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Journal of International Crisis and Risk Communication Research

Topological Generalizations of the Heisenberg Uncertainty Relation

Sohang Gandhi

Published Jan 1, 2006• 0 citations

Abstract

It is well known that the standard canonical uncertainty relation does not apply to the angular variable ? and its conjugate LZ. That is, the relation ? ø ? L Z > h/2 is false. The break down of the result has to do with difference in topology between the line and the circle. It is thus desirable to generalize the standard uncertainty relation topologically and find satisfactory results for the non-Euclidean spaces. This problem is intimately related to the issue of finding a consistent definition for quantum mechanics on "curved spaces". Just as the Heisenberg uncertainty relation was pivotal in understanding the basic structure of standard quantum mechanics, a solution to this problem should shine some light onto the proper conduct of quantum mechanics on general topological spaces. In this study we explore in detail how the standard uncertainty relation may breakdown. We also address the importance of topological considerations in quantum mechanics in general - we shall show how a change in topological character can change the nature of the quantum mechanics for a system and how the consideration of the topology of a system can greatly organize the solution of a problem and in some cases even be necessary for a. full understanding of the problem. We then discuss the derivation of satisfactory uncertainty relations for the compact, homogeneous spaces of the circle, the n-torus and the n-sphere. Finally, we draw out any implications to the issue of properly defining quantum mechanics on the non- Euclidean spaces.

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It is well known that the standard canonical uncertainty relation does not apply to the angular variable ? and its conjugate LZ. That is, the relation ? ø ? L Z > h/2 is false. The break down of the result has to do with difference in topology between the line and the circle. It is thus desirable to generalize the standard uncertainty relation topologically and find satisfactory results for the non-Euclidean spaces. This problem is intimately related to the issue of finding a consistent definition for quantum mechanics on "curved spaces". Just as the Heisenberg uncertainty relation was pivotal in understanding the basic structure of standard quantum mechanics, a solution to this problem should shine some light onto the proper conduct of quantum mechanics on general topological spaces. In this study we explore in detail how the standard uncertainty relation may breakdown. We also address the importance of topological considerations in quantum mechanics in general - we shall show how a change in topological character can change the nature of the quantum mechanics for a system and how the consideration of the topology of a system can greatly organize the solution of a problem and in some cases even be necessary for a. full understanding of the problem. We then discuss the derivation of satisfactory uncertainty relations for the compact, homogeneous spaces of the circle, the n-torus and the n-sphere. Finally, we draw out any implications to the issue of properly defining quantum mechanics on the non- Euclidean spaces.

The proof of the Heisenberg uncertainty relation is modified to produce two improvements: (a) The resulting inequality is stronger because it includes the covariance between the two observables, and (b) the proof lifts certain restrictions on the state to which the relation is applied, increasing its generality. The restrictions necessary for the standard inequality to apply are not widely known, and they are discussed in detail. The classical analog of the Heisenberg relation is also derived, and the two are compared. Finally, the modified relation is used to address the apparent paradox that eigenfunctions of the z component of angular momentum Lz do not satisfy the φ–Lz Heisenberg relation; the resolution is that the restrictions mentioned above make the usual inequality inapplicable to these states. The modified relation does apply, however, and it is shown to be consistent with explicit calculations.

The disputed question of uncertainty relations (UR) on a circle is regarded as a particular element of a more general problem which refers to the quantum description of angular observables

L_z

and

ϕ

. The improvised

L_z-ϕ

UR are found to be affected by unsourmontable shortcomings. Also in contradiction with a largely accepted belief it is proved that the usual procedures of quantum mechanics are accurately applicable for the

L_z-ϕ

pair. The applicability regards both the known circular motions and the less known non-circular rotational motions. The presented facts contribute as arguments to the following indubitable conclusions: (i) the traditional interpretation of UR must be denied as an incorrect doctrine, (ii) for a natural physical consideration of the the

L_z-ϕ

pair the results from the usual quantum mechanics are sufficient while the improvised

L_z-ϕ

UR must be rejected as senseless formulas and (iii) the descriptions of quantum measurements have to be done in a framework which is distinct and additional in respect with the usual quantum mechanics.

We consider the uncertainty relation between position and momentum of a particle on $ S^1 $ (a circle). Since $ S^1 $ is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on $ S^D $ (a

D

-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.

We consider the uncertainty relation between position and momentum of a particle on S1 (a circle). Since S1 is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on SD (a D-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.

Abstract Real Hilbert space is a Euclidean space of infinite dimensionality. Complex Hilbert space is less amenable to elementary geometrical intuitions, but these do serve to give structure to the present argument. The familiar process of finding, relative to a vector x, the expectation Ā of a self-adjoint linear operator A, and then using the deviation A — Ā to obtain a dispersion ΔA, may be interpreted as a projection leading to a unit vector e orthogonal to x. If we have n self-adjoint operators Ar, we thus form n unit vectors er orthogonal to x. From a basic property of Hilbert space, the vector c1e1 + c2e2 + ... + cnen has a non-negative norm for arbitrary complex coefficients. This demands the positive-indefinite character of a real quadratic form in 2n variables; equivalently, all its eigenvalues must be non-negative. This leads to inequalities on the dispersions ΔAr, the expectations of the commutators i[Ar, As] and other expectations. For n = 2, this process leads to Schrödinger’s improvement of the Heisenberg inequality. The case n = 3 is here worked out explicitly.