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Axioms for non-relativistic quantum mechanics

1961; Mathematical Sciences Publishers; Volume: 11; Issue: 3 Linguagem: Inglês

10.2140/pjm.1961.11.1151

1945-5844

Neal Zierler,

Quantum Mechanics and Applications

ZIERLERplausible properties imply all the others (Theorem 1.1).This work is a modification of part of a thesis submitted to the Department of Mathematics of Harvard University in partial fulfillment of the requirements for the degree of Doctor of Philosophy.1. Events and states: preliminaries• Let P be a partly ordered set with least and greatest elements 0 and 1 respectively.If the greatest lower bound or least upper bound of elements a and b of P exists in P it is denoted ab or a V b respectively.Let a->a' be an orthocomplementation in P; that is, for each a e P, a' e P and(1, a'a and a V a! exist and equal 0 and 1 respectively.Two elements a and b of P are said to be orthogonal, a _L δ, if and only if α ^ δ'.Clearly a J_ b is equivalent to b J_ α.If Q is a set of pairwise orthogonal elements of P we shall say, for short, that Q is orthogonal.It is easy to see that De Morgan's law holds in P: (ab)' = α' V V in the sense that if either ab or α/ V V exists, so does the other and the equality holds.We assume that P satisfies (LI) If {a lf a 29 •••} is orthogonal, then \fa { exists in P.It follows readily that a variety of sups and infs exists in P: e.g., δ'c', δα' and baConsider the following three properties for P. (W ) a ^ b implies b = ba' V a, (Wl) α ^ b and 6α' = 0 imply a = b, (W2) α ^ c and & J_ c imply (a V δ)c = α. 1 LEMMA 1.1.// (W) holds then a A_b implies b = (α V δ)α'.Proo/.α ^ &' so 6' = b'a f V α by (W) and 6 = (&W V a)' = (a V δ)α' LEMMA 1.2.J/ (W) holds and a, b and c are pairwise orthogonal, then (a V &)(α Vc) = α and (a V δ)(α V c)' = b.Proof, b ^ α', 6 ^ c' imply b ^ α'c' so α'c' = a'c'V V δ.Then a = α(α V c V δ) = (α V δ)δ'(α V c V δ) by Lemma 1.1 = (α V δ)(α'c'δ f V δ)' = (α V b)(a'c')' -(αV 6)(α V c).