Foundations independent of measurements
Traditional foundations of quantum mechanics all depend far too heavily
on the concept of (hypothetical, idealized) experiments. This is one
of the reasons why these foundations are still unsettled, 85 years
after the discovery of the basic equations for modern quantum mechanics.
No other theory has such shaky foundations.
The main reason is that the idealization that goes as hard core into
the usual interpretations (where they are taken very seriously) is in
reality only a didactical trick to make the formal definitions of
quantum mechanics a bit easier to swallow for the newcomer. Except in
a few simple cases, it is too far removed from experimental practice to
tell much about real experiment, and hence about how quantum mechanics
is used in real applications.
In experimental physics, measurement is a very complex thing - far
more complex than Born's rule (the usual starting point) suggests.
To measure the distance between two galaxies, the mass of the top
quark, or the Lamb shift - just to mention three basic examples -
cannot be captured by the idealistic measurement concept used there,
nor by any of the refinements of it discussed in the literature.
In each of the three cases mentioned, one assembles a lot of auxiliary
information and ultimately calculates the measurement result from a
best fit of a model to the data. Clearly the theory must already be in
place in order to do that. We don't even know what a top quark should
be whose mass we are measuring unless we have a theory that tell us
this!
And the Lamb shift (one of the most famous real observables in the
history of quantum mechanics) is not even an observable in the
traditional sense!
But to define quantum mechanics (or any other physical theory) properly,
we only need to define the corresponding calculus and then say how to
relate the quantities that can be calculated from quantum mechanical
models (or models of the theory considered) to the stuff experimental
physicists talk about.
We may then proceed as in the modern account of the oldest of the
physical sciences: Euclidean geometry, where (on laboratory scales)
there is consensus about how theory and reality correspond:
We develop a theory that simply gives a precise formal meaning to the
concepts physicists talk about. This is pure math, in case of geometry
consisting of textbook linear algebra and analytic geometry. The
identification with real life is done after having the theory
(though the theory and the nomenclature was _developed_ with the goal
to enable this identification in a way consistent with tradition):
For geometry, by declaring anything in real life resembling an ideal
point, line, plane, circle, etc., to be a point, line, plane, circle,
etc., if and only if it can be assigned in an approximate way
(determined by the heuristics of traditional measurement protocols,
whatever that is) the properties that the ideal point, line, plane,
circle, etc., has, consistent to the assumed accuracy with the
deductions from the theory. If the match is not good enough, we can
explore whether an improvement can be obtained by modifying measurement
protocols (devising more accurate instruments or more elaborate
error-reducing calculation schemes, etc.) or by modifying the theory
(to a non-Euclidean geometry, say, which uses the same concepts but
assumes slightly different properties relating them.
For quantum mechanics, by declaring anything in real life resembling
an ideal photon, electron, atom, molecule, crystal, ideal gas, etc.,
to be a photon, electron, atom, molecule, crystal, ideal gas, etc., if
and only if it can be assigned in an approximate way (determined by
the heuristics of traditional measurement protocols, whatever that is)
the properties that the ideal photon, electron, atom, molecule,
crystal, ideal gas, etc., has, consistent to the assumed accuracy with
the deductions from the theory.
This is precisely the way Callen, one of the great expositors of
thermodynamics, justifies phenomenological equilibrium thermodynamics
in his famous textbook
This identification process is fairly independent of the way
measurements are done, as long as they are capable to produce the
required accuracy for the matching, hence carries no serious
philosophical difficulties.
We need the theory already to define precisely what it is that we
observe.
On the other hand, the theory must be crafted in such a way that it
actually applies to reality - otherwise the observed properties cannot
match the theoretical description.
As a result, theoretical concepts and experimental techniques
complement each other in a way that, if a theory is reaching maturity,
it has developed its concepts to the point where they are a good match
to reality. We then say that something in real life ''is'' an instance
of the theoretical concept if it matches the theoretical description
to our satisfaction.
It is not difficult to check that this holds not only in physics but
everywhere where we have clear concepts about some aspect of reality.
If the match between theory and observation is not good enough, we can explore whether an improvement can be obtained by modifying measurement protocols (devising more accurate instruments or more elaborate error-reducing calculation schemes, etc.) or by modifying the theory (to a hyper quantum mechanics, say, which uses the same concepts but assumes slightly different properties relating them).
Then, having established informally that the theory is an appropriate
model for the physical aspects of reality, one can study the
measurement problem rigorously on this basis:
One declares that a real instrument (in the sense of a complete
experimental arrangement including the numerical postprocessing of raw
results that gives the final result) performs a real measurement of an
ideal quantity if and only if the following holds:
Modeling the real instrument as a macroscopic quantum system (with the
properties assigned to it by statistical mechanics/thermodynamics)
predicts raw measurements such that, in the model, the numerical
postprocessing of raw results that gives the final result is in
sufficient agreement with the value of the ideal quantity in the model.
Thus measurement analysis is now a scientific activity like any other
rather than a philosophical prerequisite for setting up a consistently
interpreted quantum mechanics.
The procedure outlined above is the basis of my thermal interpretation of quantum mechanics. The thermal interpretation is superior to the interpretions found in the literature, since it
The thermal interpretation is based on the observation that quantum
mechanics does much more than predict probabilities for the possible
results of experiments done by Alice and Bob. In particular,
it quantitatively predicts the whole of classical thermodynamics.
For example, it is used to predict the color of molecules, their
response to external electromagnetic fields, the behavior of material
made of these molecules under changes of pressure or temperature, the
production of energy from nuclear reactions, the behavior of
transistors in the chips on which your computer runs, and a lot more.
The thermal interpretation therefore takes as its ontological basis
the states occurring in the statistical mechanics for describing
thermodynamics (Gibbs states) rather than the pure states figuring in
a quantum mechanics built on top of the concept of a wave function.
This has the advantage that the complete state of a system completely
and deterministically determines the complete state of every subsystem
- a basic requirement that a sound, observer-independent interpretation
of quantum mechanics should satisfy.
The axioms for the formal core of quantum mechanics are those specified
in the entry ''Postulates for the formal core of quantum mechanics''
of Chapter A4 of my
In the thermal interpretation of quantum physics, the directly
observable (and hence obviously ''real'') features of a macroscopic
system are the expectation values of the most important fields Phi(x,t)
at position x and time t, as they are described by statistical
thermodynamics. If it were not so, thermodynamics would not provide
the good macroscopic description it does.
However, the expectation values have only a limited accuracy; as
discovered by Heisenberg, quantum mechanics predicts its own
uncertainty. This means that
This defines the surface ontology of the thermal interpretation.
There is also a deeper ontology concerning the reality of inferred
entities - the thermal interpretation declares as real but not directly
observable any expectation of operators with a space-time
dependence that satisfy Poincare invariance and causal commutation
relations. These are distributions that produce measurable numbers
when integrated over sufficiently smooth localized test functions.
Deterministic chaos is an emergent feature of the thermal
interpretation of quantum mechanics, obtained in a suitable
approximation. Approximating a multiparticle system in a semiclassical
way (mean field theory or a little beyond) gives an approximate
deterministic system governing the dynamics of these expectations.
This system is highly chaotic at high resolution. This chaoticity
seems enough to enforce the probabilistic nature of the measurement
apparatus. Neither an underlying exact deterministic dynamics nor an
explicit dynamical collapse needs to be postulated.
The same system can be studied at different levels of resolution.
When we model a dynamical system classically at high enough resolution,
it must be modeled stochastically since the quantum uncertainties must
be taken into account. But at a lower resolution, one can often neglect
the stochastic part and the system becomes deterministic. If it were
not so, we could not use any deterministic model at all in physics but
we often do, with excellent success.
This also holds when the resulting deterministic system is chaotic.
Indeed, all deterministic chaotic systems studied in practice are
approximate only, because of quantum mechanics. If it were not so, we
could not use any chaotic model at all in physics but we often do,
with excellent success.
The chaoticity of the semiclassical approximation is enough to explain
the randomness inherent in the measurement of quantum observables.
This gives an ensemble interpretation to the Heisenberg uncertainty
relation. The Heisenberg uncertainty principle states that the product
of the variances of p and q is bounded below by a small number.
It doesn't say what the variances represent. The general consensus is
that the variance represents an ensemble average - i.e., the result of
a statistics over many independent measurements on identically prepared
systems.
(In order to take the variance as a time average one would need to
invoke an ergodic theorem stating that the time average equals the
ensemble average. However such an ergodic theorem makes sense only
semiclassically, and is valid only for very simple systems.
Most systems are far from ergodic. Thus the interpretation of the
variance as a time average is usually not warranted. This also means
that - in contrast to what usually happens in the popular literature -
the so-called vacuum fluctuations cannot be interpreted as fluctuations
in time.)
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ Thermal Interpretation FAQ