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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Analysis of the Experimental Meaning of Coherent Superposition and the Nonexistence of Superselection Rules

Ronald Mirman

Published Jun 15, 1970• 26 citations
Publisher

Abstract

The experimental meaning of coherent superposition of state vectors is analyzed. It is shown that the involvement of the measuring instrument is of basic importance and that it is erroneous to study superposition without it. From this analysis it is found that there are no such things as superselection rules. The meaning of the sign change of spinors under $2\ensuremath{\pi}$ rotation is considered, and it is found to have experimental consequences. The nonmeasurability of the relative parity of states with different $z$ components of angular momentum is shown. An appendix discusses the application of some of the considerations of the paper to decaying states and elementary particles.

DOI
10.1103/physrevd.1.3349

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The experimental meaning of coherent superposition of state vectors is analyzed. It is shown that the involvement of the measuring instrument is of basic importance and that it is erroneous to study superposition without it. From this analysis it is found that there are no such things as superselection rules. The meaning of the sign change of spinors under 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ rotation is considered, and it is found to have experimental consequences. The nonmeasurability of the relative parity of states with different zzz components of angular momentum is shown. An appendix discusses the application of some of the considerations of the paper to decaying states and elementary particles.
The superposition principle is a very basic ingredient of quantum theory. What may come as a surprise to many students, and even to many practitioners of the quantum craft, is tha superposition has limitations imposed by certain requirements of the theory. The discussion of such limitations arising from the so-called superselection rules is the main purpose of this paper. Some of their principal consequences are also discussed. The univalence, mass and particle number superselection rules of non-relativistic quantum mechanics are also derived using rather simple methods.
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.
We discuss the possibility of observing quantum nonlocality using the so-called mode entanglement, analyzing the differences between different types of particles in this context. We first discuss the role of coherent states in such experiments, and we comment on the existence of coherent states in nature. The discussion of coherent states naturally raises questions about the role of particle statistics in this problem. Although the Pauli exclusion principle precludes coherent states with a large number of fermionic particles, we find that a large number of fermionic coherent states, each containing at most one particle, can be used to achieve the same effect as a bosonic coherent state for the purposes of this problem. The discussion of superselection rules arises naturally in this context, because their applicability to a given situation prohibits the use of coherent states. This limitation particularly affects the scenario that we propose for detecting the mode entanglement of fermionic particles.