A deterministic interpretation of the commutation and uncertainty relations of quantum theory and a redefinition of Planck’s constant as a coupling condition

Barrett, T. W.

doi:10.1007/bf02894686

Cited by 9 publications

(2 citation statements)

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“…A trajectory interpretation of quantum mechanics, or of individual particles in space-time, is a deterministic interpretation. Implied in this interpretation is the notion that the uncertainty relations represent dispersion conditions and that there is no restriction on the simultaneous physical reality of variables associated with noncommutating operators [ 1-41. In agreement with this point of view, and commencing with a consideration of the potential underlying the trajectory, a deterministic account can be made of the dynamics underlying (i) the Schrodinger equation [5], (ii) the commutation relations of quantum theory and Planck's constant [6], (iii) quantum mechanical exchange integrals [7], and (iv) the S-matrix analysis of quantum electrodynamics [8]. This stochastic analysis requires a four-parameter quantum and Thom's topological methods [9] are used in demonstrating that the Schrodinger equation describes a mapping singularity.…”

Section: Introductionmentioning

confidence: 99%

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The Transition Point Between Classical and Quantum Mechanical Systems Deterministically Defined

Barrett¹

1987

Phys. Scr.

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From a deterministic point of view of quantum theory the Schrodinger equation, considered as an equation of motion, is defined within an area of instability formed by two potential minima. This view can be related to the renormalization group theory of critical phenomena, and the location of two nontrivial fixed points can be defined with respect to those two potential minima.The renormalization group theory parameter: d, the dimensionality of space, defines the transition between classical and quantum theory. For a system in which d 2 4, classical mechanics applies, and for d < 4, quantum mechanics applies. Whereas the classical potential defining an equation of motion has one potential minimum (at the Gaussian fixed point), the quantum potential defining an equation of motion has two potential minima (at the nontrivial fixed points).An analogy exists between systems based on wave mechanics and those based on lattice systems. The underlying dynamics of that potential, the minimization of which results in an equation formally equivalent to the Schrodinger equation, can, in this formal analogy, be identified with the underlying dynamics of the exchange energy at the vertex of a lattice. There is thus a mathematical global similarity underlying the behavior of these two distinct theoretical frameworks which describes both. The principal and angular momentum quantum numbers of electron systems in this analogy are equivalent to the dimensionality of space, d, and the dimensionality of spin space, n, in the general theory of critical phenomena.The general theory describes the phase transitions of electron (wave mechanical) systems, and also a "Mendeleev's table" based on the d and n parameters constructed for those electron systems, in analogy with such a table for lattice systems.

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Section: Introductionmentioning

confidence: 99%

“…This stochastic analysis requires a four-parameter quantum and Thom's topological methods [9] are used in demonstrating that the Schrodinger equation describes a mapping singularity. When described in four-parameter form, quantum mechanical systems behave in a way identical to the hypercomplex system called quaternions by Hamilton [6].…”

Section: Introductionmentioning

confidence: 99%