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A Representation of Quantum Measurement in Order-Unit Spaces

Gerd Niestegge

Zillertalstrasse 39, 81373 Muenchen, Germany

gerd.niestegge@web.de

Abstract. A certain generalization of the mathematical formalism of quantum mechanics

beyond operator algebras is considered. The approach is based on the concept of conditional

probability and the interpretation of the Lüders - von Neumann quantum measurement as a

probability conditionalization rule. A major result shows that the operator algebras must be

replaced by order-unit spaces with some specific properties in the generalized approach, and it

is analyzed under which conditions these order-unit spaces become Jordan algebras. An

application of this result provides a characterization of the projection lattices in operator

algebras.

Key Words. Operator algebras, Jordan algebras, convex sets, quantum measurement,

quantum logic

1. Introduction

Despite of many efforts, quantum mechanics and relativity theory have been resisting their

unification for almost a century. Either both theories are universally valid and the right way

how to unify them has still not been found, or at least one of the two theories needs to be

extended to achieve their unification. Just as classical probability theory was not general

enough to cover the probabilities occurring in quantum mechanics, the current mathematical

formalism of quantum mechanics may still not be general enough for the unification, or this

may hold for relativity theory or for both theories. The present paper deals with the first case -

i.e., with a potential generalization of the mathematical formalism of quantum mechanics.

The starting point is a certain axiomatic approach to this formalism, developed by the

author in some recent papers [21-23], which is based on the concept of conditional

probabilities and the interpretation of the Lüders - von Neumann quantum measurement as a

probability conditionalization rule. This approach leads to the standard model of quantum

mechanics with the Hilbert space over the complex numbers. When the last one among the

axioms is dropped, it leads to Jordan algebras, which are only a little bit more general than the

standard Hilbert space model since most of them - but not all - have representations as

operator algebras on a complex Hilbert space. The approach provides further opportunities for

potential generalizations just by dropping some more axioms and keeping only those that

describe the basic properties of the conditional probabilities.

In the paper, it is shown how this results in a certain new mathematical structure - an

order-unit space with some specific additional properties. It still features some of the well-

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Gerd NiesteggeA Representation of Quantum Measurement in Order-Unit Spaces

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known properties of Jordan operator algebras (e.g., existence and uniqueness of the

conditional probabilities), but not all (e.g., the spectral theorem does not hold anymore for all

elements of the order-unit space). Conditions under which the order-unit space becomes a

Jordan operator algebra are studied. An application of the main results of the paper provides a

characterization of the projection lattices in Jordan operator algebras, which modifies an earlier

result by Bunce and Wright [7].

Order-unit spaces were introduced by Kadison [17] and play an important role in the study

of convex sets [1,2,6] as well as of the order structure of operator algebras [14]. Alfsen and

Shultz [2] used them in their approach to a non-commutative spectral theory. Bunce, Wright

[7] and Pulmannová [25] studied the relation of Alfsen's and Shultz's results to quantum logic.

Edwards and Rüttimann [12] investigated the conditional probabilities assuming the Alfsen-

Shultz property (i.e., every exposed face of the state space is projective). All these results are

based on the notion of P-projections introduced by Alfsen and Shultz. A similar, but more

general type of projections will play a central role in the present paper; they become identical

with the P-projections only in specific situations.

The next section gives an overview of the basic notions from the author's recent papers

[21-23] as far as needed in the present paper. A first major result is the derivation of the order-

unit space and its specific properties from these basic notions in section 3. Observables and the

relationship with Alfsen and Shultz's spectral duality are studied in section 4. Further results in

the last section concern the conditions under which the order unit spaces become Jordan

algebras and the characterization of the projection lattices in operator algebras.

2. Events, states, and conditional probabilities

Our model of events (or quantum logic [7,25,28,29]) shall be a mathematical structure

which is as simple as possible, but has enough structure for the consideration of states. This

requires an orthogonality relation and a sum operation for orthogonal events. The precise

axioms for the system E of events were presented in [21] and look as follows.

E is a set with distinguished elements 0 and 1 I, an orthogonality relation ⊥ and a partial

binary operation + such that the following conditions hold for e,f,g∈E:

(OS1)

(OS2)

(OS3)

If e⊥f, then f⊥e; i.e., the relation ⊥ is symmetric.

e+f is defined for e⊥f, and then e+f=f+e; i.e., the sum operation is commutative.

If g⊥e, g⊥f, and e⊥f, then g⊥e+f, f⊥g+e and g+(e+f)=(g+e)+f; i.e., the sum operation

is associative.

0⊥e and e+0=e for all e∈E?

For every e∈E, there exists a unique e'∈E such that e⊥e' and e+e'=1 I.

There exists d∈E such that e⊥d and e+d=f if and only if e⊥

(OS4)

(OS5)

(OS6)

′ f .

Then 0'=1 I and e''=e for e∈E. A further relation ? is defined on E via e?f iff e⊥

relation will be needed for the definition of the conditional probabilities, but note that the

above axioms do not imply that it is an order relation. We call E orthogonally σ-complete if the

sum exists for any countable orthogonal subset of E, and we call E orthogonally complete if

the sum exists for any orthogonal subset of E.

A state is a map µ:E→[0,1] such that µ(1 I)=1 and µ(e+f) = µ(e) + µ(f) for orthogonal

pairs e and f in E. Then µ(0)=0 and µ(e1+...+ek) = µ(e1)+...+µ(ek) for orthogonal elements

e1,...,ek in E. When E is orthogonally σ-complete, the state µ is called σ-additive if

µ(Σnen)=Σnµ(en) for any orthogonal sequence en in E, and when E is orthogonally complete,

the state µ is called completely additive if µ(Σf∈F f) = Σf∈Fµ(f) for any orthogonal subset F of E.

′ f . This

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Denote by So the set of all states on E, by Sσ the set of σ-additive states on E and by Sc the set

of completely additive states on E, where it is assumed that E is orthogonally σ-complete and

orthogonally complete, respectively, in the latter two cases.

With a state µ and µ(e)>0 for some e∈E, another state ν is called a conditional probability

of µ under e if ν(f) = µ(f)/µ(e) holds for all f∈E with f?e. Now let S be either So or Sσ or Sc.

We shall consider the following axioms that were introduced in [21].

(UC1)

(UC2)

If e,f∈E and µ(e)=µ(f) for all µ∈S, then e=f.

If e∈E and µ∈S with µ(e)>0, there is one and only one conditional probability of µ

under e.

If these axioms are satisfied, E is called an S-UCP space (S = So, Sσ, or Sc) - named after

the major feature of this mathematical structure which is the existence of the unique

conditional probability - and the elements in E are called events. The unique conditional

probability of µ under e is denoted by µe and, in analogy with probability theory, we also write

µ(f|e) instead of µe(f) for f∈E. The above two axioms imply that there is a state µ∈S with

µ(e)=1 for each event e≠0, that the difference d in (OS6) becomes unique, that the relation ?

is anti-symmetric (but not necessarily transitive), and that e⊥e iff e⊥1 I iff e=0 (e∈E).

Note that the following identity which will be used later holds for convex combinations of

states µ,ν∈S (0<s<1):

(sµ+(1-s)ν)e = (sµ(e)µe+(1-s)ν(e)νe)/(sµ(e)+(1-s)ν(e)).(1)

Examples of the above structure can be obtained considering Jordan algebras. The

multiplication operation in a Jordan algebra A satisfies the condition a2?(a?b)= a?(a2?b) for

a,b∈A. A JB algebra is a complete normed real Jordan algebra A satisfying ||a?b||≤||a|| ||b||,

||a2||=||a||2 and ||a2||≤||a2+b2|| for a,b∈A. A partial order relation ≤ on A can then be derived by

defining its positive cone as {a2:a∈A}. If A is unital, we denote the identity by 1 I. A JB

Algebra A that owns a predual A* (i.e., A is the dual space of A*) is called a JBW algebra and is

always unital. A JBW algebra can also be characterized as a JB algebra where each bounded

monotone increasing net has a supremum in A and a normal positive linear functional not

vanishing in a exists for each a≠0 in A (i.e., the normal positive linear functionals are

separating). A map is normal if it commutes with the supremum. It then turns out that the

normal functionals coincide with the predual. The self-adjoint part of any W*-algebra (von

Neumann algebra) equipped with the Jordan product a?b:=(ab+ba)/2 is a JBW algebra, but

not each JBW algebra is the complete self-adjoint part of a W*-algebra. Moreover, there are

exceptional JBW algebras that cannot be represented as an algebra of self-adjoint operators at

all (e.g., the algebra of hermitean 3×3 matrices over the octonions equipped with the Jordan

product).

The monographs [14] and [26] are recommended as excellent references for the theory of

JB/JBW algebras and W*-algebras, respectively. Historically important references are Jordan,

von Neumann and Wigner's work [16] which initiated the theory of Jordan algebras and covers

the finite-dimensional case, as well as Alfsen, Shultz and Størmer's achievement for the infinite-

dimensional case [5].

The idempotent elements of a JB algebra are called projections, and the projections in a

JBW algebra A form a complete lattice. These projection lattices now become examples of Sc-

UCP spaces if A does not contain a type I2 part. The conditional probability has the shape

µ(f|e)=? µ({e,f,e})/µ(e), where {a,b,c} := a?(b?c) - b?(c?a) + c?(a?b) is the Jordan triple

product, and in the case of a W*-algebra we get µ(f|e)=? µ(efe)/µ(e). This was shown in [21]

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and reveals the link to the Lüders - von Neumann quantum measurement. Note that ? µ on A is

the unique linear extension of the state µ on the projection lattice; this extension exists by

Gleason's theorem [13] and its later enhancements to W*-algebras and JBW algebras

[8,9,11,20,30,31]. A complete characterization of the projection lattices in JBW algebras

without type I2 part as Sc-UCP spaces with some further properties will be a major result at the

end of this paper.

3. Order-unit spaces

A partially ordered real vector space A is an order-unit space if A contains an order-unit 1 I

and if A is Archimedean [1,6,14]. The order-unit 1 I is positive and, for all a∈A, there is t>0

such that -t1 I ≤ a ≤ t1 I. A is Archimedean if na ≤ 1 I for all n∈IN implies a≤0. An order-unit

space A has a norm given by a = inf {t>0: -t1 I ≤ a ≤ t1 I}. Each x∈A can be written as x=a-b

with positive a,b∈A (e.g., choose a = ||x||1 I and b = ||x||1 I - x). A positive linear functional

ρ:A→IR on an order-unit space A is norm continuous with ||ρ||=ρ(1 I) and, vice versa, a norm

continuous linear functional ρ with ||ρ||=ρ(1 I) is positive. Note that unital JB algebras are

order-unit spaces.

The order-unit spaces A considered in the following are dual spaces of base-norm spaces V

such that the unit ball of A is compact in the weak-*-topology σ(A,V). For ρ∈V and x∈A

define ? ρ(x):=x(ρ); the map ρ→? ρ is the canonical embedding of V in its second dual V**=A*.

Then ρ∈V is positive iff ? ρ is positive on A. Moreover, A is monotone complete and

? ρ(sup xα)=lim? ρ(xα) holds for ρ∈V and any bounded monotone increasing net xα in A; in the

JBW/W*-algebra setting one would say that ρ∈V is normal.

For any set K in A, denote by linK the σ(A,V)-closed linear hull of K and by convK the

σ(A,V)-closed convex hull of K. For a convex set K, denote by ext K the set of its extreme

points which may be empty unless K is compact. A projection is a linear map U:A→A with

U2=U and, for a≤b, we define [a,b] := {x∈A: a≤x≤b}. Suppose that E is a subset of [0,1 I] in A

such that

(a) 1 I∈E,

(b) 1 I-e∈E if e∈E, and

(c) d+e+f∈E if d,e,f,d+e,d+f,e+f∈E.

Define e':=1 I-e and call e,f∈E orthogonal if e+f∈E. Then E satisfies (OS1),...,(OS6) such that

we can consider So as in section 2. Since [0,1 I] is monotone complete, any sum of orthogonal

elements in E σ(A,V)-converges in [0,1 I]. If these sums converge in E, we call E orthogonally

complete, and we call E orthogonally σ-complete, if only the countable sums converge in E. In

these cases, we can again consider Sσ and Sc as in section 2.

Proposition 3.1: Suppose that A is an order-unit space with order unit 1 I and that A is

the dual of the base-norm space V. Moreover, suppose that E is a subset of [0,1 I] satisfying

the three above conditions (a), (b), (c). Consider the cases S=So, S=Sσ, or S=Sc, assuming that

E is orthogonally σ-complete or orthogonally complete, respectively, in the latter two cases,

and suppose that the following two conditions hold:

(i) A = linE, and for each µ∈S there is a σ(A,V)-continuous positive linear functional ? µon

A with ? µ(e) = µ(e) for e∈E.

(ii) For each e∈E there is a σ(A,V)-continuous positive projection Ue:A→A such that Ue1 I=e,

UeA = lin{f∈E: f≤e} and ? µ=? µUe for µ∈S with µ(e)=1.

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Then E is an S-UCP space. The conditional probabilities have the shape µ(f|e) = ? µ(Uef)/µ(e)

for e,f∈E and µ∈S with µ(e)>0.

Proof. For e,f∈E with e≠f there is ρ∈V+ with ρ(e-f)≠0. The restriction of ρ/ρ(1 I) to E then

yields a state µ∈S with µ(e)≠µ(f). Therefore (UC1) holds.

Suppose e∈E and µ∈S with µ(e)>0. It is rather obvious that the map g → ? µ(Ueg)/µ(e) on

E provides a conditional probability of µ under e. Now assume that ν is a further conditional

probability of µ under e. Then ν(e)=1 and thus ? ν=? νUe. From Ueg∈lin{f∈E: f≤e} we get that

ν(g) = ? ν(Ueg) = ? µ(Ueg)/µ(e) for g∈E. Therefore, (UC2) holds as well.q.e.d.

Note the similarities between the second part of condition (i) in the preceding proposition

and the Gleason theorem. We shall now see that the situation of Proposition 3.1 is universal for

the S-UCP spaces; i.e., each S-UCP space has such a shape as described there.

Theorem 3.2: Suppose that E is a S-UCP space with S=So, S=Sσ, or S=Sc. Then E is a

subset of [0,1 I] in some order-unit space A with predual V as described in Proposition 3.1.

Proof. Define V:={sµ-tν: µ,ν∈S, 0≤s,t∈IR }, which is a linear subspace of the

orthogonally additive real-valued functions on E, and consider for ρ∈V the norm ||ρ|| :=

inf{r∈IR : r≥0 and ρ ∈ r conv(S∪-S)}. Then |ρ(e)| ≤ ||ρ|| for every e∈E. Let A be the dual

space of the base-norm space V and let ? µ be the canonical embedding of µ∈V in V**=A*. If

? µ(x)≥0 for all µ∈S, the element x∈A is called positive and we write x≥0. Equipped with this

partial ordering, A becomes an order-unit space with the order-unit1 I:=π(1 I), and the order-

unit norm coincides with the dual space norm such that sup{|? µ(x)| : µ∈S} = ||x|| for x∈A. With

e∈E define π(e) in A via π(e)(ρ) := ρ(e) for ρ∈V. Then ||π(e)|| ≤ 1, and the finite additivity, σ-

additivity, or complete additivity of π in the different cases follow immediately from this

definition. Moreover, A is the σ(A,V)-closed linear hull of π(E).

We now define Ue for e∈E. Suppose x∈A and sµ-tν∈V with µ,ν∈S and 0≤s,t∈IR . Then

define (Uex)(sµ-tν) := sµ(e)? µe(x)-tν(e)? νe(x). Here, ? µe and ? νe are the canonical embeddings

of the conditional probabilities µe and νe in A*; they do not exist in the cases µ(e)=0 or ν(e)=0

and then define µ(e)? µe(x):=0 and ν(e)? νe(x):=0, respectively. We still have to show that Ue is

well defined for sµ-tν = s'µ'-t'ν' with µ,µ',ν,ν'∈S and 0≤s,s',t,t'. Then s-t = (sµ-tν)(1 I) =

(s'µ'-t'ν')(1 I) = s'-t' and s+t'=s'+t. If s+t'=0, s=s'=t=t'=0 and Uex is well-defined. If s+t'>0, then

either sµ(e) + t'ν'(e) = s'µ'(e) + tν(e) = 0 and sµ(e) = t'ν'(e) = s'µ'(e) = tν(e) = 0, or

(sµ+t'ν')/(s+t') = (s'µ'+tν)/(s'+t) ∈ S and, calculating the conditional probability under e for

both sides of this identity by using (1), we get sµ(e)µe+t'ν'(e) ′ νe = s'µ'(e) ′

cases, Ue is well defined.

If µ(e)=1 for µ∈S, then µ=µe and ? µ(Uex) = (Uex)(µ) = ? µ(x) such that ? µ=? µUe. Thus,

(UeUex)(µ) = µ(e)? µe(Uex) = µ(e)? µe(x) = (Uex)(µ) for all µ∈S and hence for all ρ∈V such that

UeUe=Ue, i.e., Ue is a projection. Its positivity, σ(A,V)-continuity as well as Ue1 I=π(e) and

Ueπ(f)=π(f) for f∈E with f≤e follow immediately from the definition.

Therefore lin{π(f): f∈E, f≤e} ⊆ UeA. Assume Uex∉lin{π(f): f∈E, f≤e} for some x∈A. By

the Hahn-Banach theorem, there is ρ∈V with ? ρ(Uex)≠0 and ρ(f)=0 for f∈E with f≤e. Suppose

ρ=sµ-tν with µ,ν∈S and 0≤s,t∈IR . Then sµ(f)=tν(f) for f∈E with f≤e and thus sµ(e)µe(f) =

tν(f)νe(f). The uniqueness of the conditional probability implies sµ(e)µe(f) = tν(f)νe(f), i.e.,

µe+tν(e)νe. In all