The Grothedieck bound formalism is studied using “rescaling transformations,” in the context of a single quantum system. The rescaling transformations enlarge the set of unitary transformations (which apply to isolated systems), with transformations that change not only the phase but also the absolute value of the wavefunction, and can be linked to irreversible phenomena (e.g., quantum tunneling, damping and amplification, etc). A special case of rescaling transformations are the dequantisation transformations, which map a Hilbert space formalism into a formalism of scalars. The Grothendieck formalism considers a “classical” quadratic form which takes values less than 1, and the corresponding “quantum” quadratic form which takes values greater than 1, up to the complex Grothendieck constant kG. It is shown that can be expressed as the trace of the product of θ with two rescaling matrices, and can be expressed as the trace of the product of θ with two dequantisation matrices. Values of in the “ultra-quantum” region (1, kG) are very important, because this region is classically forbidden [ cannot take values in it]. An example with is given, which is related to phenomena where classically isolated by high potentials regions of space, communicate through quantum tunneling. Other examples show that “ultra-quantumness” according to the Grothendieck formalism , is different from quantumness according to other criteria (like quantum interference or the uncertainty principle).
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September 2024
Research Article|
September 11 2024
Rescaling transformations and the Grothendieck bound formalism in a single quantum system
Special Collection:
Mathematical Methods in Physics ICMMP23
A. Vourdas
A. Vourdas
a)
(Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing)
Department of Computer Science, University of Bradford
, Bradford BD7 1DP, United Kingdom
a)Author to whom correspondence should be addressed: a.vourdas@bradford.ac.uk
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a)Author to whom correspondence should be addressed: a.vourdas@bradford.ac.uk
J. Math. Phys. 65, 092101 (2024)
Article history
Received:
January 31 2024
Accepted:
August 15 2024
Citation
A. Vourdas; Rescaling transformations and the Grothendieck bound formalism in a single quantum system. J. Math. Phys. 1 September 2024; 65 (9): 092101. https://doi.org/10.1063/5.0201690
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