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arXiv:hep-th/9411173 · Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

Observation and superselection in quantum mechanics

Nicolaas P. Landsman

Published Apr 1, 1995• 37 citations
arXivHTMLPDF
DOI
10.1016/1355-2198(95)00001-a

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In [N. Friis, New J. Phys. 18, 033014 (2016)] the non-relativistic description of fermions is considered and in particular the role of the parity superselection rule in relation to the characterization of entanglement. An argument based on the spin-statistics connection is presented as a physical motivation for the parity superselection rule. Since the spin statistics connection was derived in the context of relativistic quantum mechanics it is argued that the inclusion of the parity superselection rule is motivated by Lorentz invariance. Based on this it is further argued that fermionic Quantum Information theory and the theory of special relativity are conceptually inseparable. In this comment a different, and well known, motivation for the parity superselection rule is given that does not rely on arguments from relativistic quantum mechanics, but instead uses the assumptions that the laws of physics are the same for different observers and the no-signalling principle.
We show that five foundational features of quantum mechanics—complementarity, the observability of probabilities, superselection rules, the Fresnel–Arago laws, and the location of the quantum classical boundary—emerge from a single algebraic structure: the Z2 Galois grading on the representation ring of the alternating group A5. The irreducible decomposition of the unique transitive A5-permutation representation of degree 12, 12 = 1 ⊕ 3 ⊕ 5 ⊕ 3′, admits a grading induced by σ : √5 →−√5 that partitions representations into observable (V+, dim 6) and hidden (V−, dim 6) sectors. We prove that A5 is the minimal finite simple group admitting such a partition. TheHadamard product rules V− ⊙ V− ⊆ V+ and V+ ⊙V− ⊆ V− then yield: (i) complementarity as a selection rule, (ii) the observability of |ψ|2 without postulate, (iii) a Galois superselection rule for bidding cross-sector interference, (iv) the Fresnel–Arago laws (1811) as a 200-year-old experimental confirmation of this superselection rule, and (v) a per-sector action scale ℏρ derived from the graph Laplacian, with ℏτ5 = 0—interference fringes are algebraically classical. The double slit is not a quantum mystery. The paths are the mystery.