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.