A Representation of Quantum Measurement in Nonassociative Algebras
AA Representation of Quantum Measurement in Nonassociative Algebras
Gerd Niestegge
Zillertalstrasse 39, 81373 Muenchen, [email protected]
Abstract . Starting from an abstract setting for the Lüders-von Neumann quantummeasurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in apreceding paper. This is an order-unit space with some specific properties. It becomes aJordan operator algebra under a certain set of additional conditions, but does not own amultiplication operation in the most general case. A major objective of the present paper is thesearch for such examples of the structure mentioned above that do not stem from Jordanoperator algebras; first natural candidates are matrix algebras over the octonions and othernonassociative rings. Therefore, the case when a nonassociative commutative multiplicationexists is studied without assuming that it satisfies the Jordan condition. The characteristics ofthe resulting algebra are analyzed. This includes the uniqueness of the spectral resolution aswell as a criterion for its existence, subalgebras that are Jordan algebras, associativesubalgebras, and more different levels of compatibility than occurring in standard quantummechanics. However, the paper cannot provide the desired example, but contribute to thesearch by the identification of some typical differences between the potential examples andthe Jordan operator algebras and by negative results concerning some first natural candidates.The possibility that no such example exists cannot be ruled out. However, this would result inan unexpected new characterization of Jordan operator algebras, which would have asignificant impact on quantum axiomatics since some customary axioms (e.g., power-associativity or the sum postulate for observables) might turn out to be redundant then.
Key Words . Quantum measurement, quantum logic, operator algebras, Jordan algebras,order-unit spaces
1. Introduction
Starting from an abstract setting for the Lüders-von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure [5], the author derived a certain generalization of operator algebras in a preceding paper [6], where two extreme cases were considered - the most general one without any multiplication operation and a set of additional conditions resulting in a Jordan operator algebra. The present paper studies a case between these two extremes; this is the case when a nonassociative commutative multiplication exists without assuming that it satisfies the Jordan condition x $ ( x $ y ) = x $ ( x $ y ). This case is ideally suited for studying the question whether matrix algebras over the octonions or other nonassociative rings can provide examples of the structure mentioned above.The characteristics of the resulting nonassociative algebra are analyzed. Although thecomplete algebra need not any more satisfy the Jordan condition, some subalgebras still do. Ifa spectral resolution exists, it is unique, and it exists for those elements which generate an erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 2associative subalgebra with positive squares. Moreover, different levels of compatibility areinvestigated, and there seem to be more different levels than occurring in standard quantummechanics.A major motivation of the present paper is to support the search for such examples of thestructure mentioned above that do not stem from Jordan operator algebras. However, suchexamples have not been found yet; the paper contributes to the search by the identification ofsome typical differences between the potential examples and the Jordan operator algebras andby negative results concerning some first natural candidates.The monograph [2] is recommended as reference for the theory of Jordan operatoralgebras; it also includes some basic material on order-unit spaces. A brief sketch of Jordanoperator algebras and order unit spaces as far as needed in the present paper is provided insection 2 together with a theorem by Iochum and Loupias [3] that will be used in section 3.The structure of the nonassociative algebras under consideration is presented and studied insection 3. A certain type of "small" algebras is analyzed in section 4; they turn out to beidentical with the spin factors or type I factors from the theory of Jordan operator algebras.The different levels of compatibility are considered in section 5 and the matrix algebras insection 6.
2. Power-associative algebras
An algebra is called power-associative if each element x lies in an associative subalgebra;this is equivalent to x n $ x m = x n + m for n , m ˛ IN , where x n is inductively defined via x n +1 = x $ x n .Jordan algebras are always power-associative, but a power-associative algebra need not be aJordan algebra. Jordan, von Neumann and Wigner [4] showed that the finite-dimensionalformally real power-associative commutative algebras are Jordan algebras. Iochum andLoupias [3] extended this result to the infinite-dimensional case making use of the theory ofJB and JBW algebras and order-unit spaces [2].A JB algebra is a complete normed real Jordan algebra M satisfying || a $ b || £ || a || || b ||,|| a ||=|| a || and || a || £ || a + b || for a,b ˛ M . A partial order relation £ on M can then be derived bydefining its positive cone as { a : a ˛ M }. If M is unital, we denote the identity by 1I . A JBAlgebra M that owns a predual M * (i.e., M is the dual space of M * ) is called a JBW algebraand is always unital. A JBW algebra can also be characterized as a JB algebra where eachbounded monotone increasing net has a supremum in M and a normal positive linearfunctional not vanishing in a exists for each a „
0 in M (i.e., the normal positive linearfunctionals 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. An order-unit space is a partially ordered real vector space L that contains an order-unit 1Iand is Archimedean [2]. The order-unit 1I is positive and, for all a ˛ L , there is t >0 such that - t £ a £ t
1I . L is Archimedean if na £
1I for all n ˛ IN implies a £
0. An order-unit space L has a norm given by a = inf{ t >0: - t £ a £ t
1I }. Each x ˛ L can be written as x = a - b withpositive a , b ˛ A (e.g., choose a = || x ||
1I and b = || x ||1I - x ). A positive linear functional r : L fi IRon an order-unit space L is norm continuous with || r ||= r ( 1I ) and, vice versa, a norm continuous linear functional r with || r ||= r (
1I ) is positive. Note that unital JB algebras areorder unit spaces.The order-unit space L considered in the following is the dual space of a Banach space V such that the unit ball of L is compact in the weak-*-topology s ( L , V ). We will identify r ˛ V with its canonical embedding in V** = L* . Then L is monotone complete and r (sup x a ) = erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 3lim r ( x a ) holds for r ˛ V and any bounded monotone increasing net x a in L ; in the operatoralgebra setting one would say that r ˛ V is normal.Iochum and Loupias [3] showed that, in the definition of a JB algebra, the Jordancondition can be replaced by power-associativity. This result will be used in section 3, and aslightly different proof is presented here because not only the result itself, but also a majorpart of this proof will be needed in section 3. Theorem 2.1: ( Iochum/Loupias 1985 ) Suppose that M is a power-associativecommutative normed algebra over the real numbers with unit element such that || x $ y|| £ || x |||| y ||, || x || = || x || and || x || £ || x + y || for x,y ˛ M. Then M is a Jordan algebra . Proof : Without loss of generality we may assume that M is norm complete. Define M + :={ y : y ˛ M } and denote by C ( x ) the norm closed subalgebra generated by x ˛ M and 1I .Then C ( x ) is an associative JB algebra and thus isomorphic to the algebra of continuousfunctions on some compact Hausdorff space. Therefore we get for x ˛ M with || x || £
1 that x ˛ M + if and only if ||1I - x || £
1. This implies that M + is a convex cone. Since x =- y with x,y ˛ M implies 0=|| x + y || and thus 0=|| x ||=|| x || such that x =0, M + defines a partial ordering on M making M an order-unit space with order unit 1I . Thus a linear functional r in the dual space M * with || r ||= r ( 1I ) is positive. Then we have m ( x ) ‡
0 for x ˛ M and m ˛ S :={ r ˛ M * : || r ||= r ( 1I )=1}. The Cauchy-Schwarz inequality yields( m ( x $ y )) £ m ( x ) m ( y ) for x,y ˛ M, m ˛ S , and ( m ( x )) £ m ( x ) with y = 1I . Moreover, each r ˛ M * has the shape r = s m - t m with m , m ˛ S and s , t ‡
0. Now consider the seminorms x fi m ( x ) , m ˛ S , on M and the topology defined by them on M which is called the s -topology. Normconvergence implies s -convergence. Due to the Cauchy-Schwarz inequality, s -convergenceimplies s ( M , M * )-convergence and the product x $ y is s -continuous separately in each factor.We shall now prove that the product x $ y is jointly s -continuous in both factors on boundedsubsets of M .Let x a and y b be two bounded nets in M . First suppose that the net x a s -converges to 0.Considering C ( x a ) we find that 0 £ ( ) x a £ x x a a holds for each a ; therefore the net x a s -converges to 0. If furthermore the net y b s -converges to 0, the identity x $ y =(( x + y ) - x - y )/2 implies that the net x a $ y b s -converges to 0. Now suppose that the two nets x a and y b s -converge to x o and y o , respectively. Then use the identity x o $ y o - x a $ y b =( x o - x a ) $ y o + ( x a - x o ) $ ( y o - y b ) + x o $ ( y o - y b ) to conclude that x a $ y b s -converges to x o $ y o .Furthermore, consider the second dual M ** and assume that M is canonically embedded in M ** . Let N comprise all those elements of M ** that are the s ( M , M * )-limit of a bounded net in M which is a Cauchy net with respect to the seminorms defining the s -topology. Then the product $ has an s -continuous extension to N and N is power-associative. Moreover, m ( x ) ‡ for m ˛ S , x ˛ N . Therefore, the Cauchy-Schwarz inequality again holds and, since || x ||=sup{| m ( x )|: m ˛ S } for x ˛ N , N inherits from M the properties || x || = || x || and || x + y || ‡ || x || for x,y ˛ N . Note that M ** does not automatically inherit these properties since the map x fi x is not s ( M , M * )-continuous. An element x ˛ N is positive iff m ( x ) ‡
0 for m ˛ S , or iff x = a for some a ˛ N .Again, the norm closed subalgebra C ( a ) generated by some a ˛ N and 1I is an associativeJB algebra; therefore a £ || a || a holds for positive a ˛ N . If now x a is a bounded monotone increasing net in N , then x a s ( M ** , M * )-converges to sup x a in M ** . Since ( x - y ) £ || x - y|| ( x-y ) for y £ x in N , the net x a s -converges to sup x a such that sup x a ˛ N . Therefore N is monotonecomplete and the restrictions of the positive elements of V provide a separating family ofnormal functionals. erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 4In the same way, one can conclude that the s -closed subalgebra W ( x ) ˝ N generated bysome x ˛ N and 1I is monotone complete with a separating family of normal functionals suchthat it becomes an associative JBW algebra. Then the spectral theorem holds and x can benorm-approximated by elements having the shape a = S t k e k with real numbers t k , idempotentelements e ,..., e n in W ( x ) and e k $ e l =0 for k „ l . By a result in [7], the identity e $ ( f $ y )= f $ ( e $ y )holds in any power-associative algebra for idempotent elements e and f with e $ f =0 and any y .Therefore a a y t t e e y t t e e y a a y k l k l k l k l k l l k $ $ $ $ $ $ $ $ ( ) ( ) ( ) ( ) = = =S S S S and thus x $ ( x $ y ) = x $ ( x $ y ) for x , y ˛ N and particularly for x , y in the subalgebra M . q.e.d.
3. Nonassociative algebras
The structure that we will study is motivated by the results in [6], where an order-unitspace with a specific type of positive projections was derived from an abstract setting forconditional probabilities and the Lüders-von Neumann quantum measurement. A projection isa linear map U : A fi A on the order-unit space A with U = U . Examples of this structure are theJBW algebras, the finite-dimensional version of which are the formally really Jordan algebras.In these cases the specific positive projections have the shape U e x ={ e , x , e } with an idempotentelement e , where { a , b , c } := a $ ( b $ c ) - b $ ( c $ a ) + c $ ( a $ b ) denotes the so-called tripleproduct. If e is idempotent, { e , x , e } becomes 2 e $ ( e $ x ) - e $ x . Therefore, in the JBW case,there is a close relation between the specific positive projections and the Jordan product. Theidea behind the following assumptions is to keep the connection of the positive projectionswith a nonassociative product without imposing any further restrictions; particularly theproduct need not satisfy the Jordan condition.For any set K in an order-unit space A with predual V denote by lin K the s ( A , V )-closedlinear hull of K . Assumptions 3.1: (i)
A is an order-unit space .(ii)
A is the dual of the Banach space V .(iii)
A is a real algebra with the ( not necessarily associative ) commutative multiplication $ .(iv) The element in A is the order unit and the identity for the multiplication .(v) ||x $ y|| £ ||x|| ||y|| for x,y ˛ A .(vi) The product x $ y is s ( A , V )- continuous in x with y fixed as well as in y with x fixed.We define E := { e ˛ A : e $ e = e } , U e x := { e,x,e } for e ˛ E, x ˛ A, and S := { m ˛ V : || m ||= m ( 1I )=1}. (vii) The linear map U e : A fi A is a positive projection with U e A = lin { f ˛ E : f £ e } for each e ˛ E . (viii) If m ˛ S and e ˛ E with m ( e )=1, then m is invariant under U e ( i . e ., m = m U e ). In the remaining part of the present paper, A shall always satisfy all the conditions (i) - (viii). From 0 £ U e
1I = e for e ˛ E we get that the elements of E are positive. Since 1I - e ˛ E is also positive, we have 0 £ e £
1I . With the orthocomplementation e' := 1I - e , the set E becomes an orthomodular partially ordered set. Two elements e,f ˛ E are called orthogonal if f £ e' ; then U e f =0 since 0 £ U e f £ U e e' ={ e , 1I - e , e }=0. Moreover, U e U e ' = U e ' U e =0.As already in Refs. [5] and [6], we interpret the set E consisting of the idempotent elements of A as a generalized non-Boolean event structure and call the elements of E events.This is the viewpoint of probability theory. From another viewpoint, E could also be called aquantum logic and the elements of E could be called propositions. erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 5Some more background information and motivation for the above conditions (i), (ii), (vii)and (viii) can be found in [6] where they were derived from a few very basic assumptionsconcerning events, states and particularly the conditional probabilities. The new conditions(iii), (iv), (v) and (vi) mean that the same relation to a nonassociative product $ is assumed aswe find it in the JBW algebras: The events become idempotent elements with regard to thisproduct and the positive projections have the shape U e x = { e,x,e } = 2 e $ ( e $ x ) - e $ x . However,we do neither assume here that the product satisfies the Jordan condition or that it is power-associative nor that the conditions || a ||=|| a || or || a || £ || a + b || hold for the norm.Note that the link to quantum measurement and conditional probabilities is the formula m ( f | e ) = m ( U e f )/ m ( e ) for m ˛ S , e,f ˛ E with m ( e )>0 (see [6]). Here m ( f | e ) denotes the conditionalprobability of the event f under another event e in the state m . In the quantum measurementsetting, m ( f | e ) is the probability that a second measurement provides the result f after a firstmeasurement has already been performed and has provided the result e , assuming that thephysical system under consideration is in the state m . In a special Jordan algebra (e.g., the self-adjoint part of a W*-algebra), U e f ={ e , f , e } becomes efe , which reveals the connection to theLüders - von Neumann measurement process in the customary Hilbert space model ofquantum mechanics.The structure of the algebra A is designed in such a way that it owns all those properties ofa JBW algebra that are necessary to make the map f fi m ({ e,f,e })/ m ( e ) a unique conditionalprobability within the class S of normalized positive linear functionals on A . The situation in[5,6] was a little different. There unique conditional probabilities within the normalizedpositive additive functions on E were considered, which requires a Gleason type theoremmaking sure that these functions on E have linear extensions to A .We shall now study subalgebras of A and identify conditions that make them Jordanalgebras. Since the intersection of any family of monotone closed subalgebras is a monotoneclosed subalgebra, there is a smallest monotone closed subalgebra containing any given subsetof A ; it is called the monotone closed subalgebra generated by the subset. Note that thefollowing theorem does not require the conditions (vii) and (viii) of the assumptions 3.1. Theorem 3.2:
Suppose that M is a power-associative subalgebra of A with ˛ M andx ‡ for each x ˛ M . Then M is a Jordan algebra, its norm closure is a JB algebra, and themonotone closed subalgebra that M generates is a JBW algebra . Proof . Since m ( x ) ‡ x ˛ M and m ˛ S , the Cauchy-Schwarz inequality yields ( m ( x $ y )) £m ( x ) m ( y ) for x,y ˛ M and ( m ( x )) £ m ( x ) with y = 1I . Therefore || x || £ || x || = sup{( m ( x )) : m ˛ S } £ sup{ m ( x ): m ˛ S } = || x || such that || x || = || x || for x ˛ M . Moreover || x + y || = sup{ m ( x )+ m ( y ): m ˛ S } ‡ sup{ m ( x ): m ˛ S } = || x || for x,y ˛ M . By Theorem 2.1, M is a Jordan algebra and its norm closure a JB algebra. As in the proof of Theorem 2.1 consider the s -topology again and let N comprise all those elements of A that are the s ( A,V )-limit of a bounded net in M which is a Cauchy net with respect to the seminorms defining the s -topology. As with Theorem 2.1 now conclude that N is a JBW algebra. The monotone closed subalgebra generated by M is contained in N and thus a JBW algebra as well. q.e.d. Lemma 3.3: (i)
Two elements e and f in E are orthogonal ( i.e ., f £ e' =
1I - e ) iff U e' f = f, or iff e $ f =0. They satisfy e £ f iff e $ f = e .(ii) If e n is an orthogonal sequence in E , then S e n ˛ E . erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 6
Proof. (i) If f £ e' , then U e' f = f holds since f ˛ U e' A . If U e' f = f , then U e f =0 and the identity e $ f =( f + U e f - U e' f )/2 implies e $ f =0. If e $ f =0, then 1I - e - f =( 1I - e - f ) such that 1I - e - f ˛ E and thus1I - e - f ‡ f £ e' . Moreover, we have e £ f iff e and ¢ f are orthogonal, and this is equivalent to0= e $ ¢ f = e $ ( 1I - f )= e - e $ f .(ii) The sum S e n exists in A due to the monotone completeness of A and converges withregard to the s ( A,V )-topology. By (i) the orthogonality implies e n $ e m =0 for n „ m . Thus( S e n ) $ e m = e m for each m and ( S e n ) $ ( S e n )= S e n . q.e.d.It follows from the above lemma that the algebra A is associative if and only if E is aBoolean lattice (or Boolean algebra). If A is associative, e $ f ˛ E for e , f ˛ E and E becomes aBoolean lattice with e (cid:217) f = e $ f . If E is a Boolean lattice, then any two elements e and f in E canbe decomposed as e = d + d and f = d + d with orthogonal elements d , d , d ˛ E . Then e $ f = d = e (cid:217) f . Therefore d $ ( e $ f )= d (cid:217) e (cid:217) f =( d $ e ) $ f for any d , e , f ˛ E and A becomes associativesince it is generated by E .A spectral measure X allocates to each Borel measurable subset B of the real numbers IR an idempotent element e B in E such that the map B fi e B is s -additive and e B = 1I for B = IR . If m ˛ S , then B fi m ( e B ) becomes a probability measure over IR which is denoted by m X . Thespectral measure X is called a spectral resolution of x ˛ A if the measure integral (cid:242) t d m X coincides with m ( x ) for all m ˛ S . Such an x exists in A and is uniquely determined for eachbounded spectral measure X [6]. However, not each x in A has a spectral resolution.It follows from Theorem 3.2 that elements of A that lie in a power-associative subalgebrawith positive squares have a spectral resolution and that the spectral resolution is uniquelydetermined in the generated JBW subalgebra. We shall now see that it is uniquely determinedin A . Proposition 3.4:
Suppose that an element x ˛ A has a spectral resolution . Then its spectralmeasure X is uniquely determined in A. Moreover, the norm closed subalgebra generated by xand is an associative JB algebra and the monotone closed subalgebra generated by x and is an associative JBW algebra . Proof.
Suppose that x ˛ A has a spectral resolution. The spectral measure must then bebounded and x can uniformly be approximated by elements having the shape a = S t k e k with realnumbers t k , idempotent elements e ,..., e n in A and e k $ e l =0 for k „ l . Since elements with thisshape are power-associative with a m = S t km e k , we get that x is power-associative and that (cid:242) t n d m X = m ( x n ) for all m ˛ S . Because the moments of a probability distribution m X uniquely determinethe distribution (this follows from the Fourier transformation), we get that m ( e B ) is uniquely determined for all m ˛ S and thus e B is uniquely determined in A for every Borel set B . Moreover, m (( p ( x )) ) = (cid:242) ( p ( t )) d m X ‡ m ˛ S and any polynomial p such that ( p ( x )) ‡ The subalgebra generated by x and
1I is associative and the squares of its elements are positive. As in the proof of Theorem 3.2 now conclude that its norm closure is an associative
JB algebra and that the generated monotone closed subalgebra is an associative JBW algebra. q.e.d.
Real-valued observables can be defined as spectral measures and those elements in A which own a spectral resolution can be identified with real-valued observables therefore [5,6].We shall now see that A becomes a JBW algebra if each element in A represents anobservable. erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 7
Corollary 3.5:
If each element in A has a spectral resolution, then A is a JBW algebra . Proof.
By Proposition 3.4 A is power-associative and the squares of its elements arepositive. Then apply Theorem 3.2. q.e.d.With Theorem 3.2 and Proposition 3.4, an element x in A has a spectral resolution withinthe idempotent elements of A if and only if x lies in an associative subalgebra such that y ‡ y in this subalgebra. Associative JB or JBW algebras are also called abelian andare identical with the self-adjoint parts of the abelian C*-algebra and abelian W*-algebras,respectively. The abelian subalgebras play an important role in the theory of operator algebras.In the algebra A , however, one can study two further potential properties of its subalgebras- power-associativity of a subalgebra and the positivity of the square of each element in asubalgebra. The subalgebras with both these properties become Jordan algebras, while A itselfneed not be a Jordan algebra. Concerning the search for an example that satisfies theassumptions 3.1, but is not a JB algebra, this means that such an example would have toviolate at least one of the two properties: power-associativity or positivity of the squares.
4. "Small" algebras
An event 0 „ e ˛ E is called minimal, if there is no other nonzero event f with f £ e ; then U e A = IR e . In this section, we are going to study algebras where all nontrivial events areminimal. Such algebras can be considered small in the sense that any orthogonal family ofnonzero events cannot contain more than two elements. They represent the most simple casewhich is possible with the assumptions 3.1.
Theorem 4.1:
Suppose that A and E= { e ˛ A : e =e } satisfy the assumptions and thateach element in E which differs from and is minimal . Then A is a JB algebra . Proof . Suppose that e and f are any minimal events different from 1I . Then e' and f' areminimal as well. Furthermore U e f = l e and U e ' f = l ' e ' with some l , l ' ˛ [0,1]. In a first step, weshow that l + l '=1.For any r , s , t ˛ IR , consider the linear combination x := re + se '+ tf in A . From 2 e $ f = f + U e f - U e ' f = f + l e - l ' e ' and 2 e ' $ f = f - l e + l ' e ', we get 2 f $ x =( r - s )( l e - l ' e ')+(2 r +2 s + t ) f ,2 f $ ( f $ x )=(( r - s )/2)( l ( f + l e - l ' e ')- l '( f - l e + l ' e '))+(2 r +2 s + t ) f and U f x ={ f , x , f }=2 f $ ( f $ x )- f $ x =(( r - s )/2)( l + l '-1)( l e - l ' e ')+((2 r +2 s + t )/2) f . Since U f x ˛ IR f must hold for all r , s , t , either l + l '=1 or l e = l ' e '. The second case implies l = l '=0 (multiply both sides first with e and then with e ') and e $ f = f /2. Then ¢ f $ e=( 1I - f ) $ e = e - f $ e = e - f /2, ¢ f $ ( ¢ f $ e )= ¢ f $ e and U f' e = ¢ f $ ( ¢ f $ e )- ¢ f $ e = ¢ f $ e = e - f /2 such that e ˛ IR f ¯ IR ¢ f . Therefore e = f and l =1, or e = ¢ f and l '=1, resulting in a contradiction to l = l '=0. Therefore, only the case l + l '=1 can occur. Now consider any three minimal events d , e , f - each one different from
1I - and define x := d - U e d - U e ' d , y := f - U e f - U e ' f . In a second step, it is shown that x $ y ˛ IR
1I .
From the first step, we get U e f = a e , U e ' f =(1- a ) e ', U e d = b e , U e ' d =(1- b ) e ', U f d = g f and U d f ¢ = (1- g ) ¢ f with some a , b , g ˛ [0,1]. Then x = d - b e -(1- b ) e '= d +(1-2 b ) e -(1- b ) 1I and y = f - a e -(1- a ) e ' = f +(1-2 a ) e -(1- a ) 1I . Therefore x $ y = d $ f + (1-2 a ) d $ e - (1- a ) d + (1-2 b ) e $ f + ( a + b -1) e - (1- b ) f + (1- a )(1- b ) 1I . erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 8Using the identities d $ f =( d + U f d - U d f ¢ )/2=( d + g f -(1- g ) ¢ f )/2=( d + f -(1- g ) 1I )/2, d $ e =( d + U e d - U e ' d )/2 = ( d + e -(1- b ) 1I )/2 and e $ f =( f + U e f - U e ' f )/2=( f + e -(1- a ) 1I )/2 we get x $ y = ( d + f -(1- g ) 1I )/2 + (1-2 a )( d + e -(1- b ) 1I )/2 - (1- a ) d + (1-2 b )( f + e - (1- a ) 1I )/2+ ( a + b -1) e - (1- b ) f + (1- a )(1- b ) 1I= d/2 + (1-2 a )d/2 - (1- a ) d + f/2 + (1-2 b )f/2 - (1- b ) f + (1-2 a )e/2 + (1-2 b )e/2+ ( a + b -1) e - (1- g ) 1I /2 - (1-2 a )(1- b ) 1I /2 - (1-2 b )(1- a ) 1I /2 + (1- a )(1- b ) 1I= ( a /2 + b /2 + g /2 - a b - 1/2) 1I ˛ IR
1I .In the third step, consider any two elements a , b ˛ A and define x := a - U e a - U e ' a , y := b - U e b - U e ' b . Since a and b both are linear combinations of minimal events, we get from thesecond step that x $ y ˛ IR
1I . Therefore x = l
1I with l ˛ IR . It is shown in the fourth step thatthen l ‡ l >0 for x „ l <0. Without loss of generality assume x =- 1I (if this is not the case, replace x by| l | -1/2 x ) and define f := (1-5 ) e + (1+5 ) e '+ x . From e $ x = ( x + U e x - U e ' x ) and U e x = U e ' x =0 weget e $ x = x and e ' $ x =( 1I - e ) $ x = x . Therefore f = f and U e f = (1-5 ) e . However, since f ˛ E , we have U e f ‡
0, resulting in a contradiction to (1-5 )<0.Now suppose l =0. Then x =0, and e $ x = x implies ( e + sx ) = e + sx such that e + sx ˛ E and e + sx £
1I for all s ˛ IR . Therefore sx £
1I for all s ˛ IR and x =0.Finally, we show that A is power-associative and that the elements of A have positivesquares. An application of Theorem 3.2 then yields that A is a JB algebra..Since A = U e A ¯ U e ' A ¯ { a - U e a - U e ' a : a ˛ A } = IR e ¯ IR e ' ¯ { a - U e a - U e ' a : a ˛ A }, we have A = IR ¯ H with H := IR ( e - e ') ¯ { a - U e a - U e ' a : a ˛ A }. From ( e - e ') = 1I and ( e - e ') $ x = x - x =0 for x = a - U e a - U e ' a with a ˛ A , it follows that x $ y ˛ IR
1I for x , y ˛ H . Moreover, x = l
1I with l >0 for x „ a = s
1I + x ˛ A with s ˛ IR and x ˛ H . If x =0, a lies in the associative subalgebra IR a = s
1I is positive. Now consider the case x „
0. Since x = l
1I with l >0, we can define d := ( 1I + l -1/2 x ). Then d = d ˛ E , d' = ( 1I - l -1/2 x ) and a =( s + l ) d +( s - l ) d' . Therefore a lies inthe associative subalgebra IR d ¯ IR d' and a =( s + l ) d +( s - l ) d' is positive. q.e.d.In Theorem 4.1, A is associative if E ={0, 1I } or E ={0, e , e' , 1I } with some event e ; in allother cases, A is a so-called spin factor or type I JBW factor (definitions can be found in[2]).Note that the dimension of a spin factor need not be finite; indeed there is a spin factor of dimension n for each cardinal number n ‡
3, including the infinite cardinal numbers.
5. Compatibility
The following two lemmas concern orthogonal idempotent elements and will be needed for the investigation of the different notions of compatibility in this section and for the study of the matrix algebras in the next section.
Lemma 5.1:
Under the assumptions suppose that e and f are two orthogonal elements of E. Then U U U e f e f ¢ ¢ + ¢ = ( ) = U U f e ¢ ¢ . erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 9
Proof . Suppose x ˛ [0, 1I ]. Then 0 £ U U e f ¢ ¢ x £ U U e f ¢ ¢
1I =
U f e ¢ ¢ = e' - f = ( e + f ) ' .Therefore m U U e f ¢ ¢ = 0 = m U e f ( ) + ¢ for m ˛ S with m (( e + f ) ' )=0. Now consider m ˛ S with m (( e + f ) ' )>0 and define n := m U U e f ¢ ¢ / m (( e + f ) ' ) ˛ S . From ( e + f ) ' £ e' and ( e + f ) ' £ f ' , we get U U e f ¢ ¢ ( e + f ) ' =( e + f ) ' and n (( e + f ) ' ) = 1 such that n = n U e f ( ) + ¢ . Since U e ¢ U e f ( ) + ¢ = U e f ( ) + ¢ = U f ¢ U e f ( ) + ¢ , we have n = m U e f ( ) + ¢ / m (( e + f ) ' ) and thus m U U e f ¢ ¢ = m U e f ( ) + ¢ . Therefore U U U e f e f ¢ ¢ + ¢ = ( ) . In the same way we get U U U f e e f ¢ ¢ + ¢ = ( ) . q.e.d. Lemma 5.2:
Under the assumptions suppose that e and f are two orthogonalelements of E. Then a $ ( b $ x )= b $ ( a $ x ) for a ˛ U e A , b ˛ U f A and x ˛ A.Proof . From the preceding lemma we have
U U e f ¢ ¢ = U U f e ¢ ¢ . In [6] it was shown that U e U f = U f U e =0, U e' U f = U f U e' = U f and U f ¢ U e = U e U f ¢ = U e . Therefore U e , U f , U e' , U f ¢ commutepairwise. The identities e $ y = ( y + U e y - U e' y )/2 and f $ y = ( y + U f y - U f ¢ y )/2 ( y ˛ A ) then imply e $ ( f $ x ) = f $ ( e $ x ) for x ˛ A. Since this holds for all orthogonal elements in E , we have g $ ( h $ x )= h $ ( g $ x ) for g,h ˛ E with g £ e and h £ f . Therefore a $ ( b $ x )= b $ ( a $ x ) for a ˛ U e A = lin { g ˛ E : g £ e } and b ˛ U f A =lin { h ˛ E : h £ f }. q.e.d.Each condition in the following proposition represents a certain degree of compatibility;the first conditions represent a rather weak type of compatibility and the last ones a ratherstrong type. The proposition analyzes the precise logical relations among all the differentconditions. Proposition 5.3:
Under the assumptions consider the following eleven conditions fora pair of events e,f ˛ E .(i) f = U f U f e e + ¢ ( i.e ., m ( f )= m ( e ) m ( f | e )+ m ( e' ) m ( f | e' ) for all m ˛ S ).(ii) e $ ( e $ f )= e $ f .(iii) f = U f U f e e + ¢ and e = U e U e f f + ¢ .(iv) e $ ( e $ f )= f $ ( e $ f )= e $ f .(v) U e f = e $ f = U f e .(vi) U a b = U b a for a,b ˛ { e,e',f,f' } ( i.e ., m ( a ) m ( b | a )= m ( b ) m ( a | b ) for a,b ˛ { e,e',f,f' } and m ˛ S ).(vii) U a U b = U b U a for a,b ˛ { e,e',f,f' }.(viii) e $ ( f $ x )= f $ ( e $ x ) for x ˛ A ( i.e., e and f operator-commute ). (ix) e $ f , e' $ f , e f $ ¢ and ¢ ¢ e f $ lie in E . (x) e and f lie in an associative subalgebra . (xi) There are three orthogonal elements d , d , d ˛ E such that e = d + d and f = d + d . Then the following logical relations hold among these conditions : (i) (cid:219) (ii) (cid:220) (iii) (cid:219) (iv) (cid:219) (v) (cid:219) (vi) (cid:220) (vii) (cid:219) (viii) (cid:220) (ix) (cid:219) (x) (cid:219) (xi).
Proof . From the identity
U f U f e e + ¢ = { e,f,e }+{
1I - e,f,
1I - e } = 2{ e,f,e }+ f -2 e $ f =4 e $ ( e $ f ) - 4 e $ f + f we immediately get the equivalence of (i) and (ii). The implication (i) (cid:220) (iii) is obvious. The equivalence of (iii) and (iv) follows from the one for (i) and (ii), the oneof (iv) and (v) from the identities U e f ={ e,f,e }=2 e $ ( e $ f )- e $ f and U f e =2 f $ ( e $ f )- e $ f .Now suppose (v); then we also have (iv) and e' $ ( e' $ f )= f -2 e $ f + e $ ( e $ f )= f - e $ f=e' $ f suchthat U e' f =2 e' $ ( e' $ f ) - e' $ f = e' $ f and U f e' = U f ( 1I - e )= f - U f e = f - e $ f = e' $ f = U e' f . In the same way erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 10conclude that
U e f ¢ = e f $ ¢ = U f e ¢ . Furthermore U f e ¢ ¢ = e' - U e' f = e' - e ' $ f = ¢ ¢ e f $ and U e f ¢ ¢ = ¢ - ¢ f U e f = ¢ - ¢ f e f $ = ¢ ¢ e f $ = U f e ¢ ¢ . The remaining cases for (vi) are trivial. Condition(vi) implies (iii) via U f U f e e + ¢ = U f e + U f e' = U f ( e + e' )= U f
1I = f and U e U e f f + ¢ = e in the sameway, and we have the equivalence of (iii), (iv), (v) and (vi).Condition (vii) implies (vi) via U a b = U a U b
1I = U b U a
1I = U b a , and (viii) follows from (vii) bythe identities e $ y = ( y + U e y - U e' y )/2 and f $ y = ( y + U f y - U f ¢ y )/2 ( y ˛ A ). Vice versa, (viii)implies a $ ( b $ x )= b $ ( a $ x ) for x ˛ A and a,b ˛ { e,e',f,f' }; this means that the operators T a and T b defined as T a x := a $ x and T b x := b $ x commute and therefore U a =2 T a - T a and U b =2 T b - T b commute such that we have the equivalence of (vii) and (viii).From (xi) we get (viii) by Lemma 5.2 and (x) by considering the subalgebra IR d + IR d + IR d ; this subalgebra is associative and contains e = d + d as well as f = d + d .Now suppose (x). Then e , f and 1I generate an associative subalgebra such that( a $ b ) = a $ b = a $ b for a,b ˛ { e,e',f,f' }, and we have (ix). From (ix) we get (xi) by defining d := e f $ ¢ , d := e $ f and d := e' $ f . Then d and d are orthogonal since d £ d + ¢ ¢ e f $ = ¢ f and d £ d + d = f ; d and d are orthogonal since d £ d + d = e and d £ d + ¢ ¢ e f $ = e' ; d and d areorthogonal since d £ d + d = e and d £ e' . q.e.d.The identity m ( f )= m ( e ) m ( f | e )+ m ( e' ) m ( f | e' ) is a well-known rule for classical conditionalprobabilities. However, it is not anymore universally valid in a nonclassical framework likequantum mechanics. Its validity for all states m becomes a first weak and asymmetrical notionof compatibility for a pair of events e and f . This is condition (i) and is equivalent to thealgebraic condition (ii).Its validity also for exchanged roles of e and f becomes a stronger and symmetrical notionof compatibility. This is condition (iii) of the above proposition. It is equivalent to each one ofthe conditions (iv), (v) and (vi). The latter one represents another rule for classical conditionalprobabilities which is not anymore universally valid in quantum mechanics: m ( a ) m ( b | a )= m ( b ) m ( a | b ) for a,b ˛ { e,e',f,f' } and m ˛ S .A still stronger form of compatibility is described by each one of the two equivalentconditions (vii) and (viii). While (viii) represents a purely algebraic condition, (vii) has aninteresting interpretation in quantum measurement; it means that the order of two successivemeasurements in a series of measurements does not matter when one of the two successivemeasurement tests e versus e' and the other one f versus f' . For a deeper look at this, iteratedconditional probabilities and their connection to quantum measurement must be considered(see [5]). If each pair of elements in E satisfies (vii), then A is associative by (viii) whichimplies (x) such that all conditions are satisfied in this case, resulting in a classical situation. The strongest level of compatibility for two events e and f is represented by each one of the three equivalent conditions (ix), (x) and (xi). The latter one means that e and f lie in a Boolean subalgebra of E . If A is the self-adjoint part of a W*-algebra, the weakest one among all the conditions which is (i) means f = efe + e'fe' and implies ef = efe = fe such that e and f commute and (ix) holds. Therefore, in this case, all the above conditions become equivalent and there is only one single level of compatibility coinciding with the usual concept of commuting operators in quantum mechanics. Likewise (i) implies (ix) and all conditions become equivalent if A is aJBW algebra [2]. In the more general framework of the assumptions 3.1, however, thereappear to be four different levels of compatibility. For each level there is a set of equivalentconditions describing it; these sets are: (i)-(ii), (iii)-(vi), (vii)-(viii), (ix)-(xi). The verificationthat these levels really differ still requires an example of an algebra that satisfies theassumptions 3.1, but is not a JB algebra. erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 11
Corollary 5.4:
If an associative subalgebra M of A is generated by its idempotentelements, then y ‡ for y ˛ M . Proof . Any e,f ˛ E ˙ M satisfy (x) in Proposition 5.3 and thus (ix) and (xi). Therefore eachelement y ˛ M has the shape y = S t k e k with real numbers t k and orthogonal elements e ,..., e n ˛ E ˙ M . Then y = S t e k k ‡
0. q.e.d.
6. Matrix algebras
Let R be a real *-algebra with unit 1 and define R sa :={ a ˛ R : a = a *}. Note that the productin R is neither assumed to be commutative nor associative nor alternative. Let H n ( R ) denotethe space of Hermitian, or self-adjoint, n n · matrices with coefficients in R . On H n ( R )consider the product defined by a $ b :=( ab + ba )/2. This type of Hermitian matrix algebras isstudied in the present section because they are a natural candidate for a structure satisfying theassumptions 3.1.A case of particular interest is when the *-algebra R is the octonions, since then H n ( R ) is aformally real Jordan algebra for n £
3, but not for n ‡
4 [2]. Therefore one might hope to findwith n ‡ Lemma 6.1: (i)
If R sa = IR ( note that IR is identified with IR here ), then aa *= a * a for a ˛ R . If furthermore a * a „ holds for each a „ then we have a * a ‡ for a ˛ R .(ii) If R sa = IR and if a * a „ holds for each nonzero element a in R , then H ( R) is a norm-densesubalgebra of a spin factor . Proof . (i) Assume R sa = IR and a ˛ R . Then t := a + a * ˛ IR such that a and a *= t - a commute.Now assume a * a „
0 for all a „
0 and b * b <0 for some b ˛ R . Consider the real polynomialfunction h ( s ) := ( s +(1- s ) b )*( s +(1- s ) b ) = s +s(1-s)( b + b *)+(1- s ) b * b . Since h (0)= b * b <0 and h (1)=1, there exists some s o with 0< s o <1 and h ( s o )=0 such that a * a =0 for a = s o +(1- s o ) b .Therefore s o +(1- s o ) b =0, thus b ˛ IR and b * b = b ‡ n =2, R sa = IR and a * a „ „ a ˛ R . Denote by a ij the matrix whose entry inthe i -th row and j -th column is 1 and whose all other entries are zero. Define V :={ a ( a - a )+ b a + b * a : a ˛ IR , b ˛ R } ˝ H ( R) . Then H ( R) is the direct sum of V and IR .Moreover, for x = a ( a - a )+ b a + b * a ˛ V and y = a ( a - a )+ b a + b * a ˛ V with a , a ˛ IR , b , b ˛ R , we get from (i) that x y $ = a a +( b + b )*( b + b )- b b * - b b * and thus x y $ ˛ IR , x x $ ‡ x x $ „ x „
0. Therefore, x y $ becomes an inner product on V . If the real dimension of R is finite, so is the one of V and H ( R) is a finite-dimensional spin factor. If the dimension of R is not finite, neither R nor V need be complete. In this case, let W be the completion of the pre-Hilbert space V and consider the spin factor W ¯ IR
1I . Then H ( R) becomes a norm-dense subalgebra of this spin factor. q.e.d. erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 12
Lemma 6.2:
Suppose that
A=H n ( R) and E= { e ˛ A : e =e } satisfy the assumptions If a = a holds only for a =0 and a =1 in R sa ( i.e., R sa does not contain nontrivialidempotents ), then R sa = IR .(ii) If n ‡ then R must not contain any element a with a * a = aa * ˛ IR and a * a = aa *<0. Particularly , R sa must not contain any element a with a ˛ IR and a <0. Moreover , a * a „ or aa * „ for a „ If R sa = IR , we have a * a = aa *>0 for a „ If n ‡ and R sa = IR , then R is alternative and does not contain any zero divisors .(iv) If n ‡ then R is associative ( and A=H n ( R) becomes a special Jordan algebra [2]).
Proof . Denote by a ij the matrix whose entry in the i -th row and j -th column is 1 and whoseall other entries are zero.(i) Suppose that a = a holds only for a =0 and a =1 in R sa . Consider e := a ˛ E . If d ˛ E with d £ e , then d = U e d ={ e , d , e }= a e , where a ˛ R sa is the first entry in the first row of the matrix d .Thus a e = d = d = a e and a = a such that either a =0 and d =0 or a =1 and d = e . Therefore U e A = lin { d ˛ E : d £ e }= IR e . For any a ˛ R sa now consider the matrix a e in A and conclude a ˛ IR from a e ={ e , a e , e }= U e ( a e ) ˛ IR e .(ii) Suppose n ‡ a * a = aa *<0 for an element a ˛ R . Without loss of generality assumethat a * a = aa * = -1 (if this is not the case, replace a by a /(- a * a ) ). Then consider the two matrices e := a and f := (1-5 ) a + (1+5 ) a + a a + a * a . Both matrices are idempotentand thus lie in E , but U e f ={ e , f , e }= (1-5 ) e is not positive since (1-5 )<0.Now assume n ‡
2 and a * a = aa *=0 for an element a ˛ R . Consider x := a a + a * a . Then x =0 and e x $ = x /2. Therefore ( e + sx ) = e + sx such that e + sx ˛ E and e + sx £
1I for all s ˛ IR . Thus sx £
1I for all s ˛ IR and x =0 such that a =0. The remaining part of (ii) follows from Lemma 6.1.(iii) Suppose n ‡
3 and R sa = IR . For any two elements a , b ˛ R with a * a =1 consider thefollowing four matrices: e :=( a + a + a a + a * a )/2, x := b a + b * a , d := a + a and f := a .Then e , d , f ˛ E , e £ d , and d and f are orthogonal. Therefore f and each g ˛ E with g £ e areorthogonal such that f g $ =0 by Lemma 3.3. From U e x ˛ lin { g ˛ E : g £ e } we get that f $ U e x =0. Multiplying out the matrices gives first 8 U e x =[ a *( a b )- b ] a +[ a *( a b )- b ]* a andfinally 16 f $ U e x = [ a *( a b )- b ] a +[ a *( a b )- b ]* a such that a *( a b )= b .For a „ a /( a * a ) and get a *( a b )=( a * a ) b . This identity also holdsfor a =0. Since a + a * ˛ R sa = IR , we have ( a + a *)( a b ) = (( a + a *) a ) b and thus a ( a b )+ a *( a b ) = a b +( a * a ) b . Combining this with the other identity yields a ( a b )= a b . This is the leftalternative law. Taking adjoints, we obtain the right alternative law.Since R is alternative, each pair of elements in R generates an associative subalgebra. Therefore, for any two elements a , b ˛ R , all four elements a , a *, b , b * lie in the associative subalgebra generated by a - a * and b - b * because a + a * and b + b * are real numbers. Thus ( a b )( a b )* = ( a b )( b * a *) = a ( bb *) a * = ( aa *)( bb *). If now a b =0, a =0 or b =0 must hold. (iv) Suppose n ‡
4. For a , b , g ˛ R consider the following four matrices in H n ( R) : x := a a + a * a , y := b a + b * a , z := g a + g * a , and e := a + a . Then e ˛ E , x ˛ { e,A,e }= U e A , and z ˛ { e',A,e' }= U e' A . Lemma 5.2 implies x $ ( z $ y )= z $ ( x $ y ). Multiplying out the matrices, we get a ( b g )=( a b ) g . q.e.d.When Lemma 5.2 is available, the proof of part (iv) of the above lemma becomes an exactcopy of the one of the same result for Jordan algebras [2], but the proof of (iii) is different. erd Niestegge A Representation of Quantum Measurement in Nonassociative Algebras Page 13
Theorem 6.3:
Suppose that
A=H n ( R ) and E= { e ˛ A : e =e } satisfy the assumptions with a real *-algebra R with unit and R sa = IR . Then A is a formally real Jordan algebra . Ifn =2,
A is a spin factor . If n =3,
R is the real numbers, the complex numbers, the quaternions orthe octonions . If n ‡ R is the real numbers, the complex numbers or the quaternions . Proof . If n =1, A=H ( R )= IR . If n =2, A=H ( R ) is a spin factor by Lemma 6.1 (ii); thecompleteness follows from the assumptions 3.1 since dual spaces are Banach spaces. If n =3, R is an alternative real division algebra by Lemma 6.2 (iii), and there are no other such algebrasthan the real numbers, the complex numbers, the quaternions or the octonions [1]. If n ‡ R isan associative real division algebra by Lemma 6.2 (iii) and (iv), and there are no other suchalgebras than the real numbers, the complex numbers and the quaternions [1]. In all cases, A=H n ( R ) is a formally real Jordan algebra [2], if the involution * on R coincides with theusual conjugation on these division algebras. This conjugation is characterized by linearityover IR and the requirements 1*=1 and j *=- j whenever j =-1. The identity 1*=1 follows from R sa = IR . Now suppose j =-1 and define t := j*j ˛ IR . Using the right alternative lay, we get - tj =-( j*j ) j = - j*j = j* and taking adjoints j = - tj * = t j , so that t = – j *= – j . However, j *= j isimpossible since then j ˛ IR such that j „ -1. q.e.d.We now look at some further consequences of the above lemmas. Besides the realnumbers, complex numbers, quaternions and octonions, there are some more *-algebras, thepotential relevance of which for modern physics is sometimes discussed. A natural question istherefore whether H n ( R ) satisfies the assumptions 3.1 when R is one of these *-algebras. Thebioctonions, quateroctonions, octooctonions, which are linked with the exceptional Lie groupsby the so-called magic square [1], do not satisfy the condition R sa = IR and are thus not coveredby Theorem 6.3. However, they are ruled out by Lemma 6.2 (ii). A further type of *-algebrasarises when the Cayley-Dickson construction [1] is continued beyond the octonions. This caseis covered and ruled out by Theorem 6.3 for n ‡ n =2 by Lemma6.1 (ii). The split-complex numbers, split-quaternions, split-octonions are excluded byTheorem 6.3 for n ‡ n ‡
7. Conclusions
The structure considered here is motivated by the results in [6] where an abstract settingfor conditional probabilities and the Lüders-von Neumann quantum measurement wasintroduced. The assumptions 3.1 do not represent the most general case of this setting, but amore specialized case where the conditional probabilities can be expressed by anonassociative product as it is possible in the Jordan operator algebras. This specialized case is ideally suited for studying the question whether matrix algebras can provide examples of that setting.
We have seen that an algebra A satisfying the assumptions 3.1 need not necessarily be a Jordan algebra, but still features some important properties of an operator algebra. This includes existence and uniqueness of the conditional probabilities as well as the uniqueness of the spectral resolution. However, the spectral resolution does not exist for all elements of the algebra, but only for those which generate an associative subalgebra with positive squares.