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arXiv:2307.01061

Quantizing the Quantum Uncertainty

Etera R. Livine

Published Jul 3, 2023• 0 citations
arXivHTMLPDF

Abstract

The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the $\textrm{sl}_{2}$ Lie algebra formed by the operators $\hat{x}^{2}$, $\hat{p}^{2}$ and $\widehat{xp}$. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?

DOI
10.48550/arxiv.2307.01061

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The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the sl2\textrm{sl}_{2}sl2​ Lie algebra formed by the operators x^2\hat{x}^{2}x^2, p^2\hat{p}^{2}p^​2 and xp^\widehat{xp}xp​. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?
The Heisenberg uncertainty postulates are a convenient way to describe the qualitative properties of quantum mechanics. The response from Heisenberg’s uncertainty principle is that one can create states that require a certain amount of momentum and position localization. The numerical quantities used by the old Newtonian mechanics, position, momentum, and energy are mere “shadows” of what nature describes: operators. The energy value contradicts the Newtonian notion of any energy level being necessary for a harmonic oscillator. The basic postulates of classical statistical mechanics are now applied to quantum statistical mechanics. In classical mechanics, the variables characterize a microstate that evolves according to the Hamiltonian equations. If the standard deviation relative to what we are interested in is very high, it is probably safe to use the expected value as a classical value. A quantum field theory is a relativistic invariant quantum theory where there is a particular form of quantity operators connected with the classical fields.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.
The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction).
Quantum uncertainty lies at the heart of quantum mechanics, fundamentally distinguishing it from classical physics. Traditionally quantified through variance-based measures, these approaches often rely on the expectation values of observables, providing a macroscopic statistical view rather than an intrinsic characterization of the wavefunction's inherent indeterminacy. This paper introduces a novel commutator-induced metric designed to directly quantify quantum uncertainty within the wavefunction itself, independent of specific measurement outcomes or a pair of conjugate variables. By leveraging the fundamental non-commutativity of quantum operators, this metric provides a more fundamental and intrinsic measure of the delocalization and inherent indeterminacy encoded in the quantum state. We define this metric based on the expectation value of an operator derived from the self-commutator of the wavefunction's defining operators, mapping the non-commutativity inherent in the Hilbert space onto a scalar value that reflects the degree of quantum uncertainty. This approach offers a new perspective on interpreting the uncertainty principle, providing a generalized framework that extends beyond canonical conjugate pairs and potentially offering insights into the structure of quantum information. The paper details the theoretical construction of this metric, explores its mathematical properties, and discusses its potential applications in quantum information theory, quantum measurement, and the foundational understanding of quantum mechanics.
A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.