The reality and dimension of space and the complexity of quantum mechanics

The dimension (and signature) of space is a result of distances being real numbers and quantum mechanical state functions being complex ones; it is an inescapable consequence of quantum mechanics and group theory. So nonrelativistic quantum mechanics cannot be complete (it requiresad hoc additional assumptions) and consistent (nor can classical physics), leading to relativity, quantum mechanics, and field theory. Implications of the constraints of consistency and physical reasonableness and of group theory for the structure of these theories are considered. It appears that there are simple, perhaps unavoidable reasons for the laws of physics, the nature of the world they describe, and the space in which they act.     

Generalized measure theory

It is argued that a reformulation of classical measure theory is necessary if the theory is to accurately describe measurements of physical phenomena. The postulates of a generalized measure theory are given and the fundamentals of this theory are developed, and the reader is introduced to some open questions and possible applications. Specifically, generalized measure spaces and integration theory are considered, the partial order structure is studied, and applications to hidden variables and the logic of quantum mechanics are given.     

Fundamental principles of quantum theory. II. From a convexity scheme to the DHB theory

Some classical and quantum theories are characterized within the convexity approach to probabilistic physical theories. In particular, the structure of the so-called DHB quantum theory will be analyzed. It turns out that the natural generalization of the standard Hubert space quantum mechanics, the operational one, is such a theory. The operational Hilbert space quantum theory will be reconstructed from the (weak) projection postulate and the complementarity principle. This is then used to argue that the DHB quantum theory is identical with the operational Hilbert space quantum theory.     

Understanding quantum entanglement: Qubits,rebits and the quaternionic approach

It has been recently pointed out by Caves, Fuchs, and Rungta [1] that real quantum mechanics (that is, quantum mechanics defined over real vector spaces [2–5]) provides an interesting foil theory whose study may shed some light on just which particular aspects of quantum entanglement are unique to standard quantum theory and which are more generic over other physical theories endowed with this phenomenon. Following this work, some entanglement properties of two-rebit systems are discussed and a comparison with the basic properties of two-qubit systems, i.e., the systems described by standard complex quantum mechanics, is made. The use of quaternionic quantum mechanics as applied to the phenomenon of entanglement is also discussed.     

On the quantum logic approach to quantum mechanics

A quantum logic structure for quantum mechanics which contains the concepts of a physical space, localizability, and symmetry groups is formulated. It is shown that there is an underlying Hilbert space which mirrors much of this axiomatic structure. Quantum fields are defined and shown to arise naturally from the quantum logic structure. The fields ofHaag andWightman are generalized to this theory and an attempt is made to find a local equivalence for these fields.     

An Introduction to Many Worlds in Quantum Computation

The interpretation of quantum mechanics is an area of increasing interest to many working physicists. In particular, interest has come from those involved in quantum computing and information theory, as there has always been a strong foundational element in this field. This paper introduces one interpretation of quantum mechanics, a modern ‘many-worlds’ theory, from the perspective of quantum computation. Reasons for seeking to interpret quantum mechanics are discussed, then the specific ‘neo-Everettian’ theory is introduced and its claim as the best available interpretation defended. The main objections to the interpretation, including the so-called “problem of probability” are shown to fail. The local nature of the interpretation is demonstrated, and the implications of this both for the interpretation and for quantum mechanics more generally are discussed. Finally, the consequences of the theory for quantum computation are investigated, and common objections to using many worlds to describe quantum computing are answered. We find that using this particular many-worlds theory as a physical foundation for quantum computation gives several distinct advantages over other interpretations, and over not interpreting quantum theory at all.     

Quantum Mechanics is About Quantum Information

I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion of quantum information is to be understood as a new physical primitive---just as, following Einsteins special theory of relativity, a field is no longer regarded as the physical manifestation of vibrations in a mechanical medium, but recognized as a new physical entity in its own right.     

Non-commutative supersymmetric quantum mechanics

General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two- and three-dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in physical systems with potentials involving spin and non-local interactions.     

Facts,Values and Quanta

Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian conception. The paper includes a comparison of the orthodox and Bayesian theories of statistical inference. It concludes with a few remarks concerning the implications for the concept of physical reality.     

The conceptual analysis (CA) method in theories of microchannels: Application to quantum theory. Part II. Idealizations. “Perfect measurements”

The application of the conceptual analysis (CA) method outlined in Part I is illustrated on the example of quantum mechanics. In Part II, we deduce the complete-lattice structure in quantum mechanics from postulates specifying the idealizations that are accepted in the theory. The idealized abstract concepts are introduced by means of a topological extension of the basic structure (obtained in Part I) in accord with the “approximation principle”; the relevant topologies are not arbitrarily chosen; they are fixed by the choice of the idealizations. There is a typical topological asymmetry in the mathematical scheme. Convexity or linear structures do not play any role in the mathematical methods of this approach. The essential concept in Part II is the idealization of “perfect measurement” suggested by our conceptual analysis in Part I. The Hilbert-space representation will be deduced in Part III. In our papers, we keep to the tenet: The mathematical scheme of a physical theory must be rigorously formulated. However, for physics, mathematics is only a nice and useful tool; it is not purpose.     

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.     

The physics of the Einstein-Podolsky-Rosen paradox

The Einstein-Podolsky-Rosen paradox as formulated in their original paper is critically examined. Their argument that quantum mechanics is incomplete is shown to be unsatisfactory on two important grounds. (i) The gedanken experiment proposed by Einstein, Podolsky, and Rosen is physically unrealizable, and consequently their argument is invalid as it stands. (ii) The basic assumptions of their argument are equivalent to the assumption that quantum mechanical systems are in fact describable by unique eigenfunctions of the operators corresponding to physical observables, independent of any observation or measurement. Following an argument due to Furry, it is shown that this interpretation of quantum mechanics must lead to some physical predictions at variance with those of conventional quantum mechanics. A decisive experiment has been performed by Freedman and Clauser, which rules out this interpretation, and imposes severe restrictions on any alternative theory which incorporates the Einstein, Podolsky, and Rosen concept of physical reality.     

Quantum mechanics is compatible with realism

A new paradox of quantum mechanics has recently been proposed by an author claiming that any attempt to inject realism in physical theory is bound to lead to inconsistencies. In this paper we show that the mentioned paradox is not such a one and that at present there are no reasons to reject realism.     

Interpretation of quantum mechanics

New axioms are proposed for the interpretation of quantum mechanics. They rest on a kind of calculus allowing to select meaningful physical statements and giving rules to check a given physical reasoning containing implications. Measurement theory is reformulated.