Why do the quantum observables form a Jordan operator algebra? Gerd Niestegge Zillertalstrasse 39, D-81373 Muenchen, Germany Abstract. The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive non-associative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in the present paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e. from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the type II, III von Neumann algebras. Key Words: Foundations of quantum mechanics, quantum probability, quantum logic, Jordan operator algebras 1. Introduction A non-Boolean extension of classical probabilities was presented in [10]. The main purpose of [10] was to elaborate on the interpretation of the model and on some applications to quantum measurement. For that purpose, it was sufficient to consider finitely-additive probabilities. In the present paper, countably-additive probabilities are needed to study observables which are defined in an abstract way as the analogue of the classical random variables. It is shown that, under certain conditions, a multiplication operation exists on the system of bounded real-valued observables which form a Jordan operator algebra then. The associativity of the multiplication operation is equivalent to the classical case. Since almost all Jordan operator algebras can be represented on a Hilbert space, the non-Boolean extension of the classical probabilities thus provides an axiomatic access to quantum mechanics. Other axiomatic approaches to quantum mechanics start from different postulates including either an orthomodular partial ordering on the system of events (called propositions in the quantum logical approaches; e.g. [3,11,12,16]) or a distributive multiplication operation on the system of bounded real observables (e.g. [8,14,15]) or both (e.g. [13]). Postulating the existence of the distributive multiplication operation is hard to justify from a physical or statistical point of view (when using the so-called Segal product[13,14], the distributivity becomes a problem as pointed out in [15]), and the purely logical approaches are neither able to rule out some physically irrelevant cases [9] nor to cover the physically relevant type II,III von-Neumann 1 Gerd Niestegge Why do the quantum observables form a Jordan operator algebra? algebras (which do not contain the minimal events needed for the geometrical methods these approaches use). The approach of the present paper assumes a rather weak structure for the system of events (with a simple orthogonality relation instead of an orthomodular partial ordering); more important are certain statistically interpretable properties, postulated for the conditional probabilities and the observables, where the distributive multiplication operation can be derived from. This approach is closer to Kolmogorov's measure-theorectic access to classical probability theory than the other approaches. Moreover, the type II,III cases are included and the physically irrelevant cases discovered in [9] are excluded. 2. Non-Boolean probabilities An orthospace[10] is a set * with distinguished elements 0 and 1I , a relation ⊥ and a partial binary operation + such that for D,E,F∈*: (OS1) E⊥F ⇒ F⊥E (OS2) E+F is defined for E⊥F, and E+F=F+E (OS3) D⊥E, D⊥F, E⊥F ⇒ D⊥E+F, F⊥D+E and D+(E+F)=(D+E)+F (OS4) 0⊥E and E+0=E for all E∈* (OS5) For every E∈* there exists a unique E'∈* such that E⊥E' and E+E'=1I (OS6) E⊥F' ⇔ There exists a D∈* such that E⊥D and E+D=F We say "E and F are orthogonal" for E⊥F. (OS2,4,5) imply that 0'=1I and E''=E for E∈*. A further relation % is defined on * via: E % F :⇔ E⊥F' (E,F∈*). Then E % F if and only if * contains an element D such that D⊥E and F=E+D. Moreover, we have 0 % E % 1I for all E∈*. The relation % is reflexive by (OS4) or (OS5), but is not a partial ordering since it is neither anti-symmetric nor transitive in general. Therefore, the orthospace structure is far away from what is usually considered a quantum logic and is a rather weak structure the only purpose of which is to provide the opportunity to define states as an analogue of the classical probability measures. Further postulates concerning the states and conditional probabilities will be considered below and will then provide a sufficiently rich structure. A state on an orthospace* is a map µ:* → []0,1 such that µ()1I = 1 and µ()()()E +F = µE + µ F for all orthogonal pairs E,F∈*. Then µ()0 = 0 , and µ is additive for each finite family of pairwise orthogonal elements in *. (OS6) ensures that µ()()E ≤ µ F for E % F . σ Σ ∞ Definition 2.1: (i) A -orthospace is an orthospace * such that n=1 En is defined in * for every sequence of mutually orthogonal events En. µ σ σ µ Σ ∞ Σ ∞ µ (ii) A state on a -orthospace * is called -additive if ( n=1 En)= n=1 (En) for every sequence of mutually orthogonal events En. The elements E∈* are interpreted as events and will be called so in the following. Orthogonality means that the events exclude each other. The (only partially defined) operation + is interpreted as the or connection of mutually exclusive events, E′ is the negation of E. For a state µ, the interpretation of the real number µ(E) is that of the probability of the event E in the state µ. If µ is a state on an orthospace * and E∈* with µ(E)>0 and if ν is another state such that ν(F) = µ(F) µ(E) holds for all F∈* with F % E, then ν is called a conditional probability of µ , 2 Gerd Niestegge Why do the quantum observables form a Jordan operator algebra? under E. Essential shortcomings of this conditional probability are that such a state ν may not exist at all and that, if such a state exists, it may not be unique. The requirement that unique conditional probabilities must exist guides us to the following definition of σ-UCP spaces (which is the adaptation of the UCP spaces considered in [10] to σ-additive states). Definition 2.2: A σ-UCP space is a σ-orthospace * satisfying the following two axioms: (UC1) If E,F∈* and E≠F, then there is a σ-additive state µ with µ()()E ≠ µ F . (UC2) For each σ-additive state µ and E∈* with µ(E)>0, there exists one and only one σ- additive conditional probability µE of µ under E. µE(F) is the probability of the event F in the state µ after the event E has been observed. Using the same terminology as in mathematical probability theory, we will also write µ(F|E) for µE(F) in the sequel. If µ(E)=1, then µE=µ and µ(F|E)=µ(F) for all F∈*. There is a σ-additive state µ with µ(E)=1 for each element E≠0 in a σ-UCP space, since from (UC1) we get a σ-state ν with ν(E)>0, and then choose µ=νE. (UC1) implies the uniqueness of D in (OS6): E+D1=F=E+D2, then µ(E)+µ(D1)=µ(F)= µ(E)+µ(D2) for all σ-additive states µ, hence µ(D1)=µ(D2) for all σ-additive states µ and D1=D2. Moreover, if E % F and F % E for E,F∈*, then E=F; i.e. the relation % is anti-symmetric: If F=E+D1 and E=F+D2, then µ(F)=µ(E)+µ(D1)=µ(F)+µ(D2)+µ(D1), therefore µ(D1)=µ(D2)=0 for all σ-additive states µ, and D1=D2=0 by (UC1). Note that the relation % need not be transitive so far. Furthermore: E⊥E ⇔ E⊥1I ⇔ E=0 (If E⊥E, then E⊥E+E'=1I by (OS3,5). If E⊥1I , then E⊥0' and E'⊥0, i.e. E % 0 and 0 % E, hence E=0.). In [10], the concepts of statistical predictability (state-independence of the conditional probability) and compatibility have been introduced. The adaptation of these concepts to σ-UCP spaces is straight forward, but not needed for the purpose of the present paper. 3. Observables An observable is supposed to be the analogue of a classical random variable which is a measurable point function f between two measurable spaces. With the σ-UCP space model, there are no points, but only events. A closer look at classical probability theory shows that the map X allocating the event f -1(E) to the event E is more essential to the theory than f itself. The map X is a homomorphism between the σ-algebras of events. An observable is therefore defined as a homomorphism between two σ-orthospaces; similar but less general definitions of observables can be found in [4,12,16]. Definition 3.1: A map X from a σ-orthospace + to a σ-orthospace * is called an observable if (i) X(1I )=1I , (ii) X(E)⊥X(F) in * for all pairs E,F∈+ with E⊥F, and ∞ ∞ (iii) X ()∑F = ∑ X ()F for every sequence of mutually orthogonal events F in +.