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A Study of Modifications to Quantum Mechanics

Zachary Lewis

Published Mar 5, 2013• 0 citations
OpenAlex

Abstract

In this work, the consequences of several modifications to quantum mechanics are examined. These modifications, motivated by string theory, fall into two categories: ones in which the canonical commutation relations between position and momentum are deformed and ones in which the space of states used are vector spaces over Galois fields instead of complex Hilbert spaces. The particular deformation of the canonical commutation relations used leads to a minimum value of the uncertainty in position which is interpreted as a minimum length scale. Both harmonic and anharmonic oscillators are studied in this framework with normalizable, positive energy eigenstates found in both cases. The quantum uncertainty relations and classical counterparts to these states are discussed. Creating modified quantum theories by replacing the Hilbert spaces of canonical quantum mechanics with vector spaces defined over several finite, Galois fields is accomplished. Correlation functions are calculated in these theories and the maximum values are shown to not behave as would be expected by the standard, Bell-like, bounding inequality theorems. The interpretations and implications of these theories are discussed.

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In this work, the consequences of several modifications to quantum mechanics are examined. These modifications, motivated by string theory, fall into two categories: ones in which the canonical commutation relations between position and momentum are deformed and ones in which the space of states used are vector spaces over Galois fields instead of complex Hilbert spaces. The particular deformation of the canonical commutation relations used leads to a minimum value of the uncertainty in position which is interpreted as a minimum length scale. Both harmonic and anharmonic oscillators are studied in this framework with normalizable, positive energy eigenstates found in both cases. The quantum uncertainty relations and classical counterparts to these states are discussed. Creating modified quantum theories by replacing the Hilbert spaces of canonical quantum mechanics with vector spaces defined over several finite, Galois fields is accomplished. Correlation functions are calculated in these theories and the maximum values are shown to not behave as would be expected by the standard, Bell-like, bounding inequality theorems. The interpretations and implications of these theories are discussed.
Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.
Several quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose their physical meaning. In quantum mechanics, the insurgence of such a minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on position measurements has a lower bound. The value of the minimal length is not predicted by theories and must be estimated experimentally. In this paper, we address the quantum bound to the estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra induced by the deformed commutation relations.
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.
Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra of the Hilbert subspace.
We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by $[\stackrel{^}{x},\stackrel{^}{p}]=i\ensuremath{\hbar}(1+\ensuremath{\beta}{\stackrel{^}{p}}^{2})$. This deformed commutation relation leads to the minimal length uncertainty relation $\ensuremath{\Delta}x\ensuremath{\ge}(\ensuremath{\hbar}/2)(1/\ensuremath{\Delta}p+\ensuremath{\beta}\ensuremath{\Delta}p)$, which implies that $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ at small $\ensuremath{\Delta}p$ while $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$ at large $\ensuremath{\Delta}p$. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0m>0m>0), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 (2002), only populate the $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ branch. The other branch, $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$, is found to be populated by the energy eigenstates of the ``inverted'' harmonic oscillator (m<0m<0m<0). The Hilbert space in the inverted case admits an infinite ladder of positive energy eigenstates provided that $\ensuremath{\Delta}{x}_{\mathrm{min}}=\ensuremath{\hbar}\sqrt{\ensuremath{\beta}}>\sqrt{2}[{\ensuremath{\hbar}}^{2}/k|m|{]}^{1/4}$. Correspondence with the classical limit is also discussed.
A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.