Quantum mechanics the way I see it

Hidden variables or subquantum theories¹⁰

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

statistical thermodynamics		quantum mechanics
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classical mechanics		hidden variables theory

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 l 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 l 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 l' since the `objectivity⁶¹ of l' 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 Kochen-Specker theorem. Actually, the Bohr-Einstein `(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 contextualistic-realist interpretation of observables over Einstein's objectivistic-realist 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 `derivation⁶⁰ of the Bell inequality for local hidden variables theories' and the `alleged existence of a nonlocal hidden-variables theory (viz. Bohm's theory)' it is widely supposed that any hidden-variables 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 hidden-variables 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 often-heard 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 non-existence of such quadruples.
In the following an attempt is made to sketch the outlines of a local hidden-variables 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|l)', 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 l (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 `hidden-variables theory' by distinguishing between "microstates described by a hidden variable l" and "macrostates <l> corresponding to ergodic paths in hidden-variables space L (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 l 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 hidden-variables 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 wave-like 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 particle⁶³, since only in this way does it seem possible to explain interference in a double-slit 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
  
  y = R e^iS, R,S real,
  
  the (complex) Schrödinger equation for a particle takes the form of two coupled real equations,
  
  P = R²,
  
  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 Hamilton-Jacobi 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 so-called 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(y^*╤y - y╤y^*)/2m = P╤S/m of the probability flux). This may cause the individual interpretation not only to be redundant but, possibly, even self-contradictory.
  - 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 action-at-a-distance 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 above-mentioned 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 L (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 l 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|l) that the measurement result of observable A is a_m if the hidden variable had value l (note that the deterministic theory is a special case for which
  p_A(a_m|l) = c_{L_m}(l), the characteristic function of the subset L_m of L yielding A = a_m with certainty). If r (l) is the probability that the hidden variable is prepared with value l, then the quantum mechanical probability p_m is represented in the stochastic hidden variables theory by
  
  p_m = Є_Ldl r(l) p_A(a_m|l).
  
  By the same token we find for the joint probability distribution of two (compatible) standard observables A and B:
  
  p_mn = Є_Ldl r (l) p_AB(a_m, b_n|l).
  
  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|l) satisfy the following condition of conditional statistical independence:
  
  p_AB(a_m, b_n|l) = p_A(a_m|l) p_B(b_n|l).
  
  Such a condition seems to be quite reasonable if A and B are measured in causally disconnected regions of space-time. For this reason this condition is generally applied to EPR-Bell 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 l, the stochasticity involved in the deviation of the conditional probabilities p_A(a_m|l) 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 `EPR-Bell measurements in case of strict correlation of the single-particle 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 A₁, B₁, A₂, and B₂', for which the measured probabilities can be represented by
    
    p(a_1i,b_1j,a_2k,b_2m) = Є_Ldl r (l) p_{A₁B₁A₂B₂} (a_1i,b_1j,a_2k,b_2m|l).
    
    This is quite general. For instance, it includes deterministic theories (in which the conditional probabilities are characteristic functions of regions L_ijkm of hidden variables space L). 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|l) 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|l) = p_(g₁)(a_1i, b_1j|l) p_(g₂) (a_2k,b_2m|l),
    
    where g_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 (g₁,g₂) = (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|l) = p_A(a_m|l) p_B(b_n|l) 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) = Є_Ldl r (l) p_A₁(a_1i|l) p_A₂(a_2k|l), etc..
    
    It is now easily seen that the `quadrivariate probability distribution'
    
    p(a_1i,b_1j,a_2k,b_2m) = Є_Ldl r (l) p_A₁(a_1i|l) p_B₁(b_1j|l)p_A₂(a_2k|l) p_B₂(b_2m|l)
    
    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 modelled 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|l), conditioning a microscopic (or even macroscopic) quantity a_m on a submicroscopic quantity l. 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: such expressions need not have a physical meaning.
    I will refer to theories endorsing the quantities p_A(a_m|l) (usually considered within the context of the Bell inequality) as quasi-objectivistic theories because of their suggestion that, although 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 `quasi-objectivism 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.

ааааааааа thermodynamics		аааааааа quantum mechanics
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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 l 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 l (like is done in the probability distributions given above). The dynamics of `quantum mechanical processes' might be analogous to the dynamics of `quasi-static 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|l) 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 non-ergodic theory by something like

p_m = Є_Ld<l>_erg r(<l>_erg) p_A(a_m|<l>_erg),

the integration being over some space of ergodic states <l>_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 non-ergodic 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 <l>_{erg, A_n} ╣ <l>_{erg,
B_n}, n=1,2. 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 L, or whether they are restricted to subspaces L^(A=a) of L, corresponding to well-defined 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 <l>_{erg, A=a} in L^(A=a).
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 quasi-static (equilibrium) processes, then it is to be expected that deviations from quantum mechanics will be found if it is possible to prepare a non-equilibrium state (analogous to a non-equilibrium 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 t << 10^-15s 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 l of the hidden variable (rather than an ergodic average) can be experimentally probed, so as to allow Aspect-type 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.