How arithmetic generates the logic of quantum experiments

April 4, 2010

The origin of indeterminacy within quantum physics and the mechanism of decision at measurement

Filed under: 1 — steviefaulkner @ 2:14 pm

The accompanying downloadable pdf is the full research paper resolving logical problems in Quantum Physics. Help yourself.

This study confirms that the conventionally formulated Quantum Mechanics with which we are familiar, is not the whole theory. The part that has been missing is not some undiscovered physics; it is Mathematical Logic. And in order to be entire, Physical Theory must embrace and incorporate it.

Amongst the discoveries of this research is foundation for The Quantum Logic. This is not an experimental logic or postulated ‘toy’; on the contrary, it is logic found present in the theory which Applied Mathematics cannot detect. It is found that Quantum Mechanics, conventionally prescribed by a collection of postulates, is a fragment of a larger theory, axiomatised under the Field Axioms. From first principles, these axioms derive equations of the subject, but show that different types of scalar: complex, real and rational, carry distinctly different logical validities. This is because the rational scalars exist as theorems of the Field Axioms, whereas existence of complex and real scalars can neither be proved nor disproved: they are mathematically undecidable.

In showing this, the role played by the square root of minus one is rigorously established. And in doing so, findings further establish that this research applies to all theories in Quantum Physics that employ scalar products between vectors in orthogonal spaces.

This new formalism resolves long standing logical anomalies for which the subject is notorious. It derives quantum indeterminacy, the existence of probability, Pythagorean addition of probability amplitudes and the mechanism of decision at measurement. It overturns two ‘intuitive fundamentalities’ and derives them instead: that observables are real values and entities must show particle behaviour.

The theory substantiates the 3-valued logic of Reichenbach, and via his work resolves ‘causal anomalies’ of complementarity and the EPR-paradox of action at a distance.

Theorems of Model Theory show that validities and indeterminacy of this axiomatised Quantum Mechanics go hand in hand with its theorems and undecidable sentences. These are interpreted as a causeology in Nature providing entities whose existence is caused and other entities whose existence is permitted. These mirror rational scalars, caused by the theory’s axiomatisation, and complex scalars permitted by the theory’s axiomatisation. The caused entities can be confirmed and witnessed; but the permitted entities can be neither confirmed nor denied.

What is especially interesting is, in contrast to the causative processes of classical physics where cause ascends through chains: …effect > caused-effect > caused-effect > caused-effect… ; there are quantum effects that are not caused by effects earlier in the chain but are permitted by them. In other words, there may be an effect in a chain for which no cause is traceable and for which information about the chain, upstream, is lost. To illustrate; in a wave/particle duality, all the various wavelengths that make up a wavepacket are caused effects; but probability amplitude is an uncaused permitted effect, unable to pass on the chain’s information to probability, which it in turn then causes. In addition, although all the momenta of a wavepacket are caused, and likewise so are the positions, the existence of both together is merely permitted.

Your feedback is welcome.  My email is at:

StevieFaulkner@googlemail.com

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November 7, 2009

Who’s Afraid of a Big Black Hole?- Can Self Reference help?

Filed under: 1 — steviefaulkner @ 9:22 pm

The Horizon programme, “Who’s Afraid of a Big Black Hole?” was broadcast on 3rd November 2009. In a very clear and lucid account, the programme brings us up-to-date, on the state of science regarding the densest concentrations of mass in our Universe. After explaining the major advances, the programme focuses on difficulties in the theory relating to the singularity: that point at the centre of a black hole into which matter relentlessly cascades. Because it is an infinitesimal point, gravitation converges to infinity there; and as it stands, physical theory breaks down.

The converging gravitational field and the singularity was predicted by Einstein’s General Theory of Relativity, but because the singularity is a microscopic object, in its vicinity, Quantum Mechanics must also be satisfied. Nevertheless, for some decades, theorists have struggled without success to unify these two theories, and the programme concludes with those eminent scientists contributing to the programme, expressing their bemused but determined puzzlement as to where they should turn for inspiration on how to form one theory from the two.

While watching the programme, I knew I must make cosmologists aware of the idea that motivated my own studies, an idea of some merit and substance, because that idea has resulted in a thesis that derives the logic of Quantum Mechanics. It was this: questions along the lines: “What Physical Laws brought about The Beginning?” are misplaced on grounds of logic. No true beginning can have laws that precede it. Otherwise, what rules generated those laws… and so on, ad infinitum.

It was this thought, in 1997, that led me to consider the possibility of self-reference in physics. A friend then pointed me in the direction of Kurt Gödel’s theorems and they naturally led to a search for mathematical undecidability in Quantum Mechanics.

In the Horizon programme, parallels are drawn between the singularities of black holes and another singularity with similar conditions: the one from which the Big Bang exploded. Now of course, the singularity of the Big Bang may not have been The Beginning; but if it was we should not expect to find Physical Laws to have brought it about.

A logical alternative to this is that the Matter, the Spacetime and the Physical Laws all came into existence together in a self referent loop where each initiates the other: a loop that would otherwise have been hovering in some undecidable state.

Since 2006 I have had extremely good success in finding mathematical undecidability at the core of quantum mathematics. This has a logical form where products of orthogonal vectors are not axiomatic; yet the Field Axioms, the rules of addition and multiplication for scalar components of all mathematical objects, neither forbid nor accommodate these products.

Quantum Mechanics then develops into a theory that tells us where indeterminacy comes from, what happens to it at measurement, why the world we witness is real, the reason for particles, and why probability is not indeterminate. There are also signs that the indeterminacy of chaos follows from a related but quite separate undecidability. Most remarkably, once seen in terms of its logical form, Quantum Mechanics becomes an elegantly intuitive theory.

Mathematical undecidability lies within systems whose operation has no option but to follow a deficient set of rules. My suggestion is that at the singularity there is such deficiency and that this is a good candidate worthy of investigation.

September 26, 2009

Causeology in Quantum Physics

My work finds that hidden within Nature there is a causeology where there are entities whose existence is CAUSED and other entitiies whose existence is PERMITTED.  The caused entities can be confirmed and witnessed; but the permitted entities can be neither confirmed nor denied.
What is interesting is that unlike the causative processes with which we are accustommed in everyday experience, cause acends through chains:   cause, effect…cause, effect…cause,effect…; these quantum causes derive and materialise, not from caused entities up the chain, but permitted entities that are not caused. To illustrate, in a wave/particle duality, all the various wavelengths that make up a wavepacket are caused entities; but the interfering sums of these waves are permitted.  From these permitted interfernces derive caused probabilities.  The interfernces constitute a break in the causal chain. This claim is born out in the mathematics using Model Theory.
In the mathematics, this regime has its origin in the way numbers add and multiply.  The rules are the Field Axioms. Some numbers are caused by these axioms: the rational numbers exist as theorems; others are permitted. Juxtaposed against this system, are  products between orthogonal vectors which force the existence of certain permitted numbers. These orthogonal vectors are objects that, when multiplied with themselves give one, and, when multiplied with each other give zero.
The results lead to a 3-valued logic of true, false and indeterminate.  This was anticipated by Hans Reichenbach.  His 1944 book details the effects of this logic.  It does indeed resolve the paradoxes in quantum physics.  What Reichenbach had no idea of is where his logic came from…so it has been generally ignored.

My work finds that hidden within Nature there is a causeology where there are entities whose existence is caused and other entities whose existence is permitted.  The caused entities can be confirmed and witnessed; but the permitted entities can be neither confirmed nor denied.

What is interesting is that in contrast to the causative processes in classical physics where cause ascends through chains:   cause, effect…cause, effect…cause, effect…; there are quantum entities that derive, not from caused entities up the chain, but permitted entities that are not caused. To illustrate; in a wave/particle duality, all the various wavelengths that make up a wavepacket are caused entities; but the interfering sums of these waves are permitted.  From these permitted interferences derive caused probabilities.  So the interferences constitute a break in the causal chain. This claim is born out in the mathematics using Model Theory.

In the mathematics, this causal regime has its origin in the way numbers add and multiply.  The rules for this adding and multiplying are the Field Axioms. Some numbers are caused by these axioms: the rational numbers exist as theorems; other numbers with imaginary components, are permitted. Juxtaposed against this axiomatic system, are products between orthogonal vectors which force the existence of imaginary numbers which are merely permitted. The orthogonal vectors are objects that, when multiplied with themselves give one, and, when multiplied with each other give zero

The full causeology is 3-valued: cause, deny, permit.  These go hand in hand with a 3-valued logic of true, false and indeterminate.   This 3-valued logic was anticipated by Hans Reichenbach.  His 1944 book details the effects of this logic.  It does indeed resolve the paradoxes in quantum physics.  What Reichenbach had no idea of was where his logic came from…so it has been generally put to one side; now it has foundation.

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