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Comprimento mínimo em mecânica quântica via modificação da álgebra de Heisenberg

Gláuber C. Dorsch

Published May 31, 2010• 0 citations
OpenAlex

Abstract

Although gravity could not be quantized yet, there are good theoretical evidences that the unification of General Relativity and Quantum Mechanics should lead to the existence of a minimal observable length, a length scale below which the very notion of length looses meaning. Such effect proves to be worth some further analysis, for it acts as a natural regulator parameter, thus avoiding the divergences that plague Quantum Field Theories. In order to investigate some of the physical consequences of the existence of this peculiar effect, we propose to include it in the framework of Quantum Mechanics by modifying the Heisenberg algebra (i.e., the commutation relation of position and momentum operators, Xˆ and Pˆ), so that a minimum non-zero value for the uncertainty ?x emerges, which, being a limitation to the localizability of particles, acts as a minimal length. The Hilbert space of the theory must be modified accordingly. As we will see, the changes are not merely quantitative. On the contrary, our main result is that the familiar concept of “position measurement” must be reformulated, as well as other concepts related to it. We present a proposal for such reformulation.

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The different proposals for quantum gravity, almost in their entirety, lead to the existence of a minimum length. The introduction of a minimum length causes radical changes in the mathematical description, as well as the physical concepts involved in a theory. In a quantum theory the implementation of a minimum length scenario can be achieved by imposing a minimum uncertainty on the position, obtained through a generalization of the Heisenberg uncertainty principle (GUP). There are different proposals for GUPs. As a result of the minimal uncertainty in position, the operator’s eigenstates are not physical states, which makes the use of position space representation inappropriate. The most natural alternative is to use the moment-space representation. However, to retrieve the information about the position, the quasi-position representation obtained by projecting the state vector onto the states of maximum location is used. The objective of this work is, then, to make a functional analysis of these representation spaces for the main proposals of GUPs, in particular to determine the representations of the position and moment operators in the quasi-position space.

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Although gravity could not be quantized yet, there are good theoretical evidences that the unification of General Relativity and Quantum Mechanics should lead to the existence of a minimal observable length, a length scale below which the very notion of length looses meaning. Such effect proves to be worth some further analysis, for it acts as a natural regulator parameter, thus avoiding the divergences that plague Quantum Field Theories. In order to investigate some of the physical consequences of the existence of this peculiar effect, we propose to include it in the framework of Quantum Mechanics by modifying the Heisenberg algebra (i.e., the commutation relation of position and momentum operators, Xˆ and Pˆ), so that a minimum non-zero value for the uncertainty ?x emerges, which, being a limitation to the localizability of particles, acts as a minimal length. The Hilbert space of the theory must be modified accordingly. As we will see, the changes are not merely quantitative. On the contrary, our main result is that the familiar concept of “position measurement” must be reformulated, as well as other concepts related to it. We present a proposal for such reformulation.
Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations that are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows us to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min and Shannon entropies. We compute the corresponding entropic minimal lengths and find that the minimal length in terms of min entropy is exactly 1 bit.
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.
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 construct the states of maximal localization taking into account a modification of the commutation relation between position and momentum operators to all orders of the minimum length parameter. To first-order, the algebra we use reproduces the one proposed by Kempf, Mangano and Mann. It is emphasized that a minimal length acts as a natural regulator for the theory, thus eliminating the otherwise ever appearing infinities. So, we use our results to calculate the first correction to the Casimir effect due to the minimal length. We also discuss some of the physical consequences of the existence of a minimal length, culminating in a proposal to reformulate the very concept of "position measurement."
Candidate theories of quantum gravity predict the presence of a minimal measurable length at high energies. Such feature is in contrast with the Heisenberg Uncertainty Principle. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. The effects of such modification are expected to be relevant at large energies/small lengths. One first consequence is that position eigenstates are not included in such models due to the presence of a minimal uncertainty in position. Furthermore, depending on the particular modification of the position–momentum commutator, when such models are considered from momentum space, the position operator is changed, and a measure factor appears to let the position operator be self-adjoint. As a consequence, the (quasi-)position representation acquires numerous issues. For example, the position operator is no longer a multiplicative operator, and the momentum of a free particle does not correspond directly to its wave number. Here, we will review such issues, clarifying aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, we will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems.
The different proposals for quantum gravity, almost in their entirety, lead to the existence of a minimum length. The introduction of a minimum length causes radical changes in the mathematical description, as well as the physical concepts involved in a theory. In a quantum theory the implementation of a minimum length scenario can be achieved by imposing a minimum uncertainty on the position, obtained through a generalization of the Heisenberg uncertainty principle (GUP). There are different proposals for GUPs. As a result of the minimal uncertainty in position, the operator’s eigenstates are not physical states, which makes the use of position space representation inappropriate. The most natural alternative is to use the moment-space representation. However, to retrieve the information about the position, the quasi-position representation obtained by projecting the state vector onto the states of maximum location is used. The objective of this work is, then, to make a functional analysis of these representation spaces for the main proposals of GUPs, in particular to determine the representations of the position and moment operators in the quasi-position space.