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arXiv:2203.04827 · Foundations of Physics

Three-Space from Quantum Mechanics

László B. Szabados

Published Sep 5, 2022• 4 citations
arXivHTMLPDF

Abstract

Abstract The spin geometry theorem of Penrose is extended from SU(2) to E(3) (Euclidean) invariant elementary quantum mechanical systems. Using the natural decomposition of the total angular momentum into its spin and orbital parts, the distance between the centre-of-mass lines of the elementary subsystems of a classical composite system can be recovered from their relative orbital angular momenta by E(3)-invariant classical observables. Motivated by this observation, an expression for the ‘empirical distance’ between the elementary subsystems of a composite quantum mechanical system, given in terms of E(3)-invariant quantum observables, is suggested. It is shown that, in the classical limit, this expression reproduces the a priori Euclidean distance between the subsystems, though at the quantum level it has a discrete character. ‘Empirical’ angles and 3-volume elements are also considered.

DOI
10.1007/s10701-022-00617-2

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Abstract The spin geometry theorem of Penrose is extended from SU(2) to E(3) (Euclidean) invariant elementary quantum mechanical systems. Using the natural decomposition of the total angular momentum into its spin and orbital parts, the distance between the centre-of-mass lines of the elementary subsystems of a classical composite system can be recovered from their relative orbital angular momenta by E(3)-invariant classical observables. Motivated by this observation, an expression for the ‘empirical distance’ between the elementary subsystems of a composite quantum mechanical system, given in terms of E(3)-invariant quantum observables, is suggested. It is shown that, in the classical limit, this expression reproduces the a priori Euclidean distance between the subsystems, though at the quantum level it has a discrete character. ‘Empirical’ angles and 3-volume elements are also considered.
Abstract Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.
Complex techniques of general relativity are used to determine all the states in two- and three-dimensional momentum spaces in which the equality holds in uncertainty relations for non-commuting basic observables of Euclidean invariant elementary quantum mechanical systems, even with non-zero intrinsic spin. It is shown that while there is a 1-parameter family of such states for any two components of the angular momentum vector operator with any angle between them, such states exist for a component of the linear and angular momenta only if these components are orthogonal to each other, and hence, the problem is reduced to the two-dimensional Euclidean invariant case. We also show that analogous states exist for a component of the linear momentum and of the center-of-mass vector only if the angle between them is zero or an acute angle. No such state (represented by a square integrable and differentiable wave function) can exist for any pair of components of the center-of-mass vector operator. Therefore, the existence of such states depends not only on the Lie algebra but on the choice of its generators as well.
Quantum systems whose states are tightly distributed among several invariant subspaces (variable spin systems) can be described in terms of distributions in a four-dimensional phase-space T∗S2 in the limit of large average angular momentum. The cotangent bundle T∗S2 is also the classical manifold for systems with E(3) symmetry group with appropriately fixed Casimir operators. This allows us to employ the asymptotic form of the star-product proper for variable (integer) spin systems to develop a deformation quantization scheme for a particle moving on the two-dimensional sphere, whose observables are elements of e(3) algebra and the corresponding phase-space is T∗S2. We show that the standard commutation relations of the e(3) algebra are recovered from the corresponding classical Poisson brackets and the explicit expressions for the eigenvalues and eigenfunctions of some quantized classical observables (such as the angular momentum operators and their squares) are obtained.
The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator L^z=ℏı (x∂∂y−y∂∂x){\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )L^z​=ℏ​(x∂y∂​−y∂x∂​) is not that it is the operatorfor the zzz-component LzL_{z}Lz​ of the angular momentum L{\mathbf{L}}L, but rather the expression of the operator for the angular momentum L{\mathbf{L}}L when it is aligned with the zzz-axis. Hence what we are used to note (erroneously) as L^z{\hat{L}}_{z}L^z​ is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.
In the operational approach to general probabilistic theories one distinguishes two spaces, the state space of the "elementary systems" and the physical space in which "laboratory devices" are embedded. Each of those spaces has its own dimension- the minimal number of real parameters (coordinates) needed to specify the state of system or a point within the physical space. Within an operational framework to a physical theory, the two dimensions coincide in a natural way under the following "closeness" requirement: the dynamics of a single elementary system can be generated by the invariant interaction between the system and the "macroscopic transformation device" that itself is described from within the theory in the macroscopic (classical) limit. Quantum mechanics fulfils this requirement since an arbitrary unitary transformation of an elementary system (spin-1/2 or qubit) can be generated by the pairwise invariant interaction between the spin and the constituents of a large coherent state ("classical magnetic field"). Both the spin state space and the "classical field" are then embedded in the Euclidean three-dimensional space. Can we have a general probabilistic theory, other than quantum theory, in which the elementary system ("generalized spin") and the "classical fields" generating its dynamics are embedded in a higher-dimensional physical space? We show that as long as the interaction is pairwise, this is impossible, and quantum mechanics and the three-dimensional space remain the only solution. However, having multi-particle interactions and a generalized notion of "classical field" may open up such a possibility.