Quantum Logic, Conditional Probability, and Interference

Quantum logic originated with the Birkhoff-von Neumann paper of 1936.2 The authors argued that the transition from classical to quantum mechanics involves the transition from a propositional calculus forming a Boolean algebra to a propositional calculus which constitutes “a sort of projective geometry”.* For von Neumann, the significance of this thesis for the foundational problems of the theory lay in the possibility of regarding the projective structure as the logic of a quantum set theory -- a non-Boolean theory of sets for properties of quantum mechanical systems in which dimensionality plays the role of cardinality and transition probabilities between events are induced by automorphisms of the logic.** Thus, probabilities are logical: they are not defined, as in the classical case, by measures over events represented as sets of maximal truth value assignments, parametrized by the points of a phase space. Indeed, the quantum sets have no “points” in the general case; there are no “hidden variables” underlying the statistics associated with a quantum logic or quantum set theory. This is a radically different account of the “irreducibility” of the quantum statistics from that provided by the Copenhagen interpretation.

Since the pioneering works of Feynman 1) and Deutsch 2) using elements following the laws of quantum mechanics, interest has been growing in the area of so-called quantum computation. Among other things, a quantum teleportation scheme of the type proposed by Bennett et a1. 3) is of great interest from the measurement theoretical point of view. The original teleportation scheme of an unknown state iq'», which consists of the story of Alice and Bob, however, is insufficient due to the unclear treatment of the EPR 4) problem and the assumption of von Neumann's projection postulate. At the same time, the former unclear treatment follows from the latter assumption. This teleportation scheme, based on the existence of non­ local long range correlations between EPR pairs of particles relevant to the notion of wave function collapse, necessarily leads to epistemological discussion on the nature of physical reality. Unfortunately, tentative solutions to this problem always depend on what the authors decide to define as realism. 5 ) Therefore, such philosophical arguments concerning the conceptual aspects of EPR phenomenon are beyond the realm of physics. We shall show that a quantum teleportation can be achieved within the framework of present-day quantum mechanics without resorting to von Neumann's projection postulate. Today, the naive Copenhagen interpretation is criticized for various reasons. For example, in Machida and Namiki's theory of measurement, 5),6) which does not rely upon von Neumann's projection postulate, making use of the statistical operator as the description of the quantum state and introducing the state of the apparatus system, they showed that the resolution of the difficulty concerning the wave function collapse does not consist of a demonstration of the disappearance of other branch waves, but follows from the concept of decoherence. Their theory is widely applicable, even to incomplete measurements. 5) According to these lines of thought, one of the present authors (A. M.) attempted to analyze the process of decoherence. 7) In this paper, reviewing what is a correct understanding of the EPR problem, we intend to show how we can obtain a correct teleportation scheme without resorting to von Neumann's projection postulate. A reasonable explanation of the EPR problem was given by Machida. 8) In order

In defence of Copenhagenism

We argue that the Copenhagen interpretation of quantum mechanics cannot be applied to quantum cosmology. Among the alternative interpretations, we choose to apply the Bohm-de Broglie interpretation of quantum mechanics to canonical quantum cosmology. For minisuperspace models, we show that there is no problem of time in this interpretation, and that quantum effects can avoid the initial singularity, create inflation and isotropize the Universe. For the general case, it is shown that, irrespective of any regularization or choice of factor ordering of the Wheeler-DeWitt equation, the unique relevant quantum effect which does not break spacetime is the change of its signature from lorentzian to euclidean. The other quantum effects are either trivial or break the four-geometry of spacetime. A Bohm-de Broglie picture of a quantum geometrodynamics is constructed, which allows the investigation of these latter structures.

This article deals with the Copenhagen interpretation of quantum mechanics and its epistemological discussion by Ernst Cassirer and Richard Hönigswald. The starting point is Cassirer's treatise Determinism and Indeterminism in Modern Physics, published in Stockholm. Both philosophers and several physicists were involved in the subsequent exchange of letters. From a physical point of view, the Copenhagen interpretation was particularly criticised by v. Laue and Einstein. Both demanded a revision of the foundations of quantum mechanics or a critical examination of essential physical concepts. The epistemological implications of the Copenhagen interpretation were rejected, in particular that the interactions between observation, the observed and the observer in quantum physics experiments should lead to a renunciation of the concept of causality or an objectively unambiguous description of natural processes. Cassirer argues here that quantum mechanics fulfils the epistemological requirement of transforming ontological “concepts of things” into epistemological concepts of relations and is therefore compatible with the neo-critical approach he advocates. At the same time, Hönigswald was working on two major treatises on the structure of physics and the concept of causality. Here he undertakes an epistemological foundation of physics as a whole. A transcendental foundation of the concept of experience and its specifications in principles such as contemplation, observation or experiment appears to be called for. Hönigswald therefore essentially rejects the Copenhagen interpretation, as it draws epistemological consequences by physical means and thus inadmissibly reverses the transcendentally necessary relations of justification. These arguments are examined and could be used for a current discussion on the epistemological foundations of quantum mechanics and physics as a whole. A post-Neo-Kantian, transcendental philosophical foundation of scientific knowledge, especially of modern physics, seems possible and necessary.

G. Gr¨atzer’s gives the following example in his monograph «Universal algebra». Let A be a universal algebra (with some family of operations Σ). Let us take an arbitrary set B ⊆ A. For all of the operations f ∈ Σ (let n be the arity of f) let us look how f transformas the elements of Bn. It is not necessary that f(B) ⊆ B, so in the general case B is not a subalgebra of A. But if we define partial operation as mapping from a subset of the set Bn into the set B. then B be a set with a family of partial operations defined on it. Such sets are called partial universal algebras. In our example B will be a partial universal subalgebra of the algebra A, which means the set B will be closed under all of the partial operations of the partial algebra B. So, partial algebras can naturally appear when studying common universal algebras. The concept of congruence of universal algebra can be generalized to the case of partial algebras. It is well-known that the congruences of a partial universal algebra A always from a lattice, and if A be a full algebra (i.e. an algebra) then the lattice of the congruences of A is a sublattice of the lattice of the equivalence relations on A. The congruence lattice of a partial universal algebra is its important characteristics. For the most important cases of universal algebra some results were obtained which characterize the algebras A without any congruences except the trivial congruences (the equality relation on A and the relation A2). It turned out that in the most cases, when the congruence lattice of a universal algebra is trivial the algebra itself is definitely not trivial. And what can we say about the algebras A whose equivalence relation is, vice versa, contains all of the equivalence relations on A? It turns out, in this case any operation f of the algebra A is either a constant (|f(A)| = 1) or a projection (f(x1, ..., xi, ..., xn) ≡ xi). Kozhukhov I. B. described the semigroups whose equivalence relations are one-sided congruences. It is interesting now to generalize these results to the case of partial algebras. In this paper the partial n-ary groupoids G are studied whose operations f satisfy the following condition: for any elements x1, ..., xk−1, xk+1, ..., xn ∈ G the value of the expression f(x1, ..., xk−1, y, xk+1, ..., xn) is defined for not less that three different elements y ∈ G. It will be proved that if any of the congruence relations on G is a congruence of the partial n-ary groupoid (G, f) then under specific conditions for G the partial operation f is not a constant.

Why has de Broglie's theory been rejected?

How certain is Heisenberg’s uncertainty principle? Heisenberg’s uncertainty principle is at the heart of the orthodox or Copenhagen interpretation of quantum mechanics. We first sketch the history that led up to the formulation of the principle. Then we recall that there are in fact two uncertainty principles, both dating from 1927, one by Werner Heisenberg and one by Earle Kennard. Finally, we explain that recent work in physics gives reason to believe that the principle of Heisenberg is invalid, while that of Kennard still stands.

The quantum measurement effect of interaction without interaction for an atomic beam

The non-relativistic quantum theory has been interpreted causally by de Broglie, David Bohm, and others, where a quantum entity is viewed as a particle with a definite position and momentum. This interpretation contrasts with the Copenhagen orthodox interpretation, which conveys uncertainty through the commutator relationship between position and momentum operators. This is further exacerbated by the de Broglie-Bohm interpretation, which uses no operators corresponding to these observables. We reconcile these opposing viewpoints by introducing mathematical nonlinear transforms in Bohmian mechanics for the momentum observable. While these nonlinear operator-like transforms cannot be easily represented as matrices in the Hilbert space, we show that they can be utilized in the de Broglie-Bohm interpretations of quantum theory.

For the last decade I have been demonstrating that many of the so-called paradoxes generated by the Copenhagen interpretation of quantum mechanics have less puzzling analogs in nonlinear dynamics and chaos theory. This raises questions about the possibility of nonlinearities inherent in the foundations of quantum theory. Since most of us do not think intuitively with nonlinear logic, I take this opportunity to dwell on several peculiarities of nonlinear dynamics and chaos: nonlinear logic itself and the possible connection of infinite nonlinear regressions with free will. Superficially, nonlinear dynamics can be every bit as counterintuitive as quantum theory; yet, its apparent paradoxes are more amenable to logical analysis. As a result, using nonlinear dynamics to resolve quantum paradoxes winds up being more straightforward than many of the alternative interpretations currently being formulated to replace the orthodox interpretation. Chaos theory could be a candidate for bridging the gap between the determinism so dear to Einstein and the statistical interpretation of the Copenhagen School: For deterministic chaos is indeed deterministic; however, intrinsic limitations on precision in measuring initial conditions necessitate analyzing it statistically. Einstein and Bohr both could have been correct in their debates.

We claim that quantum collapse, as per the Copenhagen interpretation of quantum mechanics, follows naturally from the energetics of measurement. We argue that a realistic device generates an interaction energy that drives a random walk in Hilbert space and generates the probabilistic interpretation of Born.

This paper examines the three central theses of the Copenhagen interpretation of quantum mechanics. (1) Quantum mechanics must make use of concepts of classical physics; (2) descriptions which in classical physis are mutually exclusive are complementary descriptions of the same objects in quantum mechanics; (3) the uncertainty relations are natural laws which determine the limits of the application of classical concepts to micro-physical objects. Usually, these central theses of the Copenhagen interpretation are interpreted in a realistic-ontological manner. This paper criticizes this position, and, by means of a comparative investigation of the function of models in classical physics and quantum mechanics, proposes a model-theoretic interpretation. This interpretation lays special stress on the fictional status of models in quantum mechanics.

The general properties of signals permit a nonaxiomatic reconstruction of the quantum “probability” formalism independent of the standard Copenhagen interpretation of quantum mechanics. Performance standards are specified for candidate clock, signaller, and reflector devices, and it is shown that the resulting formalism forces identification of a “probability”- or “intensity”-like structure as the absolute square of an amplitude, the relative phases of amplitudes appearing explicitly in the “probability” composition law. Inequalities are produced which on one interpretation reduce to the Heisenberg relations, but it is pointed out that this reconstruction disqualifies position as an observable property of a signal.

Multi-dimensional Cantor sets in classical and quantum mechanics