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Minimal Quantum Theory

Is there more to quantum theory than quantum mechanics?
The former is the departure from the continuity scheme of Newtonian mechanics to one in which certain physical quantities only take certain restricted discrete values; the latter is the mathematical theory used to predict values of measurable quantities.

From one perspective, it can be seen that quantum physics has a third part. The first being the theory of discrete physical quantities, the second, the mathematics in the form of the wave function and the probabilities that pertain to the result of measurements and the third is the set of symmetries which attach to the processes which can be expressed as:

Representation —> Transformation —> Invariance

These elements are uncontroversial in so far as they do not stand in the way of the acceptance of the various interpretations of quantum theory. They are consistent with the Copenhagen interpretation, the many worlds interpretation, the transactional interpretation, Bohm's pilot wave theory etc.

By contrast, the mysteries or weirdness which derive from quantum discreteness are controversial. They include the quantum jump, wave particle duality and the measurement problem; to name just three. These are so difficult to deal with, or accommodate in the theory, that they have generally been side-stepped. The problem, for instance, of quantum tunnelling has to be set aside even though as a physical fact it is a necessary part of nuclear theory.

History is consistent with the idea that the question of how physical theory can successfully move beyond the standard model cannot be answered without first disentangling the physical meaning of the quantum mysteries and then incorporating them in a new foundation for the way ahead. The last hundred years of struggling to understand the atomic nucleus in a framework of continuous motion of quantum objects in curved space has failed. The continuity model is bankrupt.

The Born interpretation

It was Max Born, in 1926, who linked the quantum wave function with empirical meaning. He discovered that squaring the modulus of the wave function gives the density of the probability of an occurrence, such as observing an electron. A diffracted electron can be found in any space for which the wavefunction is not zero. If the meaning of the wavefunction is confined to what the square of the modulus refers to then what is happening between the diffraction and the ensuing detection of the electron—the intervening period—is irrelevant. This is crucially important to any realistic theory of the microscopic realm.

What then of the intervening period? Born's discovery is consistent with either the electron being somewhere, everywhere or nowhere in the intervening period. Perhaps even more possibilities can be imagined! What is important is to discover the importance or even the relevance of the difference between the three possibilities. What we do know is that the question has no relevance for the probability of the electron being somewhere, when it is observed. This is why what is going on in the intervening period can form the basis for any number of different quantum interpretations, provided the wavefunction and its importance are never lost sight of. That way, every interpretation has the same mathematical outcome—they are all equally successful, at least they are frequently claimed to be so.

The issue of the intervening period becomes somewhat more complicated when it is realized that the wavefunction is relevant to an abstract configuration space, not physical three dimensional space or four dimensional spacetime. This leaves open the possibility that what is going on in configuration space and in physical space are different, because during the intervening period their mutual relevance is undetermined. What is happening in configuration space is mathematical and not physical. It is this question of relevance which is the foundation of the perceived incompleteness which plagues quantum mechanics.

A minimalist interpretation
A consequence of the apparent equivalence of the different interpretations of quantum mechanics is that experimental differentiation between them is not just round the corner. Thus, we have no reason to expect that one or more interpretations will fail some experimental test of their correctness. Despite this, all may not be lost in the quest for the interpretation that matches the real world—if, indeed, one of them does. It is generally the case that in addition to predicting the outcome of experimental results, theories  provide explanations of formerly unexplained phenomena, whether or not they are dependent upon mathematical quantification or description. Explanations derivative of a theory may embrace phenomena, formerly considered unrelated, in a manner which unifies them under a single principle.

In the latter half of the twentieth century, unification of the four forces of Nature under a single principle was a centrally important question. As things stand at the beginning of the twenty-first century, quantum mechanics and special relativity are less than seamlessly unified. And quantum field theory (QFT) is plagued with renormalisation problems whose underlying causes have a distinctly unnatural flavour about them. Thus, one of the interpretations may pull ahead of the field in the race to explain the real microscopic world not experimentally but by way of explanatory power.

QFT provides the framework for several theories of microscopic objects in modern physics. It is the project to combine special relativity and quantum mechanics so as to make the latter comply with the rules of the former. The result is relativistic quantum theory but it is not fully satisfactory and many would claim it remains incomplete. Any project to extend quantum mechanics must take account of the views of Bohr and Heisenberg on what quantum mechanics actually means, not just what it predicts about microscopic phenomena. This is especially important if the extension entails a reliance upon more than the mathematical formalism. And such an extra formalistic reliance underlies QFT. A simply appeal against the non reality of quantum mechanics accompanied by an insertion of classicality to correct the deficiency runs the risk of fatally damaging the whole project.

When the foundations of QFT are examined in the light of Bohr's complementarity interpretation, the logic of one manoeuvre stands out as a source of unease. The manoeuvre is the insistence that evolving quantum systems have properties in opposition to Bohr's claim that unmeasured systems lack properties. From the point of view of the theory of the inheritance of order, QFT extends Bohr's concept of the unobserved system beyond the scope of its natural limit. QFT adopts the contrary postulate: "The property of a quantum system in the position representation is the wavefunction and not a specific location as in the classical case."1

The quantum state is a description containing definite values of a set of physical properties. QFT assumes that the description is applicable to the system. Such an assumption is unwarranted. The extrapolation of the description from the quantum state to the quantum system is not a simple matter. And how it differs from the relations between the set of odds for the different horses in the next race and the winner is also unclear.

More minimalism still
The present event scheme adopts a minimalist interpretation of quantum mechanics in which nothing classical is attributed to the domain of quantum objects. Nothing meaningful can be inferred about that domain, and isolated evolving quantum objects have no properties—properties only appear at measurements (or unmeasured interactions). And the properties which appear are relevant to the interaction or type of instrument the system collides with. Quantum mechanics is consistent with evolving systems, such as an electron, not being at any particular position, even though there is some probability that it might be observed at every point in space. That is sometimes interpreted to mean that the system is really at all the positions, and when observed, its being at all positions reduces instantly to being at just one of them. The evolving quantum system is somehow smeared out among all possible positions, and they mathematically all contribute to their being finally at one position.

That is not the present analysis, according to which, if it is not at a particular or definite position the electron might as well be at no position and we have no reason to go beyond that conclusion. We need to set aside all reliance upon classicality, which is sometimes called common sense. Being at no particular or definite position does not warrant any conclusion as to where the system is; or whether it is anywhere! The minimalist analysis adopted here agrees that the electron is at no definite or particular position and draws no conclusions beyond that fact. Of course, this runs counter to commonsense--but then common-sense is what tells us the earth is flat!

The analysis of the logic underlying the electron being smeared out among all the positions, at which there is some probability of its being found, depends upon two implied propositions: One is that the world contains the electron and the other is that the world includes all the positions at which it might be found in the future. However, if we adopt the mode of coupling space and time in the formation of spacetime which does not reduce the time axis to a space-like axis, and separate the concept of action from its representation, then we can question the idea that future spacetime loci are present before they occur! Put simply, if space and time are merely relations among objects and if the object has not yet arrived then perhaps neither has the space and time which relate it to other objects. If the world is discrete instead of continuous, then perhaps it only sometimes “includes all the positions at which the electron might be found.” If the positions at which the electron might be found are not in existence in the intervening period--what then? Such an analysis of spacetime is consistent with the electron being at no definite or particular position during the intervening period.

The evolution of the quantum system is inscrutable—it is private because it has no extrinsic or classical aspect. And theory says nothing about what is going on in the quantum domain. It only predicts what happens when quantum objects interact with the classical domain—when what happens is always classical. The simplest position to take is that nothing classical is going on in the quantum domain—nothing that can be counted, measured or observed.

The discrete electron and chemical periodicity
The application of this interpretation to the Dirac electron leads to a qualitative physical model of the components of the atom which is discrete in both time and space. Three of the unreal physical consequences of the Direc equation for the electron2 arise naturally in a scheme that is dependent upon special relativity and Bohr's complementarity interpretation of quantum mechanics. The theory which is developed in three papers is non-quantitative, but has considerable explanatory power.

For the abstracts of the three papers and links to the full text, as pdf files, click here.

A Discrete Model of the Dirac Electron  download pdf

On the Electrodynamics of Motionless Events  download pdf

On the Discrete Electron and the Periodicity of the Monatomic Gases  download pdf

The discrete electron and quantum gravity
This minimalist interpretation of quantum mechanics when applied to the Dirac electron enables the problem of quantum gravity to be approached from a novel direction, with surprising results.

In brief, when space and time are understood as geometric relations for the individual electron, the behaviour of the electron is both quantum mechanical and gravitational. Gravity curves the space-time trajectory of the electron, rather than a space-time that is independent of the objects that it relates.

On the Quantum Gravity of the Electron

The pre-print abstract is here.        For the full text, as a pdf file, click here

1.  S. Y. Auyang, How is Quantum Field Theory Possible? ( New York: OUP, 1995 ), p161.

2. The three consequences of the Dirac equations that have proved to be problematic when interpreted in a framework of continuity of space, time and motion are: (1) negative energy electrons, (2) an equal role for time and space in the description of the electron and (3) its motion at the speed of light.

Comments and questions are welcome to pjf@it.net.au

© Peter Fimmel 2002-2009

Last page update 02/06/09