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Alternatives in Quantum Theory

Richard C. Heyser

Published Jan 1, 1983• 0 citations
OpenAlex

Abstract

Quantum mechanical descriptions in terms of momentum and position are identified as alternatives under the condition of equal complex-vaulted Lebesgue square integrability. While this does not change any of the formal results obtained in quantum mechanics, it does shed a different interpretive light on the steps that lead up to these results. Instead of being independent, even in concept, momentum and position are identified as being the same thing, merely seen from different views. Neither is required to complement the descriptive capability of the other, since each forms a complete alternative in its own right. Apparent complementarity, as well as mutual indeterminacy of codescription, comes about whenever an attempt is made to overlap these two descriptions to form a third codescription which tries to maintain both the terminology and the net integration values of the separate alternatives. Noncommutation of associated operators is a direct consequence of the alternative nature of momentum and position, and the local to global character of the particular map between these alternatives leads to an apparent point-wave duality in their descriptions. It is suggested that challenges to the contemporary interpretation of quantum theory should begin with the assumptions that lead to its formulation involving alternative representations.

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Quantum mechanical descriptions in terms of momentum and position are identified as alternatives under the condition of equal complex-vaulted Lebesgue square integrability. While this does not change any of the formal results obtained in quantum mechanics, it does shed a different interpretive light on the steps that lead up to these results. Instead of being independent, even in concept, momentum and position are identified as being the same thing, merely seen from different views. Neither is required to complement the descriptive capability of the other, since each forms a complete alternative in its own right. Apparent complementarity, as well as mutual indeterminacy of codescription, comes about whenever an attempt is made to overlap these two descriptions to form a third codescription which tries to maintain both the terminology and the net integration values of the separate alternatives. Noncommutation of associated operators is a direct consequence of the alternative nature of momentum and position, and the local to global character of the particular map between these alternatives leads to an apparent point-wave duality in their descriptions. It is suggested that challenges to the contemporary interpretation of quantum theory should begin with the assumptions that lead to its formulation involving alternative representations.
In this thesis we shall demonstrate that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the formalism used entails a description that is non-local in that it requires all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary to specify quantum states completely. The remainder of the thesis, will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic field found by Susskind. It will be argued that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, with an interpretation in terms of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions make explicit the additional structure which is encoded more subtly in the non-local description. Additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory.
The ideas in Heisenberg's paper on quantum mechanics in 1925 mark the beginning not only of a new phase in modern physics, but also of Heisenberg's own philosophical journey. This thesis examines that journey between 1925 and the late 1950s by situating Heisenberg's philosophy of quantum mechanics in the context of his encounters with his contemporaries as well as the context of various strands of thought in the German-speaking world at the time. Heisenberg's early philosophical critique of the 'classical' viewpoint between 1925 and 1927 bears the decisive influence of Einstein's theory of relativity, more specifically, the positivism he saw as underpinning Einstein's emancipation from Newtonian physics. The positivist influence on Heisenberg's early attitude to quantum mechanics is evident in three ways: (a) his invocation of an observability principle in 1925 to justify the renunciation of the concept of the electron orbit, (b) an instrumentalist conception of understanding, which characterised Heisenberg's response to Schrodinger's demand for classical visualisation in space and time in 1926-7, and (c) the introduction of an operational definition of concepts such as position and velocity in 1927, in an attempt to replace the concepts of classical physics. But after discussions with Bohr and Einstein in 1926-7, Heisenberg soon recognised what we might term his `empiricist' viewpoint was problematic. In 1927 Heisenberg's thought undergoes a shift away from the `empiricist' viewpoint that had underpinned his early philosophy of quantum mechanics. The nature and scope of this transformation, which forms the central theme in this thesis, has, up until now, been poorly understood and often completely neglected. Through his discussions with Bohr, Heisenberg came to the realisation that despite their limitations, classical concepts were conditions for the possibility of the description of all experience. This marked the abandonment of his earlier attempt to replace classical concepts with quantum concepts. The recognition of the primacy of classical language forms the point of departure for much of Heisenberg's later thought, which brought him into contact with the attempts in the German-speaking world in the 1920s to reconstruct Kantian epistemology. By the mid-1930s, Heisenberg advocated a 'pragmatic transformation' of Kantian philosophy, in which classical concepts were held to be a priori in the sense that they remained the conditions for the possibility of experience, but were no longer held to be necessary or universal in a strict Kantian sense. After 1940 Heisenberg saw the paradoxes of quantum mechanics under the aegis of what can be termed a 'transcendental conception of language', according to which language is not a mere tool, but actively shapes, gives form, and objectifies, our 'reality'. The limits of a classical 'description' in quantum mechanics therefore came to signify for Heisenberg, the limits of 'objective reality'. While Bohr exerted perhaps the most important philosophical influence on Heisenberg, their intellectual relationship was characterised by disagreement and misunderstanding. This is most strikingly displayed in their respective views on wave-particle duality and complementarily. While after 1927 Heisenberg accepted Bohr's basic insight that our knowledge of the quantum world is mediated through classical language, he did not share Rohr's interpretation of complementarity. While Heisenberg certainly used terms such as 'complementarity' and wave-particle duality' in his writings, a close reading reveals that these terms had very different meanings for the two physicists. This is particularly evident in the contrast between Heisenberg's notion of wave-particle equivalence and Bohr's idea of complementarity. In bringing to light these divergences between Bohr and Heisenberg, this thesis lends further weight to the view - already advocated by scholars such as John Hendry and Mara Beller - that the so-called Copenhagen interpretation of quantum mechanics should not be thought of as a unified philosophical position, but actually comprises a number of different strands.
Niels Bohr's interpretation of quantum mechanics is often cast as positivist and sometimes explicitly claimed to be influenced by logical positivists due to some similarities in their thinking. While it is certainly the case that some logical positivists attempted to recruit Bohr, this paper argues that Bohr had interests of his own in the logical positivists. Bohr's interpretation of quantum mechanics focuses on observation, the use of classical concepts in quantum mechanics, and indeterminacy of quantum processes as opposed to uncertainty of measurement. His view thereby shares some common ground with the logical positivists' views on verification, the observation language, and anti-metaphysics. But Bohr also emphasized complementarity: that certain pairs of concepts - such as position and momentum - are mutually exclusive in quantum mechanics since they, according to Bohr, are only meaningful relative to different experimental arrangements. Bohr believed that complementary brought a general epistemological lesson for all of science that an objective description of nature is not separable from the observational and experimental conditions under which we explore nature. Motivated by the common ground between himself and logical positivism, Bohr tried to persuade the logical positivists and Carnap in particular to adopt and champion complementarity as well as part of their unity of science program. Though his efforts ultimately proved in vain, Bohr's attempts to influence logical positivism disprove the claim that his engagement with them was reluctant and purposefully limited.
With this second part we shall start more formal developments that allow us to improve our understanding of quantum theory. In particular, we shall deal with the two most basic quantum mechanical observables, position and momentum, which are crucial for understanding the state and the dynamics of a quantum system. Different from classical mechanics, quantum mechanics tells us that it is not possible to measure both of the position and momentum simultaneously. This is a consequence of the non-commutativity of these two observables. Moreover, in this chapter we shall learn the basic formalism of integration and differentiation, a fundamental mathematical tool if one is really interested in understanding quantummechanics.