Hidden variables or subquantum theories^{10}

Preamble

Hidden variables or subquantum
theories deal with
reality behind the phenomena,
the phenomena being described by quantum mechanics. They are
inspired by the analogy (in the following table represented by ~)

statistical thermodynamics


quantum mechanics

↓

~

↓

classical mechanics


hidden variables theory



(in which the arrow ↓ indicates that the upper theory is to be reduced to the lower one).
Quantum mechanics might be incomplete in the wider sense and be
consistent with a (possibly deterministic) subquantum theory explaining
individual measurement results by assuming that
subquantum elements of physical reality exist, determined by
a hidden variable λ describing states of the `reality behind the quantum phenomena',
analogously to the way states (q,p) of classical mechanics are describing the `reality behind
the thermodynamic phenomena'.
This, presumably, was Einstein's idea when telling us
that ``God does not play dice.'' However, by associating
his `elements of physical reality' with `quantum mechanical
measurement results a_{m}' Einstein unfortunately
did not sufficiently distinguish between different levels of physical description: whereas measurement results a_{m}
refer to macroscopic phenomena (viz. pointer positions) are hidden variables supposed to describe the (sub)microscopic behaviour of
(sub)microscopic objects. Hence, analogously to the difference between the `classical mechanical description by means of the state (q,p)' and
a `(macroscopic) thermodynamic description', hidden variable λ is supposed to describe
reality on a (submicroscopic) level, different from the level of the `phenomena of quantum mechanics'.
By equating his `element of physical reality' with a `quantum mechanical
measurement result' Einstein
became vulnerable to Bohr's objection of ambiguity of its definition,
an objection that could not have been substantiated if the `element of physical reality' would have been presented
as `corresponding to a value of a hidden variable λ' since
the `objectivity^{61} of λ' might have remained
as unchallenged as the alleged `objectivity of the classical state (q,p)'.

The idea of `hidden variables theories' has met with quite a bit of resistance.
Logical positivism/empiricism considered hidden
variables metaphysical because it was thought to be impossible to
operationally control them. Moreover, there was von Neumann's
famous 1932 `no go' theorem "proving" the impossibility of hidden variables. The
combination of the popularity of logical positivism/empiricism and von
Neumann's authority was sufficient to make it unattractive for
physicists to deal with hidden variables theories. Thus, de
Broglie switched to more "respectable" subjects, to return to
hidden variables only after such theories were revived by Bohm
(1952).

The general feeling that quantum mechanics
should be incompatible with the existence of hidden variables may
have been enhanced by Bohr's alleged "victory" over Einstein in the
`(in)completeness debate', interpreted by many as `defeating hidden
variables'. However, it must be realized
that in this debate only a very restricted notion of
hidden variables is at stake, viz. hidden variables represented
by values of quantum mechanical observables (compare the
EPR elements of physical reality),
considered as `properties the individual objects
objectively possess prior to measurement', and explaining by determinism
individual measurement results found later. This class of
hidden variables theories is already ruled out by the
KochenSpecker theorem.
Actually, the BohrEinstein `(in)completeness debate' can be completely cast into
quantum mechanical terms, and therefore is a debate on the
`interpretation of quantum mechanics' (resulting in a
victory of Bohr's contextualisticrealist interpretation of observables over Einstein's objectivisticrealist one)
rather than on `hidden variables theories proper' (compare the difference
between `completeness in the restricted sense' and `completeness in the wider sense').
Nevertheless, Bohr's victory has for a long time been interpreted as a refutation of
generic hidden variables theories.
Only after Bell's refutation in the 1960s of von Neumann's 1932 ``impossibility proof'' and his
derivation of the
Bell inequality^{0}
(see also Stochastic hidden variables theories) have `hidden variables
theories' lost their odium of `being metaphysical' and have experimenters been induced to perform measurements
testing such theories.
It, indeed, seems that at present the `assumption of the impossibility of hidden variables'
is as metaphysical as is the `assumption of their existence'.

As a result of Bell's `derivation^{60} of the
Bell inequality for local hidden variables theories' and the `alleged existence
of a nonlocal hiddenvariables theory (viz. Bohm's theory)'
it is widely supposed that any hiddenvariables theory reproducing the results of quantum mechanics
should be nonlocal.
However, this supposition is unfounded. It, indeed, is not difficult to see that
`violation of the Bell inequality within quantum mechanics' is a local affair. Hence, Bell's
expectation that such violation would be related to `nonlocality' is unwarranted. Indeed, Bell's expectation
was nurtured by
i) Einstein's observation in the EPR discussion that Bohr's assumption of
`completeness of quantum mechanics' entails
nonlocality (which `nonlocality', however, could be
evaded
by the assumption of `incompleteness of quantum mechanics'),
ii) the `nonlocality of
Bohm's hiddenvariables theory' (which theory, however, meets
objections
making the theory unattractive as a subquantum theory).
It seems evident by now that `EPR nonlocality' is
as unobservable as the world aether expelled by Einstein for reasons similar to
the ones inducing him to speak of spooky action at a distance.

The above conclusion seems to contradict the oftenheard statement that `EPR nonlocality has been proven
experimentally by experiments corroborating violation of the Bell inequality' (like the experiments performed
by Aspect and collaborators). However, this statement neglects the fact that `violation of the Bell
inequality is a consequence of incompatibility of observables that are involved'
(compare). This points at the possibility that
there may have been employed in the derivations of the Bell inequalities by Bell and others
additional assumptions probably being more essential to the final result than the `assumption of locality'.
Within quantum mechanics one such assumption has been identified as the
possessed values principle, in case of hidden variables theories
liable to be translated into an `assumption of determinism'.
However, since such a derivation is possible for (local)
stochastic hidden variables theories as well, it follows
that `determinism' is not essential to a `derivation of the Bell inequality'.

Although, hence, the generalization from `deterministic' to `stochastic hidden variables theories'
does not yield a direct answer to the question of the relevance of the `locality assumption'
to the derivation of the Bell inequality, does it indicate in which direction to look.
As demonstrated here within `stochastic hidden variables theories'
the Bell inequality can be derived without assuming locality. Nor is there any reason to
suppose that the details of the measurement procedure have any influence on derivability of the
Bell inequality. However, there is one circumstance that seems to be important for that purpose,
viz. the availability of a quadruple of measurement results at each individual preparation
of an EPR particle pair. Whenever such quadruples exist the Bell inequality can be derived
in the ways discussed here and here.
Hence, violation of the Bell inequality is associated with the nonexistence of such
quadruples.

In the following an attempt is made to sketch the outlines of a local hiddenvariables theory
that may be able to reproduce the predictions of quantum mechanics. It starts from the observation
that the `conditional probabilities
p_{A}(a_{m}λ)',
usually considered in such theories, are to be distrusted because they relate concepts of
different theories, thus being liable to all misunderstandings translations between
different languages may induce. Indeed, it is doubtful whether such quantities have any physical meaning
since they condition a `quantum mechanical measurement result a_{m} (corresponding to a
property of a macroscopic pointer of a measuring instrument)' on a `value of a hidden
variable λ (assumed to yield an instantaneous description of a microscopic object)'.
The attempt hinges on two issues:
i) an assumed analogy between `quantum mechanics' and `statistical thermodynamics', implemented into `hiddenvariables theory' by
distinguishing between "microstates described by a hidden variable λ" and "macrostates
<λ> corresponding to ergodic paths in hiddenvariables space Λ
(describing subquantum states of local equilibrium)",
the latter states replacing the microstates as `states on which quantum mechanical
measurement results should be conditioned';
ii) the conjecture that within the contexts of `mutually exclusive measurement arrangements'
one and the same microstate λ gives rise to different macrostates.

Although no account has been given of a reconstruction of quantum mechanics on the basis of these assumptions,
does it seem that the structure of the hiddenvariables theory is sufficiently similar
to the structure of quantum mechanics that such a reconstruction be possible. However,
whereas `ergodicity' may be an answer to the diffusive character of the Schrödinger equation,
does it not seem to be directly related to its wavelike character (implemented by the
superposition principle), which presumably has a different ontological basis.
Here it is important to
remember that
solutions of the Schrödinger equation do not refer to an `individual particle' but to an `ensemble'.
Nevertheless de Broglie may have been right when associating a wave with an
individual particle^{63},
since only in this way does it seem possible to explain interference in a doubleslit experiment. Such a wave
may be analogous to a ship's `bow wave', passing through both slits while the ship passes through one of the
slits only, afterwards being influenced by the `interfering parts of the bow
wave'^{64}.

Not distinguishing between different levels of description (in general requiring different theories
for describing `phenomena' and `reality behind the phenomena'),
de Broglie probably made an analogous mistake (by requiring the Schrödinger equation to
govern both) as was made by Einstein when the latter was identifying his `element of physical reality'
with a quantum mechanical quantity (compare).
Both de Broglie and Einstein were applying the `methodology of classical physics' in which the
realist interpretation is the natural one (tending to disregard the
distinction between ontology and epistemology, although Einstein,
by advocating a statistical interpretation,
showed to be aware of the distinction to a certain extent).
For both it would have been quite natural to
acknowledge that there is no need to assume that the additional elements they introduced (viz.
the `associated wave' (de Broglie) and the `element of physical reality' (Einstein)) might require
a subquantum theory for their description, these `additional elements' possibly being unobservable at
the level of the quantum mechanical description like atomic vibrations are at
the level of the classical theory of rigid bodies (compare).
In this respect both were hampered by the wide
applicability of (standard) quantum mechanics, thus withholding any experimental clue as to the direction
into which should be looked for experiments transcending the domain of quantum mechanics.

Bohm's hidden variables theory

In view of the great popularity of logical positivism/empiricism,
and the large authority of von Neumann, is Bohm's achievement not
a minor one. On the other hand, the large step made by Bohm may
be not large enough, because, at best, his hidden variables
theory is of the restricted sort referred to above,
essentially being an
interpretation of the quantum mechanical formalism rather than a genuine hidden variables theory. Thus, by putting
ψ = R e^{iS}, R,S real,
the (complex) Schrödinger equation for a particle takes the form of two coupled real equations,


P = R^{2}, 
These equations are very suggestive. The first has the
form of the continuity equation, expressing conservation
of the quantity P, with P∇S/m
the corresponding flux vector. The second equation is even more
suggestive, because it has the appearance of a HamiltonJacobi
equation, which is one possible way of dealing with classical
mechanics. By putting p = ∇S(x)
it is possible to attribute to a particle at position x
also a value of momentum. This allows to calculate
trajectories of microscopic objects (thought to be
impossible by the Copenhagen interpretation).
There is only one difference with classical
mechanics: apart from the classical potential V there is an
additional term Q, the socalled quantum potential.
Therefore trajectories of a quantum mechanical particle are
different from the classical ones.

Critique of Bohm's hidden variables theory

One criticism was already soon advanced by Einstein. He observed that the
momentum, attributed to the particle by p = ∇S,
cannot equal the quantum mechanical measurement result found in a
momentum measurement (e.g. Publ. 19).
This problem was solved by Bohm by assuming
measurement disturbance, to the effect that the quantity p
= ∇S
would be the "real" momentum, which, however, due to measurement
disturbance is not registered in a momentum measurement. Note
that this differs from Heisenberg's disturbance theory of measurement, in
which only observables are disturbed that are incompatible
with the measured one. In Bohm's theory only the position observable is
thought to be measured in an undisturbed way.
Note also that Heisenberg's disturbance
theory is corroborated by the generalized formalism.

Since the mathematical
formalism of Bohm's theory does not differ from the quantum
mechanical one, the domain of application of Bohm's theory does
not transcend the domain of quantum mechanics. Bohm's theory
cannot yield any information not contained in quantum mechanics.
The attribution of a momentum value to an individual
particle by the relation p = ∇S
does not escape the status of an interpretation of a
certain mathematical relation. Moreover, such an interpretation
would attribute a double meaning to the quantity
∇S,
since within quantum mechanics it already has another meaning,
viz. a statistical one (compare the expression i(ψ^{*}∇ψ − ψ∇ψ^{*})/2m = P∇S/m
of the probability flux). This may cause the individual
interpretation not only to be redundant but, possibly, even
selfcontradictory.

If applied to a system of two
particles, Bohm's theory exhibits a remarkable feature of
nonlocality, the quantum potential not decreasing with
increasing distance between the particles. All objections against
the reality of EPR nonlocality
apply also here. The nonlocality is a consequence of a certain interpretation of the
mathematical formalism of quantum mechanics. If the quantum
mechanical formalism is interpreted in an empiricist sense, then the nonlocal
correlations between measurement results of measurements
performed on distant particles can be attributed to
past preparation rather than to instantaneous interaction (compare). Experimentally
corroborated locality can be seen as a problem also for Bohm's
theory.
The nonlocality of Bohm's theory could be compared with the actionatadistance
of Newton's theory of gravity. The latter problem has been resolved by developing
general relativity theory and other local field theories as subtheories to
Newton's nonlocal one. It is argued here why,
contrary to a widespread belief, it is reasonable to think that
a similar resolution with respect to quantum mechanics is not
excluded by the Bell inequality.

Notwithstanding the abovementioned problems,
Bohm's theory has many supporters. Some of them can be found at
the following URLs: Goldstein,
Dürr

Stochastic hidden variables theories

In stochastic hidden variables theories the analogy with classical statistical
theories is exploited in the following way. A hidden variables
space Λ (to be compared with classical
phase space) is introduced. It is assumed that a particle can at
each instant be characterized by a certain value λ of the hidden variable.
It is assumed that the domain of the hidden variables theory is containing the domain of
quantum mechanics. Hence, also quantum mechanical measurements
should be described by it.

In a deterministic hidden variables theory the value of the hidden variable uniquely
determines the measurement result. Within the domain of
standard quantum mechanics
this would imply the unique determination of the measurement
results of all quantum mechanical standard observables. However,
since this would amount to the
possessed values principle, such
deterministic theories are not very well possible. Instead
we consider stochastic hidden variables theories, in which
a quantum mechanical measurement process is taken as a stochastic
process, characterized by the conditional probability
p_{A}(a_{m}λ)
that the measurement result of observable A is
a_{m} if the hidden variable had value λ. Note that the deterministic theory is a
special case for which p_{A}(a_{m}λ) = χ_{Λm}(λ),
χ_{Λm}(λ) the characteristic function of the
subset Λ_{m} of Λ (which is 1 if λ ∈ Λ_{m}, 0 if λ∉ Λ_{m}).
If ρ(λ) is the probability that the hidden variable is prepared with value λ, then the
quantum mechanical probability p_{m} is
represented in the stochastic hidden variables theory by
p_{m} = ∫_{Λ}dλ ρ(λ)
p_{A}(a_{m}λ).
By the same token we find for the joint probability distribution of two (compatible) standard observables
A and B:
p_{mn} = ∫_{ Λ}dλ ρ(λ)
p_{AB}(a_{m}, b_{n}λ).
In order that a joint measurement
of A and B be mutually nondisturbing it seems that
the measurement processes should be independent. For this to be
the case it is sufficient that the conditional probability
p_{AB}(a_{m}, b_{n}λ)
satisfy the following condition of conditional statistical independence:
p_{AB}(a_{m}, b_{n}λ) =
p_{A}(a_{m}λ)
p_{B}(b_{n}λ).
Such a condition seems to be quite reasonable if
A and B are measured in causally disconnected
regions of spacetime. For this reason this condition is
generally applied to EPRBell experiments, and referred to as a locality condition.
 These expressions can easily be generalized for
application to generalized observables. In that case the
nonideality of the measurement procedures can be taken into
account in the conditional probabilities. In particular, it is
not to be expected that the condition of conditional statistical
independence will be satisfied in a
joint nonideal measurement of incompatible observables.

Critique of stochastic hidden variables theories

Unfortunately, if the
quantum mechanical probabilities are represented by the
expressions given above,
stochastic hidden variables theories are not successful in performing the
task of explaining
a measurement result of a quantum mechanical observable by
referring to a specific value of the hidden variables.
Given an individual preparation described by λ, the stochasticity
involved in the deviation of the conditional
probabilities p_{A}(a_{m}λ) from 0 or 1 either does not have an
explanation at all (which would bring us back to the Copenhagen
probabilistic interpretation), or
it is explained by measurement disturbance due to `stochastic influences exerted by
of the measurement process'. Such stochasticity can explain
the deviation from ideality in nonideal measurements of standard
observables. However, certain quantum mechanical measurements
behave deterministically (viz. `ideal measurements of standard observables in case the
quantum mechanical state vector is one of the observable's
eigenvectors', and `EPRBell measurements in case of strict correlation of the singleparticle observables').
Under the reasonable assumption that the `nonideality of the measurement process' does not depend on the
`prepared state of the microscopic object', such a determinism
seems to be inconsistent with `stochasticity induced by the`measurement process'.

The Bell inequality in stochastic hidden variables theories
It is possible to give a
simple and general derivation of the Bell inequality on the
basis of the existence of a `quadrivariate probability distribution'.
Within the context of a stochastic hidden variables theory this boils down
to the existence of quadrivariate conditional probabilities.
Thus, the Bell inequality is satisfied by any `experiment
jointly measuring the four observables A_{1},
B_{1}, A_{2}, and
B_{2}', for which the measured probabilities can be
represented by
p(a_{1i},b_{1j},a_{2k},b_{2m})=
∫_{Λ}dλ ρ(λ)
p_{A1B1A2B2}
(a_{1i},b_{1j},a_{2k},b_{2m}λ).
This is quite general. For instance, it includes
deterministic theories (in which the conditional probabilities
are characteristic functions of regions Λ_{ijkm} of hidden variables space
Λ). It is also independent
of the questions of (non)locality, contextuality, and the presence
or absence of mutual disturbance in case of incompatibility of
the observables, in the sense that the conditional probabilities
p_{A1B1A2B2}
(a_{1i},b_{1j},a_{2k},b_{2m}λ) may reflect
such properties of the measurement procedure if present.
Thus, in a hidden variables description the
generalized Aspect measurement
would satisfy the Bell inequality while obeying
a locality condition for measurements performed in different arms of the
interferometer, but not for the measurements performed jointly within one arm:
p_{A1B1A2B2}
(a_{1i},b_{1j},a_{2k},b_{2m}λ) =
p_{(γ1)}(a_{1i},
b_{1j}λ) p_{(γ2)}
(a_{2k},b_{2m}λ),
where γ_{n} is the
transmissivity of the mirror in arm n, n=1,2 (compare figure 11),
determining the mutual
disturbance in the joint nonideal measurement of
A_{n} and B_{n}. For the same reasons
as in the quantum mechanical
description we are not at all urged to assume that for the
standard Aspect measurements of figure 10 a Bell inequality could be
derived from the four `quadrivariate probability distributions' of
the experiments corresponding to the four arrangements
(γ_{1},γ_{2}) =
(1,1), (1,0), (0,1) or (0,0).

However, in a
hidden variables theory we have more resources than in quantum
mechanics. The fact that the `quadrivariate probability distributions' of the generalized Aspect measurements are not
useful to derive a Bell inequality for the standard Aspect measurements, does not imply that there cannot exist
another `quadrivariate probability distribution' that
can do the job. Actually, such a probability distribution can be
constructed (at least formally) if the locality condition
p_{AB}(a_{m}, b_{n}λ) =
p_{A}(a_{m}λ)
p_{B}(b_{n}λ)
is satisfied for the standard Aspect measurements. Then the bivariate joint probability distributions
of the four standard Aspect measurements are given by
p(a_{1i},a_{2k}) =
∫_{Λ}dλ ρ(λ)
p_{A1}(a_{1i}λ)
p_{A2}(a_{2k}λ), etc..
It is now easily seen that the `quadrivariate probability distribution'
p(a_{1i},b_{1j},a_{2k},b_{2m}) =
∫_{Λ}dλ ρ(λ)
p_{A1}(a_{1i}λ)
p_{B1}(b_{1j}λ)p_{A2}(a_{2k}λ)
p_{B2}(b_{2m}λ)
yields the bivariate distributions of all standard
Aspect measurements as marginals. Therefore, these experiments
appear to have to satisfy the Bell inequality on the
basis of the assumption of the applicability of a local
hidden variables theory.

This, precisely, was
Bell's message: local hidden variables theories are
incompatible with quantum mechanics, be they deterministic,
stochastic, contextual, or noncontextual. Hence, if the
probability distributions of quantum mechanical measurements are
to be represented by the expressions of the stochastic
hidden variables theory given above,
then reality underlying quantum mechanics seems to
be necessarily nonlocal. However, it is questionable
whether quantum mechanical measurements can be modeled in
this way. In the following
reasons are advanced putting this assumption into doubt.
It may be questioned whether this kind of hidden variables theory
is really the most general one, or whether, perhaps, there exist
still other such theories being better suited to underpin quantum mechanics.
Such doubts might be raised by the rather suspicious character of an expression
like p_{A}(a_{m}λ),
conditioning a microscopic (or even macroscopic) quantity a_{m} on a submicroscopic
quantity λ. Such an expression is comparable to a phrase consisting of, e.g., English and Chinese words, liable to be meaningless because there may not even exist an exact English equivalent of a Chinese word.
Does it make sense to condition a quantum mechanical measurement
result, even in a stochastic sense, on an `instantaneous value of a hidden variable'?
Our experience with (statistical) thermodynamics suggests a negative answer: for instance,
an expression conditioning `temperature' on a phase space point (representing the instantaneous values of positions and momenta of
all particles of a gas) need not have a physical meaning, the thermodynamic notion of `temperature' not referring to a single phase space point because `temperature measurement' is too slow to yield a value of an instantaneous property of the gas.
I will refer to theories endorsing the quantities p_{A}(a_{m}λ)
(usually considered within the context of the Bell inequality) as quasiobjectivistic theories
because of their suggestion that, even if a measurement result is no longer considered as an `objective
property of the microscopic object', it is yet thought to be conditioned on an `objective submicroscopic quantity'.
The `quasiobjectivism of these theories' may be the real source of the impossibility to obtain a submicroscopic underpinning
of quantum mechanics, violating the Bell inequalities.

Quantum mechanics as a theory of equilibrium processes

Let us exploit an analogy (cf. Publ. 46) slightly different from the
one introducing hidden variables, to be referred to as
the thermodynamic analogy, viz.

thermodynamics


quantum mechanics

↓

~

↓

classical statistical mechanics


stochastic hidden variables theory



The important point is that thermodynamic quantities are
not instantaneous properties of the system. It does not
make sense within thermodynamics to consider pressure or temperature as a function of
the instantaneous positions and momenta of the particles. They
are, at best, time averaged properties, time averaging
being taken in the sense of ergodic theory (it might seem to be
even more appropriate to consider temperature as a pointer
position of a thermometer, compare the empiricist interpretation).
Thermodynamics can only be applied if a condition of local equilibrium (molecular chaos,
ergodicity) is satisfied by the object system, warranting
sufficiently chaotic motion of the atoms.
The dispersive character of the solutions of the Schrödinger equation
suggests a physical analogy of quantum mechanical and
thermodynamic systems. In any case, if the hidden variable λ
would be a fluctuating stochastic
variable, the characteristic time of the fluctuations being much
shorter than the duration T
of the measurement interaction, then it would
hardly make sense to attribute the value of a quantum mechanical
observable to a specific value of λ
(like is done in the probability distributions given above).
The dynamics of
`quantum mechanical processes' might be analogous to the dynamics
of `quasistatic thermal processes', in which state changes
proceed from one state of (local) equilibrium to another.

If the analogy between quantum mechanics and thermodynamics
is a valid one, then the conditional probabilities
p_{A}(a_{m}λ)
are not applicable within the domain of application of quantum
mechanics. Instead, quantum mechanical measurement results should
be conditioned on `ergodic (time) averages of the hidden
variables'. Hence we should replace the expression of the quantum
mechanical probability p_{m} of the nonergodic theory
by something like
p_{m} =
∫_{Λ}d<λ>_{erg} ρ(<λ>_{erg})
p_{A}(a_{m}<λ>_{erg}),
the integration being over some space of ergodic states <λ>_{erg}.

By itself the introduction of ergodicity is
not sufficient to prevent a derivation of the Bell inequality
in the way done in the local nonergodic case.
There is, however, one additional aspect,
also deriving from the thermodynamic analogy, preventing the Bell
inequality to be derived in the ergodic case. This is the
aspect of `contextuality of the ergodic states'. Ergodic
paths will depend on the `interaction between object and measuring
instrument during the measurement' (for instance, the canonical
state e^{−H/kT}/Z of a volume of gas is dependent on the
orientation of the container since H is dependent on it!).
Because of the mutual exclusiveness of
measurement arrangements of incompatible quantum mechanical
observables such a contextuality may be even more pregnant in
quantum mechanics than it is in thermodynamics. Due to this
`contextuality' it is not possible to condition on the same
ergodic state in measurement arrangements of incompatible
observables. Hence, for the standard Aspect measurements of figure
10 we have
<λ>_{erg, An}
≠ <λ>_{erg, Bn}, n=1,2,
since A_{n} and B_{n} are incompatible.
The `impossibility to condition
measurement results of incompatible observables on the same hidden variables state' blocks the construction
of the `quadrivariate probability distribution' based on the
locality condition, and, hence,
the corresponding derivation of a Bell inequality.

An interesting question would be whether in a measurement of observable A
ergodic paths extend over the whole hidden variables space Λ, or whether
they are restricted to subspaces Λ^{(A=a)} of
Λ, corresponding to welldefined values a_{m} of observable A.
The first possibility could yield an explanation of the
probabilistic interpretation of the
Born rule, no value of the observable being attributable
to the microscopic object even in a contextual sense. The second possibility would be
consistent with a statistical interpretation of the Born rule, Einstein's untenable assumption of
`quantum elements of physical reality'
being replaced by the assumption
of `subquantum elements of physical reality' described by the ergodic paths
<λ>_{erg, A=a} in Λ^{(A=a)}.

Note that, since the ergodic paths are (co)determined by the measurement arrangement,
in a generalized quantum mechanical measurement
the path should be labelled by the corresponding POVM.
Thus, for instance, in a joint measurement we have
<λ>_{erg, Rmn} as an ergodic path in Λ^{(Rmn)}.

The thermodynamic analogy suggests where to look for the boundaries
of the domain of application of quantum mechanics. If quantum
mechanics, like thermodynamics, is a theory of quasistatic
(equilibrium) processes, then it is to be expected that
deviations from quantum mechanics will be found if it is possible
to prepare a nonequilibrium state (analogous to a
nonequilibrium state of a gas, in which, for instance, all
molecules are at one side of the container, all having the same
velocity), and preparation and measurement are carried out
faster than the relaxation time of the subquantum processes
establishing a state of (local) equilibrium. Considering the
constant h/2m in the Schrödinger equation as the
characteristic constant of a diffusion process, for atomic
processes a relaxation time τ << 10^{−15}s
can be estimated. At this moment the
femtosecond time scale is only coming into reach. So, even in
this advanced experimental field no deviations from quantum
mechanics are to be expected. From the present point of view it
is not at all surprising that Aspect's 1982 switching experiment
(switching frequency 50 MHz) did corroborate quantum mechanics.
However, it is possible that in the future we will be able to
perform experiments so fast that the instantaneous value λ
of the hidden variable (rather than an
ergodic average) can be experimentally probed, so as to allow
Aspecttype experiments to satisfy the Bell inequality. Since,
however, we do not have any experimental clue about the
properties of the subquantum world, is it equally possible that
subquantum theories will turn out to be still more different from
classical theory than is quantum mechanics. Therefore, at this
moment it does not seem to be very fruitful to try to develop
subquantum theories for other reasons than just demonstrating
that such theories are possible in principle, even if they have to be local.

