Incompleteness of Quantum theory

John O. Campbell

Perhaps physicists more than other scientists have a penchant for thinking they are close to knowing practically everything that can be known. This obsession with arriving at the end of knowledge has been a constant under-current within the physics community almost since its inception as a science.

Nowhere has this hubris caused more problems than within quantum theory. On the one hand many physicists believe they are close to unveiling ‘the final theory’ or ‘the theory of everything’ on the other hand the consensus interpretation of quantum theory is a mishmash that, as Feynman noted, nobody understands. Trying to reconcile these polar attitudes towards our state of quantum knowledge has resulted in some laughable assertions. 

Most physicist believe that quantum theory is a complete theory and that there is little more to be added. The fact that we don’t understand what it means is often attributed to nature being weirder than we can imagine. Frequently it is claimed that we should not expect quantum theory to make sense, that it is not an explanation of nature only a method of calculating predictions. Following this line of reasoning some have cast quantum theory as a method of human reasoning (1), others have asserted that human consciousness is required in order for quantum processes to take place. As the Nobel laureate Eugene Wigner wrote (2):

It was not possible to formulate the laws (of quantum theory) in a fully consistent way without reference to consciousness.

I attribute these flights of fancy to the resistance amongst us all, but especially it seems amongst physicists, to recognize the huge extent of our ignorance. Rather than admit that our thinking about nature is confused we often conclude that nature acts in a confused manner.  Edwin Jaynes noted this glitch common to scientific thinking and dubbed it the ‘Mind Projection Fallacy’ (3):

it is essential to recognize that propositions at two different levels are involved. In physical prediction we are trying to describe the real world; in inference we are describing only our state of knowledge about the world. A philosopher would say that physical prediction operates at the ontological level, inference at the epistemological level. Failure to see the distinction between reality and our knowledge of reality puts us on the Royal Road to Confusion; this usually takes the form of the Mind Projection Fallacy

Jaynes reserved his most biting witticisms concerning the Mind Projection Fallacy for the consensus interpretation of quantum theory:

The current literature of quantum theory is saturated with the Mind Projection Fallacy. Many of us were first told, as undergraduates, about Bose and Fermi statistics by an argument like this: 

‘You and I cannot distinguish between the particles; therefore the particles behave differently than if we could.’ Or the mysteries of the uncertainty principle were explained to us thus: ‘The momentum of the particle is unknown; therefore it has a high kinetic energy.’ 

A standard of logic that would be considered a psychiatric disorder in other fields, is the accepted norm in quantum theory. But this is really a form of arrogance, as if one were claiming to control Nature by psychokinesis.

Despite its many short-comings quantum theory is the most accurate theory ever devised by science; it is used to make many solid predictions and thereby provides insight into nature. However the consensus interpretation of quantum theory has wrongly concluded from the theories numerous uncertainties and missing physical processes that nature is uncertain and is missing the usual physical mechanisms which allow the application of logic to a full explanation.

This situation remains the same today as it was almost a hundred years ago when Einstein pursued one of science’s great debates with other pioneers of quantum theory. His insistence on the incompleteness of quantum theory fell largely on deaf ears presumably due to an inability to accept the extent of our ignorance. It would seem the correct thing to do but the consensus chose instead to cover up the theories ‘incompleteness’ in a number of blatant denials and instead claimed that the theory was in fact complete.

We might wonder about the exact extent of scientific ignorance. One approach would be to imagine a contrasting model of the entire universe that contained complete scientific knowledge. This ultimate model would be able to predict the construction of the exact current state of the universe from its most fundamental entities. Scientific ignorance would then be the entropy of our current scientific models as compared to the complete knowledge of this ultimate model; the amount of information in bits required to bring our current scientific knowledge to completion. 

In estimating the extent of this entropy a couple of things may be worth considering. First we do not know what the fundamental entities making up the universe are. New ‘fundamental’ particles have been identified ever since the concept first arose within atomic theory and there is little reason to expect this to end any time soon. A new generation is expected if advanced theories such as super-symmetry or preons come to fruition.  In fact the current ‘fundamental’ sub-atomic particles are at the order of 10^-15 meters and we can expect new physics all the way down to the Planck scale at 10^-35 meters. This one arena of our complete ignorance is vast. Secondly and at a larger scale it is likely that complex macro processes at least partially analogous to biology and culture are common in the universe. A complete theory would be able to predict and describe all instances. 

While I will not attempt a numeric estimation of scientific ignorance it is clear that it is an enormous quantity and that we are vastly distant from any ultimate theory. It is a wonder that in the face of this immense ignorance the scant scientific knowledge we do have is sufficient to predict much of the universe we observe. This is likely due to our limited observational abilities at least as much as to the extent of our knowledge.

We can expect many uncertainties with quantum theory to be resolved and missing mechanisms found as science becomes better able to experimentally probe towards the Planck scale. However we should also expect that the current aspects of quantum theory leading to its many accurate and repeatable predictions will be incorporated within that full theory.

The concept of information and its transfer is essential to all of the axioms of quantum theory and yet the physical representation of this information is not described or identified. This must be considered a major hole in quantum theory as Zurek, building on the work of others, has conclusively demonstrated that there can be ‘no information without representation in a physical state’ (4). The consensus answer to this objection is to deny it and claim that quantum theory does not serve as a model for physical reality, it is only a calculational devise. In other words that the information of quantum theory does not have an independent existence outside of the human mind. 

Recent research seems to indicate that his consensus view is incorrect on theoretical grounds (5). It is also amusing to note that our proclivity to conclude that successful scientific models do not describe ‘real’ entities has a long history over which it has consistently proved wrong. 

Boltzmann attempted to base thermodynamics on the findings of atomic theory and the statistical kinetic motion of vast numbers of molecules. As atomic theory was not yet widely accepted he suffered many attacks from the pillars of the physics community which may have eventually influenced his decision to commit suicide. The preeminent German physics journal would not allow Boltzmann to refer to atoms as actual phenomena but only as convenient theoretical constructs. To facilitate this ruse he adopted Hertz’s theory that atoms were ‘Bilder’; merely models or pictures and were not ‘real’ (6).

This tendency to interpret effective models as only ‘calculational devices’ also arose in biology.  Mendel’s quantitative rules for predicting the frequency of parental characteristics which are inherited by their off-spring caused great excitement amongst biologists but many questions remained. There was much controversy whether these ‘genetic’ rules supported or challenged Darwin’s theory. On one thing there was a consensus; ‘genetics’ was not a physical process but merely a means of making calculations. The philosopher of science David Hull describes the situation (7):

As much as Bateson might disagree with Pearson and Weldon about the value of Mendelian genetics, he agreed with them that it was unscientific to postulate the existence of genes as material bodies. They were merely calculation devices.

The plain fact of the matter is that effective scientific models would not be effective unless they accurately described physical reality. The principle of naturalism at the foundations of the scientific world view tells us that there is nothing but physical reality. If our model is only mathematical and we are ignorant of the underlying physical situation which gives rise to the mathematics that does not mean there is no underlying physical situation, it only means that we have some remaining ignorance, that there is more to find out. 

We should not be ashamed of our ignorance; it is a noble state and vastly superior to a state of false knowledge.

If we accept that quantum states have physical forms just as real as genes we are left with the fundamental unanswered question: ‘In what form is the essential information of quantum theory physically recorded?’ 

Gerard t’Hooft has made one of very few attempts to answer this question with his cellular automata interpretation of quantum theory (8) where he speculates that the information of quantum theory is recorded within a physical medium near the Planck scale. He provides support for the idea that the rules of quantum mechanics emerge from a form of cellular automata where each bit of information is updated through simple rules depending only on the state of nearest neighbours.

This interpretation is analogous with the fact that the information of biology is recorded in a medium many order of magnitude smaller than the phenotype. Even though its small scale meant DNA was difficult for us to identify it did not rule out its physical existence. We should expect that with the many examples found in nature involving a duality between an informational model and a physical system the physical presence of the informational model will be more difficult to identify due to its smaller scale. In fact quantum physics may well be in an historical situation similar to the one in which biology found itself between Darwin’s publication of Origins of Species and the discovery almost a hundred years later of DNA, the physical form of biology’s informational model.

Bibliography

1. QBism: The frontier of quantum Bayesianism. Fuchs, C. 2010, Preprint.

2. Wigner, Eugene. Symmetries and Reflections: Scientific Essays. s.l. : MIT Press., 1970.

3. Jaynes, Edwin T. Clearing up the mysteries - the original goal. [book auth.] John Skillings. Maximum Entropy and Bayesian Methods. 1989.

4. Decoherence, einselection and the existential interpretation (the rough guide). Zurek, Wojciech H. 1998, Philosophic Transactions of the Royal Society; vol. 356 no. 1743, pp. 1793-1821.

5. Are quantum states real? Hardy, Lucien. 2013, International Journal of Modern Physics B.

6. Wikipedia. Ludwig Boltzman. Wikipedia. [Online] [Cited: June 7, 2014.] http://en.wikipedia.org/wiki/Ludwig_Boltzmann.

7. Hull, David L. Science as a Process: An Evolutionary Account of the Social and Conceptual Development of Science. Chicago and London : The University of Chicago Press, 1988.

8. The Cellular Automaton Interpretation of Quantum Mechanics. 't Hooft, Garard. s.l. : Arxiv preprint, 2014, Vols. arXiv:1405.1548 [quant-ph].