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.
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 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 in 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.
(Traditional interpretations of Copenhagen flavor cannot apply to the
finite time quantum field theory of our planetary system since there
is no classical apparatus for measuring this system. These
interpretations apply only to the results of infinite-time few-particle
scattering calculations derived from quantum field theory.)
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
σ_{A}, the square root of <(A-Ã)^{2}>,
is much less than |Ã| has the value Ã within an uncertainty
of σ_{A}.
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
σ_{A}, 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 σ_{A}
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.
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 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
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 assumption.
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
Classical models for quantum light,
Classical models for quantum light II,
Slides of lectures given on April 7-8, 2016 at the Zentrum für
Oberflächen- und Nanoanalytik of the University of Linz.
Themes related to (the 2016 version of) the thermal interpretation can also be found in parts of the following discussions on PhysicsForums:
Happy Reading!