Continuing the development [4] of an aspect of the approach to the axiomatization of quantum mechanics of G. W. Mackey [3], we consider here the real linear space X of signed measures on the set P of events generated by the states, and the set F0 of linear functionals on X which are induced in a natural way by the bounded observables. A necessary and sufficient condition for two events to be simultaneously measurable is found in terms of the order structure of F0, with the following consequence: if F0 is a lattice, P is deterministic. At the opposite extreme, F0 is said to be an anti-lattice1 if the greatest lower bound exists only for comparable pairs of its elements and we show in this case that the center of P is trivial. Our results extend those of R. V. Kadison [l], in which F0 and P are the self-adjoint operators and projections respectively in a uniformly closed self-adjoint operator algebra. While the framework and plan of the proofs were inspired by Kadison's work, almost none of the apparatus used by him is available here with the result that, in detail, our techniques are quite different from his. Let P be a weakly modular partially ordered set (see [4]). A function x from P to the non-negative real numbers and + oo is said to be a measure if x(0) =0 and x is countably additive in the sense that whenever {a,} is a pairwise orthogonal sequence of elements of P, then x(Ua?) = 2Zxiad- If x is a measure and {bt} EP is an increasing (decreasing) sequence with supremum (infimum) b, then x(&?) —*x(6). A countably additive function x from P to the extended real numbers is a signed measure if x(0) =0 and x takes on at most one of the values + =o and — oo ; x is finite if x(l) is finite. Define the functions 5 and i on the signed measures on P by six) = sup(x(a) : aEP\, fx) =inf {x(a): aEP} and set ||x|| =s(x)—?(x). Clearly ||x|| < oo if and only if x is finite. It iseasy toseethat||x|| = sup {x(a)— xia'):aE P} ■ Lemma 1. Let X be a real linear space of finite signed measures on P. Then the function || || defined above is a norm for X and, under this norm and its natural partial ordering, X is a partially ordered normed linear space. That is,2
Continuing the development [4] of an aspect of the approach to the axiomatization of quantum mechanics of G. W. Mackey [3], we consider here the real linear space X of signed measures on the set P of events generated by the states, and the set F0 of linear functionals on X which are induced in a natural way by the bounded observables. A necessary and sufficient condition for two events to be simultaneously measurable is found in terms of the order structure of F0, with the following consequence: if F0 is a lattice, P is deterministic. At the opposite extreme, F0 is said to be an anti-lattice1 if the greatest lower bound exists only for comparable pairs of its elements and we show in this case that the center of P is trivial. Our results extend those of R. V. Kadison [l], in which F0 and P are the self-adjoint operators and projections respectively in a uniformly closed self-adjoint operator algebra. While the framework and plan of the proofs were inspired by Kadison's work, almost none of the apparatus used by him is available here with the result that, in detail, our techniques are quite different from his. Let P be a weakly modular partially ordered set (see [4]). A function x from P to the non-negative real numbers and + oo is said to be a measure if x(0) =0 and x is countably additive in the sense that whenever {a,} is a pairwise orthogonal sequence of elements of P, then x(Ua?) = 2Zxiad- If x is a measure and {bt} EP is an increasing (decreasing) sequence with supremum (infimum) b, then x(&?) —*x(6). A countably additive function x from P to the extended real numbers is a signed measure if x(0) =0 and x takes on at most one of the values + =o and — oo ; x is finite if x(l) is finite. Define the functions 5 and i on the signed measures on P by six) = sup(x(a) : aEP, fx) =inf {x(a): aEP} and set ||x|| =s(x)—?(x). Clearly ||x|| < oo if and only if x is finite. It iseasy toseethat||x|| = sup {x(a)— xia'):aE P} ■ Lemma 1. Let X be a real linear space of finite signed measures on P. Then the function || || defined above is a norm for X and, under this norm and its natural partial ordering, X is a partially ordered normed linear space. That is,2
In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra can equivalently be described as certain real-valued functions on the projection lattice of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote, are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory. Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper, we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∊S(H) and P∊P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∊E(H) for A∊E(H) and P∊P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B∊S(H).
Let F : ℬ(ℝ)→ℱ(ℋ) be a commutative positive operator-valued measure from the Borel σ-algebra of the reals to the space of bounded positive operators on the Hilbert space ℋ. It is well known that there exist a self-adjoint operator A and a Markov kernel μ(·)(λ) such that F(Δ) = μΔ(A), where Δ ∈ ℬ(ℝ). We prove that, for any Δ ∈ ℬ(ℝ), if F(Δ) is a discrete operator, then A is a linear combination of the moments of F. This result allows us to characterize the quantum observable represented by A by means of the Naimark dilation of F. The result is connected with the following properties of the sequences of linear functionals. Consider an infinite sequence of linear functionals {Ti}i ∈ ℕ, such that Tif = ∫ f(t) dμt(i), corresponding to an infinite sequence of probability measures {μ(·)(i)}i ∈ ℕ on the Borel σ-algebra ℬ([0, 1]) such that μ(·)(i) ≠ μ(·)(j), where i, j ∈ ℕ and i≠j. There exists a real bounded one-to-one continuous function f such that T i f = ∫ f ( t ) d μ t ( i ) ≠ ∫ f ( t ) d μ t ( j ) = T j f , i , j ∊ ℕ , i ≠ j .
ZIERLER plausible 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) (α')' = α, (2) a < b if and only if V < a\ (3) a! is a complement of a; i.e., 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 ba f V a if b J_ c and α ^ 6; if 6 X <£ & 2 ^ then V^ = 6χ V b 2 b[ V 6 3 &2 V .Consider 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
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.