The thermal interpretation of quantum mechanics

Arnold Neumaier (Vienna)

The thermal interpretation of quantum mechanics (considered as an approximation of quantum field theory) allows a consistent quantum description of the universe from the smallest to the largest levels of modeling, including its classical aspects. Essential use is made of the fact that everything physicists measure is measured in a thermal environment for which statistical thermodynamics is relevant. This is reflected in the characterizing adjective ''thermal'' for the interpretation.

These foundations are easily stated and motivated 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 state of our planetary system, when the latter is modeled by quantum fields, completely specifies what happens in any small space-time region within the planetary system - namely through the n-point correlation functions with arguments restricted to this neighborhood. There is nothing else in quantum field theory; what we can observe is contained in the least oscillating contributions to this correlations. The spatial and temporal high frequency part is unobservable due to the limited resolution of our instruments.

(Regarding quantum field theory, traditional interpretations of Copenhagen flavor apply only to the results of infinite-time few-particle scattering calculations derived from quantum field theory. They cannot apply to the finite time quantum field theory of our planetary system since there is no classical apparatus for measuring  this system.)

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 (and in case of a quantum field theory of our planetary system never) 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 to describe a single thermodynamic system such as a single piece of metal. 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>:=trρA defining the uncertain value (generally called expectation value) of the operator A, and for its interpretation the following simple rule generalizing statistical intuition:

Uncertainty principle: A Hermitian quantity A whose uncertainty σA, the square root of <(A-Ã)2>, is much less than |Ã| has the value à within an uncertainty of σA.

From this rule one can derive under appropriate conditions the following

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 a system 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 usually without further justification. The rule applies universally. No probabilistic interpretation is needed, so it applies also to single systems. Born's famous rule turns out to be derivable under special circumstances only, namely those where the Born rule is indeed valid in practice. (Though usually invoked as universally valid, Born's rule has severe limitations. It neither applies to position measurements nor to photodetection, nor to measurement of energies, just to mention the most conspicuous misfits.)

Actually the above measurement rule should be considered as a definition of what it means to have a device measuring A. As such it creates the foundation of measurement theory. In order that a macroscopic quantum device qualifies for the description ''it measures A'' it must either be derivable from quantum mechanics, or checkable by experiment, that the property claimed in the above measurement rule is in fact valid. Thus there is no circularity in the foundations.

Most 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 via some version of quantum state tomography. Except in very simple situations, the result is a mixed state described by a density operator. Thus in the thermal interpretation, any realistic state is fully 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; see Chapter 10.5 in my online book Classical and Quantum Mechanics via Lie algebras.)

Although only a coarse-grained description of a macroscopic system is can be explicitly known, this doesn't mean that the detailed state doesn't exist. The existence of an exact state for large objects has always been a metaphysical but unquestioned assumption. 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.

In quantum optics experiments, both sources and beams are extended macroscopic objects describable by quantum field theory and statistical mechanics, and hence have (according to the thermal interpretation) associated nearly classical observables - densities, intensities, correlation functions - computable from quantum mechanics in terms of expectations.

The sources have properties independent of measurement, and the beams have properties independent of measurement. These are objects described by quantum field theory. For example, the output of a laser (before or after parametric down conversion or any other optical processing) is a laser beam, or an arrangement of highly correlated beams. These are in a well-defined state that can be probed by experiment. If this is done, they are always found to have the properties ascribed to them by the preparation procedure. One just needs sufficient time to collect the information needed for a quantum state tomography. The complete state is measurably in this way, reproducibly. Neither the state of the laser nor of the beam is changed by a measurement at the end of the beam. Thus these properties exist independent of any measurement - just as the moon exists even when nobody is looking at it!

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47 (1935), 777-781 defined the so-called EPR criterion: ''If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.'' This criterion is satisfied by the traditional quantum properties ascribed to stationary (or sufficiently slowly varying) optical sources and arrangements of beams.

But quantum particles do not satisfy this criterion. We cannot say that the particles in the beams always have properties independent of measurement, since not all properties can be simultaneously measured, and since experiments checing bell inequality violations seem to imply the violation of the EPR criterion. The deeper reason for this is that the particle concept is a derived, approximate concept that makes intuitive sense only under very special circumstances - namely in those where a system actually behaves like particles do.

Therefore in the EPR sense, sources and beams are much more real than particles. The former, not the latter, are the real players in solid foundations. That's why an inappropriate focus on the particle aspect of quantum mechanics creates the appearance of mystery.

It is a historical accident that one continues to use the name particle in the many microscopic situations where it is grossly inappropriate if one thinks of it with the classical meaning of a tiny bullet moving through space. Restrict the use of the particle concept to where it is appropriate, or don't think of particles as ''objects'' - in both cases all mystery is gone, and the foundations become fully rational.

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 (which, in a preliminary German version, dates back to 2006) 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 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!

A theoretical physics FAQ

Arnold Neumaier (