• [0,1] which i s d i s p e r s i v e w i t h respect to every P £ US^, , a hidden s t a t e induces or i s a s s o c i a t e d w i t h a d i s p e r s i o n - f r e e p r o b a b i l i t y measure which somehow assigns an exact 0 or 1 value to such P f. US^ . . And so i n an HV 164 r e c o n s t r u c t i o n o f quantum mechanics, the presumed incompleteness o f quantum mechanics i s r e f l e c t e d by the f a c t t h a t the set of quantum Exp^, measures i s a proper subset o f a l a r g e r set of measures which i n c l u d e s the d i s p e r s i o n - f r e e HV measures. The d i s p e r s i o n - f r e e measures added by an HV theory may be c l a s s i c a l p r o b a b i l i t y measures on some Boolean s t r u c t u r e proposed by the HV theory, or they may be some s o r t of g e n e r a l i z e d p r o b a b i l i t y measures defined on the quantum P A ^ ( o r on substructures o f PQ^)* Von Neumann, Jauch-Piron and Gleason-Kochen-Specker prove the i m p o s s i b i l i t y o f three kinds o f ge n e r a l i z e d d i s p e r s i o n - f r e e measures on the quantum P A ^ s t r u c t u r e s , as described i n Chapter V(D); they thus r u l e out three kinds of HV t h e o r i e s , as s h a l l be elaborated below. But besides these three proposed but impossible kinds of HV t h e o r i e s , c o n t e x t u a l HV t h e o r i e s whose d i s p e r s i o n - f r e e measures avoid the above i m p o s s i b i l i t y proofs have a l s o been proposed. In a l l , f o u r cases, each quantum Exp^ measure i s represented i n the proposed HV theory as a mixture or complex, e.g., a convex sum or weighted i n t e g r a l , of d i s p e r s i o n - f r e e HV measures. And a l l f o u r kinds o f HV proposals impose a s t a t i s t i c a l c o n d i t i o n r e q u i r i n g t h a t the complexes which represent the quantum Exp^, measures i n the HV theory must y i e l d s t a t i s t i c a l r e s u l t s which reproduce the r e s u l t s given by the quantum Exp^, measures (and so f a r observed by experiment) (Kochen-Specker, 1967, p. 59; B e l i n f a n t e , 1973, p. 9). However, as Kochen-Specker argue, the i m p o s i t i o n of t h i s s t a t i s t i c a l c o n d i t i o n alone does not yet take i n t o c o n s i d e r a t i o n the s t r u c t u r a l and f u n c t i o n a l r e l a t i o n s among the quantum magnitudes (and p r o p o s i t i o n s ) . These r e l a t i o n s are embodied i n the a l g e b r a i c s t r u c t u r e of the quantum magnitudes, and concordantly, i n the P A M s t r u c t u r e o f the 165 quantum p r o p o s i t i o n s . Von Neumann, Jauch-Piron, Gleason, and Kochen-Specker do take t h i s c o n s i d e r a t i o n i n t o account by r e q u i r i n g t h a t some or a l l o f the operations and r e l a t i o n s of P must be preserved i n an HV r e c o n s t r u c t i o n of quantum mechanics. Such requirements may be c a l l e d s t r u c t u r a l c o n d i t i o n s . As shown a t l e n g t h i n Chapter V(D), each of these authors imposes a s t r u c t u r a l c o n d i t i o n which b o i l s down to the requirement that d i s p e r s i o n - f r e e HV measures, qua g e n e r a l i z e d p r o b a b i l i t y measures on the quantum P , must preserve the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s of PQ^ ( i . e . , P ^ ^ - p r e s e r v a t i o n ) , or i n other words, proposed d i s p e r s i o n - f r e e HV measures must be b i v a l e n t homomorphisms() i n Chapter V(D). The domain of an ex p e c t a t i o n - f u n c t i o n i s the set of quantum magnitudes as represented by operators on a H i l b e r t space. And the f u n c t i o n a l r e l a t i o n s among the magnitudes are given by the f u n c t i o n a l r e l a t i o n s among the operators, t h a t i s , by the a l g e b r a i c s t r u c t u r e of the operators. So i n terms of the quantum p r o p o s i t i o n s qua idempotent magnitudes, a necessary c o n d i t i o n f o r an HV r e c o n s t r u c t i o n o f quantum mechanics i s , i n von Neumann's view, the existence • 1 of d i s p e r s i o n - f r e e Exp^ f u n c t i o n s on the quantum s t r u c t u r e s . As mentioned i n Chapter V(D), usi n g h i s trace-formalism, von Neumann proves t h a t no such d i s p e r s i o n - f r e e E xp w e x i s t ; I r e f e r r e d to t h i s r e s u l t as von Neumann's i m p o s s i b i l i t y proof. In a d d i t i o n , von Neumann proves that homogeneous ex p e c t a t i o n - f u n c t i o n s do e x i s t and i n f a c t correspond to the quantum Exp^ f u n c t i o n s induced by the pure quantum ty s t a t e s . So the quantum Exp^, cannot be represented as mixtures o f d i s p e r s i o n - f r e e Exp^ , f i r s t because the quantum Exp^, are themselves homogeneous (where by d e f i n i t i o n a homogeneous Exp cannot be represented as a weighted sum of 168 d i f f e r e n t E x p - f u n c t i o n s ) , and second because the d i s p e r s i o n - f r e e E x P w d o not e x i s t (von Neumann, 1932, p. 324). I t i s thus t h a t an HV r e c o n s t r u c t i o n of quantum mechanics i s impossible, according to von Neumann. In 1966, B e l l d i s c r e d i t e d von Neumann's i m p o s s i b i l i t y proof by arguing that i t r e s t s upon an i n c o n s i s t e n c y between the requirement t h a t HV exp e c t a t i o n - f u n c t i o n s s a t i s f y (vNjS) and the requirement t h a t HV expe c t a t i o n - f u n c t i o n s be d i s p e r s i o n - f r e e . For (vN&) r e q u i r e s the a d d i t i v i t y of the expectation values of incompatible magnitudes and incompatible p r o p o s i t i o n s qua idempotent magnitudes, and the d i s p e r s i o n - f r e e e x p e c t a t i o n value of any magnitude or p r o p o s i t i o n i s an eigenvalue of the magnitude or p r o p o s i t i o n . But sin c e the eigenvalues of incompatible magnitudes or p r o p o s i t i o n s are not a d d i t i v e , an HV e x p e c t a t i o n - f u n c t i o n which s a t i s f i e s (vN(&) and i s d i s p e r s i o n - f r e e i s impossible ( B e l l , 1966, p. 449). The Kochen-Specker v e r s i o n of von Neumann's i m p o s s i b i l i t y proof shows c l e a r l y how (vN#) i s the c u l p r i t i n the proof and so f u r t h e r s u b s t a n t i a t e s B e l l ' s c r i t i c i s m (Kochen-Specker, 1967, pp. 81-82). Such HV proposals whose imposed c o n d i t i o n s are i n c o n s i s t e n t w i t h each other are c a l l e d HV t h e o r i e s of the zero-th k i n d by B a l i n f a n t e ; t h e i r i m p o s s i b i l i t y i s not s u r p r i s i n g . B e l l a l s o appeals to the problem of measurement i n t e r a c t i o n i n order to argue t h a t HV measures (or ex p e c t a t i o n - f u n c t i o n s ) need not s a t i s f y (vn^). The r e s u l t o f a measurement of the sum + P^ o f two incompatible p r o p o s i t i o n s cannot be c a l c u l a t e d by simply adding together the r e s u l t s of separate measurements f o r P ^ > P 2 ' F ° r 3 S e x e m P l i f i e d by von Neumann (1932, p. 310), a measurement of a sum P^ + P^ o f incompatibles i n v o l v e s an experimental arrangement which i s e n t i r e l y d i f f e r e n t from the arrangements by which P^ and P Q are each measured sep a r a t e l y . Now although the 169 expectation-value assigned to + P^ by any quantum Exp^. always does equal the sum of the expectation-values assigned by Exp^, to each of the P^ , P j s e p a r a t e l y , t h i s i s not a t r i v i a l or necessary f e a t u r e of the quantum Exp^ measures. Rather, i t i s a very p e c u l i a r f e a t u r e of the quantum Exp^, measures, e s p e c i a l l y when, as B e l l suggests, one remembers wi t h Bohr "the i m p o s s i b i l i t y of any sharp d i s t i n c t i o n between the behavior of atomic o b j e c t s and the i n t e r a c t i o n w i t h measuring instruments which serve to d e f i n e the c o n d i t i o n s under which the [quantum] phenomena appear" (Bohr quoted by B e l l , 1966, p. 447). B e l l concludes t h a t there i s no reason to demand that proposed d i s p e r s i o n - f r e e HV measures must be a d d i t i v e w i t h respect to incompatible magnitudes and p r o p o s i t i o n s , as (vN&) r e q u i r e s . So when von Neumann imposes h i s c o n d i t i o n and then proves t h a t d i s p e r s i o n - f r e e HV measures are impossible and thus proves that an HV r e c o n s t r u c t i o n of quantum mechanics i s i m p o s s i b l e , he i s open to the charge of begging the HV question since (vNi$) i s u n j u s t i f i e d . Furthermore, von Neumann's i m p o s i t i o n of (vNsK) on proposed d i s p e r s i o n - f r e e HV measures begs the HV question i n another way. One of the c o n d i t i o n s which von Neumann incorporates as p a r t of h i s l i s t o f c o n d i t i o n s d e f i n i n g an e x p e c t a t i o n - f u n c t i o n Exp i n general i s the f o l l o w i n g , which he l a b e l s (E): (E) I f A,B,... are a r b i t r a r y magnitudes, then there i s an a d d i t i o n a l magnitude A + B + ••• (which does not depend on the choice of the e x p e c t a t i o n - f u n c t i o n ) , such that Exp(A + B + •••) = Exp(A) + Exp(B) + ••• (von Neumann, pp. 309, 311). With t h i s c o n d i t i o n ( E ) , von Neumann l e t s the 170 e x p e c t a t i o n - f u n c t i o n s d e f i n e the sum of incompatible magnitudes, e.g., the sum of A, B i s that magnitude which s a t i s f i e s (E) f o r a l l e x p e c t a t i o n - f u n c t i o n s . Von Neumann motivates t h i s d e f i n i t i o n by two f a c t s : A A The sum of the operators A, B (re p r e s e n t i n g the magnitudes A, B) i s i t s e l f a s e l f - a d j o i n t operator which can represent a quantum magnitude; and f o r a l l quantum Exp^ e x p e c t a t i o n - f u n c t i o n s , Exp^(A + B) = Exp^(A) + Exp^(B). Now i f we assume that d i s p e r s i o n - f r e e HV E x P w e x p e c t a t i o n - f u n c t i o n s do e x i s t , then the sum of A, B as defined by a l l the quantum Exp^ and HV Exp w may be d i f f e r e n t from the sum of A, B as defined by j u s t a l l the A A quantum Exp^ . And f o r example, although the operator A + B does represent the magnitude which i s the sum of A, B as defined by a l l the A A quantum Exp^, , the operator A + B may not represent the sum of A, B as defined by a l l the quantum Exp^, and HV E x P w » ± n which case A A A A Exp (A + B) i Exp (A) + Exp (B), c o n t r a r y to von Neumann's (vN^) c o n d i t i o n , w w w Of course, i f the d i s p e r s i o n - f r e e HV E X P W a r e impossible, then the two sums are the same. However, von Neumann imposes (vN#) which presumes th a t the two sums are the same (and so presumes th a t d i s p e r s i o n - f r e e HV E x P w d o not e x i s t ) and which r e q u i r e s proposed d i s p e r s i o n - f r e e HV Exp to s a t i s f y w Exp (A + B) = Exp (A) + Exp (B), and then von Neumann proves t h a t the W W W proposed d i s p e r s i o n - f r e e HV E x P w a r e impossible. Thus von Neumann i s begging the HV question because the i m p o s i t i o n o f c o n d i t i o n (vN&) presumes what i s being proved, namely, the i m p o s s i b i l i t y or non-existence of 2 d i s p e r s i o n - f r e e HV E x P w f u n c t i o n s . As mentioned i n Chapter V(D), u s i n g the s t r u c t u r a l c o n d i t i o n (JBiO, Jauch-Piron prove i n t h e i r C o r o l l a r y 1 that d i s p e r s i o n - f r e e measures are impossible on any i r r e d u c i b l e orthomodular l a t t i c e . T h i s , they say, i s 1 7 1 von Neumann's o l d r e s u l t , i . e . , von Neumann's proof of the i m p o s s i b i l i t y of d i s p e r s i o n - f r e e measures, proven without the contentious c o n d i t i o n (vN^). However, Jauch-Piron argue t h a t the quantum s u p e r s e l e c t i o n r u l e s ensure t h a t the quantum orthomodular l a t t i c e P Q M L s t r u c t u r e s are not i r r e d u c i b l e but ra t h e r are r e d u c i b l e l a t t i c e s w i t h n o n - t r i v i a l c e n t r e s . So C o r o l l a r y 1 does not r u l e out d i s p e r s i o n - f r e e measures on the quantum P Q ^ • Now according to Jauch-Piron, a quantum P which does admit QML hidden-variables i s c h a r a c t e r i z e d by the f o l l o w i n g property: Every measure on a PQML which admits hidden-variables can be represented as a weighted sum o f d i s p e r s i o n - f r e e measures on » ^ n p a r t i c u l a r , every quantum Exp^ measure on P Q ^ l c a n be so represented. Then i n t h e i r C o r o l l a r y 3 and again i n t h e i r Theorem 2, Jauch-Piron prove t h a t an orthomodular l a t t i c e admits hidden-variables only i f a l l i t s elements are mutually compatible, i . e . , o n ly i f the l a t t i c e i s Boolean. So any quantum P Q ^ which contains incompatible elements does not admit h i d d e n - v a r i a b l e s , and hence hidden-variables are impossible i n quantum mechanics (Jauch-Piron, 1 9 6 3 , pp. 8 3 5 - 8 3 7 ) . Bub's e l u c i d a t i o n of Jauch-Piron's work shows c l e a r l y how c o n d i t i o n (JP&) i s the c u l p r i t i n t h e i r i m p o s s i b i l i t y p r o o f ( s ) . For Bub shows how the quantum Exp^, measures on a P Q J ^ cannot be represented as weighted sums of d i s p e r s i o n - f r e e measures on Pq^L when the d i s p e r s i o n - f r e e HV measures are r e q u i r e d to s a t i s f y (JP£>) (Bub, 1974-, pp. 6 1 - 6 2 ) . For example, consider a quantum Exp^. which assigns values to two incompatible atoms P^ , P^ ° F 3 P Q M L 3 3 f o l l o w s : E x P y ( p Y > = 1» E x P y ( V = h*M € (0»1)» and s i n c e P^ A p^ = 0 , Exp^(P^ A P^) = Exp^ (0 ) = 0 . According to the Jauch-Piron c h a r a c t e r i z a t i o n o f a hidden-variables p r o p o s a l , i f ? n admits QML 172 hidden-variables then t h i s Exp^, measure on P A M L can be represented as a weighted sum 2 \.w. , where 2 \. = 1 and each w. i s a 4 3 . 1 1 . i i I l d i s p e r s i o n - f r e e (HV) measure on PQ M L • N O W ± n order to reproduce the assignment Exp^(P^,) = 1, each w^ must a s s i g n the value 1 to P^ . , i . e . , f o r every w^ i n the sum r e p r e s e n t i n g Exp^, , w ^ ( p ^ . ^ = ^ s o t h a t 2 X.w.(P.) = 1 = Exp, ( P . ) . And s i n c e P. A P = 0, w.(P,» A P ) = w.(0) ^ l l Y Y V V cp l Y cp i = 0, f o r every w^ i n the sum re p r e s e n t i n g Exp^ . Moreover, none o f the w^ can a s s i g n the value 1 to P^ because by ( J P ^ ) , w . ^ P Y ^ = ^ A N D w.(P ) = 1 y i e l d s w.(P, A P ) = 1, which c o n t r a d i c t s w.(P, A P ) = 0; i c p J 1 Y 9 1 V ) r e q u i r e s a proposed d i s p e r s i o n - f r e e HV measure to a s s i g n the value 1 to P, A P i f i t assigns the value.- 1 to each P. , P , Y cp Y
), e s p e c i a l l y when we r e c a l l the problem of measurement i n t e r a c t i o n . So when Jauch-Piron impose t h e i r s t r u c t u r a l c o n d i t i o n (JP, i . e . , the quantum ge n e r a l i z e d Exp^, p r o b a b i l i t y measures on the non-Boolean P s t r u c t u r e of the quantum phase space H ( H i l b e r t space), i n terms o f a c l a s s i c a l measure space , i . e . , c l a s s i c a l u. p r o b a b i l i t y measures on the HV Boolean P M , s t r u c t u r e of a po s t u l a t e d HV c l a s s i c a l phase space 2 (Kochen-Specker, 1967, pp. 62, 75). Thus an HV theory may be s a i d to be a 3 Boolean r e c o n s t r u c t i o n o f quantum mechanics. More e x p l i c i t l y , as described by Kochen-Specker, a Boolean HV r e c o n s t r u c t i o n o f quantum mechanics can be formulated as f o l l o w s . L i k e the formalism o f c l a s s i c a l mechanics described i n Chapter I I I , an HV theory p o s i t s a c l a s s i c a l phase space 2', each p o i n t w € 2 represents a pure hidden s t a t e , and each r e a l - v a l u e d ( B o r e l ) f u n c t i o n f . : 2 -*• R represents a magnitude i n the HV theory. The idempotent f u n c t i o n s on 2, or e q u i v a l e n t l y , the B o r e l subsets o f 2, form a Boolean s t r u c t u r e which may be l a b e l e d ? ^ . L i k e the P C M s t r u c t u r e , t h i s ? ^ i s regarded as the 175 p r o p o s i t i o n a l s t r u c t u r e of the HV theory. That i s , an idempotent f u n c t i o n fp : & ->• {0,1}, or corresponding B o r e l subset c Q} represents a p r o p o s i t i o n i n the HV theory. Each pure hidden s t a t e w induces a d i s p e r s i o n - f r e e c l a s s i c a l p r o b a b i l i t y measure u. : P ^ -»• {0,1} which i s a b i v a l e n t homomorphism on P0, 7 . The HV r e c o n s t r u c t i o n of quantum mechanics proceeds by r e p r e s e n t i n g or a s s o c i a t i n g each of the quantum magnitudes A,B,... w i t h a r e a l - v a l u e d f u n c t i o n f , f ,... on the HV phase space 2. Each quantum p r o p o s i t i o n P, qua idempotent magnitude, i s l i k e w i s e a s s o c i a t e d w i t h an idempotent f u n c t i o n f on 2 or corresponding B o r e l subset W o f 2. That i s , quantum P P p r o p o s i t i o n s are a s s o c i a t e d w i t h the elements o f ; and l e t % l a b e l HV t h i s a s s o c i a t i o n . Kochen-Specker take the s t r u c t u r e o f the quantum p r o p o s i t i o n s to be a p a r t i a l - B o o l e a n algebra P n M f l 5 t h i s f a c t i s f u r t h e r discussed below. Next, each quantum pure s t a t e ty i s represented i n the HV r e c o n s t r u c t i o n as a mixed s t a t e which induces a d i s p e r s i v e c l a s s i c a l p r o b a b i l i t y measure [i^ : P ^ -»• [0,1] on the Boolean P ^ s t r u c t u r e . In the HV theory, these d i s p e r s i v e u-^ measures represent the quantum Exp^. measures. And these u.^ measures are r e q u i r e d to s a t i s f y the s t a t i s t i c a l c o n d i t i o n , which Kochen-Specker give as f o l l o w s : For any quantum ty and f o r any quantum P, f p ( w ) du.^({w}) = Exp^.(P) (Kochen-Specker, 1967, pp. 61, 75). Now by d e f i n i t i o n , f o r any fp on 2 and f o r any hidden s t a t e w 6 2, f p ( w ) = 1 i f w € Wp and f (w) = 0 i f w € WpX . So by s u b s t i t u t i o n , the s t a t i s t i c a l c o n d i t i o n reduces t o : Exp^(P) = = J 1 d^C {»} )• + / t f 0 d(^({w}) = u.^(Wp), where tfp = f ^ C {!})•. Thus f o r a quantum system (or ensemble of quantum systems) whose s t a t e i s given 176 by y i n quantum mechanics, the p r o b a b i l i t y that the quantum proposition P i s true i s equal to the p r o b a b i l i t y that the pure hidden state w of the quantum system i s a member of that subset Wp c <2 of hidden states with respect to which the HV representative of P, namely, fp , has the value 1. Besides the s t a t i s t i c a l condition, Kochen-Specker also impose the following s t r u c t u r a l condition: The as s o c i a t i o n % of the quantum propositions with the elements of P U „ must be an imbedding(o) which nv preserves the pQjyjA structure of the quantum propositions. That i s , an imbedding(o) % : P Q ^ P^y i s a necessary condition f o r an HV reconstruction of quantum mechanics, according to Kochen-Specker. The arguments by which Kochen-Specker motivate t h i s imbedding(A) condition are further discussed below. Next, Kochen-Specker prove i n t h e i r Theorem 0, discussed i n Chapter IV(F), that an imbedding() % : P A M . -»• P e x i s t s IFF a complete QMA nV c o l l e c t i o n of bivalent homomorphisms(c>) h : PQ^ A {°»1} e x i s t s . We can better understand the " i f " h a l f of t h i s b i c o n d i t i o n a l by noting that the c l a s s i c a l p r o b a b i l i t y measures u. : P I T T I -»• {0,1} and L L , : P „ „ -»• [0,1] of J w HV \|r HV the HV reconstruction can also be regarded, v i a the imbedding(d>) % : > a s generalized p r o b a b i l i t y measures on the quantum PQ M A • The r e l a t i o n s h i p s among these mappings can be schematized as follows: 177 The equivalence between the quantum Exp^. : PQ M A -»• [0,1] and the composition u-^ p % : PQJ^ [0,1] i s ensured by the s t a t i s t i c a l c o n d i t i o n . And f o r every pure hidden s t a t e w, the composition u^o % : PQJJA -*• {0,1} i s a g e n e r a l i z e d d i s p e r s i o n - f r e e HV measure on PQJJA which preserves the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f PQ^ A > o r i n other words, each composition u^o % i s a b i v a l e n t homomorphism(cb) on PQ^a • Moreover, as described i n Chapter I V ( F ) , an imbedding i s by d e f i n i t i o n an i n j e c t i v e mapping, i . e . , f o r any ? ± ? ? 2 i n P Q M , i % ( P 2 ) . And by the s e m i - s i m p l i c i t y property o f the Boolean s t r u c t u r e P , f o r any HV f., t f„ i n P , there i s a b i v a l e n t homomorphism on P , namely, a r 2 HV HV c l a s s i c a l d i s p e r s i o n - f r e e p r o b a b i l i t y measure u.^ : P -»• {0,1} f o r some w, such t h a t p, ( f _ ) * u. ( f ). So i f the imbedding(cb) % : P . U . •+ P U „ W W r 2 yMA nv e x i s t s , then f o r every pure hidden s t a t e w, the composition \irp % i s a w we b i v a l e n t homomorphism(A) on P.... . And f o r any P„ i P„ i n P„ QMA J 1 2 QMA can be sure t h a t f p = % ( P 1 ) t %(P 2> = f p i n P H V , and we can be sure 178 t h a t f o r some w, u. (%(P.)) 4 M- ( % ( P 0 ) ) , t h a t i s , we can be sure t h a t the W 1 W 2 c o l l e c t i o n o f b i v a l e n t homomorphisms(i) on PQJ^ ^ s complete. Conversely, i f a complete c o l l e c t i o n o f b i v a l e n t homomorphisms(i>) e x i s t on P ^ ^ , then as described i n Chapter I V ( F ) , P ^ ^ can be imbeddedCci) i n t o a C a r t e s i a n product Boolean s t r u c t u r e . For example, any 2 2 two-dimensional H i l b e r t space P.... ( o r P - „ T ) can be imbedded(A) i n t o QMA QML the C a r t e s i a n product Boolean s t r u c t u r e ( Z 2 ^ *?i o r e 9 . u i v a l e n t T y , r 2 II (Z^). > where r i s the c a r d i n a l i t y o f the set o f maximal Boolean 1 1 2 substructures o f the . This C a r t e s i a n product Boolean s t r u c t u r e can be taken to be the Boolean P s t r u c t u r e o f a proposed HV r e c o n s t r u c t i o n HV 2»r of quantum mechanics, e.g., (Z_) can be regarded as the P „ „ of a 2 HV 2 2 proposed HV r e c o n s t r u c t i o n o f the quantum mechanics o f . Or i n other words, as described by Bub, the c l a s s i c a l measure space = X which provides a Boolean HV r e c o n s t r u c t i o n o f the quantum 2 2 mechanical s t a t i s t i c a l r e s u l t s given by can be regarded as a C a r t e s i a n product measure space X = n X. where i ranges over the set 2 1 1 of maximal Boolean substructures o f P . „ and each X. = <2.,P t T„,u.> i s a QM i a/ HV ^ i c l a s s i c a l measure space introduced f o r each maximal Boolean subst r u c t u r e ,2 QM mBS o f (Bub, 1974, p. 145). Since each mBS. of pj? i s 1 WM 1 QM 2 2 isomorphic t o (Z ) , each i s isomorphic t o (Z ) , and so 2 HV. 2 1 r 2 P „ „ = II .'P„„ i s the Ca r t e s i a n product FI , (Z„). mentioned above. HV . nv. 2 1 1 1 1 Now the Kochen-Specker proof o f the i m p o s s i b i l i t y o f such a proposed r e c o n s t r u c t i o n o f the quantum mechanics o f HV proceeds i n two stages. F i r s t i n Theorem 1, which i s t h e i r v e r s i o n o f Gleason's i m p o s s i b i l i t y proof, Kochen-Specker show that b i v a l e n t homomorphisms(i>), i . e . , g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measures, are 179 impossible on any (and t h i s r e s u l t a l s o a p p l i e s to PQJJJ^ ' T N E N f o l l o w s by Kochen-Specker's Theorem 0 t h a t an imbedding(o) o f any PQ^A i n t o any proposed Boolean P ^ s t r u c t u r e o f an HV r e c o n s t r u c t i o n i s impossible, and hence, s i n c e such an imbedding(i) i s a necessary c o n d i t i o n f o r an HV r e c o n s t r u c t i o n , an HV r e c o n s t r u c t i o n o f the quantum mechanical s t a t i s t i c a l r e s u l t s o f i n terms of some c l a s s i c a l HV QM Y measure space <8,P r,u> i s impossible; t h i s i s the second stage of the HV Kochen-Specker proof o f the i m p o s s i b i l i t y o f an HV r e c o n s t r u c t i o n o f quantum mechanics. So while Gleason's i m p o s s i b i l i t y proof and Kochen-Specker's Theorem 1 j u s t show the i m p o s s i b i l i t y o f b i v a l e n t homomorphisms(cb), i . e . , g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measures, on P^~ , Kochen-Specker's Theorem 0 and imbedding(A) c o n d i t i o n connect t h i s r e s u l t w i t h the f u r t h e r question of the p o s s i b i l i t y o f a type of HV r e c o n s t r u c t i o n o f n v quantum mechanics. For the imbedding(i) c o n d i t i o n , according to which an imbedding(o) % : Pjjy ^ s a n e c e s s a r v c o n d i t i o n f o r such an HV r e c o n s t r u c t i o n , ensures t h a t proposed c l a s s i c a l d i s p e r s i o n - f r e e HV measures '"w : PHV ~* preserve the PQ^a s t r u c t u r e o f the quantum p r o p o s i t i o n s so t h a t , f o r each hidden s t a t e w, the composition u. o % i s a g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measure on P-. M A • And Theorem 0, which QMA b i c o n d i t i o n a l l y connects the existence o f a complete c o l l e c t i o n o f such measures on a P.,,. w i t h the existence o f an imbedding(<4) % : P „ „ A -»• P I T „ , QMA QMA HV thus e n t a i l s t h a t the existence o f a complete c o l l e c t i o n o f g e n e r a l i z e d , d i s p e r s i o n - f r e e Gleason measures i s a necessary c o n d i t i o n f o r a < S 2 , P „ „ , | J > HV type o f HV r e c o n s t r u c t i o n . In t h i s way, Kochen-Specker apply Gleason's r e s u l t to the c o r r e c t l y formulated HV q u e s t i o n ; w h i l e i n c o n t r a s t , the . von Neumann and the Jauch-Piron proofs do not even address the HV question 180 as so formulated. L i k e the s t r u c t u r a l c o n d i t i o n s (vN$) and (JP&), the s t r u c t u r a l c o n d i t i o n P ^ ^ - p r e s e r v a t i o n , whose i m p o s i t i o n upon proposed d i s p e r s i o n - f r e e HV measures i s e n t a i l e d by the i m p o s i t i o n o f the (vN6) c o n d i t i o n , or the (JPo) c o n d i t i o n , or the (Ga) c o n d i t i o n , or the Kochen-Specker imbedding() c o n d i t i o n , has a l s o been subject to the three s o r t s of c r i t i c i s m s l i s t e d i n the Preface above. These c r i t i c i s m s w i l l be elaborated next. But i n S e c t i o n B, another p e r s p e c t i v e on the problem o f hidden-variables i s introduced, according to which the s t r u c t u r a l c o n d i t i o n of P ^ ^ - p r e s e r v a t i o n emerges unscathed by these c r i t i c i s m s . In B e l i n f a n t e ' s view, the type of HV theory proved impossible by Gleason and Kochen-Specker i s l i k e the types proved impossible by von Neumann and Jauch-Piron; they are a l l HV t h e o r i e s of the z e r o t h k i n d whose i m p o s s i b i l i t y i s due to an i n c o n s i s t e n c y between the c o n d i t i o n s which the proposed d i s p e r s i o n - f r e e HV measures are r e q u i r e d to s a t i s f y ( B e l i n f a n t e , 1973, p. 17). However, the s t r u c t u r a l c o n d i t i o n of P ^ ^ - p r e s e r v a t i o n i s not simply i n c o n s i s t e n t w i t h the d i s p e r s i o n - f r e e c o n d i t i o n , i n the way t h a t i s . Nor i s P ^ ^ - p r e s e r v a t i o n i n c o n s i s t e n t w i t h the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h the s t a t i s t i c a l c o n d i t i o n , i n the way t h a t (JP&) i s . On the c o n t r a r y , the s t r u c t u r a l c o n d i t i o n o f P ^ ^ - p r e s e r v a t i o n f o l l o w s from the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h (Ga), or the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h ( v N i ) , or the d i s p e r s i o n - f r e e c o n d i t i o n together w i t h (JP) and (JP^>). In f a c t , the t r o u b l e w i t h P ^ ^ - p r e s e r v a t i o n has to do w i t h the overlap p a t t e r n s among the mBS's o f a s t r u c t u r e , and as B e l i n f a n t e p o i n t s out, the t r o u b l e w i t h P ^ ^ - p r e s e r v a t i o n has to do w i t h the assumption t h a t HV 181 measures are noncontextual, as s h a l l be described below. B e l l ' s c r i t i c i s m of Gleason's i m p o s s i b i l i t y proof (and thus of Kochen-Specker's Theorem 1) hinges upon the d i f f e r e n c e between what are sometimes c a l l e d c o n t e x t u a l and noncontextual HV t h e o r i e s , though B e l l does not use these terms. B e l l presents a v e r s i o n o f Gleason's proof which focuses upon two s t r u c t u r a l c o n d i t i o n s which B e l l d e r i v e s from Gleason's a d d i t i v i t y (Ga). Both c o n d i t i o n s are subsumed by the s t r u c t u r a l c o n d i t i o n o f P„„„-preservation, which l i k e w i s e f o l l o w s from (Ga). B e l l shows how the QMA second c o n d i t i o n which he d e r i v e s from (Ga) r u l e s out ( g e n e r a l i z e d ) d i s p e r s i o n - f r e e HV measures on the set of a l l p r o j e c t o r s or subspaces o f a three-or-higher dimensional H i l b e r t space; and as s h a l l be described s h o r t l y , t h i s second c o n d i t i o n ensures t h a t ( g e n e r a l i z e d ) d i s p e r s i o n - f r e e HV measures are i n f a c t noncontextual. B e l l c r i t i c i z e s the i m p o s i t i o n of t h i s second c o n d i t i o n upon HV measures because the second c o n d i t i o n r e l a t e s i n a n o n t r i v i a l and u n j u s t i f i e d way the r e s u l t s o f measurements which cannot be performed simultaneously. Although Gleason's proof r e f e r s to an i n f i n i t e set o f subspaces (or p r o j e c t o r s ) o f three-or-higher dimensional H i l b e r t space, i n order to understand B e l l ' s e x p l i c a t i o n and c r i t i q u e o f Gleason*s proof we need only 3 consider the f o l l o w i n g twelve-element fragment of ^ which i n c l u d e s two overlapping mBS's: 182 One maximal Boolean substructure mBS^ i s generated by the three mutually orthogonal ( i . e . , compatible) atoms {P^,P 2,P 3) and the other mBS^ i s 3 generated by {P3,P^,P,-}. A g e n e r a l i z e d measure |_L on P which s a t i s f i e s (Ga) assigns values to these f i v e atoms as f o l l o w s : M-(P1) + ki(P 2) + n(P 3). = p.(P1 V P 2 VP 3) = (i-(l) = 1. and l i ( P 3 ) + n ( P 4 ) + [ii? ) = |j,(P V P 4 v P g ) = n ( l ) = 1. I t f o l l o w s t h a t i f [i (P„) = 1 then (j, (P. ) = [i ( P 0 ) = 0; and s i m i l a r l y , i f u. ( P 0 ) = 1 then w 3 w 1 w 2 J "w 3 [x(P. ) = i x ( P _ ) = 0. (The s u b s c r i p t w i s added because a measure which w 4 w 5 e assigns 0, 1 values i s d i s p e r s i o n - f r e e . ) These two c o n d i t i o n a l s are instances o f the f i r s t c o n d i t i o n which B e l l d e r i v e s from (Ga) and which he l a b e l s (A) ( B e l l , 1966, p. 450). B e l i n f a n t e argues t h a t because B e l l ' s (A) r e f e r s to only one t r i a d of mutually orthogonal atoms at a time, we cannot y et conclude t h a t , i f u. (P ) = 1 then [i. (P. ) = [i (P.) = u. (P„) = u. (P_) = 0' ( B e l i n f a n t e , 1973, W O W l W 2 W 4 w o p. 65). But such a c o n c l u s i o n is_ guaranteed by the second c o n d i t i o n which B e l l derives from (Ga) and which he l a b e l s ( B ) . An instance of (B) i s : I f a w ( P 1 ) = M-W(P2) = 0 then, f o r any other P < P 1 V P 2 , M- (P) = 0. Thus i f [i (P„) = 1, then by ( A ) , u. (P ) = (x (P^) = 0, and then, s i n c e W O W 1 W 2 P 4 ~ P l V P 2 a n d P 5 ~ P l V P 2 ' b y ( B ) M'w ( P4 ) = M'w ( P5 ) = °* These two c o n d i t i o n s (A) and (B) which B e l l d e r i v e s from (Ga) correspond to the two c o n d i t i o n s (KS1) and (KS2) s t a t e d i n Chapter V(B) and to the two c o n d i t i o n s l a b e l e d (61b) and (64) i n B e l i n f a n t e ' s d e s c r i p t i o n of Kochen-Specker's work ( B e l i n f a n t e , 1973, pp. 39, 41). The f i r s t c o n d i t i o n of each p a i r , namely, ( A ) , (KS1), (61b), ensure t h a t the assignment o f 0, 1 values to the atoms i n a given mBS of a P^ M preserve the Boolean operations and r e l a t i o n s , i . e . , the Boolean s t r u c t u r a l f e a t u r e s , o f the mBS. And the second c o n d i t i o n o f each p a i r , namely, ( B ) , (KS2), (64), ensure t h a t 183 the assignment of 0, 1 values to the atoms in any overlapping mBS's in a preserve the overlap patterns among the mBS's. Both sorts of conditions are subsumed by the structural conditions of P^^-preservation, which i t s e l f has two aspects: F i r s t , i t ensures that the Boolean structural features of each mBS in a P are preserved; second, i t ensures that the partial-Boolean structural features of the entire P are pre-served; in particular, i t ensures that the overlap patterns among the mBS's in a P Q J^ a r e preserved. So Bell rightly points to the second condition (B), which he derives from (Ga), as the crucial part of Gleason's impossibility proof. For as described in Chapter V(B), i t is the preservation of the overlap patterns which makes bivalent homomorphisms(<4 ) impossible in Kochen-Specker's Theorem 1 version of Gleason's proof. Bell argues that proposed dispersion-free HV measures need not be required to satisfy (B). For any proposition P which i s less than or equal to P^ V P^ is incompatible with each of P^ , P^ , unless P = P^ or P = P„ . And i f P p 2 ' s n°uld be the same as the result of a 184 measurement o f Pg together w i t h P^ , P 5 ( B e l l , 1966, p. 451). An HV theory which allows i t s d i s p e r s i o n - f r e e HV measures to as s i g n d i f f e r e n t 0, 1 values to a given element P € P depending upon which other elements are measured together w i t h P have been c a l l e d c o n t e x t u a l HV t h e o r i e s . And the t a c i t assumption mentioned by B e l l i s the assumption that-an HV theory i s non-contextual; i . e . , i t s d i s p e r s i o n - f r e e HV measures a s s i g n a unique 0 or 1 value to a given element P € P M^ r e g a r d l e s s o f which other elements are measured together w i t h P. Now i n quantum mechanics, the outcome of a measurement of any magnitude A (which i s always one of A's eigenvalues) or of any idempotent magnitude P (which i s always one of P's 0 or 1 eigenvalues) i s determined by the quantum s t a t e ty, though i f ty i s incompatible w i t h any of A's or P's e i g e n s t a t e s , then as described i n the Preface, the quantum formalism a t best determines the p r o b a b i l i t y o f any one of A's or P's eigenvalues being the outcome o f a measurement and determines the average value ( i . e . , expectation-value) of A or P f o r a l a r g e number o f the same measurements of A or P on many quantum systems whose s t a t e i s described by f- In a c o n t e x t u a l HV theory, the outcome of a measurement o f A or P i s determined by the hidden s t a t e and the context o f measurement. A hidden s t a t e , l a b e l e d w above, i s s p e c i f i e d i n a c o n t e x t u a l HV theory by the quantum s t a t e ty together w i t h the hidden v a r i a b l e ( s ) g; so h e r e a f t e r , a hidden s t a t e o f a c o n t e x t u a l HV theory s h a l l be designated by ty, And the context i s taken to be the set of a l l p o s s i b l e outcomes of the measure-ment as s p e c i f i e d by a complete, orthogonal set o f eigens t a t e s o f the measured magnitude. As mentioned i n Chapter IV(A), the eigens t a t e s of any magnitude, as represented by p r o j e c t o r s {P,-},^ T T 1^ a w o n a H i l b e r t space, 185 are orthogonal and s a t i s f y J P. = I . In order t h a t the set o f eigenst a t e s i o f a magnitude be complete, i t s u f f i c e s t h a t each P. i s a one-dimensional p r o j e c t o r on ff; i . e . , an atom i n the P ^ s t r u c t u r e o f 5 H. Thus the context o f a measurement o f a magnitude represented by an operator on an n dimensional H i l b e r t space HU i s s p e c i f i e d by a s e t of n orthogonal one-dimensional p r o j e c t o r s on tfn, i . e . , by a s e t of n mutually orthogonal atoms i n the s t r u c t u r e o f H N . And sin c e a set of n mutually orthogonal atoms i n P ^ .generates a unique maximal Boolean s u b s t r u c t u r e o f P ^ M , ^ the context o f a measurement o f a magnitude represented by an operator on H N can eq u a l l y w e l l be s p e c i f i e d by an mBS i n the P ^ s t r u c t u r e of tfn, as suggested by Gudder (1970, p. 432). In p a r t i c u l a r , when we consider any idempotent magnitude P, which i s * n n represented by the p r o j e c t o r P on H and so i s an element i n the P ^ s t r u c t u r e o f H N , P, qua element o f P ^ , i s i t s e l f a member of any o f the mBS's i n P ^ which s p e c i f y p o s s i b l e contexts o f measurement o f P. A. For P i s i t s e l f a member (or a sum o f members) of any set of n orthogonal, one-dimensional p r o j e c t o r s on Hn r e p r e s e n t i n g a complete, orthogonal s et of eigenstates o f the idempotent magnitude P and so s p e c i f y i n g the context o f a measurement o f P; thus P, qua element of PQ^ , i s i t s e l f a member (or a j o i n of members) of any set o f n mutually orthogonal atoms i n P N ^ s p e c i f y i n g the context of a measurement o f P; and so P, qua element of P N „ , i s i t s e l f a member o f any mBS i n P N „ — — QM J QM s p e c i f y i n g the context o f a measurement o f P. In s h o r t , i n a co n t e x t u a l HV theory, the outcome of a measurement o f any P € P ^ i s determined by the hidden s t a t e y, £ and the context o f measurement, s p e c i f i e d by an mBS i n P . „ w i t h P € mBS. QM The f a c t s t a t e d i n the l a s t sentence can be and has been 186 f o r m a l i z e d i n any number of ways. Most a b s t r a c t l y , s i n c e the outcome o f a measurement o f P i s always one of P's 0 or 1 eigenvalues, we may t a l k o f a c o n t e x t u a l HV theory proposing contextually-dependent 0, 1 value assignments to the elements o f P^ . For example, B e l i n f a n t e t a l k s o f a c o n t e x t u a l HV theory, which he r e f e r s to as a " r e a l i s t i c " HV theory, i n t r o d u c i n g , f o r a given hidden s t a t e ty, £, a b i v a l e n t mapping v whose arguments are quantum p r o p o s i t i o n s and which depend not only upon ty, £, but a l s o upon the context of measurement ( B e l i n f a n t e , 1973, pp. 40-42). Less a b s t r a c t l y , s i n c e i n t h i s chapter and i n Chapters I I I and VI we have described how i n c l a s s i c a l mechanics, quantum mechanics, and proposed HV t h e o r i e s , 0, 1 value assignments to the elements i n P0„„ , Pn., , P„„ *> CM QM HV s t r u c t u r e s are preformed by various kinds of state-induced d i s p e r s i o n - f r e e p r o b a b i l i t y measures, we can i n a s i m i l a r v e i n say t h a t the hidden s t a t e s of a c o n t e x t u a l HV theory induce d i s p e r s i o n - f r e e HV measures which a s s i g n 0, 1 values to elements of P i n a c o n t e x t u a l l y dependent manner. For example, Bub t a l k s i n t h i s way (1974, pp. 146-147; 1973, p. 51). While according to Gudder's way of f o r m a l i z i n g the c o n t e x t u a l HV p r o p o s a l , a hidden s t a t e of a c o n t e x t u a l HV theory induces a d i s p e r s i o n - f r e e HV measure on only an mBS of P so t h a t the c o n t e x t u a l dependence o f the measure i s at l e a s t p a r t l y handled by r e s t r i c t i n g i t s domain to one context, i . e . , one mBS (Gudder, 1970, p. 433). We s h a l l focus upon the n o t i o n o f the hidden s t a t e s o f a c o n t e x t u a l HV theory inducing d i s p e r s i o n - f r e e HV measures which a s s i g n 0, 1 values to the elements of P M^ i n a c o n t e x t u a l l y dependent manner. The contextual-dependence of the d i s p e r s i o n - f r e e measures may be and has been formulated i n two equivalent ways. One way i n v o l v e s c o n t e x t u a l i z i n g proposed g e n e r a l i z e d , d i s p e r s i o n - f r e e HV measures M^ , ^ on P^ by having 187 the domain o f each a, „ be the cross-product o f P and the set o f mBS's i n P„w so that the value which a, „ assigns to an element P € P™ QM y, g QM depends upon which mBS co n t a i n i n g P i s being considered ( i . e . , depends upon the context i n which P i s being measured). Thus a hidden s t a t e y, £ induces a c o n t e x t u a l i z e d , g e n e r a l i z e d , d i s p e r s i o n - f r e e HV measure \H 1 PQM X { m B S i } i €lndex > { 0 , 1 } S U C h t h a t ' f ° r e x a m P l e ' ^ , C ( < P 3 ' m B S l > ) need not equal u, (). According to Bub, the Bohm 1952 HV ^y,g 3 4 proposal i s such a co n t e x t u a l HV theory. However, one would be hard pressed to f i n d anything l i k e t h i s u.. : PAvt x {mBS.}.._ , * {0,1} i n Bohm's J & Y»C QM l lCI-ndex work or even i n Bub's d e s c r i p t i o n o f Bohm's work (Bub, 1973, p. 51). For again, the above n o t i o n o f an a. measure i s an a b s t r a c t i o n , which helps make sense o f Bub's d e s c r i p t i o n o f Bohm's work and which was suggested to me by B e l i n f a n t e ' s method o f c o n t e x t u a l i z i n g h i s b i v a l e n t v mappings (w i t h R. E. Robinson suggesting the cross-product f o r m u l a t i o n ) . Now the a l t e r n a t i v e way in v o l v e s proposing that a Boolean P s t r u c t u r e be the HV domain o f proposed c l a s s i c a l d i s p e r s i o n - f r e e HV measures u.^ ^ : P^ y -»• {0,1} induced by the hidden s t a t e s , w i t h a c o n t e x t u a l i z e d a s s o c i a t i o n of the elements of P_.w w i t h the elements o f the PTT„ . Thus we have a QM HV co n t e x t u a l i z e d a s s o c i a t i o n % : P^„ x {mBS.}.- T ,—*" Pt7„ such t h a t , f o r QM I i€Index HV example, %() need not equal %(), and so u.. (%(
)) need not equal (j, (%(
)). The Bohm-Bub 1966 y>s 3 i ys5 3 " HV proposal i s such a cont e x t u a l HV theory. According to Bub, both ways o f form u l a t i n g the con t e x t u a l HV pro p o s a l , e i t h e r i n terms of c o n t e x t u a l i z e d measures on P.„ or i n terms o f a c o n t e x t u a l i z e d a s s o c i a t i o n o f P.,, w i t h QM QM P^v , are f o r m a l l y equivalent (Bub, 1973, p. 51). C l e a r l y , both have the same e f f e c t , namely, the proposed d i s p e r s i o n - f r e e HV measure induced by a hidden s t a t e i n a con t e x t u a l HV theory does not a s s i g n a unique 0 or 1 188 value to a given element P € P^ M when P i s a member of more than one mBS i n P^^ , i . e . , when P i s a member of two or more overlapping mBS's l n PQM ' Thus the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s of a con t e x t u a l HV theory e s p e c i a l l y break up the overlap patterns among the mBS's of any i n "the manner suggested i n Chapter V(B), namely, by as s i g n i n g d i f f e r e n t values to a s i n g l e element which i s i n more than one mBS of P Q M . So, f o r example, although P g .= Pg i n the twelve-element 3 fragment of P ^ diagrammed above, and although Exp^(Pg) = Exp^(Pg) f o r 3 every quantum Exp^, on P^^ , ne v e r t h e l e s s , i n a c o n t e x t u a l HV theory, Pg may be assigned d i f f e r e n t v a l u e s , as e x e m p l i f i e d i n the previous paragraph. In t h i s sense, the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a c o n t e x t u a l HV theory do not preserve the r e l a t i o n Pg = Pg . That i s , they do not preserve the = r e l a t i o n o f P , and so i t c l e a r l y f o l l o w s t h a t w i t h respect to elements i n overlapping mBS's i n PQJ^ , the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s of a cont e x t u a l HV theory do not preserve any o f the operations and r e l a t i o n s of PQ^ . A c o n t e x t u a l HV theory and i t s d i s p e r s i o n - f r e e HV measures thereby avoid HV i m p o s s i b i l i t y p roofs. Or as Bub puts i t , i n terms o f the second fo r m u l a t i o n o f the co n t e x t u a l HV proposal which i n c l u d e s a Boolean P . and n v c l a s s i c a l d i s p e r s i o n - f r e e HV measures on P , a c o n t e x t u a l HV theory i s a type of Boolean r e c o n s t r u c t i o n o f quantum mechanics (Bub, 1974, p. 146) which avoids the Kochen-Specker i m p o s s i b i l i t y proof by l e t t i n g the a s s o c i a t i o n o f the elements o f P.,, w i t h the elements o f P T T T T be a QM HV c o n t e x t u a l i z e d mapping which breaks up the overlap patterns among the mBS's of P Q ^ 3 r a t h e r than demanding, as Kochen-Specker do, t h a t t h i s a s s o c i a t i o n be an imbedding(i) which preserves P Q J^ » i . e . , preserves a l l the 189 p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f P Q ^ » i n c l u d i n g the < r e l a t i o n (and thus the = r e l a t i o n ) and i n c l u d i n g the overlap p a t t e r n s among the mBS's. Now to continue w i t h the t h i r d s o r t o f c r i t i c i s m , l a b e l e d ( i i i ) i n the Preface, o f the i m p o s i t i o n o f the s t r u c t u r a l c o n d i t i o n of Pg^-preservation and o f the development o f the Gleason, Kochen-Specker HV i m p o s s i b i l i t y p r o o f s . I f , as B e l l argues, there i s no reason why the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f P , i n p a r t i c u l a r , the overlap patterns among the mBS's of P ^ , must be preserved by non-contextually a s s i g n i n g the same unique 0 or 1 value t o , say, P^ r e g a r d l e s s of whether P i s measured i n the context mBS or i n the context mBS , then the Gleason and Kochen-Specker HV i m p o s s i b i l i t y proofs beg the HV question. For these proofs r e s t upon c o n t r a d i c t i o n s caused by r e q u i r i n g t h a t 0, 1 values be assigned to the elements of a P^ i n a non-contextual, P ^ ^ - p r e s e r v i n g manner which i s not j u s t i f i e d . Moreover, these proofs do not r u l e out a cont e x t u a l HV r e c o n s t r u c t i o n of quantum mechanics, and so they do not r u l e out hidden v a r i a b l e s , as they purport to do. Kochen-Specker's work i s e s p e c i a l l y v u l n e r a b l e to the above c r i t i c i s m because of the f o l l o w i n g ambiguity, pointed out by Bub, i n the manner i n which Kochen-Specker ground the p a r t i a l - B o o l e a n algebra o f quantum p r o p o s i t i o n s which they r e q u i r e an HV theory to preserve: (a) On the one hand, Kochen-Specker regard the quantum p r o p o s i t i o n a l s t r u c t u r e as simply given by the p a r t i a l - B o o l e a n algebra o f p r o j e c t o r s or subspaces o f H i l b e r t space, which has been l a b e l e d P Q J ^ • For according to Kochen-Specker, i t i s a "basic t e n e t " o f quantum mechanics th a t quantum magnitudes are represented by operators on a H i l b e r t space and 190 s i m i l a r l y , quantum p r o p o s i t i o n s , qua idempotent magnitudes, are represented by p r o j e c t o r s on a H i l b e r t space (Kochen-Specker, 1967, p. 65). That i s , the quantum p r o p o s i t i o n a l s t r u c t u r e i s a P^ s t r u c t u r e , i n p a r t i c u l a r , a PQMA s t r u c t u r e , o f p r o j e c t o r s or subspaces of a H i l b e r t space. And f o r example, two p r o p o s i t i o n s are equivalent i n P Q J ^ i f "they are represented by the same p r o j e c t o r (or subspace). (b) But on the other hand, Kochen-Specker d e f i n e a p a r t i a l - B o o l e a n algebra of quantum p r o p o s i t i o n s w i t h respect to a_ set of s t a t e s and measures; the defined s t r u c t u r e s h a l l be l a b e l e d pBA to d i s t i n g u i s h i t from the above P Q J ^ • The d e f i n i t i o n o f pBA may y i e l d , f o r example, t h a t two p r o p o s i t i o n s are equivalent i n pBA i f t h e i r expectation-values are equal f o r a l l quantum Exp^ measures. I f the quantum p r o p o s i t i o n a l s t r u c t u r e which Kochen-Specker r e q u i r e an HV theory to preserve i s such a pBA defined w i t h respect to the quantum measures, then Kochen-Specker's i m p o s s i b i l i t y proof, which r e s t s upon the requirement t h a t the quantum p r o p o s i t i o n a l s t r u c t u r e be preserved, begs the HV question. For there i s no reason why proposed HV measures must preserve such a pBA, and i n p a r t i c u l a r preserve the equivalence = P^ i n pBA, i f P^ = P^ i n pBA only because Exp^(Pg) = Exp^,(P,p f o r a l l quantum Exp^ measures. Moreover, i f d i s p e r s i o n - f r e e HV measures do e x i s t , then the pBA defined w i t h respect to the quantum measures and the HV measures may be d i f f e r e n t from the pBA defined w i t h respect to j u s t the quantum Exp^, measures. These c r i t i c i s m s o f Kochen-Specker's defined pBA are s i m i l a r to the c r i t i c i s m s o f . von Neumann's use of h i s c o n d i t i o n (E) to d e f i n e the sums o f incompatibles. This ambiguity between (a) and ( b ) , and the way i n which (b) leads to a misunderstanding o f Kochen-Specker's work and makes the Kochen-Specker 191 i m p o s s i b i l i t y r e s u l t appear to be e s p e c i a l l y v u l n e r a b l e to B e l l ' s c r i t i c i s m , are described by Bub (1974, pp. 84-88). Bub concludes t h a t Kochen-Specker are best understood r e f e r r i n g to P n M f l r a t h e r than pBA, t h a t i s , Kochen-Specker should have used j u s t the (a) n o t i o n and not discussed the (b) n o t i o n a t a l l . Moreover, as s h a l l be described i n S e c t i o n B ', from Bub's pe r s p e c t i v e on the problem o f h i d d e n - v a r i a b l e s , the ambiguity between (a) and (b) i s not s u b s t a n t i a l l y important, though i t i s confusing and leads to a misunderstanding o f Kochen-Specker's work, and so the ambiguity i s worth c l a r i f y i n g . In the r e s t o f t h i s s e c t i o n , Kochen-Specker's (b) d e f i n i t i o n o f pBA i s elaborated, and a reason why Kochen-Specker may have been motivated to develop t h i s (b) d e f i n i t i o n i s given. According to Kochen-Specker, a p h y s i c a l theory l i k e c l a s s i c a l mechanics or quantum mechanics or a proposed HV theory c o n s i s t s o f a s e t o f magnitudes {A,...}, a set o f s t a t e s . .. },• and a set o f ( c l a s s i c a l ) p r o b a b i l i t y measures {p ,...} on the real-number l i n e R, or more ty,A e x a c t l y , on the Boolean s t r u c t u r e B^ o f B o r e l subsets o f R. For any B o r e l subset R c R, f o r any magnitude A, and f o r any s t a t e ty, Pty € [0,1] i s the p r o b a b i l i t y t h a t the r e a l - v a l u e o f A i s a member of R. These p measures on B D are r e l a t e d to the more f a m i l i a r ty , A K e x p e c t a t i o n - f u n c t i o n s Exp and are r e l a t e d to the HV measures p, of a Kochen-Specker type o f HV r e c o n s t r u c t i o n o f quantum mechanics, by equations given below. Now Kochen-Specker argue t h a t the magnitudes of a p h y s i c a l theory are not independent o f each other but r a t h e r are f u n c t i o n a l l y r e l a t e d , e.g., 2 2 the magnitude A i s c l e a r l y a f u n c t i o n of A. And the f u n c t i o n A o f the magnitude A can be measured by simply measuring A and squaring the r e s u l t i n g value. That i s , the r e a l value o f any ( B o r e l ) f u n c t i o n g(A) of 192 any magnitude A i s c a l c u l a t e d by simply a p p l y i n g t h a t f u n c t i o n g to the r e a l value o f A. The l a s t sentence i s a statement o f what may be regarded as an uncontentious general p r i n c i p l e which a p p l i e s to the magnitudes o f any p h y s i c a l theory. Kochen-Specker a l s o assume that the magnitudes o f a p h y s i c a l theory are determined by the p^ A measures i n the f o l l o w i n g sense: (*) For any magnitudes A, B, i f p^ A ( R ) = p^ g ( R ) f o r every s t a t e ty and any B o r e l subset R c R, then A = B. With (*), the above general p r i n c i p l e suggests the f o l l o w i n g d e f i n i t i o n , which Kochen-Specker l a b e l ( 3 ) , f o r a f u n c t i o n g(A) of any magnitude A: (3) For any A and any B o r e l f u n c t i o n g, Pty g ( A ) ^ = Pty A^ S f ° r a n y s t a t e Y a n d a n v R - R (Kochen-Specker, 1967, pp. 61, 63). In f a c t , (3) can be regarded as a restatement of the uncontentious general p r i n c i p l e . For i f the r e a l value o f A i s a member of some B o r e l subset R £ R, then by the general p r i n c i p l e , the r e a l value o f g(A) i s a member of the B o r e l subset g(R) £ R. Likewise, i f the r e a l value of g(A) i s a member o f some R c R 5 then by the general p r i n c i p l e , the r e a l value o f A i s a member of the B o r e l subset g (R) c R. So assuming t h a t Pty g ( A ) ^ R ^ 1 S t n e P r°b ability t h a t the r e a l value of g(A) i s i n R, and assuming t h a t the p^. A measures determine the magnitude o f a p h y s i c a l theory i n the above (*) sense, then by the general p r i n c i p l e we can be sure t h a t P V , g ( A ) ( R ) = P Y i A ( s " 1 ( R ) ) -Moreover, w i t h respect to a Kochen-Specker type o f HV r e c o n s t r u c t i o n o f quantum mechanics, i n which each quantum magnitude A i s 193 represented by a f u n c t i o n f : S2 -»• R on the HV phase space S2 and the r e a l value of A f o r any hidden s t a t e w 6 8 i s ^ A( w)» t n e general p r i n c i p l e y i e l d s the i d e n t i t y : f o r any w € S2, ^ g ( A ) ^ W ^ = 2^ A^ W'^ * ^° i n a Kochen-Specker type of HV r e c o n s t r u c t i o n , the f u n c t i o n s { f ^ , . . . } re p r e s e n t i n g the quantum magnitudes i n the proposed HV theory must s a t i s f y the f o l l o w i n g s t r u c t u r a l c o n d i t i o n l a b e l e d (4) by Kochen-Specker: (4) For any quantum magnitude A and any B o r e l f u n c t i o n g, f g ( A ) = * ( f A > ' Kochen-Specker aim t o show that an HV r e c o n s t r u c t i o n o f quantum mechanics which s a t i s f i e s (4) i s impossible. But f i r s t Kochen-Specker r e p l a c e (4) by a more t r a c t a b l e s t r u c t u r a l c o n d i t i o n as f o l l o w s . Using (*) and ( 3 ) , Kochen-Specker d e f i n e the r e l a t i o n o f commeasurability, i . e . , c o m p a t i b i l i t y , among the magnitudes of a p h y s i c a l theory as s t a t e d i n Chapter IV(B). Then u s i n g (*) and (3) again, Kochen-Specker d e f i n e the r i n g operations + and • among commeasurable magnitudes as f o l l o w s : For any magnitudes A^ , A^ , i f , A^ are commeasurable, then f o r some magnitude B and B o r e l f u n c t i o n s g^ , g^, k± = g 1(B) and A 2 = g 2 ( B ) , and then (5) A 1 + A 2 = ( g l + g 2 ) ( B ) , A 1 • A 2 = ( g l • g 2)(B). With + and • so defined among compatible magnitudes, the set of magnitudes of a p h y s i c a l theory acquires the s t r u c t u r e o f a p a r t i a l - a l g e b r a , or i n the terminology o f Chapter 1(D), a p a r t i a l - r i n g - w i t h - u n i t . And thus the set o f p r o p o s i t i o n s o f a p h y s i c a l theory, qua idempotent magnitudes, i . e . , qua idempotent elements o f a p a r t i a l - r i n g - w i t h - u n i t , has the s t r u c t u r e of a p a r t i a l - B o o l e a n algebra. In p a r t i c u l a r , by (*), ( 3 ) , and ( 5 ) , the 194 mutually compatible magnitudes o f c l a s s i c a l mechanics form a commutative-r i n g - w i t h - u n i t , which i s a s p e c i a l case of a p a r t i a l - r i n g - w i t h - u n i t , namely, the case where a l l elements are mutually compatible, as described i n Chapters I I I ( B ) and 1(D). And the p r o p o s i t i o n s o f c l a s s i c a l mechanics form a Boolean a l g e b r a , which again i s the s p e c i a l case o f a p a r t i a l - B o o l e a n algebra where a l l elements are mutually compatible. Li k e w i s e , the magnitudes of a proposed Kochen-Specker type o f HV theory form a commutative-ring-with-u n i t , and the p r o p o s i t i o n s o f such an HV theory form a Boolean algebra. And f i n a l l y , by (*), ( 3 ) , and (5) the magnitudes o f quantum mechanics form a p a r t i a l - r i n g - w i t h - u n i t , and the p r o p o s i t i o n s of quantum mechanics form a p a r t i a l - B o o l e a n a l g e b r a , l a b e l e d pBA. This completes the Kochen-Specker d e f i n i t i o n o f a pBA of quantum p r o p o s i t i o n s , which s h a l l be f u r t h e r discussed s h o r t l y . Kochen-Specker then note t h a t t h e i r c o n d i t i o n (4) i m p l i e s t h a t the p a r t i a l - o p e r a t i o n s + and •, which are def i n e d among ( j u s t ) compatible quantum p r o p o s i t i o n s { p i > p 2 > ' a n d a m o n § "'-he compatible HV re p r e s e n t a t i v e s {f , f ,•••} o f quantum p r o p o s i t i o n s by the c o n d i t i o n 1 2 ( 5 ) , are preserved by the mapping % which a s s o c i a t e s the quantum p r o p o s i t i o n s w i t h t h e i r HV r e p r e s e n t a t i v e s , i n t h i s case % : pBA P^v . For example, as elaborated by Bub, f o r any compatible , P 0 » which are by the d e f i n i t i o n o f c o m p a t i b i l i t y B o r e l f u n c t i o n s o f some common P, say P 1 = g l ( P ) and P 2 = g 2 ( P ) , we have: f ^ = f ^ p ^ p j = (by (5)) f(g1+g2)(P) = ( b y C 4 ) ) ( g l + g 2 ) ( f P ) = ( b y C 5 ) ) g l ( f P ) + g 2 ( V = ( b y M ) f , , + f , . = f + f (Bub, 1974, p. 87). So, f o r example, i f % ( P 1 ) = f p and % ( P 2 ) = f p and % ( P 1 + P 2 ) = f p + p , then by (4) and 195 (5) we have: % ( P 1 + P ) = %(P ) + % ( P 2 ) . Thus the mapping % which a s s o c i a t e s quantum p r o p o s i t i o n s w i t h t h e i r HV r e p r e s e n t a t i v e s preserves the p a r t i a l - o p e r a t i o n + among compatible quantum p r o p o s i t i o n s . S i m i l a r l y , i t can be shown t h a t , by (4-) and ( 5 ) , % preserves the p a r t i a l - o p e r a t i o n among compatible quantum p r o p o s i t i o n s . And so w i t h the operation and the p a r t i a l - o p e r a t i o n s A, V defined i n terms of +, • as u s u a l , the mapping % : pBA -»• P m r preserves these A, V, operations s i n c e i t HV preserves the +, • opera t i o n s . In other words, % i s an imbedding(i>) (Kochen-Specker,. 1967, pp. 63-66). However, as pointed out by Bub, i t i s c l e a r t h a t i n t h i s (b) d e f i n i t i o n o f pBA, Kochen-Specker r e l y upon the ^ measures to d e f i n e , b y ' ( * ) , the equivalence o f quantum p r o p o s i t i o n s , and to d e f i n e , w i t h (*) and ( 3 ) , the f u n c t i o n a l r e l a t i o n s and the c o m p a t i b i l i t y r e l a t i o n s among the quantum p r o p o s i t i o n s . That i s , the pBA s t r u c t u r e o f quantum p r o p o s i t i o n s which Kochen-Specker r e q u i r e an HV r e c o n s t r u c t i o n to preserve i s defined w i t h respect to the ^ measures. These measures on 8^ are r e l a t e d to ex p e c t a t i o n - f u n c t i o n s Exp by the equation: For any magnitude A and any s t a t e v, Exp.(A) = f r dp, . ( { r } ) , r 6 P. And the p. . measures are R Y J - ° ° ty,A y,A r e l a t e d to the p measures o f a Kochen-Specker type of HV r e c o n s t r u c t i o n by the equation: For any magnitude A, any s t a t e y, and any B o r e l subset R c R, p^ a(R) = [^(f" A (R)) (Kochen-Specker, 1967, p. 61). Now so f a r , A and Y designate any magnitude and any s t a t e i n any p h y s i c a l theory. So w i t h respect to the issu e o f a proposed HV r e c o n s t r u c t i o n o f quantum mechanics, i t i s not c l e a r whether the set of \|/ s t a t e s , which v i a the p measures y, A d e f i n e s pBA, i n c l u d e s j u s t the quantum s t a t e s , which are u s u a l l y designated by Y» °r in c l u d e s both the quantum s t a t e s and the hidden s t a t e s proposed 196 by an HV r e c o n s t r u c t i o n . And as suggested above, the pBA defined w i t h respect to j u s t the quantum s t a t e s may be d i f f e r e n t from the pBA defined w i t h respect to both the quantum and the hidden s t a t e s ; i n p a r t i c u l a r , w h i l e the former i s isomorphic to P Q ^ A » t n e l a t t e r might not be. I f Kochen-Specker mean the set of s t a t e s which d e f i n e , v i a the ^ measures, t h e i r pBA to i n c l u d e both quantum and hidden s t a t e s , then they are presuming t h a t hidden s t a t e s e x i s t and they thus beg the HV question i n a t r i v i a l way. I f i t does not matter whether the set of s t a t e s i n c l u d e s j u s t the quantum s t a t e s or i n c l u d e s both quantum and hidden s t a t e s , then Kochen-Specker beg the HV question i n the sense t h a t they presume t h a t the pBA defined w i t h respect to the quantum y s t a t e s i s the same as the pBA defined w i t h respect to both quantum y and hidden w s t a t e s ; i n p a r t i c u l a r , they presume th a t \i (P.) = u. (P„) j u s t as Exp (P ) = Exp (P ). w o w o y o y d But i n a c o n t e x t u a l HV theory, an element P 3 which i s a member of two or more overlapping mBS's i n pBA i s not assigned a unique value f o r a given hidden s t a t e w s p e c i f i e d by y, £, e.g., u\. (
) may not equal y , c , o 1 [i.^ ^ (). And f i n a l l y , i f Kochen-Specker mean the s e t o f s t a t e s which d e f i n e t h e i r pBA to in c l u d e j u s t the quantum s t a t e s , then they beg the HV question i n the manner described on page 190. For then by quantum s t a t e s determine the i d e n t i t y of the quantum magnitudes and quantum p r o p o s i t i o n s ; i . e . , f o r any quantum p r o p o s i t i o n s P ^ ' P 2 ' P l = P 2 ^ ^y P ~ ^y P e v e r y 9. u a n t u m s " t a t e V a n d any B o r e l subset R c R. Or i n other words, by the above equation connecting Exp^ w i t h p^ A we have: For any quantum p r o p o s i t i o n s P^ , P^ , p ^ = p 2 i f Exp^,(P^) = Exp^CP^) f o r every quantum s t a t e . But there i s no reason why proposed d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a 197 proposed HV theory must preserve t h i s equivalence which i s defined with respect to the quantum states and measures. Thus contextual HV measures which do not preserve the equivalences i n pBA may be proposed e s p e c i a l l y i n order to avoid the Kochen-Specker HV i m p o s s i b i l i t y proof. So I f Kochen-Specker had only the (b) d e f i n i t i o n of pBA, then t h e i r c r u c i a l imbedding(A) condition would u n j u s t i f i a b l y demand the preservation of a structure defined with respect to maybe just the quantum states and measures. However, Kochen-Specker have not only the defined pBA but also the (a) PQJ^ given by the basic tenets of quantum mechanics. And as Bub argues, both the Kochen-Specker HV i m p o s s i b i l i t y proof and the contextual HV counter-proposal are best understood i f we give Kochen-Specker the benefit of the doubt and resolve t h e i r ambiguity between (a) and (b) i n favour of the (a) PQ^a • The very f a c t that Kochen-Specker require that the quantum p r o p o s i t i o n a l structure be preserved i n an HV reconstruction suggests that they regard i t as something more than a merely s t a t i s t i c a l structure defined with respect to the dispersive quantum states and measures. Moreover, Kochen-Specker s p e c i f i c a l l y declare t h e i r Theorem 1 to be a f i n i t e version of Gleason's i m p o s s i b i l i t y proof, which r e f e r s to the projectors or subspaces of H i l b e r t space. Thus Kochen-Specker*s f i n i t e version of Gleason's proof may likewise be understood as r e f e r r i n g to the PQJ^ structure of projectors or subspaces of H i l b e r t space rather than r e f e r r i n g to the pBA structure. Kochen-Specker may have been motivated to develop t h e i r (b) d e f i n i t i o n of pBA i n order that t h e i r contentious imbedding(A) condition should f o l l o w from the uncontentious general p r i n c i p l e as described above. But then Kochen-Specker should have used the general p r i n c i p l e only to 198 support, via their imbedding(c>) condition rather than to help define, via (*)» (3), (4), (5), a pBA of quantum propositions. For example, we may take the quantum propositional structure to be a PQJ^ °f projectors or subspaces of Hilbert space; so = i f p^ = P2 ' a n d t h e partial-operations +, • are defined among compatible propositions as projector addition and multiplication. Then we may s t i l l argue that in a proposed HV reconstruction of quantum mechanics, where any quantum proposition P is represented by an idempotent function f p : Si -> {0,1} on the HV phase space Si and any Borel function g(P) is correspondingly represented by the idempotent function ^g(p) » t n e uncontentious general principle requires that the 0, 1 values issued by ^g(p) m u s " t be g-functions of the 0, 1 values issued, by f p . And the fulfillment of this requirement is best ensured by making fg(p) = §(fp)» f ° r a nY p a n d any Borel function g. Thus we have condition (4), from which the imbedding (A) condition follows as described above. In other words, the crucial Kochen-Specker imbedding(A) condition, which requires that PQJJA be preserved in any proposed HV reconstruction of quantum mechanics, _is_ supported by the uncontentious general principle which i t seems no c r i t i c of Kochen-Specker's HV impossibility proof could reasonably object to. However, without realizing or disregarding the above elaborated connection between the general principle and the imbedding(i) condition, c r i t i c s of the Kochen-Specker proof may argue that even i f Kochen-Specker are understood as referring to PQJ^ rather than pBA, their proof begs the HV question because their imbedding(A) condition, which requires pQ^-preservation and which rules out hidden-variables, is not j u s t i f i e d . In other words, c r i t i c s may argue that there i s no reason why a proposed HV 199 r e c o n s t r u c t i o n must preserve even t h i s PQ^A given by the fundamental p o s t u l a t e s o f quantum mechanics. In f a c t , B e l l must be understood as making t h i s f u r t h e r argument, f o r he addresses h i m s e l f to the Gleason i m p o s s i b i l i t y proof and thus to an (a) type of s t r u c t u r e r a t h e r than a (b) type of s t r u c t u r e . Bub rescues the Gleason, Kochen-Specker proofs from t h i s c r i t i c i s m , as described i n the next s e c t i o n . S e c t i o n B. E i t h e r P ^ ^ - p r e s e r v a t i o n or Boolean Reconstruction Bub argues t h a t the concept of an HV r e c o n s t r u c t i o n of quantum mechanics does not make sense unless the quantum p r o p o s i t i o n a l s t r u c t u r e i s preserved. For according to Bub, quantum mechanics i s a p r i n c i p l e theory r a t h e r than a c o n s t r u c t i v e theory. The d i s t i n c t i o n i s due to E i n s t e i n and i s d escribed by Bub as f o l l o w s . C o n s t r u c t i v e t h e o r i e s "aim to reduce a wide c l a s s of d i v e r s e systems to component systems of a p a r t i c u l a r k i n d (e.g., the molecular hypothesis o f the k i n e t i c theory of gases)." In c o n t r a s t , p r i n c i p l e t h e o r i e s "introduce a b s t r a c t s t r u c t u r a l c o n s t r a i n t s t h a t events are h e l d to s a t i s f y , " e.g., s p e c i a l and general r e l a t i v i t y can be viewed as p r i n c i p l e t h e o r i e s of space-time s t r u c t u r e (Bub, 1974, pp. v i i , 14-2). Bub regards quantum mechanics and c l a s s i c a l mechanics as p r i n c i p l e t h e o r i e s of l o g i c a l s t r u c t u r e because, . . . they introduce c o n s t r a i n t s on the way i n which the p r o p e r t i e s of a p h y s i c a l system are s t r u c t u r e d . The l o g i c a l s t r u c t u r e of a p h y s i c a l system i s understood as imposing the most general k i n d of c o n s t r a i n t on the occurrence and non-occurrence of events. (Bub, 1974, p. 149) The l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of a p h y s i c a l system i s given by the p r o p o s i t i o n a l s t r u c t u r e as determined by the mathematical formalism of the 2 0 0 p h y s i c a l theory d e s c r i b i n g the system, namely, the c l a s s i c a l P and the quantum P . So at the very core o f quantum mechanics i s the non-Boolean P s t r u c t u r e , which Bub and Kochen-Specker e x p l i c i t l y and Gleason i m p l i c i t l y t a k e ( s ) to be a P n M. • And according to Bub, (1) the question of the completeness o f quantum mechanics must be posed w i t h respect to PQMA ' ^ e q u a n ' t u m p r o b a b i l i t y measures are defined on P Q J ^ a n d the s t a t i s t i c a l r e s u l t s of quantum mechanics make sense w i t h respect t o P Q J ^ » and ( i i i ) any HV r e c o n s t r u c t i o n o r extension o f quantum mechanics must preserve the quantum P Q J ^ • Now as shown by Kochen-Specker and by Gudder, a Boolean HV r e c o n s t r u c t i o n of quantum mechanics which does not preserve the P Q ^ A s t r u c t u r e i s always p o s s i b l e . By a t r i v i a l c o n s t r u c t i o n , Kochen-Specker show that i s i s always p o s s i b l e to introduce a c l a s s i c a l measure space = X which reproduces the quantum s t a t i s t i c s but does not preserve nv PQMA ( K o c n e n ~ S p e c k e r , 1 9 6 7 , p. 6 3 ) . And Gudder proves t h a t i t i s always p o s s i b l e t o introduce a cont e x t u a l HV Boolean r e c o n s t r u c t i o n which reproduces the quantum s t a t i s t i c s and preserves the Boolean s t r u c t u r a l f eatures of the mBS's o f P Q ^ a but which breaks up the overlap patterns among the mBS's and so does not preserve P Q ^ (Gudder, 1 9 7 0 , pp. 4 - 3 4 — 4 3 6 ) . However, as Bub argues: The c o n t r i b u t i o n o f Kochen-Specker l i e s i n showing that the problem o f hidden v a r i a b l e s i s not th a t o f f i t t i n g a t h e o r y — i . e . , a c l a s s o f event s t r u c t u r e s — t o a s t a t i s t i c s . This can. always be done i n an i n f i n i t e number of ways; i n p a r t i c u l a r , a Boolean r e p r e s e n t a t i o n i s always p o s s i b l e . Rather, the problem concerns the ki n d o f s t a t i s t i c s d e f i n a b l e on a given c l a s s o f event s t r u c t u r e s . (Bub, 1 9 7 4 , p. 8 8 ) . The event s t r u c t u r e s given by the fundamental p o s t u l a t e s of quantum mechanics 201 are the non-Boolean P s t r u c t u r e s , i n p a r t i c u l a r , the s t r u c t u r e s . So the problem o f the completeness o f quantum mechanics and the concordant problem of hidden v a r i a b l e s i s c o r r e c t l y addressed w i t h respect to the quantum PQJ^ » a s done by Gleason and Kochen-Specker. In Bub's view, Gleason's completeness proof shows t h a t the quantum formalism generates a l l _n>3 p o s s i b l e ( g e n e r a l i z e d ) p r o b a b i l i t y measures on the s t r u c t u r e s o f three-or-higher dimensional H i l b e r t space. That i s , w i t h respect to P Q ^ » the quantum mechanics o f three-or-higher dimensional H i l b e r t space i s complete. And i t f o l l o w s as a c o r o l l a r y t h a t , f o r PQ^ a » d i s p e r s i o n - f r e e ( g e n e r a l i z e d ) p r o b a b i l i t y measures which preserve the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s of PQJ^ a r >e impossible. And so by Kochen-Specker's Theorem 0, an imbedding(i) o f P N M F L i n t o a Boolean s t r u c t u r e i s impossible. Thus a Boolean HV r e c o n s t r u c t i o n o f quantum mechanics which preserves P Q ^ i s impossible. That i s , w i t h respect to PQ^A > a n HV r e c o n s t r u c t i o n o f the quantum mechanics o f three-or-higher dimensional H i l b e r t space i s impossible. The above i n t e r p r e t a t i o n o f Gleason and Kochen-Specker's work a c t u a l l y depends upon our acknowledging the p r i o r i t y o f the PQ^a s t r u c t u r e as the core, or at l e a s t p a r t o f the core, of quantum mechanics which must be preserved. For d i s p e r s i o n - f r e e HV measures and a Boolean HV r e c o n s t r u c t i o n which do not preserve PQ^a a r e always p o s s i b l e . So i f F*QMA were not r e q u i r e d to be preserved, then i n s p i t e o f Gleason's completeness proof, the f a c t t h a t a l l the measures generated by the quantum formalism are d i s p e r s i v e would s i g n a l the incompleteness o f quantum mechanics r e l a t i v e to a p o s s i b l e Boolean HV r e c o n s t r u c t i o n which i n c l u d e d d i s p e r s i o n - f r e e HV measures. 202 Now as Bub mentions, the completeness of quantum mechanics with respect to PQJ^ » i . e . , the f a c t that the quantum formalism generates a l l possible (generalized) p r o b a b i l i t y measures on any 9 guarantees that the PQJ^ structure given by the fundamental quantum postulates and the pBA structure defined with respect to the quantum measures are isomorphic (Bub, 1974, p. 45). So the ambiguity, described i n Section KAJ, i n Kochen-Specker's notion of the quantum pr o p o s i t i o n a l structure as a (a) given PQJ^ a n d a (b) defined pBA i s not harmful but merely confusing. In p a r t i c u l a r , we can be sure that i f Exp^P^) = Exp^(P 2) f o r a l l quantum Exp. , then P, = P. i n P.„. . * f 1 2 QMA Moreover, i f we acknowledge the p r i o r i t y of the PQJJa structure i n quantum mechanics, then the s t r u c t u r a l condition of PQ^a preservation and the Gleason, Kochen-Specker HV i m p o s s i b i l i t y proofs emerge unscathed by the three sorts of c r i t i c i s m s described i n the previous section. In p a r t i c u l a r , P^^-preservation must s t i l l be required of a proposed HV theory i n s p i t e of the f a c t that t h i s condition leads to contradictions which make the HV theory impossible and of the zeroth kind, i n Belinfante's terminology. And P^^-preservation must be required of the proposed dispersion-free measures of an HV theory i n s p i t e of the considerations of measurement i n t e r a c t i o n which B e l l r a i s e s i n order to dissuade our imposing t h i s condition. And f i n a l l y , the Gleason, Kochen-Specker proofs cannot be charged with begging the HV question because they impose the Pg^-preservation condition, f o r the question of an HV reconstruction of quantum mechanics does not even make sense except with respect to the quantum PQJ^ structure, which must be preserved. In contrast, an HV advocate may choose to regard to quantum P^ 203 structure, whether P Q ^ A O R ^ Q M L ' a s n°"t worthy of preservation when considered with respect to the l a r g e r enterprise of providing a c l a s s i c a l , Boolean reconstruction or r e - i n t e r p r e t a t i o n of quantum mechanics, e s p e c i a l l y because such a reconstruction i s possible i f the quantum P N ^ i s not preserved. So rather than affirming the p r i o r i t y of the quantum structure i n the i n t e r p r e t a t i o n of quantum mechanics, an HV advocate may instead a f f i r m that ( i ' ) the problem of the completeness of any phys i c a l theory only makes sense when posed or framed with respect to a Boolean logical-property-event structure, ( i i ' ) the p r o b a b i l i t y measures of any s t a t i s t i c a l theory l i k e quantum mechanics are to be defined on a Boolean structure, and ( i i i 1 ) a Boolean HV reconstruction of quantum mechanics need not preserve the quantum P N ^ structure. As described by Bub, i f we acknowledge the p r i o r i t y of a Boolean HV reconstruction of quantum mechanics by aff i r m i n g these three primed conditions, then quantum mechanics i s incomplete and an HV reconstruction i s possible and completes quantum mechanics. Most simply, a Boolean structure always admits dispersion-free measures., yet quantum mechanics lacks dispersion-free measures. So with respect to a Boolean logical-property-event structure, quantum mechanics i s incomplete; and quantum mechanics i s completed when reconstructed as a Boolean HV theory which includes dispersion-free measures. Moreover, i f we acknowledge the p r i o r i t y of a Boolean HV reconstruction of quantum mechanics, then the ambiguity i n the Kochen-Specker notion of the quantum p r o p o s i t i o n a l structure i s again not harmful but merely confusing; f o r neither the (a) given P nor the (b) defined pBA need be preserved. I t also follows from the above acknowledgement that the s t r u c t u r a l condition of P „ „ .-preservation succumbs QMA 204 to the three s o r t s o f c r i t i c i s m s described i n the previous s e c t i o n , as do the s t r u c t u r a l c o n d i t i o n s (vn£>) and (JP&). In p a r t i c u l a r , s i n c e there i s no reason why an HV r e c o n s t r u c t i o n must s a t i s f y any of these s t r u c t u r a l c o n d i t i o n s , the von Neumann, the Jauch-Piron, the Gleason and the Kochen-Specker i m p o s s i b i l i t y proofs do beg the HV question s i n c e each r e s t s upon c o n t r a d i c t i o n s caused by the i m p o s i t i o n of an u n j u s t i f i e d c o n d i t i o n . B e l l ' s c o n s i d e r a t i o n s o f measurement i n t e r a c t i o n lend f u r t h e r support to the r e j e c t i o n o f the s t r u c t u r a l c o n d i t i o n s as u n j u s t i f i e d . And s i n c e the s t r u c t u r a l c o n d i t i o n s l e a d to c o n t r a d i c t i o n s , i n other words, s i n c e HV t h e o r i e s which i n c l u d e these s t r u c t u r a l c o n d i t i o n s are of the zeroth k i n d and are imp o s s i b l e , we can be sure t h a t the s t r u c t u r a l c o n d i t i o n s are p r e c i s e l y what a proposed HV r e c o n s t r u c t i o n o f quantum mechanics must not be r e q u i r e d t o s a t i s f y . So there are these two ways of i n t e r p r e t i n g quantum mechanics: E i t h e r the s t r u c t u r e i s regarded as the core o f quantum mechanics which must be preserved, i n which case quantum mechanics i s complete (as proved by Gleason) and a Boolean HV r e c o n s t r u c t i o n o f quantum mechanics i s impossible (as proved by Kochen-Specker). Or the p o s s i b i l i t y o f a Boolean r e c o n s t r u c t i o n o f quantum mechanics i s regarded as the most important consi&eratd'onrintthe interpretarion'-.ofc quantum:.mechanics,-' in-'which case a co n t e x t u a l Boolean HV r e c o n s t r u c t i o n which does not preserve P_,,. i s c QMA p o s s i b l e and quantum mechanics i s incomplete r e l a t i v e to t h i s r e c o n s t r u c t i o n . The a r t i c u l a t i o n o f t h i s dichotomy i s Bub's d e c i s i v e c o n t r i b u t i o n to the i n t e r p r e t a t i o n o f quantum mechanics and the problem of hi d d e n - v a r i a b l e s (see, e.g., Bub, 1973, p. 48). And n o t i c e t h a t t h i s dichotomy undercuts the three s o r t s o f arguments described i n Sectio n '.A . For re g a r d l e s s o f 205 the i n c o n s i s t e n c y and question begging c l a i m s , and re g a r d l e s s of B e l l ' s c o n s i d e r a t i o n s o f measurement i n t e r a c t i o n , the s t r u c t u r a l c o n d i t i o n s and the HV i m p o s s i b i l i t y proofs e i t h e r stand or f a l l depending upon which s i d e of the dichotomy one favours. In f a c t , which s i d e of the dichotomy one favours a l s o determines whether the i n c o n s i s t e n c y and question begging claims stand or f a l l . In the r e s t o f t h i s s e c t i o n , some arguments i n favour o f the P ^ ^ - p r e s e r v a t i o n s i d e o f t h i s dichotomy are described. One might a l s o consider regarding the orthomodular l a t t i c e P A M T r a t h e r than the QML p a r t i a l - B o o l e a n algebras P N M » as the core o f quantum mechanics which must QMA be preserved; some arguments ag a i n s t r e g a r d i n g p . T as the core o f quantum QML 7 mechanics are suggested by various p o i n t s made throughout t h i s t h e s i s . Both sides o f the dichotomy imply the i m p o s i t i o n o f s t r u c t u r a l c o n d i t i o n s on a proposed Boolean HV r e c o n s t r u c t i o n o f quantum mechanics. C l e a r l y , on the P ^ ^ - p r e s e r v a t i o n s i d e , the Boolean s t r u c t u r a l f e a t u r e s of each mBS i n a P^ M and the overlap p a t t e r n s among the mBS's i n a P^~ must be preserved. And on the Boolean r e c o n s t r u c t i o n s i d e , the Boolean s t r u c t u r a l f e a t u r e s o f each mBS i n a P^ may be preserved but, by v i r t u e o f the Gleason, Kochen-Specker r e s u l t s , the overlap p a t t e r n s among the mBS's i n a PQJ^3 cannot be preserved. So i t i s not the case t h a t one s i d e o f the dichotomy imposes s t r i n g e n t s t r u c t u r a l c o n d i t i o n s w h i l e the other s i d e does not. Rather, both sides impose e q u a l l y s t r i n g e n t c o n d i t i o n s : e i t h e r the overlap p a t t e r n s among the mBS's must be preserved, or the overlap patterns must be v i o l a t e d . The simple proposal that i n a proposed Boolean r e c o n s t r u c t i o n , the operations and r e l a t i o n s among compatibles ought to be preserved w h i l e 206 the operations and r e l a t i o n s among incompatibles ought to be ignored, does not help decide between the two sides of the dichotomy. A l l the elements i n an mBS of a P are mutually compatible; and f o r any non-overlapping mBS. , mBS. of P.,, , every element P. € mBS. (except the distinguished 0, 1 elements) i s incompatible with every element P^ € mBS_. (except the distinguished 0, 1 elements). But the elements i n any overlapping mBS's n>3 of P n^ are i n e x t r i c a b l y compatible and incompatible with each other i n the following sense. On the one hand, i f the operations and r e l a t i o n s among compatibles are preserved, then the overlap patterns are preserved, and then i t follows, as B e l l r i g h t l y argues, that some r e l a t i o n s among incompatibles are also preserved. On the other hand, i f the overlap patterns are not preserved, then these r e l a t i o n s among incompatibles are not preserved, but also some r e l a t i o n s among compatibles are not preserved. For example, consider the r e l a t i o n P^ < P^ V P,_ among the elements P^ , P^ , P i n the two overlapping mBS's of the twelve-element Pn^ diagrammed i n Section ''A.. I f the overlap pattern between mBS^ and mBS^ i s preserved, then t h i s r e l a t i o n i s preserved even though P. P^ V P , t h i s r e l a t i o n i s not preserved i n the sense that, f o r example, i n a contextual HV theory, f o r a given hidden state f , £, P^ may be assigned a value which i s not less-than-or-equal-to the value assigned to P^ v P 5 , i . e . , () £ g ( < I V v p 5 » m B S 4 > ^ I n short, r e l a t i o n s among compatible elements i n overlapping mBS's cannot be preserved without also preserving r e l a t i o n s among incompatibles, and r e l a t i o n s among incompatible elements i n overlapping mBS's cannot be ignored without also ignoring r e l a t i o n s among compatibles. 207 The c o n t e x t u a l HV proposals are the se r i o u s contenders on the Boolean r e c o n s t r u c t i o n s i d e o f the dichotomy. As Gudder makes c l e a r , c o n t e x t u a l HV t h e o r i e s preserve the Boolean s t r u c t u r a l f e a t u r e s of the mBS's i n a P s t r u c t u r e (Gudder, 1970, p. 435). But as described i n S e c t i o n A , the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a cont e x t u a l HV theory v i o l a t e the overlap patterns among the mBS's by as s i g n i n g d i f f e r e n t 0 or 1 values to a given element P € P^ when P i s a member of overlapping mBS's. That i s , the value assigned to P when considered i n the context o f one mBS may be d i f f e r e n t from the value assigned to P when considered i n the context o f another mBS. In t h i s sense, the i d e n t i t y P = P i s v i o l a t e d ; so c l e a r l y , any other o p e r a t i o n or r e l a t i o n among elements i n overlapping mBS's may be v i o l a t e d . And, f o r example, the d i s p e r s i o n - f r e e HV measures of a con t e x t u a l HV theory cannot even preserve j u s t the -1- o p e r a t i o n and the S. r e l a t i o n of P because, as shown i n Chapter V(C), j u s t x , 2 p r e s e r v a t i o n i s s u f f i c i e n t to ensure that a l l operations and r e l a t i o n s among compatible and incompatible elements i n the u l t r a s u b s t r u c t u r e s o f P.,, are preserved, where an u l t r a s u b s t r u c t u r e QM n>3 i n a PQ^ i s a union of overlapping mBS's; thus the overlap patterns among mBS's i n the u l t r a s u b s t r u c t u r e s o f PQ^ 3 a r e preserved i f •J~ , < are preserved. Quantum mechanics i t s e l f i s the se r i o u s contender on the Pq^-preservation s i d e o f the dichotomy; HV proposals which preserve PQJ^ are impossible. The quantum Exp^, measures on a P^ do preserve the 5, = r e l a t i o n s and the ^ op e r a t i o n o f , and they do preserve the A, v operations among any compatible p a i r s o f elements i n P^ (even though Exp^ may not be b i v a l e n t w i t h respect to every element i n PQ M)J thus the Exp^ measures do preserve a n d do preserve the overlap 208 p a t t e r n s among the mBS's. And i n p a r t i c u l a r , w i t h respect to the domain US^ where each Exp^ i s b i v a l e n t , Exp^ preserves a l l the operations and r e l a t i o n s among a l l (compatible and incompatible) elements i n the overlapping mBS's i n US, , as shown i n Chapter V I ( B ) . t Tutsch gives an example o f how the quantum mechanical o r d e r i n g o f p r o p o s i t i o n s , i . e . , the 5 r e l a t i o n of and thus the = r e l a t i o n , i s not preserved i n the Bohm-Bub co n t e x t u a l HV theory. In t h i s theory, once the hidden s t a t e \|/, £ and the context o f measurement are s p e c i f i e d , the outcome of a measurement o f a magnitude A (which i s an eigenvalue o f A) i s determined by the s o - c a l l e d polychotomic a l g o r i t h m , h e r e a f t e r c a l l e d the HV alg o r i t h m . Tutsch's example shows how according to the HV a l g o r i t h m , f o r a given hidden s t a t e y, g» "the outcome o f a measurement o f the magnitude /S /, the absolute value of s p i n - 1 i n the z d i r e c t i o n , i s the z eigenvalue 0 , w h i l e the outcome o f a measurement of the magnitude S , z s p i n - 1 i n the z d i r e c t i o n , i s the eigenvalue - 1 . C l e a r l y , /S / i s a Z f u n c t i o n o f S ; each magnitude i s represented by an operator on z three-dimensional H i l b e r t space; and both magnitudes share the eigenstate y n a s s o c i a t e d w i t h t h e i r 0 eigenvalues and represented by the one-dimensional p r o j e c t o r P Q = I Y Q > < Y Q I « l"n both quantum mechanics and a co n t e x t u a l HV theory, the outcome 0 f o r a measurement o f /S / or f o r a z measurement o f S^ i s connected w i t h the assignment o f the value 1 to the element P n € which qua p r o j e c t o r represents the eigenstate YQ • That i s , i n quantum mechanics, f o r a given quantum s t a t e Y» "the outcome o f a measurement o f /S / i s the eigenvalue 0 ( i . e . , Exp (/S /) = 0) IFF z z Exp i(P.) = 1 ; and l i k e w i s e , the outcome of a measurement o f S i s the Y u z eigenvalue 0 ( i . e . , Exp^.(Sz) = 0) IFF Exp^,(P Q) = 1 . S i m i l a r l y , i n a 209 con t e x t u a l HV theory, f o r a given hidden s t a t e y, and f o r any context mBS, the outcome o f a measurement o f /S / i s the eigenvalue 0 IFF Z LL, >,() = 1: and l i k e w i s e , the outcome o f a measurement of S i s *Y>£ 0' z the eigenvalue 0 IFF |j, () = 1. Furthermore, i n the quantum 3 p r o p o s i t i o n a l s t r u c t u r e P of three-dimensional H i l b e r t space, the element P^ , which qua p r o j e c t o r represents the eigenstate YQ , represents both o f the f o l l o w i n g p r o p o s i t i o n s : "The eigenvalue o f /S / z i s 0." "The eigenvalue of S i s 0." Since both p r o p o s i t i o n s are z 3 represented by the same element P^ = P^ i n P , each p r o p o s i t i o n i m p l i e s the other i n the sense t h a t P^ < P^ and P^ > P^ (where the 2 of i s i n t e r p r e t e d as l o g i c a l i m p l i c a t i o n ) . And s i n c e , f o r every quantum s t a t e w, Exp,(/S /) = 0 IFF Exp,(P ) = 1 IFF Exp,(S ) = 0, T Y z Y 0 Y Z each p r o p o s i t i o n i m p l i e s the other i n the sense t h a t , f o r any quantum s t a t e Y» i f the outcome o f a measurement o f /S / i s the eigenvalue 0, then z the outcome of a measurement o f S i s the eigenvalue 0, and z 8 conversely. But i n s p i t e of the f a c t t h a t quantum mechanically, each o f the above p r o p o s i t i o n s i m p l i e s the other ( i n both senses o f i m p l i e s ) , Tutsch gives an example of how i n a co n t e x t u a l HV theory, the p r o p o s i t i o n "The eigenvalue o f /S / i s 0." need not imply ( i n e i t h e r sense) the z p r o p o s i t i o n "The eigenvalue o f S i s 0." z According to the d e s c r i p t i o n of c o n t e x t u a l HV t h e o r i e s given i n Section ;A , such a d e v i a t i o n from quantum mechanics i s p o s s i b l e because PQ i s a member of ( a t l e a s t ) two over l a p p i n g mBS's s p e c i f y i n g p o s s i b l e contexts of measurements o f /S /, S , described as f o l l o w s . Besides the z z 0 eigenvalue-j the magnitude S has two other eigenvalues, 1, -1, each z ass o c i a t e d w i t h an eigenstate represented by a one-dimensional p r o j e c t o r 210 P. = l Y i ^ V i I a n d P -I = Iv i ^ V I j but the magnitude /S / has only J. J L J L "~" J. ~~ J. —• J- Z one other eigenvalue besides 0, namely, the eigenvalue 1 a s s o c i a t e d w i t h an eigenstate represented by a two-dimensional p r o j e c t o r , say _ i = P l v P _ i (i«e., the eigenvalue 1 o f /S^/ i s degenerate). Though P = P V P , i t i s e q u a l l y t r u e t h a t P = P V P f o r J L j J L JL JL 1 )~i 3. J J any orthogonal P ,P, which s a t i s f y P v P. = P. V P . . So the set a b a b 1 -1 {P n,P^,P_^} ( i . e . , the mBS^ generated by th a t set) may be the context o f a measurement o f /S / as w e l l as the set ( i . e . , the mBS Z U cl JD 3. generated by that s e t ) ; but only the set {FQ,F^,F_^} ( i . e . , mBS^) may be the context of a measurement o f S z . And c l e a r l y , mBS^ overlaps w i t h mBS s i n c e both share the element P„ . Now as described i n Section A , a 0 ' f o r a unique hidden s t a t e y»5» i t i s p o s s i b l e t h a t y, (
) y ,5 0 a 4 u, „(
).L So given the connection between the outcome 0 f o r a y,g 0 1 measurement o f /S / or S and the assignment o f the value 1 to the z z element P. described above, t h i s p o s s i b i l i t y : u,, (
) 0 Y>£ 0 a f- u. „() means t h a t i t i s p o s s i b l e that i f /S / i s measured i n y,£ o i z the context mBS and S i s measured i n the context mBS„ , then f o r a a z 1 unique hidden s t a t e y,£, the outcome of the measurement of /S / i s the z eigenvalue 0 (which occurs IFF u, () = 1 ) , w h i l e the outcome y,£ 0 a of the measurement of S i s one o f S 's other eigenvalues not equal to z z 0 (which occurs IFF (i (
) ^ 1 ) . In h i s example, Tutsch gives a y s s " i hidden s t a t e which, according to the HV a l g o r i t h m , assigns values which exemplify t h i s p o s s i b i l i t y . In p a r t i c u l a r , h i s hidden s t a t e y i e l d s the outcome 0 f o r /S / but the outcome -1 f o r S . And although Tutsch z z does not e x p l i c i t l y s t a t e t h a t i n h i s example, /S / i s measured i n a z context d i f f e r e n t from the context i n which S i s measured, Tutsch does z ' 211 conclude t h a t h i s example could mean t h a t the two p r o p o s i t i o n s : "The eigenvalue o f /S / i s 0." and "The eigenvalue o f S i s 0." r e f e r to . z z " p r o p e r t i e s o f the system pl u s apparatus and hence, d i f f e r e n t apparatus may produce d i f f e r e n t r e s u l t s . " This c o n c l u s i o n suggests that i n h i s example, /S / i s measured i n a context d i f f e r e n t from the context i n which S i s z z measured (Tutsch, 1969, pp. 1118-1119). B e l i n f a n t e speaks o f Tutsch's example as an example o f a paradox 9 which i s derived from the HV al g o r i t h m and which i s r e l a t e d to but i n f a c t worse than the Kochen-Specker t r o u b l e s (which motivate the c o n t e x t u a l HV proposals) i n th a t such paradoxes are "much l e s s ( i f a t a l l ) j u s t i f i a b l e as a ' r e s u l t o f the i n f l u e n c e of the measuring arangement'" ( B e l i n f a n t e , 1973, p. 135). Assuming that the above a n a l y s i s of Tutsch's example i s c o r r e c t , the example i s not an example o f a paradox. For /S / and S are z z measured i n d i f f e r e n t contexts ( i n v o l v i n g d i f f e r e n t experimental arrange-ments), and as Bub makes c l e a r , we must expect the d i s p e r s i o n - f r e e measures induced by the hidden s t a t e s of a c o n t e x t u a l HV theory to a s s i g n d i f f e r e n t values even to the same magnitude when measured i n d i f f e r e n t contexts. Gudder, who was i n contact w i t h Tutsch at the time o f the p u b l i c a t i o n o f each o f t h e i r papers, l i k e w i s e understands Tutsch's example as i n v o l v i n g measurements i n d i f f e r e n t contexts (Gudder, 1970, p. 436). Moreover, the "paradoxes" e x e m p l i f i e d by Tutsch's example are r e l a t e d to the Kochen-Specker t r o u b l e s only i n the sense that such "paradoxes" are feat u r e s o f a cont e x t u a l HV theory which are necessary i n order to avoid the Kochen-Specker HV i m p o s s i b i l i t y proof. For i f the d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a contextual HV theory d i d not a s s i g n d i f f e r e n t values to the same magnitude when measured i n d i f f e r e n t contexts and ins t e a d 212 assigned unique values to an element even though P Q i s a member of overlapping mBS's, then such measures would be r u l e d out by the Kochen-Specker proof. Gudder suggests that the s o r t of c o n t e x t u a l HV d e v i a t i o n s from quantum mechanics which are e x e m p l i f i e d by Tutsch's example may be candidates f o r experimental v e r i f i c a t i o n or f a l s i f i c a t i o n . In a d d i t i o n to Tutsch's s o r t o f d e v i a t i o n s , there i s a l s o the f o l l o w i n g s o r t of c o n t e x t u a l HV d e v i a t i o n s from quantum mechanics which has been the subject of experimental t e s t . In a c o n t e x t u a l HV theory, a pure quantum s t a t e ty ^ s t r e a t e d as a mixed s t a t e w i t h respect to the p o s s i b l e hidden s t a t e s (each represented by Y together w i t h some g ) , and the mixed s t a t e ty describes an ensemble of hidden s t a t e s w i t h a s o - c a l l e d e q u i l i b r i u m d i s t r i b u t i o n o f hidden v a r i a b l e s . In order t h a t the s t a t i s t i c a l c o n d i t i o n , mentioned i n the Preface to t h i s chapter, be s a t i s f i e d , t h i s e q u i l i b r i u m d i s t r i b u t i o n of hidden v a r i a b l e s together w i t h ty must reproduce, v i a the HV a l g o r i t h m , the s t a t i s t i c a l r e s u l t s of quantum mechanics given by the Exp^, measures ( B e l i n f a n t e , 1973, p. 136); or i n other words, the s t a t i s t i c a l r e s u l t s of quantum mechanics are derived from the HV a l g o r i t h m by assuming t h a t the hidden v a r i a b l e s are i n , an e q u i l i b r i u m d i s t r i b u t i o n . For example, as described by B e l i n f a n t e , consider a l a r g e number o f quantum systems whose quantum s t a t e i s ty on which we perform measurements of a magnitude A and, whenever the outcome i s a p a r t i c u l a r eigenvalue a.. , we f o l l o w up by measuring a d i f f e r e n t magnitude B. In quantum mechanics, the average value of B i s determined not by Exp,(B) but r a t h e r by Exp , ( B ) , where ty. i 3 i s the eigenstate of A a s s o c i a t e d w i t h the eigenvalue a.. ; t h a t i s , the f i r s t measurement of A i s assumed to have reduced the i n i t i a l s t a t e t 213 Y to the eigenstate • °^ A - In a co n t e x t u a l HV theory, i n order t h a t the HV al g o r i t h m reproduce t h i s quantum mechanical r e s u l t Exp, ( B ) , besides the r e d u c t i o n o f Y t° Yj > ^ m u s t a l s o be assumed t h a t the hidden v a r i a b l e s , which together w i t h y_. describe!, the hidden s t a t e s of the ensemble o f quantum systems a f t e r the measurement o f A, are i n an e q u i l i b r i u m d i s t r i b u t i o n before the measurement o f B occurs ( B e l i n f a n t e , 1973, pp. 139-140). However, a f t e r the measurement o f A, the HV al g o r i t h m c a l c u l a t i o n s y i e l d a non-equilibrium or biased d i s t r i b u t i o n o f hidden-v a r i a b l e s . I t i s assumed th a t such biased d i s t r i b u t i o n s of hid d e n - v a r i a b l e s very r a p i d l y r e l a x to the e q u i l i b r i u m d i s t r i b u t i o n which reproduces the Exp (B) r e s u l t . But i f the measurement o f B i s performed before the Yj biased d i s t r i b u t i o n r e s u l t i n g from the measurement o f A has r e l a x e d t o the e q u i l i b r i u m d i s t r i b u t i o n , then the biased d i s t r i b u t i o n p r e d i c t s v i a the HV algor i t h m s t a t i s t i c a l r e s u l t s f o r B which d i f f e r from what quantum mechanics p r e d i c t s v i a i t s Exp (B) formalism ( B e l i n f a n t e , 1973, p. 163). Yj These s o r t s o f d e v i a t i o n s from quantum mechanics which are connected w i t h non-equilibrium d i s t r i b u t i o n s o f hidden v a r i a b l e s a f t e r measurement are d i f f e r e n t from the Tutsch s o r t o f d e v i a t i o n s which are connected w i t h d i f f e r e n t contexts o f measurement. For as described by Bohm-Bub (1966, p. 466), the non-equilibrium s o r t o f d e v i a t i o n s occur f o r measurements o f magnitudes represented by operators on two-dimensional H i l b e r t space. But c l e a r l y , the c o n t e x t u a l s o r t of d e v i a t i o n s can occur f o r measurements only o f magnitudes represented by operators on a three-or-higher dimensional H i l b e r t space s i n c e only P Q ^ s t r u c t u r e s have overlapping mBS's. The existence o f these d e v i a t i o n s make i t a t l e a s t i n p r i n c i p l e p o s s i b l e to experimentally v e r i f y o r f a l s i f y the p r e d i c t i o n s o f the HV 214 a l g o r i t h m and thus to decide between quantum mechanics and the proposed c o n t e x t u a l , Boolean HV r e c o n s t r u c t i o n s of quantum mechanics. Experiments t e s t i n g f o r the non-equilibrium d i s t r i b u t i o n s o r t of d e v i a t i o n s w i t h a time -13 i n t e r v a l of l e s s than 10 seconds between the measurements of d i f f e r e n t magnitudes ( l i k e the measurements o f A and B described above) have so f a r found no d e v i a t i o n s from the p r e d i c t i o n s of quantum mechanics and have thus f a l s i f i e d the p r e d i c t i o n s of the HV a l g o r i t h m . However, HV advocates may argue t h a t the time i t takes a b i a s e d d i s t r i b u t i o n of hidden v a r i a b l e s -13 to r e l a x to the e q u i l i b r i u m d i s t r i b u t i o n i s l e s s than the 10 second i n t e r v a l between the measurements o f the experiments mentioned above. For as described by B e l i n f a n t e , " i t has not yet been e s t a b l i s h e d how f a s t one may t h e o r e t i c a l l y expect biased h i d d e n - v a r i a b l e d i s t r i b u t i o n s to r e l a x . . . .' So even i f the HV a l g o r i t h m i s f a l s i f i e d at an even s h o r t e r time i n t e r v a l i n some f u t u r e experiment, HV advocates may nevertheless continue to argue t h a t the s h o r t e r time i n t e r v a l i s not yet short enough to capture the b i a s e d d i s t r i b u t i o n of h i d d e n - v a r i a b l e s before i t r e l a x e s to the e q u i l i b r i u m d i s t r i b u t i o n which reproduces the quantum mechanical p r e d i c t i o n s . Thus w h i l e experiments have so f a r f a l s i f i e d and may continue to f a l s i f y the HV a l g o r i t h m , i t may be that no experiment w i l l ever c o n c l u s i v e l y decide between quantum mechanics and the proposed c o n t e x t u a l HV t h e o r i e s ( B e l i n f a n t e , 1973, pp. 88, 100). Nevertheless, quantum mechanics i s so f a r supported by.experimental evidence. And as pointed out by B e l i n f a n t e , the formalism of quantum mechanics i s simpler than the formalism of the c o n t e x t u a l HV t h e o r i e s . So by the usual c r i t e r i a o f experimental evidence and formal s i m p l i c i t y , quantum mechanics i s a b e t t e r theory of quantum phenomena than i s a c o n t e x t u a l HV 215 theory. So why i s quantum mechanics s t i l l challenged by the c o n t e x t u a l HV proposals? Four reasons are d e s c r i b e d , to the end o f t h i s s e c t i o n . 1. One reason quantum mechanics i s v u l n e r a b l e to a con t e x t u a l HV proposal i s because even i f i t i s granted t h a t the c l a s s i c a l n o t i o n o f a p r o b a b i l i t y measure defined on a Boolean s t r u c t u r e may be g e n e r a l i z e d so as to be defined on the non-Boolean quantum P N ^ s t r u c t u r e s , the notion o f a ge n e r a l i z e d measure on P N ^ i s open w i t h regard to the i s s u e o f which operations and r e l a t i o n s o f P A ^ ought to be r e q u i r e d to be preserved by the g e n e r a l i z e d measures. As described i n Chapter IV(A), Bub and Jauch-Piron each d e f i n e two d i f f e r e n t s o r t s o f g e n e r a l i z e d measures on P 0 ^ . The cont e x t u a l HV measures can be regarded as a t h i r d s o r t o f g e n e r a l i z e d p r o b a b i l i t y measure on P N ^ (even though the domain o f a c o n t e x t u a l HV measure i s P_„ x {mBS.}.- _ , ). A l l three s o r t s 'of g e n e r a l i z e d measures QM 1 i J i € I n d e x 6 preserve the Boolean s t r u c t u r a l f e a t u r e s of the (maximal) Boolean substructures o f P A M . In a d d i t i o n , Bub, Gleason, Kochen-Specker, and Jauch-Piron r e q u i r e t h a t a g e n e r a l i z e d p r o b a b i l i t y measure s a t i s f y Gleason's a d d i t i v i t y c o n d i t i o n (Ga) which ensures t h a t d i s p e r s i o n - f r e e g e n e r a l i z e d p r o b a b i l i t y measures preserve the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f P N ^ , i n p a r t i c u l a r , preserve the overlap patterns among the mBS's of P Q ^ -Jauch-Piron f u r t h e r r e q u i r e t h e i r g e n e r a l i z e d measures t o s a t i s f y (JP&). An argument ag a i n s t the i n c l u s i o n of (JP^) as pa r t o f the c o n d i t i o n s d e f i n i n g a g e n e r a l i z e d p r o b a b i l i t y measure i s given i n the note below. X^ Here we consider whether or not (Ga), which e n t a i l s P n ^ A ~ p r e s e r v a t i o n , ought to be included. The d i s p e r s i o n - f r e e HV measures induced by the hidden s t a t e s o f a c o n t e x t u a l HV theory do not and cannot s a t i s f y (Ga) because together w i t h 216 the d i s p e r s i o n - f r e e c o n d i t i o n , (Ga) y i e l d s P ^ ^ - p r e s e r v a t i o n and (Ga) a l s o y i e l d s the c o n d i t i o n l a b e l e d (B) by B e l l , n e i t h e r o f which i s s a t i s f i e d by co n t e x t u a l HV d i s p e r s i o n - f r e e measures. However, the i n c l u s i o n of at l e a s t Gleason's a d d i t i v i t y c o n d i t i o n as par t of the c o n d i t i o n s d e f i n i n g a gen e r a l i z e d p r o b a b i l i t y measure on a PQ^ i s s t r o n g l y supported by the precedent t h a t i n c l a s s i c a l p r o b a b i l i t y theory, c o n d i t i o n (ua) i s inc l u d e d among the c o n d i t i o n s d e f i n i n g a c l a s s i c a l p r o b a b i l i t y measure on a Boolean s t r u c t u r e , as s t a t e d i n Chapter I I I ( C ) . (see f o r example, Kolmogorov, 1933, p. 2). Since elements i n a P^^ are d i s j o i n t IFF they are orthogonal, or i n other words, o r t h o g o n a l i t y i s the quantum analogue o f d i s j o i n t e d n e s s , c o n d i t i o n (Ga), which r e q u i r e s t h a t a g e n e r a l i z e d p r o b a b i l i t y measure on a PQ^ be a d d i t i v e w i t h respect to orthogonal elements o f P^ M , i s the quantum analogue o f c o n d i t i o n ( p a ) , which r e q u i r e s t h a t a c l a s s i c a l p r o b a b i l i t y measure on a Boolean s t r u c t u r e be a d d i t i v e w i t h respect to d i s j o i n t elements. Or i n other words, (Ga) i s simply the c o n d i t i o n (p,a) as a p p l i e d to the quantum P^ M s t r u c t u r e s . So i t i s arguable that because (pa) i s one of the c o n d i t i o n s d e f i n i n g a c l a s s i c a l p r o b a b i l i t y measure, (Ga) ought to be one o f the c o n d i t i o n s d e f i n i n g a g e n e r a l i z e d p r o b a b i l i t y measure. Moreover, as elaborated at the end of S e c t i o n A , the c o n d i t i o n of p Q ^ - p r e s e r v a t i o n which f o l l o w s from (Ga) i s independently supported by the uncontentious general p r i n c i p l e according to which the r e a l value of say B o r e l f u n c t i o n o f any magnitude i n any p h y s i c a l theory i s c a l c u l a t e d or determined by simply applying t h a t B o r e l f u n c t i o n to the r e a l value o f the magnitude. Since any magnitude i s compatible w i t h any B o r e l f u n c t i o n o f i t s e l f , the general p r i n c i p l e r e f e r s to the p r e s e r v a t i o n o f f u n c t i o n a l r e l a t i o n s among compatible magnitudes ( o r p r o p o s i t i o n s ) . So i n a c o n t e x t u a l 217 HV theory, w h i l e the f u n c t i o n a l r e l a t i o n s among compatible elements i n any mBS o f P ^ are preserved, the f u n c t i o n a l r e l a t i o n s among compatible elements i n overlapping mBS's o f P^ M are not preserved s i n c e the P Q M A s t r u c t u r e i s not preserved and thus the general p r i n c i p l e which e n t a i l s p Q ^ - p r e s e r v a t i o n i s not s a t i s f i e d i n a c o n t e x t u a l HV theory. For example, as suggested by Gudder, i f one considered two d i f f e r e n t mBS's c o n t a i n i n g P and g(P) r e s p e c t i v e l y , then one would get independence of the re p r e s e n t i n g f u n c t i o n s f„ , f , r a t h e r than the f u n c t i o n a l r e l a t i o n f / T, N = g ( f _ ) . P g(P) g(P) P And the excuse given by c o n t e x t u a l HV advocates f o r t h i s v i o l a t i o n of the general p r i n c i p l e i s that the c o n s i d e r a t i o n of two d i f f e r e n t mBS's i n v o l v e s two separate measurements w i t h d i f f e r e n t experimental arrangements, and so i n such cases one would expect to o b t a i n independent r e s u l t s f o r P and g(P) (Gudder, 1970, p. 4-35). This excuse ignores o r makes l i g h t o f the f a c t t h a t , as determined by quantum mechanics and as (so f a r ) experimentally observed, the r e s u l t s o f any measurements o f P, g ( P ) , are not independent but r a t h e r are f u n c t i o n a l l y r e l a t e d i n accordance w i t h the general p r i n c i p l e . 2. As suggested again i n the previous paragraph, quantum theory i s v u l n e r a b l e to the co n t e x t u a l HV proposals i f measurement i n t e r a c t i o n o r measurement disturbance i s regarded as the cause or b a s i s o f the non-c l a s s i c a l p e c u l i a r i t i e s o f quantum mechanics and as ( a t l e a s t p a r t o f) the reason why the von Neumann, Jauch-Piron, and Kochen-Specker type of HV proposals are impossible. For example, according to Heisenberg's v e r s i o n of the Copenhagen i n t e r p r e t a t i o n of quantum mechanics, one reason why quantum ensembles cannot be r e s o l v e d i n t o subensembles which are d i s p e r s i o n - f r e e (as re q u i r e d i n von Neumann's HV proposal) i s because quantum systems are dis t u r b e d by measurement. And f o r an example o f how measurement c o n s i d e r a t i o n s 218 support c o n t e x t u a l HV proposals, we have of course B e l l ' s argument, from the pe r s p e c t i v e o f Bohr's v e r s i o n of the Copenhagen i n t e r p r e t a t i o n , t h a t s t r u c t u r a l c o n d i t i o n s l i k e P n M A ~ p r e s e r v a t i o n which r e f e r even i n d i r e c t l y to measurements of incompatible magnitudes must not be imposed upon the proposed d i s p e r s i o n - f r e e measures of an HV theory because of the i n e x t r i c a b l e wholeness o f quantum phenomena and measuring devices. Now the outcome of a measurement at best determines an assignment of 0, 1 values to a maximal Boolean subst r u c t u r e o f elements i n a P A ^ . For a measurement o f any magnitude A can at best be a measurement of what i s c a l l e d a complete s e t of commuting magnitudes i n c l u d i n g A (and i n c l u d i n g j u s t A i f none of A's eigenvalues are degenerate) whose eigens t a t e s ^ i ^ i € I n d e x ' a s r e P r e s e i r t e d by (n) orthogonal atoms ^ i ^ ^ I n d e x i n a P Q ^ s t r u c t u r e which generate a unique maximal Boolean subst r u c t u r e mBS^ of P Q ^ » s p e c i f y the context o f the measurement o f A. And the outcome o f the measurement, which i s an eigenvalue a^ of A a s s o c i a t e d w i t h an eigenstate \|/\ i n the set ^ i ^ i ' g i n d e x ' determines v i a E x p ^ i n quantum mechanics and v i a u. i n a c o n t e x t u a l HV theory, an assignment o f 0, 1 values to the elements i n th a t mBS. . The c o n t e x t u a l HV measure does no A more without changing i t s 0, 1 value assignments to the members o f mBS . However, without changing i t s value assignments to the members of mBSA , the quantum measure Exp : US ->• {0,1} i n a d d i t i o n assigns 0, 1 values ^ j * j , to every element i n the u l t r a s u b s t r u c t u r e US^ = {P € P Q M : P o P.} 2 m B S A > where (unless US happens to equal mBS.) US i s a union o f ¥j A Yj overlapping mBS's i n c l u d i n g mBS . A These a d d i t i o n a l 0, 1 value assignments by the quantum measure Expr" mean the f o l l o w i n g . Let B be any magnitude which shares the 219 eigenstate t y w i t h A even though B k> A ( i . e . , B and A do not share a l l t h e i r e i g e n s t a t e s ) . E i t h e r alone ( i f none o f B's eigenvalues are degenerate) o r as p a r t o f a complete set o f commuting magnitudes, B s p e c i f i e s a unique maximal Boolean substructure mBS^ of P n^ which c l e a r l y overlaps w i t h mBSA s i n c e the atom P , which qua p r o j e c t o r represents the eigenstate ty. , i s a member of both mBS and mBS . 3 A And s i n c e every element i n mBS^ i s compatible w i t h P^ , c l e a r l y mBSg c US, . Thus Exp, assigns 0, 1 values to every element i n mBSg . And these value assignments by Exp mean tha t i f B i s measured a f t e r Yj A i s measured, or i f B, in s t e a d o f A, had been measured w i t h the outcome b^ , then the outcome o f the measurement o f B, namely, the eigenvalue b_. a s s o c i a t e d w i t h the eigenstate , determines an assignment o f 0, 1 values to the elements i n mBS such that the values assigned to the common elements i n mBS fl mBS match the value assignments A 15 determined by the outcome o f the ( f i r s t ) measurement of A. Thus the 0, 1 value assignments by the quantum measure Exp to every element i n both mBS. c US, and mBS,, c US, are determined by J A ty. B ty. J the outcome o f one measurement y e t r e f e r to the outcomes o f more than one measurement. For A and B cannot be measured simultaneously, i . e . , A & B. (And i f A i B, then mBS^ = mBSg i n P^ .) In other words, the f a c t t h a t a quantum Exp^ measure assigns 0, 1 values to overlapping mBS's of elements i n a manner which preserves the overlap p a t t e r n s says something about d i f f e r e n t measurements of incompatible magnitudes. S i m i l a r l y , i f proposed d i s p e r s i o n - f r e e HV measures are r e q u i r e d to a s s i g n 0, 1 values to o v erlapping mBS's of elements i n a manner which preserves the overlap p a t t e r n s , then t h i s requirement does r e f e r to d i f f e r e n t measurements of 2 2 0 incompatible magnitudes, as the c o n t e x t u a l HV advocates argue. For example, 3 the 1 9 2 atoms contained i n 1 1 8 overlapping mBS's i n the P Q ^ a considered by Kochen-Specker i n the Theorem 1 p a r t o f t h e i r HV i m p o s s i b i l i t y proof represent the eigenst a t e s of 1 1 8 incompatible magnitudes which cannot a l l be measured together, y e t Kochen-Specker r e q u i r e proposed d i s p e r s i o n - f r e e HV measures to preserve the overlap p a t t e r n s among the eigenstates o f these magnitudes. This requirement, which i s p a r t o f the P ^ ^ - p r e s e r v a t i o n c o n d i t i o n , i s very hard t o motivate i f measurement i n t e r a c t i o n , as des c r i b e d by Bohr and B e l l , o r measurement d i s t u r b a n c e , as described by Heisenberg w i t h h i s Uncertainty P r i n c i p l e , are t r e a t e d as c e n t r a l i n the i n t e r p r e t a t i o n o f quantum mechanics, as the cause o f the n o n - c l a s s i c a l p e c u l i a r i t i e s o f quantum mechanics, and as the reason why hidden-v a r i a b l e s are e i t h e r impossible or e l s e dependent upon the context o f measurement. In c o n t r a s t , i f the non-Boolean P_„. s t r u c t u r e a b s t r a c t e d from QMA the fundamental p o s t u l a t e s o f quantum mechanics i s t r e a t e d as c e n t r a l i n the i n t e r p r e t a t i o n o f quantum mechanics, then the n o n - c l a s s i c a l p e c u l i a r i t i e s of quantum mechanics are regarded as due to the non-Boolean character o f the P „ „ . s t r u c t u r e r a t h e r than due to measurement i n t e r a c t i o n o r QMA d i s t u r b a n c e . X x And as Kochen-Specker and Bub make c l e a r , c o n s i d e r a t i o n o f measurement i n t e r a c t i o n or disturbance are beside the p o i n t i f the problem of hidd e n - v a r i a b l e s i s c o r r e c t l y understood as posing the question o f whether the s t a t i s t i c a l r e s u l t s o f can be r e c o n s t r u c t e d i n QMA terms o f a c l a s s i c a l measure space <®,P „,u> i n a manner which preserves HV the core s t r u c t u r e of quantum mechanics. For example, i n s p i t e o f Heisenberg's U n c e r t a i n t y P r i n c i p l e , the s t a t i s t i c a l r e s u l t s o f 2 2 can be c l a s s i c a l l y r e c o n s t r u c t e d , as Kochen-Specker 221 demonstrate by producing an HV theory f o r t h a t p a r t of quantum mechanics which i n v o l v e s j u s t two-dimensional H i l b e r t space (Kochen-Specker, 1967, pp. 75-80, 86). 3. Metaphysical p r e j u d i c e s , l i k e the " r e l i g i o u s b e l i e f t h a t 'nature must be d e t e r m i n i s t i c ' . . ." mentioned by B e l i n f a n t e (1973, p. 18) make quantum mechanics e s p e c i a l l y v u l n e r a b l e to the c o n t e x t u a l HV proposals. As described by Bub, the main reason why quantum mechanics i s v u l n e r a b l e to c o n t e x t u a l HV proposals i s because o f the p r e s u p p o s i t i o n t h a t the l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of r e a l i t y and of any p h y s i c a l theory about any p o r t i o n of r e a l i t y is_ and can only be a Boolean s t r u c t u r e . Bub argues that behind the a f f i r m a t i o n of the three primed c o n d i t i o n s (i')» ( i i ' ) , ( i i i ' ) d e s c r i b i n g the Boolean r e c o n s t r u c t i o n s i d e o f the dichotomy i n the i n t e r p r e t a t i o n o f quantum mechanics i s the (metaphysical) p r e s u p p o s i t i o n t h a t the l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of quantum phenomena must be a Boolean s t r u c t u r e , l i k e the Boolean l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of c l a s s i c a l phenomena and c l a s s i c a l mechanics. In c o n t r a s t , behind the a f f i r m a t i o n of the three un-primed c o n d i t i o n s ( i ) , ( i i ) , ( i i i ) d e s c r i b i n g the P ^ ^ - p r e s e r v a t i o n s i d e o f the dichotomy, there i s an open acceptance of the n o t i o n of a non-Boolean l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of quantum phenomena and quantum mechanics (Bub, 1973, p. 54; 1974, p. 144). This acceptance i s motivated by the f o l l o w i n g analogy. The l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e o f c l a s s i c a l phenomena as described by c l a s s i c a l mechanics i s i d e n t i f i e d w i t h (or i s considered to be isomorphic with) the Boolean p r o p o s i t i o n a l s t r u c t u r e P^ a b s t r a c t e d from the formalism of c l a s s i c a l mechanics. The non-Boolean p r o p o s i t i o n a l s t r u c t u r e P N ^ , i n p a r t i c u l a r , the P Q M A s t r u c t u r e , i s a b s t r a c t e d from the formalism of quantum mechanics 222 i n a manner e x a c t l y analogous to the way i n which P i s a b s t r a c t e d from the c l a s s i c a l formalism. So the l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e of quantum phenomena, as s u c c e s s f u l l y described by quantum mechanics, may be and ought to be i d e n t i f i e d w i t h (or considered to be isomorphic with) the P Q ^ A s t r u c t u r e r a t h e r than any proposed Boolean P s t r u c t u r e . HV Now i f the quantum P Q J ^ could be imbedded() i n t o a Boolean s t r u c t u r e , then the s t a t i s t i c a l r e s u l t s o f quantum mechanics could be reconstructed i n terms o f a c l a s s i c a l measure space w i t h a HV Boolean s t r u c t u r e a t i t s core, and thus quantum mechanics could be regarded as a r a t h e r baroque e l a b o r a t i o n o f what i s e s s e n t i a l l y a c l a s s i c a l s t a t i s t i c a l theory. For example, Pq^A c a n be imbedded(i>) i n t o a Boolean s t r u c t u r e , 2 2 and the s t a t i s t i c a l r e s u l t s can be recon s t r u c t e d i n terms QMA r Y of a c l a s s i c a l measure space, as demonstrated by Kochen-Specker. So i f quantum mechanics made use of j u s t two-dimensional H i l b e r t space r a t h e r than any higher dimensional H i l b e r t spaces, then quantum mechanics would i n f a c t be a c l a s s i c a l s t a t i s t i c a l theory s i n c e a l l o f i t s s t a t i s t i c a l r e s u l t s c o u l d be c l a s s i c a l l y r e c o n s t r u c t e d . However, the quantum P Q ^ s t r u c t u r e s cannot be imbedded(A) i n t o a Boolean s t r u c t u r e , and the < W N _ ^ ' P Q ^ ' ^ P ^ s t a t i s t i c a l r e s u l t s cannot be recon s t r u c t e d i n terms o f a c l a s s i c a l measure space. For Kochen-Specker and f o r Bub, t h i s f a c t demarcates quantum 12 mechanics from c l a s s i c a l mechanics. As Bub says: I have argued t h a t the t r a n s i t i o n from c l a s s i c a l t o quantum mechanics i s to be understood as a g e n e r a l i z a t i o n o f the Boolean p r o p o s i t i o n a l s t r u c t u r e s o f c l a s s i c a l mechanics to a p a r t i c u l a r c l a s s of non-Boolean s t r u c t u r e s . (1974, pp. 149-150) So the f a c t t h a t a P ~. i s not imbeddable(3 i s the p r e s e r v a t i o n o f the p a r t i a l - B o o l e a n s t r u c t u r a l f e a t u r e s o f P Q J ^ > i n p a r t i c u l a r , the p r e s e r v a t i o n o f the overlap p a t t e r n s among the mBS's i n n-3 P Q M A , which must be given up i n order to make a Boolean HV r e c o n s t r u c t i o n of quantum mechanics, i n p a r t i c u l a r , a con t e x t u a l HV theory, p o s s i b l e lends f u r t h e r support to l o c a t i n g the P Q J ^ s t r u c t u r e a t the core of quantum mechanics which must be preserved. As suggested above, the p r e s e r v a t i o n o f the P Q ^ a s t r u c t u r e i s f u r t h e r motivated by regarding P Q J ^ a s t n e l o g i c a l s t r u c t u r e o f quantum mechanics and as the l o g i c a l space, i n a W i t t g e n s t e i n i a n sense, o f micro-events, as Bub does (Bub, 1973, p. 52; W i t t g e n s t e i n , 1921, p. 35). Now whether or not the P.,,. (or the P „ „ T ) s t r u c t u r e i s accepted as a QMA QML r new quantum l o g i c depends upon one's views o f what l o g i c i s and of what r o l e l o g i c plays i n a p h y s i c a l theory. Bub argues t h a t the s t r u c t u r e o f l o g i c a l space i s not " p a r a s i t i c on the s y n t a c t i c p r o p e r t i e s o f a f o r m a l i z e d language," i s not con v e n t i o n a l , and i s not "a p r i o r i i n the sense that the laws o f l o g i c s c h a r a c t e r i z e necessary f e a t u r e s o f any l i n g u i s t i c framework s u i t a b l e f o r the d e s c r i p t i o n and communication of experience." Rather, " l o g i c i s about the world, not about language"-(Bub, 1973, pp. 52-53). And: "The l o g i c a l s t r u c t u r e of a p h y s i c a l system i s understood as imposing the most general k i n d o f c o n s t r a i n t on the occurrence and non-occurrence o f events" (Bub, 1974, p. 149). Moreover, as f i r s t suggested by Putnam, j u s t as geometry play s an explanatory r o l e i n r e l a t i v i s t i c mechanics, e.g., the curved geometry o f space-time " e x p l a i n s " g r a v i t y , s i m i l a r l y , quantum l o g i c p l a y s an explanatory r o l e i n quantum mechanics, e.g., the f a c t t h a t the l o g i c a l core o f quantum mechanics i s the non-Boolean P Q ^ s t r u c t u r e 2 2 4 " e x p l a i n s " the n o n - c l a s s i c a l p e c u l i a r i t i e s o f quantum mechanics (Bub, 1 9 7 3 , p. 5 2 ; Putnam, 1 9 6 9 ) . 4 . F i n a l l y , as suggested at the end o f p o i n t ( 2 ) above, an inadequate or i n c o r r e c t view o f the problem o f hidden- v a r i a b l e s and the problem o f the completeness of quantum mechanics makes quantum mechanics vul n e r a b l e to the c o n t e x t u a l H V proposals. As Bub argues, the n o t i o n o f a completion o r extension o f a p h y s i c a l theory o n l y makes sense w i t h respect to the u n d e r l y i n g l o g i c a l - p r o p e r t y - e v e n t s t r u c t u r e as given by the p r o p o s i t i o n a l s t r u c t u r e determined by the theory's formalism. So a co n t e x t u a l H V theory which does not preserve the quantum P Q J ^ s t r u c t u r e a b s t r a c t e d from the fundamental p o s t u l a t e s o f quantum mechanics i s not a completion o f quantum mechanics but r a t h e r i s an e n t i r e l y separate theory of quantum phenomena which w i l l have to stand on i t s own f e e t (Bub, 1 9 7 4 , p. 1 4 7 ) . Considering the experimental f a l s i f i c a t i o n s o f the c o n t e x t u a l H V d e v i a t i o n s from quantum mechanics, we may conclude w i t h Stapp t h a t quantum mechanics i s complete, i n at l e a s t the pragmatic sense t h a t . . . no t h e o r e t i c a l c o n s t r u c t i o n can y i e l d [or has so f a r y i e l d e d ] experimentally v e r i f i a b l e p r e d i c t i o n s about atomic phenomena th a t cannot be e x t r a c t e d from the quantum t h e o r e t i c d e s c r i p t i o n . (Stapp, 1 9 7 2 , p. 1 1 0 8 ) 1 3 Notes: 1 This statement i s corroborated by Kochen-Specker ( 1 9 6 7 , p. 8 1 ) and Gudder ( 1 9 7 0 , p. 4 3 2 ) . 2 B e l i n f a n t e makes a s i m i l a r p o i n t ( 1 9 7 3 , pp. 2 5 - 2 6 ) . 3 Bub uses t h i s phrase i n reference to c o n t e x t u a l H V t h e o r i e s , as w i l l be discussed below (Bub, 1 9 7 4 , p. 1 4 6 ) . 225 4 As mentioned above, B e l l claims t h a t Gleason's i m p o s s i b i l i t y -proof r e s t s upon the t a c i t assumption that d i s p e r s i o n - f r e e HV measures are non-contextual. B e l i n f a n t e s u b s t a n t i a t e s B e l l ' s c l a i m by e l a b o r a t i n g how the c o n d i t i o n B e l l l a b e l s (B) f o l l o w s from the c o n d i t i o n B e l l l a b e l s (A) together w i t h the non-contextual assumption ( B e l i n f a n t e , 1973, p. 65). 5 ~ I f any of the P. re p r e s e n t i n g the eigenstates o f A are two-or-higher dimensional p r o j e c t o r s , which obtains i f any of A's eigenvalues are degenerate, then c o n t e x t u a l HV t h e o r i e s have a procedure, e.g., Tutsch's r u l e , f o r augmenting the set of eigenstates so th a t the set i s complete and unique, and so s p e c i f i e s a context ( B e l i n f a n t e , 1973, pp. 132-133). See note 4 of Chapter V, and the d i s c u s s i o n of measurement under poin t .2. at the end of the next S e c t i o n B.. 7 Various p o i n t s made throughout t h i s t h e s i s suggest the problematic character of the l a t t i c e d e f i n i t i o n s o f A, V among incompatibles and so favour the p a r t i a l - B o o l e a n algebra. P • . f o r m a l i z a t i o n of the quantum p r o p o s i t i o n a l s t r u c t u r e s . As described i n Chapter IV, von Neumann f i r s t developed, i n 1932, something l i k e a p a r t i a l - B o o l e a n algebra o f quantum p r o p o s i t i o n s . A l a t t i c e o f quantum p r o p o s i t i o n s was developed by von Neumann fou r years l a t e r w i t h the c o l l a b o r a t i o n o f B i r k h o f f , who had j u s t founded l a t t i c e theory and so no doubt had the idea of a l a t t i c e , w i t h 5 i n t e r p r e t e d as l o g i c a l i m p l i c a t i o n , s t r o n g l y i n mind. But B i r k h o f f and von Neumann immediately recognized the problematic c h a r a c t e r o f the meets and j o i n s o f incompatible p r o p o s i t i o n s , which they s a i d could not be i n t e r p r e t e d as experimental p r o p o s i t i o n s . Moreover, the d e f i n i t i o n of A , v among incompatible p r o p o s i t i o n s qua p r o j e c t o r s cannot be given simply i n terms o f + and • as u s u a l , r a t h e r , Jauch had to create d e f i n i t i o n s i n v o l v i n g the l i m i t s of i n f i n i t e products. I t has been s a i d t h a t the l a t t i c e d e f i n i t i o n s o f A , V among incompatibles r e s u l t s i n misinterpretations•of"theeelements- o f P N M . And the l a t t i c e d e f i n i t i o n s o f A , v among incompatibles can cause a p r o l i f e r a t i o n o f l a t t i c e elements, as e x e m p l i f i e d i n Chapter V I ( C ) , and do cause t r u t h - f u n c t i o n a l i t y ( o , & ) problems which are p e c u l i a r to the Pnm and are avoided by the PQ^A s t r u c t u r e s , as described i n Chapter V. Also see notes 10 and 12 below f o r f u r t h e r c r i t i c i s m s of the ai 8. orthomodul r l a t t i c e P.... f o r m a l i z a t i o n o f the quantum P „ „ QML ^ QM 'When Tutsch t a l k s o f the "ordinary h e u r i s t i c sense" o f i m p l i c a t i o n , I understand him to be r e f e r r i n g to the l a t t e r sense o f i m p l i c a t i o n which i n v o l v e s the outcomes o f two measurements. The two measurements may be successive measurements, or the two measurements may be a l t e r n a t e measurements performed on two systems i n the same prepared s t a t e . In the former case, i t i s assumed that the f i r s t measurement i s " r e p r o d u c i b l e , " which l o o s e l y speaking means that the measurement can be f o l l o w e d up by another measurement (e.g., the measured system has not been a n n i h i l a t e d ) and which more s t r i c t l y speaking means t h a t the measurement can serve as 226 what has been c a l l e d a s t a t e p r e p a r a t i o n ( B e l i n f a n t e , 1973, p. 6; B a l l e n t i n e , 1970, p. 366). 9 B e l i n f a n t e p o i n t s to the HV al g o r i t h m and the s o - c a l l e d Tutsch's r u l e as the b a s i s o f the "paradox" e x e m p l i f i e d by Tutsch's example ( B e l i n f a n t e , 1973, pp. 142, 217). As mentioned i n note 5 above, Tutsch's r u l e i s a r u l e by which a unique and complete context o f measurement i s determined f o r a measurement o f magnitude which has degenerate eigenvalues. This r u l e may determine, f o r example, the context {P n> p »P^} ( i . e . , mBS ) U 3. JD 3. f o r a measurement o f the magnitude /S / whose eigenvalue 1 i s degenerate. X < ^ I n c l a s s i c a l p r o b a b i l i t y theory, a c l a s s i c a l p r o b a b i l i t y measure i s d efined as a f u n c t i o n p, : B -»• [0,1] s a t i s f y i n g : ( i ) Gleason's a d d i t i v i t y c o n d i t i o n (Ga), where o r t h o g o n a l i t y i s equivalent to d i s j o i n t e d n e s s ( i i ) n(0) = 0 and (j,(l) = 1 (see Chapter 111(C)) And i t i s easy t o show th a t u. s a t i s f i e s c o n d i t i o n (JP&), i . e . , f o r any b,c € B, i f u.(b) = |i(c) = 1 then |i(b Ac) = 1 (Jauch, 1976, pp. 136-137). Now Jauch defi n e s a g e n e r a l i z e d p r o b a b i l i t y measure as a f u n c t i o n M- : P Q M L -> [0,1] s a t i s f y i n g : ( i ) (Ga) ( i i ) u.(0) = 0 and u.(l) = 1 ( i i i ) (JPfc) And Jauch remarks t h a t property ( i i i ) "must be po s t u l a t e d s i n c e i t cannot be d e r i v e d from the other two as i n the c l a s s i c a l p r o b a b i l i t y c a l c u l u s " (Jauch, 1976, pp. 135, 136). In order to help motivate the i n c l u s i o n of (JP&) as part of the c o n d i t i o n s d e f i n i n g a g e n e r a l i z e d p r o b a b i l i t y measure, Jauch mentions h i s passive f i l t e r i n t e r p r e t a t i o n o f quantum p r o p o s i t i o n s , according to which the c o n j u n c t i o n P j _ A P 2 ' "^or P l ^ P 2 ' 1 S i n " t e r P r ' e ' t e d - a s a n i n f i n i t e , a l t e r n a t i n g sequence o f f i l t e r s r e p r e s e n t i n g P^ , P^ . (This passive f i l t e r i n t e r p r e t a t i o n , described i n greater d e t a i l i n Jauch's (1968, pp. 74-76), has always seemed suspect to me; Jauch proposes i t i n order to make sense of the l a t t i c e d e f i n i t i o n of A among incompatibles.) And us i n g Gleason's Completeness r e s u l t , Jauch give s a d e r i v a t i o n o f ( i i i ) , i . e . , (JPjfe), from ( i ) and ( i i ) f o r the case o f pn>3 . So although Gleason QML does not i n c l u d e (JP&) as par t o f h i s d e f i n i t i o n o f a g e n e r a l i z e d p r o b a b i l i t y measure, Jauch uses Gleason's r e s u l t t o help make the i n c l u s i o n of (JP&) as par t o f Jauch's d e f i n i t i o n o f a g e n e r a l i z e d p r o b a b i l i t y measure more p a l a t a b l e . However, Jauch adds: 227 This i s The only example known to me of a p r o b a b i l i t y measure on a l a t t i c e which does not s a t i s f y ( i i i ) i s i n a l a t t i c e w i t h a maximal chain o f three elements [ i . e . , p2 1 of course p r e c i s e l y the case t h a t i s excluded by the hypothesis o f Gleason's theorem t h a t dim H>3. In view of t h i s f a c t i t would be of considerable i n t e r e s t to prove property ( i i i ) i n the l a t t i c e - t h e o r e t i c s e t t i n g . No such proof i s known to me.' (Jauch, 1976, p. 139) Thus, si n c e there i s no d e r i v a t i o n o f ( i i i ) , i . e . , (JP,#) c o n d i t i o n considered i n Chapters IV and V ensures the p r e s e r v a t i o n o f the feat u r e s of P . T which d i s t i n g u i s h P_ T from P 0 M A J namely, the meets and j o i n s o f incompatibles. But as f a r as I know, no author who has considered the problem o f how best to define the no t i o n o f a g e n e r a l i z e d p r o b a b i l i t y measure on the quantum PQJ^ s t r u c t u r e has advocated the i n c l u s i o n o f a c o n d i t i o n as strong as t r u t h - f u n c t i o n a l i t y ^ , & > ) . ''""''Some examples o f how the non-Boolean char a c t e r o f the quantum PQMA s t r u c t u r e i s the b a s i s o f the n o n - c l a s s i c a l p e c u l i a r i t i e s o f the quantum s t a t i s t i c a l r e s u l t s are described by Bub (1974, pp. 149, 120-122, 125-127). 12 Bub's poi n t about the demarcation between c l a s s i c a l mechanics and quantum mechanics suggests the f o l l o w i n g c r i t i c i s m o f the orthomodular l a t t i c e P Q M L f o r m a l i z a t i o n of the quantum P^ M s t r u c t u r e . As described i n Chapters IV(F) and V, from the P _ M T p e r s p e c t i v e , the p e c u l i a r l y n o n - c l a s s i c a l f e a t u r e which d i s t i n g u i s h e s the quantum p r o p o s i t i o n a l s t r u c t u r e s from the c l a s s i c a l ones and which, f o r example, makes a c l a s s i c a l HV r e c o n s t r u c t i o n o f quantum mechanics im p o s s i b l e , i s the 228 presence of incompatible elements. This i s made c l e a r i n the statement o f Theorem A (Chapter V(A)) and, f o r example, i n Jauch-Piron's concluding remark about t h e i r HV i m p o s s i b i l i t y proof: "To r u l e out hidden v a r i a b l e s i t s u f f i c e s to e x h i b i t two p r o p o s i t i o n s o f a p h y s i c a l system which are not compatible" (Jauch-Piron, 1963, p. 837). In c o n t r a s t , from the P_„. p e r s p e c t i v e , i t i s not the presence of incompatibles but r a t h e r the presence of overlapping mBS's ( f o r which the presence of incompatibles i s necessary but not s u f f i c i e n t ) which d i s t i n g u i s h e s the quantum p r o p o s i t i o n a l s t r u c t u r e s from the c l a s s i c a l ones and makes a c l a s s i c a l HV r e c o n s t r u c t i o n of quantum mechanics impossible. However, as described i n Chapter I V ( F ) , any PQ^ whose mBS's happen to not overlap can be imbedded(i) i n t o a Boolean s t r u c t u r e and so i s c l a s s i c a l , i n a Kochen-Specker sense. And i n p a r t i c u l a r , the mBS's o f a „2 s t r u c t u r e do not o v e r l a p , p 2 can be QM QM imbedded((i») i n t o a Boolean s t r u c t u r e , and the quantum mechanics of two-dimensional H i l b e r t space does admit a c l a s s i c a l HV r e c o n s t r u c t i o n . Thus from the P^MA p e r s p e c t i v e , there i s a classical/quantum demarcation between P A ^ s t r u c t u r e s w i t h non-overlapping mBS's and P N ^ s t r u c t u r e s w i t h overlapping mBS's. And i n p a r t i c u l a r , from the P^MA p e r s p e c t i v e , there i s a classical/quantum demarcation between p2 and „n>3 s t r u c t u r e s . QM QM These classical/quantum demarcations, which are not recognized from the P Q^ p e r s p e c t i v e , are i n f a c t corroborated by the l a t t i c e -t h e o r e t i c i a n Jauch as f o l l o w s . Jauch argues t h a t although P N M T s t r u c t u r e s may be i r r e d u c i b l e o r may be r e d u c i b l e , the quantum s u p e r s e l e c t i o n r u l e s ensure t h a t every quantum mechanically r e l e v a n t POMT i s r e d u c i b l e (Jauch, 1968, p. 109). Thus Jauch makes a non-quantum/quantum demarcation between i r r e d u c i b l e P M T and r e d u c i b l e P A „ T . Now i t i s i n t u i t i v e l y obvious t h a t i f a P^ W T i s r e d u c i b l e , QML J QML then P N M T contains overlapping mBS's. For i n a r e d u c i b l e P_„. , there i s a t l e a s t one element Prt 4 0,1, such t h a t P ^ o P y f o r every P € P_„ T ; thus P. must be a member of more than one J QML 0 mBS of PQJ^ ' C o n t r a p o s i t i v e l y , i f none of the mBS's i n a PQ^L o v e r l a p , then the only elements shared by any mBS's are the 0, 1 elements, and so P . „ T must have a t r i v i a l c e n t r e , i . e . , P - „ T i s i r r e d u c i b l e . So QML QML i f we consider a r e d u c i b l e P N M T (which f a l l s on the quantum-side of Jauch's demarcation), s i n c e a r e d u c i b l e P N M T contains overlapping mBS's, a r e d u c i b l e - P N M T a l s o f a l l s on the quantum-side of the P N M A demarcation. And i f we consider a two-dimensional H i l b e r t space 2 ' - P P^,,T (which f a l l s on the c l a s s i c a l - s i d e o f the QMA demarcation), s i n c e QML 2 - • " 2 the mBS's i n a P R T W T do not o v e r l a p , a P „ „ T i s i r r e d u c i b l e and so f a l l s ' QML QML on the non-quantum-side of Jauch's demarcation. This suggests, i n broad o u t l i n e , a c o r r e l a t i o n between Jauch's demarcation and the POMA demarcations, even though the l a t t e r are demarcations between classical/quantum w h i l e the former i s a demarcation between non-quantum/ quantum. So although the P ^ demarcations between P ^ w i t h overlapping 229 2 n^3 mBS's and P ^ w i t h non-overlapping mBS's and between P ^ and PQ~ are not recognized from the PQ^L p e r s p e c t i v e , they are r e f l e c t e d i n Jauch's demarcation between i r r e d u c i b l e and r e d u c i b l e s t r u c t u r e s , which suggests 13, t h a t the d marcations are worth r e c o g n i z i n g . 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