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Logical Types in Quantum Mechanics

Andrew Soltau

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Abstract: Barbour shows that time does not exist in the physical world, and similar conclusions are reached by others such as Deutsch, Davies and Woodward. Every possible configuration of a physical environment simply exists in the universe. The system is objectively static. Observation, however, is an inherently transtemporal phenomenon, involving actual or effective change of the configuration, collapse. Since, in a static environment, all possible configurations exist, transtemporal reality is of the logical type of a movie. The frame of a movie film is of one logical type, an element of a set of frames, the movie, itself of a second logical type. In a static nocollapse universe, the configurations are of the first logical type, transtemporal reality of the second. To run, the movie requires iteration, a third logical type. Phenomenal consciousness is subjectively experienced as of this third logical type with regard to physical configurations. Everett's formulation clearly describes the transtemporal reality of an observer, which follows the physical in the linear dynamics, but departs from it on observation, giving rise to the the appearance of collapse, and the alternation of

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Abstract: Barbour shows that time does not exist in the physical world, and similar conclusions are reached by others such as Deutsch, Davies and Woodward. Every possible configuration of a physical environment simply exists in the universe. The system is objectively static. Observation, however, is an inherently transtemporal phenomenon, involving actual or effective change of the configuration, collapse. Since, in a static environment, all possible configurations exist, transtemporal reality is of the logical type of a movie. The frame of a movie film is of one logical type, an element of a set of frames, the movie, itself of a second logical type. In a static nocollapse universe, the configurations are of the first logical type, transtemporal reality of the second. To run, the movie requires iteration, a third logical type. Phenomenal consciousness is subjectively experienced as of this third logical type with regard to physical configurations. Everett's formulation clearly describes the transtemporal reality of an observer, which follows the physical in the linear dynamics, but departs from it on observation, giving rise to the the appearance of collapse, and the alternation of
Abstract: Everett demonstrates the appearance of collapse, within the context of the unitary linear dynamics. However, he does not state clearly how observers are to have determinate measurement records, hence 50 years of debate. This, however, is inherent. He defines the observer as the record of observations, which, naturally, is the record of correlations established with the physical environment. As in Rovelli's Relational Quantum Mechanics, the correlations record is the sole determinant of the effective physical environment, here the quantum mechanical frame of reference: due to multiple realisation of the functional identity of the observer, the physical environment is a simultaneity of all the physical environments in which it is instantiated, a 'universe superposition', in which only the environment correlated with the observer by observations is determinate. This effects a discrete and idiosyncratic physical environment for each version of an observer, in which determinate measurement records are recorded. Quantum mechanics is on this view fully relational, demonstrated as not only viable but necessary by Rovelli & Laudisa. The quantum mechanical frame of reference is Everett's 'Relative State', and on Tegmark's 'inside view', the time evolution follows the standard von NeumannDirac formulation. Thus observers get precisely the measurement records predicted by the standard formulation, but since objectively there is only the appearance of collapse, there is neither a measurement problem nor a disparity with relativity. The linear dynamics and the collapse dynamics are directly experienced, as the passage of time and the making of observations, respectively. 1
A non-relativistic quantum mechanical theory is proposed that describes the universe as a continuum of worlds whose mutual interference gives rise to quantum phenomena. A logical framework is introduced to properly deal with propositions about objects in a multiplicity of worlds. In this logical framework, the continuum of worlds is treated in analogy to the continuum of time points; both “time” and “world” are considered as mutually independent modes of existence. The theory combines elements of Bohmian mechanics and of Everett’s many-worlds interpretation; it has a clear ontology and a set of precisely defined postulates from where the predictions of standard quantum mechanics can be derived. Probability as given by the Born rule emerges as a consequence of insufficient knowledge of observers about which world it is that they live in. The theory describes a continuum of worlds rather than a single world or a discrete set of worlds, so it is similar in spirit to many-worlds interpretations based on Everett’s approach, without being actually reducible to these. In particular, there is no splitting of worlds, which is a typical feature of Everett-type theories. Altogether, the theory explains (1) the subjective occurrence of probabilities, (2) their quantitative value as given by the Born rule, and (3) the apparently random “collapse of the wavefunction” caused by the measurement, while still being an objectively deterministic theory.
The quasi-classical world of modern physics is a class of classical worlds superposed, literally a 'metaworld'. The measurement problem is itself evidence for the two different types of world, at the different levels of logical type. The enactment of linear dynamics in the classical worlds alters the class invariant, and thus induces physical collapse of the metaworld to a subclass. No new physics is required. The complex formalism that seems deeply counterintuitive is class arithmetic. Superposition and interference: properties of classes of worlds. Measurement: set-selection of a subclass compatible with the observed outcome. The uncertainty principle: incompatible set-selections. Fourier transforms: switching set-selection. Operators: enactment of set-selection. Decoherence: loss of coherence between worlds. Normalisation is required because all worlds are equally real, thus the probability of an outcome is essentially the proportion of worlds that instantiate it. True probability arises through phase, which is the degree to which the rest of each world aligns internally with this outcome. This is the second-logical-type phenomenon of interference: the degree to which worlds cohere in the classical actions that give rise to the outcome.This transforms the familiar structures of the formalism from postulates to inherent attributes of the class structure. The Born rule, unitary evolution, the complex structure of amplitudes, the probabilistic character of prediction, and the wave-mechanical form of the theory emerge as outputs of the ontology rather than independent axioms. The wavefunction is real. It is the ontological architecture of the metaworld and the reality of the quasi-classical world: this is the quantum realm. The classical world, quantum mechanics and relativistic physics are unified in logical types.
The nature and topology of time remains an open question in philosophy, both tensed and tenseless concepts of time appear to have merit. A concept of time including both kinds of time evolution of physical systems in quantum mechanics subsumes the properties of both notions. The linear dynamics defines the universe probabilistically throughout space-time, and can be seen as the definition of a block universe. The collapse dynamics is the time evolution of the linear dynamics, and is thus of different logical type to the linear dynamics. These two different kinds of time evolution are respectively tensed and tenseless. Ascribing tensed semantics to the collapse dynamics is problematic in the light of special relativity, but this difficulty does not apply to a relational quantum mechanics. In this context, while the linear dynamics is the time evolution of the universe objectively, the collapse dynamics is the time evolution of the universe subjectively, applying solely in the functional frame of reference of the observer.
A perspective on Everett's relative state formulation is proposed leading to a relational quantum mechanics. There are inevitably a large number of different versions of the universe in which a specific observer could exist, and in the universe of the unitary wave function they are all existing and coincident. If these different versions of the universe are superposed the result is a universe in which the superposition of all of the identical copies sums to a single observer. The effective universe in the functional frame of reference of this observer would be highly indeterminate but determinate where observed by this observer. This would naturally relativise the universe of the conventional view since each observer would inhabit an effective universe in which different aspects were determinate. Although the identity of the observer as a physical body does not readily fit this concept, it appears to apply inevitably to the functional identity of an observer as depicted by Everett. In this relational quantum mechanics a collapse dynamics applies only to the functional frame of reference of the observer and raises no incompatibility with the linear dynamics.
QBism explicitly takes the subjective view: probabilities of events are defined solely by past experiences, i.e. the record of observations. As shown by the authors (Fuchs et al, 2013), this: “... removes the paradoxes, conundra, and pseudo-problems that have plagued quantum foundations for the past nine decades”. It is criticised for its lack of ontology and anthropocentric nature. However, if Everett's (1957) formulation is taken at face value, exactly the features of QBism are the result, and the ontology is inherent. The anthropocentric nature of the solution is simply an indication that the quantum state is relative, as is central to Everett. Problems of measurement and locality do not arise. As described by Tegmark (1997), Everett defines the relationship between two fundamentally different views of the world, different types of frame of reference. The outside view is defined by the unitary wave function, the inside view by the record of observations. As Lockwood (1989) states, the preferred basis problem is fully resolved if the latter is taken as the basis. Taken literally, only that defined by the record of observations is determinate in the physical world of this inside view. The reality of QBism is the result. The same can be derived from first principles in the no-collapse universe. Taking the outside view to define all possible decoherent quasi-classical worlds, the physical world of the inside view is the superposition of all such worlds in which it is instantiated. In this quantum mechanical frame of reference only that observed is determinate. The quantum state as ontology is questioned because it defines only the linear dynamics and cannot account for collapse. No deeper reality is required, however: the inside view, and the collapse dynamics that operates in this type of frame of reference, are emergent properties of the system, and operate at a different level of logical type. The world operates as described by QBism, but is nonetheless defined by the quantum state.