing this state amounts to having represented all physics accessible to us by the study of its subsystems. Cosmology studies this state in a very coarse (and partly conjectured) approximation where even details at the level of galaxies are averaged over. Only for observables localized in the solar system we have a much more detailed knowledge.


In more detail, the basics of my present views on the thermal interpretation are presented in the following topics from my theoretical physics FAQ:

I have also written some technical documents about the thermal interpretation, see and Chapter 10 of my online book

Themes related to (the 2016 version of) the thermal interpretation can also be found in parts of the following discussions on PhysicsForums:


Older, partially outdated but in many respects more detailed discussions

Happy Reading!


Arnold Neumaier (Arnold.Neumaier@univie.ac.at)