Appendix A SET THEORY AND LATTICES
The objective of the Appendices is to present the main definitions and propositions of some of the mathematical structures used in the text. With a few exceptions, no attempt was made to present proofs of the propositions, for otherwise the length of the Appendices would be greater than the main text. Any serious student who needs to improve his knowledge about the topics treated below must consult mathematical texts as, e.g., [1-6] and also the following excellent texts on quantum theory[7-10].
A.1 MAIN DEFINITIONS There are two primitive concepts in set theory, the notion of set itself1 and the notion of pertinence as used, e.g., in sentences like "x is an element of the set S." We write xES if x is a member of a given set S and write x ~ S otherwise. In general a set S is determined by some property P shared by its members and we denote this fact writing S = {xIP(x)}. definition 1. Given any property P, the empty (or null, or void) set is the set
0= {xIP(x) is false}. (A. 1)
definition 2. Let A and B be two sets such that Vx E A => x E B. We say that A is a subset of B and denote this fact, writing A ~ B or B ;2 A. If it happens that A ~ B and B ;2 A, then both sets are equal and we write A = B. If A ~ B or A :f:. B we say that A is a proper subset of B and write sometimes A C B.
The symbol ~ (and also C) is called set theoretical inclusion. We observe that given any set S we have 0 ~ S (or better 0 C S).
definition 3. Let A ~ S. The mapping
A -+ S, (A.2) 1 if x E A, XA(X) = 0 if x ~ A. 187 188 NONLOCALITY IN QUANTUM PHYSICS is called the characteristic function of the set A. definition 4. Given two sets A and B, their union is the set denoted A U B such that
Au B = {xix E A or x E B}. (A.3)
definition 5. Given two sets A and B, their intersection is the set denoted A n B such that
AnB = {xix E Aandx E B}. (A.4)
If An B = 0, then A and B are said to be disjoint. definition 6. The complement of A ~ S is the set denoted AC (or S \ A) such that
A C = {x ~ A, XES}. (A.5)
definition 7. Let A, B ~ S. The difference of A and B is the set denoted A - B such that
A - B = An BC= {xix E A and x ~ B}. (A.6)
The difference of A and B is also called the relative complement of B in A. definition 8. The symmetric difference of A and B (A, B ~ S) is the set denoted Al:::,.B such that
Al:::,.B = (A - B) U (B - A). (A.7)
definition 9. Let J be a set whose elements we will call indexes. Given a set X, an indexedfamity of elements of X with indexes in J is a mapping x : J -t X, a t-+ XO. The indexed family is denoted {Xo}oEJ or simply {xo } when it is clear who the set J is. When J = N = {1, 2, ... }, the indexed family is said to be a sequence of elements ofX. definition 10. An indexed family {Ao} of subsets of A, with indexes in J is said to be a partition of A if \la, (3 E J, a :I (3, Ao n A,B = 0 and UOEJAo = A definition 11. The union and the intersection of the Ao are respectively the sets
UOEJAo {xix E Ao for at least one E J}, noEJAo = {xix E Ao for all a E J}. (A.8)
definition 12. Given a set S, the class of all its subsets is said to be the power set of S and is denoted by 28 . definition 13. The Cartesian product of the n sets AI, A2 , . .. An is the set
When Ai = A, i = 1,2, ... n we write the Cartesian product as An. APPENDIX A: SET THEORY AND LATTICES 189
A.2 PO SETS definition 14. A (binary) relation in a set 8 is a specified subset R ~ 8 x 8. If (x, y) E R we say that x stands in relation with y and write xRy. definition 15. Let R ~ 8 x 8 be a relation in 8. It is said to be: (i) reflexive if xRx, 'Vx E S, (ii) symmetric if xRy ::::} yRx, (iii) transitive if xRy, yRz ::::} xRz, (iv) antisymmetric if xRy and yRx ::::} x = y. definition 16. An equivalence relation in a set 8 is a relation in 8 which is reflexive, symmetric and transitive. We often use the symbol x'" y to denote that x, y E 8 are equivalent. proposition 17. An equivalence relation in S leads to a unique partition of S. Conversely, any given partition of 8 defines an equivalence relation in 8. definition 18. Let'" be an equivalence relation in S. Let xES be an arbitrary but fixed element. Consider the collection of all y such that x '" y. This collection denoted [xl ~ 8 is called the equivalence class of x. We have,
[xl = {y E 8 and y '" x}. (A 10)
definition 19. Let E ~ S x 8 be an equivalence relation in 8. The quotient set of S modulo E, denoted by SI E is the class of all distinct equivalence classes induced in 8 by E. We have,
81E = {[xli x E 8}. (All)
definition 20. The mapping 71' : S -t S IE; x t-+ [x Jis called the canonical mapping (or projection) of S onto SI E. definition 21. A relation in a set S is called an order relation if it is reflexive, anti symmetric and transitive. The symbol::; is used to denote in what follows the order relation. If x, y, z E 8 we have,
(i) x < x, 'Vx E 8, (ii) x < y and y ::; x ::::} x = y, (iii) x < y and y ::; z ::::} x ::; z. (AI2)
Sometimes instead of x ::; y (which reads x is less than or equal to y, or x is contained in y) we write y ~ x. If x::; y and x f:. y we write x < y. When x < yand there is no z such that x < z < y we say that y covers x and write x -< y. definition 22. Given a set 8 and an order relation::; in 8, the pair (8,::;) is called a poset (a short form for partially ordered set).2 definition 23. Let (8,::;) be a po set. If there exists 0 E 8 such that 0 ::; x, 'Vx E 8, then 0 is said to be the first (or smallest, or least) element of 8. If there exists 1 E 8 such that x ::; 1, 'Vx E 8, then 1 is said to be the last (or largest, or greatest) element of S. 190 NONLOCALITY IN QUANTUM PHYSICS
proposition 24. The first element of 5 when it exists is unique. The last element of 5 if it exists, is unique. definition 25. Let (5, ::;) be a poset. m E 5 is called a minimal element of 5 if there is no element in 5 which is strictly smaller than m, i.e., if x ::; m, then x = m. An element m E 5 is said to be the maximal element of 5 if there is no element of 5 that is strictly greater than m, i.e., if m ::; x, then m = x. definition 26. Let (5, ::;) be a poset and A ~ 5 (eventually A = 5). l E 5 is said to be a lower bound of of A if and only if Va E A :::} l ::; a. An element u E 5 is called an upper bound of A if and only if Va E A :::} u :::: a. It is important to have in mind that A ~ 5 need not have necessarily any lower bound and even if it does, let us say b, in general b ¢ A. In this way it is clear that A may have many different lower bounds. An analogous comment is valid regarding upper bounds. definition 27. Let (5,::;) be a poset and A ~ 5. Denote by AL the set of all lower bounds of A. If AL has a last (greatest) element, then it is called the infimum (or the greatest lower bound) and is denoted inf(A). Denote the set of all upper bounds of A by Au. If Au has a first (smaller) element then it is called the supremum (or least upper bound) denoted sup(A). The notations VaEAa and AaEAa are often used respectively for inf(A) and sup(A). When A is a finite countable (infinite countable) set we write for the infimum Vi=l a (V~l a). For the supremum we write A;;' 1 a (A~l a). proposition 29. Let (5, ::;) be a poset and x, y, z E 5. Then,
(i) inf{x,{y,z}} inf{x,y,z} (ii) sup{ x, {y, z}} sup{x,y,z}
A.3 LATTICES definition 30. A poset C is called a lattice if: (i) :1 0 E C, :1 1 E C and 0 =j:. 1, (ii) each pair x, y E C, x =j:. y, has a supremum (also called union, join or dis• junction) and denoted by x V y and an infimum (also called intersection, meet or conjunction) denoted by x A y. definition 31. If every finite or infinite countable subset A ~ C has a join and a meet, then C is said to be a a-complete lattice. If for any nonempty set A ~ C, VaEAa and AaEAa exist, then the lattice is said to be complete. If C is finite countable, then it is said to be finite. definition 32. A lattice C is said to be atomic (or atomistic) if Vx E C :::} x 2 a, where 0 =j:. a E 1: is an atom, i.e., 0 -< a. proposition 33. Let C be a lattice and x, y, z E C. The following properties are true: (i) idempotency: x V x = x, x A x = x, (ii) commutativity: x V y = y V x, x A Y = Y A x, (iii) associativity: x A (y A z) = (x A y) A z, x V (y V z) = (x V y) V z, (iv) convolution or absorptivity: x A (x V y) = x, x V (x A y) = x, (v) the statements: (a) x ::; y, (b) x V y = y, (c) x A Y = x are equivalent, APPENDIX A: SET THEORY AND LATTICES 191
(vi) x :s y => x V z :s y V z and x 1\ z :s y 1\ z. definition 34. A lattice £ is said to be complemented if Vx E £ has at least one complement, i.e., :3 y E £Ix 1\ y = 0 and x V y = 1. £ is said to be uniquely complemented if Vx E L has only one complement. In what follows we denote a complement of x E C as x' (x' E C). proposition 35. Let C be a lattice and x, y, z E £. Then,
(i)xV(yl\z) < (xVy)l\(xVz), (ii)xl\(yVz) > (xl\y)V(xl\z). (A.13)
proposition 36. Let C be a lattice and x, y, z E C and x :s z. Then,
x V (y 1\ z) :s (x V y) 1\ z. (A.14) definition 37. A distributive lattice £ is a lattice such that Vx, y, z E £, the distributive laws hold, i.e.,
(i)xV(yl\z) = (xVy)l\(xVz), (ii)xl\(yVz) (xl\y)V(xl\z) (A.lS)
definition 38. A lattice £ is said to be modular ifVx, y, z E £ and x :s z, we have
x V (y 1\ z) = (x V y) 1\ z. (A.l6)
proposition 39. If £ is distributive it is also modular. definition 40. A complemented distributive lattice B is called a Boolean algebra. If B is a O"-lattice it is called a Boolean O"-algebra. proposition 41. Every element in a Boolean algebra (or O"-algebra) B has only one complement. Also, if x, y, z E J3 and x V y = x V z, x 1\ Y = x 1\ z => y = z. proposition 42. Vx, y E J3 (a Boolean lattice) the following properties hold: (i) 0' = 1; (ii) l' = 0; (iii) (X')' = x; (iv) x = y ¢:> x' = y'; (v) (x V y)' = x' 1\ y'; (vi) (x 1\ y)' = x' V y'; (vii) x :s y => y' :s x'; (viii) the statements: (a) x :s y, (b) x 1\ y' = 0, (c) x'V y = 1, are equivalent. definition 43. Let £1 and £2 be two lattices. A surjective (onto) mapping is called an epimorphism (or homomorphism) of £1 into £2 if for any at most countable sequence {xd, i = 1,2 ... of £1,
(A. 17)
If f is an injective (one-to-one) mapping, it is called a monomorphism. If f is bijective, it is called an isomorphism. Since Vx E £1, X V 0 = x, x 1\ 0 = 0, x VI = 1, x 1\ 1 = x it follows that for any f as in definition 43,
f(O) = 0, f(l) = 1 (A.l8)
definition 44. An isomorphism d : £1 -+ £2 is called dual if Vx, y E £1 with x :s y then d(y) :s d(x). 192 NONLOCALITY IN QUANTUM PHYSICS
definition 45. An isomorphism C ~ C is called an automorphism. A dual automorphism a: C ~ C such that \/x E C, a(a(x)) = x is called involutive. proposition 46. If a : C ~ C is an involutive dual automorphism then,
(i) a(x V y) = a(x) 1\ a(y); (A. 19) (ii) a(x 1\ y) = a(x) Va(y) and the statements
(iii) x ~ a(x) :::} x = 0, (iv) \/x E C :::} x 1\ a(x) = 0, (A.20) (v) \/x E C:::} x V a(x) = 1 are equivalent. definition 47. A lattice C together with an involute dual automorphism C ~ C is said to be orthocomplemented. The involutive dual automorphism is called an orthocomplementation and is denoted by .L: C 3 x I-t x.L E C. x.L is said to be the orthocomplement of x. In what follows, an orthocomplemented lattice will be denoted by Co. Since x.L is a complement (definition 34 ) we denote in the text x.L by Xl when no confusion arises. definition 48. We say that x, y E Co are orthogonal (or disjoint) and write x .L y if x :S yl.. It follows immediately from definition 48 that x .L y :::} Y .L x. definition 49. A weakly modular lattice denoted Cw in what follows is an ortho• complemented lattice, such that \/x, y E Cw then
x ~ y :::} Y = x V (Xl 1\ y). (A.21)
definition 50. A quasi-modular lattice is an orthocomplemented lattice, such that \/x,y,z E C then
x ~ y ~ Zl :::} X = (x V z) 1\ y. (A.22)
definition 51. An orthomodular lattice denoted Com in what follows is an ortho• complemented lattice such that \/y, z E L then \/x ~ z we have y .L z ==> xV (y I\z) = (xVy)l\z. proposition 52. An orthocomplemented lattice C is weakly modular if and only if \/x,y E C then
x ~ y :::} x = Y 1\ (x V yl). (A.23)
proposition 53. For an orthocomplement lattice C we have that:
Orthomodularity ¢} quasi-modularity ¢} weak modularity.
proposition 54. Every orthocomplemented lattice is orthomodular. APPENDIX A: SET THEORY AND LATTICES 193
proposition 55. Let x, Y E .co. Then
(x!\ y) V (x !\ Y') ~ x. (A.24) definition 56. Let x, Y E .co. We say that x is compatible with Y and write x ++ Y if
(x !\ y) V (x !\ Y') = x. (A.25) definition 57. Let x, Y E .co. We say that x is commensurable with Y and write x ...... Y if x V (x !\ Y') = Y V (Y' !\ x). (A.26) definition 58. Let x, y E .co. We say that x is commeasurable with y and write x '" y if 3 Xl, Yl, Z E .co with Xl ..l YI, Xl ..l Z, Yl ..l z such that x = Xl V z and Y = Yl V z. proposition 59. Let x, Y E .co. Then, x ++ Y => x ++ y'. proposition 60. Let x, Y E .com. Then, x ++ Y => Y ++ x and x ~ Y => x ++ y. proposition 61. Let x, Y E .co. If x ~ y => x ++ y then .co is a .com. proposition 62. Let x, y E .co. If x ...... y => y ...... x.
proposition 63. Each one of lattice relations ++ or ...... or f'V in .com are such that it implies the other two. definition 64. A subset S ~ .c is said to be a sub lattice of .c if S is itself a lattice with respect to the lattice operations of .c. proposition 65. Let {.cOl.} be an indexed family of subsets of .c. Then the set theoretical intersection nOl..cOl. is a sublattice of .c. definition 66. Let S ~ .c be an arbitrary subset of .c. We call the lattice generated by S, denoted spanS, the set theoretical intersection of all the lattices which contain S.
proposition 67. Let x, y E .co. Then x f'V y if and only if the lattice generated by {x, x', y, y'} is a Boolean (algebra) lattice.
definition 68. An element c E .co is said to be central if c f'V x, 't/x E .co. The set (A.27) is called the center of .co. If the center is trivial, i.e., C(.co) = {O, I}, then .co is said to be irreducible. If C(.co) ::j:. {O, I}, then .co is said to be reducible. definition 69. Let.cl , .c2, ..• .cn be lattices. The Cartesian product.c = .cl x .c2 x .. ..c n is said to be the direct union of the .ci if it is equipped with the order relation
(Xl,X2, ... xn) ~ (Yl,Y2, ···Yn) if Xi ~ XiH,Yi ~ Yi+l (A.28) and if the meet and join are given by
(XI,X2, ... ,xn)!\ (YI,Y2, ···Yn) = (Xl !\ YI,X2!\ Y2, ... , Xn !\ Yn), (XI,X2, ... ,xn) V (YI,Y2, ···,Yn) = (Xl VYl,X2 VY2, ... ,Xn VYn). (A.29) proposition 70. If a lattice .c is a direct union, then its center is non trivial, i.e., the lattice is reducible. 194 NONLOCALITY IN QUANTUM PHYSICS
A.4 THE BOOLEAN ALGEBRA OF SETS
proposition 71. Let A, B, C E 25 . The following properties hold true:
idempotent laws :A U A = A, A n A = A, associative laws :(A U B) U C = Au (B U C), (A n B) n C = An (B n C), commutative laws :A U B = B u A, A n B = B n A, distributive laws :A U (B n C) = (A u B) n (A u C), An (B u C) = (A n B) u (A n C), identity laws :A U 0 = A, Au S = S, An 0 = 0, An S = A, complement laws :A U AC= S, (AC)C = A, An AC= 0, SC = 0, 0c = S De Morgan laws :(A U B)C = ACnBc, (A n B)C = ACu BC (A.30)
proposition 72. Any cJass C of slJbsets of X eqlJipped with the operations U and II ",\len that
A, BEe:::} Au BEe A, BEe:::} A n BEe A E C:::} A C E C (A.31 ) is a Boolean algebra. The proof of proposition 72 is trivial; we only need to recall the definition of Boolean algebra given above and identify the set operations U, n , ~ and the set complementation c with the lattice operations V, A , ::; and'. Also we identify 0 with 0,1 with S.
A.S BOREL SETS, MEASURES AND INTEGRATION A basic fact about Boolean algebras is that these structures can be identified with a certain class of rings, which enables us to study Boolean algebras using the well developed theory of rings. definition 73. Let 9't be a non empty class of subsets of a given set S. 9't is said to be a ring of sets if the following conditions hold:
VA, B E 9't :::} A~B E 9't, VA, BE !n:::} An BE !n. (A.32) From definition 63, it is easy to verify that for any finite collection of subsets Ai,i = 1,2, ... nwehave (A.33)
definition 74 A ring of sets !n is a Boolean ring if it is a ring with identity, where the ring operations + (sum) and· (product, denoted by juxtaposition) are such that APPENDIX A: SET THEORY AND LATTICES 195
A + B = A.6.B (A.34) AB =AnB and such that 'VA E 9l::} A2 = A We can easily show that starting with a Boolean ring 9l and using the definitions A 1\ B = AB and A V B = A + B + AB we can convert it into a Boolean algebra. definition 75. If for every countable sequence {Ad, i = 1,2, ... , Ai E 9l it follows that
(A.35) then 9l is said to be a a-ring. proposition 76. If a a-ring 9l is such that S E 9l, it has a structure of a Boolean a-algebra. proposition 77. Let C be any class of sets and let {RaJ be an indexed family of all rings containing C. Then 9l(C) = noRa is also a ring called the ring generated by C. The ring generated by C is the smallest ring containing the class C and it is unique. definition 78. Let X be a set. A Borel field Q3 on X is a class of subsets of X called Borel sets such that the following axioms hold: (i) If Bi E Q3, i = 1,2, ... , then all finite union Uf=1 Bi and even the countable union U~1 Bi belong to Q3. (ii) If B E Q3 then B' E Q3. definition 79. A pair (X, Q3) is called a Borel space. As an important example of Borel sets, consider the real line 'R = {x I - 00 < x < oo} and let C be the class of all bounded semi-closed intervals
la,b) = {xl- a:S x < b} (A.36)
We can verify that the class generated by the a-ring 9l(C) is a Borel field on 'R. Then {'R,91(C)} is said to be the Borel space of the reals. It is important to have in mind that there are many other kinds of Borel sets of the reals(3). definition 80. A measurable space is a pair (X,!m) where!m is a a-ring on the set X with the property that X is the union (not necessarily countable) of sets taken from the class 2!m. The elements of !m are said to be the measurable sets of X. definition 81. Let (X, Q3x) and (Y, Q3y) be two Borel spaces. A mapping
(A.37) is said to be a Borel function if and only if for 'V By E Q3y we have that 1-1 (By) E Q3x. We often write eq.(A.37) as 1: Q3x ~ Q3y. definition 82. Let (X,!m) be a measurable space. A measure on!m is a mapping
(A.38) satisfying the following properties: (i) 'VB E !JR, JL(B) 2: 0, 196 NONLOCALITY IN QUANTUM PHYSICS
(ii) J-L(0) = 0, (iii) If Bi E m, i = 1, 2... and Bi n B j = 0 for i =I- j then J-L(U'~1 Bi) = E~l J-L(Bi). Since the codomain of J-L is n we may have sets whose measure is +00. Also it may happen that E~l p,(Bi) is infinite. definition 83. A set B E m is said to be of zero measure or a null set with respect to J-L if J-L(B) = O. Also a set BE m is said to have finite measure if J-L(B) =I- 00. In what follows unless explicitly stated we suppose that all measures are finite, i.e.,V B Em, p,(B) =I- 00. definition 84. A triple (X, m, J-L) where (X, m) is a measurable space and J-L is a measure on m, is said to be a measure space. definition 85. Take X = n (the real line). A Lebesgue measure J-L is a measure where m is the a-ring ~(C) whose elements are the Borel sets of the real line and J-L{la, b)} = b - a (A.39) definition 86. Take X = n (the real line). A Lebesgue-Stieltjes measure J-L is a measure where m is the a-ring ~(C) whose elements are the Borel sets of the real line and p,{la, b)} = cp(b) - cp(a) (A.40) where cp : n ~ n is such that lime-+o cp(>. + c) = cp(>.). Another important measure used in this book is the probability measure on the lattice of propositions of a quantum mechanical system whose possible states are elements of a Hilbert space 1{ which we describe in Chapter 9. Also the concept of measure space permits us to introduce the Lebesgue and Lebesgue-Stieltjes integrals of functions which are more general than the Riemann integral of elementary calculus and which are crucial for the definition of the function realization of a Hilbert space and other function spaces that are fundamental ingredients of quantum theory. definition 87. Let (X, m, p,) be a measurable space. A measurable function on X is a mapping
f:x~n (A.41) such that for every Borel set B E ~(C) (on the real line), the set rl(B) E m is measurable. We see at once that the characteristic function XA (see definition 3 ) of a set A E m is measurable since for every Borel set B E ~(C) we have
-l(B)-{ A ifxEA (A.42) XA - 0 if x ¢ A
Let (X, m, J-L) be a measurable space and let j, 9 : X ~ n be two arbitrary measurable functions. We verify at once that the functions f + 9 and f 9 are measurable. definition 88. Let (X, m, J-L) be a measurable space. An integrable simple/unction is a mapping f : X ~ n such that APPENDIX A: SET THEORY AND LATTICES 197
n f(x) = L aiXAi (x), ai En, Ai E 9)1 and Ai n Aj = 0, i i- j (A.43) i=1 It is trivial to verify that any simple function is measurable. definition 89. Let (X, 9)1, p,) be a measurable space and let f : X -+ n any integrable single function like in eq.(A.43). By definition the integral of f is
j fdp, = t aiP,(Ai) < 00. (A.44) x i=1 definition 90. Let (X, 9.n, p,) be a measurable space and let f : X -+ n be a measurable function. We say that a sequence U;}, i = 1,2 ... , of measurable finite simple functions on X is a Cauchy sequence in measure (or converge in the measure) to the measurable function f if for 'tIx E X if 'tiE > 0
lim p,({x: If(x) - j;(x)1 ~ E) = O. (A.45) l-tOO
proposition 91. Let (X, 9)1, p,) be a measurable space and let f : X -+ n be a a finite value measurable function. f is said to be integrable if there exists a sequence U;}, i = 1, 2 ... ,of integrable simple functions on X which converge in the measure to f and
(A.46)
We observe that it is easy to verify that the rules for integration of the sum (f + g) and product (f g) of finite value measurable functions are equal to the corresponding ones in Riemann integration. Also,
(A.47)
definition 92. If X ~ n, the real line and oot = !n( C) and p, is the Lebesgue measure then the integral in eq.(A.46) is called the Lebesgue integral of f. definition 93. If X ~ n, the real line and 9)1 = !n( C) and if the Lebesgue-Stieltjes measure is generated by a non decreasing function p(x) then eq.(A.45) is denoted as
Ix fdp,p or Ix f(x)dp,p(x) or Ix fdp(x), (A.48) and is called the Lebesgue-Stieltjes integral of f. definition 94. If f is a complex measurable function on (X, 9)1, p,) , i.e., the real (Ref) and imaginary (1m!) parts of f are measurable functions, then its integral of f is given by
r fdp, = r Refdp, + i r Imfdp,. (A.49) Js~x Js~x Js~x 198 NONLOCALITY IN QUANTUM PHYSICS
definition 95. Let (X, 9Jl, fJ,) be a measurable space. We call L1 the function space of all complex integrable functions I : X -+ n. proposition 96. £1 is a linear manifold, i.e., VI, 9 E L1 and c E C (the complex field) 1+ 9 E L1 and cl E L1. (ASO) definition 97. Let lEU. The norm of I is by definition II I 11= Ix I/ld{1. (ASl) definition 98. Let I, 9 E L1. The distance between I and 9 is II I - 9 II, if II I - 9 11= 0, I is said to be equivalent to g. They differ at most on a set of zero measure. definition 99. A space function is said to be complete if for every sequence Ud that has a limit, i.e., if II Ii - 9j 11-+ 0 when i,j -+ 00 there exists an integrable function I such that lim II - Iii = O. (A52) '-+00 definition 100. Let (X, 9Jl, {1) be a measurable space. We call L 2 (X,9Jl,{1) the function space of all complex integrable functions I : X -+ n which are such that the square of its modulus is integrable. We define the norm of I E L2 (X, 9Jl, {1) by
II I 11= Ix 1/1 2 dfJ, < 00. (A53) When the space L2 (X, 9Jl, {1) is equipped with a positive inner product defined for VI, h E L2 (X, 9Jl, {1) by (I, h) = Ix f* hd{1, (A.54) satisfying the properties in (i) of definition 1 of Appendix B it becomes a Hilbert space as the reader can verify without difficulties. We end this appendix by calling the reader's attention to the fact that besides the generalization of the Riemann integral presented above, there also exists a generaliza• tion of the concept of an indefinite integral. This permits us to introduce the so called Radon-Nikodym derivative of an integrable function. We are not going to present the corresponding definitions here since they are not needed in the main text. However, we emphasize that any serious student of quantum theory must know this concept well which together with the other topics of this appendix may be found in many excellent books, some of them quoted in the references.
Notes
I. We call the reader's attention to the fact there are naive definitions of sets like the one given by Cantor: "A set is a collection of definite and distinct objects of our intuition or of our thought", which lead to paradoxes[ll]. In particular, an arbitrary collection is not necessarily a set. In the use made of set theory in this book, all sets involved occur in the context of a single (universal) set, from which they are parts. Then we can use the word set as synonymous for aggregate, family, collection. The word class is reserved for a collection of sets. APPENDIX A: SET THEORY AND LATTICES 199
2. When the meaning of the order relation ~ is clear we simply write S for the poset (S, ~).
References
[1] G. Birkhoff, Lattice Theory, AMS Colloquium Pub!. 25, third edition (AMS, Providence, Rhode Island, 1993).
[2] Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick, Analysis. Mani• folds and Physics, revised edition (North Holland Pub!. Co. Amsterdam,1997).
[3] P. Roman, Some Modern Mathematics/or Physicists and other Outsiders, vols.l and 2 (Pergamon Press Inc., New York, 1975).
[4] I. R. Porteous, Topological Geometry, second edition (Cambridge Univ. Press, Cambridge, 1981).
[5] G. F. Simmons, Introduction to Topology and Modern Analysis, (McGraw-Hill Book Co., Singapore, 1963).
[6] A. E. Taylor, Introduction to Functional Analysis (1. Wiley & Sons, New York, 1958).
[7] N. N. Bogolubov, A. A. Logunov and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory (w. A. Benjamin Inc., Reading, MA, 1975).
[8) J. M. Jauch, Foundations o/Quantum TheO/y (Addison-Wesley Pub!. Co., Read• ing, MA, 1968).
[9] M. Jammer, The Philosophy a/Quantum Mechanics (1. Wiley & Sons, New York, 1974).
[10] G. W. Mackey, Mathematical Foundations o/Quantum Mechanics (w. A. Ben• jamin, New York, 1963).
[11] V. S. Varadarajan, Geometry o/Quantum Theory, vols. 1 and 2 (D. van Nostrand Co. Inc., Princeton, 1968).
[12] R. Rucker, Infinity and the Mind (Bantan Books, New York, 1983). Appendix B HILBERT SPACES
References for Appendix B include all the ones of Appendix A and some other specific ones.
B.1 DEFINITION OF HILBERT SPACE definition O. A linear (vector) space V over a commutative field J( is a non empty set whose elements are called vectors, denoted by boldface letters like v, w, ... , etc, 1 and such that 'Vv, w, z E V and 0, 1, k, m E J( there exists a mapping (called scalar multiplication)2,
J( x V ~ Vj (k, v) t-+ kv = vk, such that the following properties hold:
(i) v + w = w+v, (ii) (v + w) + z = v + (w + z), (iii) 3 0 E V such that 0 + v = v, (iv) 3 (-v -Iv) EVsuchthatv+(-v) =0, (v) 1 V = V, (vi) k(mv) = (km)v, (vii) (k + m)v = kv + mY, (viii) k(v + w) = kv+ kw. Note also that Ov = 0, 'Vv E V. definition 1. A Hibert space (denoted 1-£) is a complex linear space (i.e., J( = C) for which the following axioms hold: (i) There exists in 1-£ a strictly positive inner product (sometimes called scalar product), i.e., a mapping
(,): 1-£ x 1-£ -t C, (B.1) 201 202 NONLOCALITY IN QUANTUM PHYSICS satisfying the properties (Ix), Iy») = (Iy), Ix»)*, (B.2) (Ix), Iy) + Iz») = (Ix), Iy») + (Ix), Iz»), (B.3) (Ix), ely») = e(lx), Iy»), "Ie E C. (B.4)
(ii) There exists a norm function in 11., i.e., a mapping
1111: H -r n+ + {O}, (B.5) such that
II Ix) 11= J(lx), Ix»), (B.6) and II Ix) 11= 0 => Ix) = 0, (iii) 11. is separable, i.e., there exists a sequence {14>i)}.i = 1,2 .... , l4>i) E 11. dense in 11. in the following sense: given '114» E 11. and c: > 0, there exists at least one l4>i) in the sequence such that
(B.7)
(iv) 11. is complete, i.e., any sequence such that
(B.8) defines a unique l4>i) E 11. such that .lim I1I4>i) - 14» 11= o. (B.9) t--+oo definition 2. When eq.(B.9) holds we write l4>i) -r 14» and we say that the sequence {14>i)} is strongly convergent. We can easily prove the following inequalities,
1(14)), 11/1»)1 ~1I14» 111111/1) II, (B.IO) II <14» + 11/1») II ~1I14» II + 1111/1) II, (B.1l) which are known respectively as the Schwartz and Minkowski inequalities. definition 3. 14» and 11/1») are said to be orthogonal if (I¢), 11/1») = O. We sometimes write 14» 1. 11/1). A sequence {14>i)}, i = 1,2 .... , l4>i) E 11. is said to be orthonormal if
(B.12)
proposition 4. Given I¢) E 11. and an orthonormal sequence {14>i)}, i = 1,2 .... , l4>i) E 11. we have
00 L 1(I4>i), 14»)1 2 ~III¢) 112 . (B. 13) i=1 APPENDIX B: HILBERT SPACES 203 which is called Bessel's inequality. definition 5. When the equality sign holds in eq.(B.l3), the sequence {I¢i)} is said to be complete. proposition 6. When the sequence {I¢i) } is complete the partial sums given by the functions
00 IIPi) = ~)I¢i)' I¢) )I¢i), (B.l4) i=l converge strongly to I¢) and we write
00 I¢) = ~)I¢i)' 1¢))I¢i). (B.l5) i=l definition 7. In any linear vector space V over a (commutative) field K a finite or infinite sequence of vectors {I¢i)}, l¢i)E 11. is called linearly independent if and only if L ai I¢i) = 0 => ai = 0, for all i in the index set and ai E K. If L ad¢i) = 0 and not all ai = 0 then the set {I¢i)} is said to be linearly dependent. definition 8. The maximal number n of linearly independent vectors in V is called its dimension. When n is finite, the space is said to be of finite dimension. When n = 00 the space is said to be of infinite dimension. Hilbert spaces can be of finite or infinite dimension. When n < 00 the separability and completeness of 11. follows from properties (i) and (ii) in definition l, but when n = 00, this is not the case.
B.2 LINEAR MANIFOLDS AND SUBSPACES definition 9. Let M ~ 11. be a subset of 11.. M is said to be a linear manifold if
(i) 'VI¢) EM=> cl¢) E M, 'Vc E C, (ii) 'VI¢), I7/!) EM=> I¢) + 17/!)) E M. (B.l6)
definition 10. Let S ~ 11. be an arbitrary set of vectors. The smallest linear manifold containing S (which exists and is unique) will be denoted by Ms or [S1 and will be called the linear manifold spanned by S. In an arbitrary Hilbert space 1i, it may happen that all sequences with only a finite number of non zero components are a linear manifold. However these sequences are not complete, and then the linear manifold is not in general a subspace of 11.. definition 11. A limit vector of a linear manifold M is a vector I¢) E 1i such that a sequence {I¢i)}, i = 1,2 .... , I¢i) E M exists such that I¢i) -+ I¢). definition 12. A closed linear manifold (or subspace) of 1/. is a linear manifold M in which every limit vector I¢) of M is such that I¢) EM. Adding to a linear manifold M all its limit vectors we obtain its closure M = M which is the smallest subspace containing M. definition 13. Given any set S ~ 1/., the set of all vectors orthogonal to S is called the orthogonal complement of S and is denoted
S1- = {I¢) I (I¢), 17/!)) = 0, 'VI7/!) E S}. (B.l7) 204 NONLOCALITY IN QUANTUM PHYSICS
We can easily verify that S1. is a subspace of 1£ and directly from eq.(B 17) we get {IO)}1. = 1£, 1£1. = {IO)}, (B.18) s n S1. = {IO)}, (B.19) SI ~ S2 => sf ~ st, 'rIS1,S2 ~ 1£. (B.20)
Observe that since M = M ;2 M, then M1. ~ M1. and also M1.1. ~ M1.1. = M, from where it follows that M 1.1. = M since M is the smallest subspace containing M. definition 14. Let M, N be two subspaces of 1£. The algebraic sum of M and N, denoted by M + N is the set of all vectors of the form I¢) + 1'1/1) withl¢) EM, 1'1/1) E N. The set M + N is clearly a linear manifold, and indeed it is the smallest linear manifold containing both M and N , i.e., (using the notation of definition 10),
M+N=[MuN]. (B.2l) However, M + N is not, in general, a subspace of 1£ because it may happen that it is not closed. definition 15. Let M and N be subspaces of 1£. The closure of M + N, which is a subspace of 1£ denoted by M ltl N is said to be the algebraic union of M and N. We have
MltlN=[M+N] (B.22)
Definition 15 generalizes for any countable (even infinite) union of subspaces, i.e., if {Mil is a countable (finite or infinite) family of subspaces of 1£, we define their algebraic union by
(B.23) where Li Mi is the (algebraic) sum of the Mi. proposition 16. Let M, N be subspaces of 1£. The set theoretical intersection of M and N , denoted as usual by M n N is a subspace of 1£. It is the largest subspace contained in both M and N. proposition 17. Let 1£1, H2, 1£3~ 1£ be arbitrary subspaces. Then, (1£1 n 1£2) n 1£3 = 1£1 n (1£2 n 1£3) (1£1 ltl1£2) ltl1£3 = 1£1 ltl (1£z ltl1£3) (B.24) definition 18. Let M, Nbe subspaces of1£. If M nN ={IO)}, the linear manifold M + N is said to be the direct sum of M and N and is denoted by M EEl N . It is important to have in mind that M EEl N is not in general a subspace of1£. When M is a subspace of 1£, then using the definition of M 1. we get immediately that M n M 1. = {IO)}. Propositions 20, 21 and 22 below are important steps for proving that 1£ = M EElM1.. APPENDIX B: HILBERT SPACES 205
definition 19. Let I¢» E Hand S be any subset of H. The distance from I¢» to S is defined by
d(I¢», S) = inf{III¢» -I'l/J) II} : I'l/J) E S, (B.25) Le., d is the greatest lower bound of the distances from I¢» to the points of S. proposition 20. If M is a proper subspace of H (i.e., M :j:. 10), M :j:. H), then 31¢>o) E H, I¢>o) :j:. 0 such that I¢>o) .1 M (i.e., I¢>o) is orthogonal to all vectors of M). proposition 21. Let M,N be subspaces ofH such that M .1 N. Then, the linear manifold M + N is a subspace. proposition 22. If M is a subspace of H, then H = M ffi M1.. corollary 23. IfH = M ffi M1. then M I±JM1. = M ffiM1.. (B.26) Proposition 22 can easily be generalized for a sequence of mutually orthogonal subspaces {Hil, i = 1,2 ... , of H (Hi n Hj = {IO), i :j:. j, i, j = 1,2 ... }, i.e., the sequence spans H (B.27)
Equation (B.27) implies that VI¢» E H can be written in a unique way as
00 I¢» = L I¢>i), I¢>i) E Hi (B.28) i=1 where the infinite sum means 2::1 I¢>i) -+ I¢».
definition 24. Let HI be any subspace of H. The (set) mapping
.1: HI -+ Ht (B.29) is called orthocomplementation. definition 25. Let HI, H2 be two subspaces of H. They are said to be compatible and we denote this fact by writing HI +-t H2 if and only if (B.30)
It is easy to show that from eq.(B.30) it follows that
(B.31) which shows that compatibility is a symmetric relation, justifying the use of the symbol +-t . However, in general, the compatibility relation is not transitive, i.e., if HI, H2, H3 are three subspaces of H and if HI +-t H2, H2 +-t H3 it does not follows that HI +-t H3. definition 26. Let HI, H2 be two subspaces of H. We say that they are disjoint if HI ~ Hi- and we write HI .1 H2 . 206 NONLOCALITY IN QUANTUM PHYSICS
Definition 26 implies that Hz ~ Ht, showing that the disjoint relation expressed through the set inclusion operation ~ is symmetric. This gives us another possibility of expressing the compatibility relation. proposition 27. Let HI, Hz be two subspaces of H. They are compatible if there exists three mutually disjoint subspaces M, Nand ;C such that
(B.32)
Observe that from eq.(B.32) it follows immediately that
;C = HI n Hz, M = HI n;cJ., N = Hz n;CJ.. (B.33)
proposition 28. Let HI, Hz, H3 be three subspaces of H which are pairwise compatible. Then they satisfy the distributive laws
HI n (Hz I±J H3) = (HI n H2) I±J (HI n H3), HI I±J (Hz n H3) = (HI I±J Hz) n (HI I±J H3). (B.34)
The proof of relations (B.34) is trivial. It is very important to have in mind that given three subspaces HI, Hz, H3 of H, in general the distributive laws do not hold. example 29. As an illustrative example, consider a two-dimensional Hilbert space H and two one- dimensional subspaces, Hland Hz = Ht of H. Let, moreover H3 :j:. HI, Hz be an one dimensional subspace of H. Then
(B.35)
On the other hand, since
(B.36) it follows that
(B.37)
proposition 30. Let H be a finite dimensional Hilbert space and let HI, Hz, H3 be three subspaces of H, such that HI ~ H3. Then the so called modular law holds, i.e.,
(B.38)
The proof of eq.(B.38) can be done by verifying that HI I±J (Hz n H3) ~ (HI i±J H2) n H3 and that (HI I±J Hz) n H3 ~ HI i±J (H2 n H3)' proposition 31. Let H be an infinite dimensional Hilbert space and let HI, Hz, H3 be three subspaces ofH, such that HI ~ H3. Then we have in general that
(B.39)
Proof: We give a proof of proposition 31 for the simple case where HI n Hz = {IO)}. Let
(BAO) APPENDIX B: HILBERT SPACES 207 and suppose that it is a limit of vectors of the form 1(1)1) + 12)' with 11) E 1{1, 12) E 1{2, but suppose that I1) E 1{1. But this is impossible unless I1/!) = 10) and then 1{2 n 1{3 = {IO)}. Then, (B.42) and
(1{1 I:!:J 1{2) n 1{3 = 1{3' (B.43)
Eqs.(B.42-43) and the hypothesis 1{1 ~ 1{3 imply eq.(B.39).
B.2.1 THE LATTICE OF SUBSPACES
Recalling the definitions of lattice theory given in Appendix A and the properties proved above, we have the following: proposition 32. The set of subspaces of an infinite dimensional Hilbert space 1{ is an atomic, complete, orthocomplemented and non modular lattice. This lattice, called the lattice of subspaces of 1{ will be denoted £(1{). Proof: Let C be an abstract lattice with the properties stated above. All we need for proving the proposition is: (i) to recall definition 44 of Appendix A of lattice isomorphism, i : C -+ £(1{), (ii) to interpret the operations V , 1\ , '(=-1-, see definition 47 of Appendix A ) and ::; as the corresponding the operations i±J, n, -1- (see definition 24) and ~ in £(1{). corollary 33. If 1{ is a finite dimensional Hilbert space, then £(1{) is a modular lattice.
B.3 DUAL AND CONJUGATE DUAL SPACES definition 34. Let V be a linear (vector) space over a commutative field K. The dual space of V, denoted by V* is by definition the linear space (over K) of all linear forms over V, i.e.,
V* = {FIF(v) E K, Vv E V}. (B.44)
The elements of V* are usually called covectors or forms and if F, F1, F2 E V*, v, w E V and k E K, the following properties hold F(v + w) = F(v) + F(w), F(kv) = kF(v), (B.4S) (F1 + F 2 )(v) = F1(V) + Fz(v), (kF)(v) = k(F(v)). (B.46) For a Hilbert space 1{ the covectors are also called linear functionals or bra vectors (or simply bras). We denote bras by symbols like (FI, (xl, etc ... and write the action of (FI E 1{* on I
(PI: H -+ C, I
definition 35. Let (PI E 1/.*. (PI is said to be a bounded bra (or bounded functional) if VI rfJ) E 1/. there exists a real number m < 00 such that
1(PlrfJ) I < m IIlrfJ) II, m < 00 (B.48)
definition 36. The norm of (FI E 1/.* denoted II (PI II is the greatest lower bound of all numbers m satisfying eq.(B.48). proposition 37. Let (1/11 E 1/.* be a bounded bra. Then, VlrfJ) E 1/. there exists a unique 11/1 ) E 1/. such that
(1/IIrfJ) = (11/1), IrfJ)), (B.49) II (1/1111 =1111/1) II . (B.50)
Proposition 37 is known as the Riesz representation theorem and eq.(B.49) is the reason for the physicist's representation of the scalar product used in the main text. observation 38. The set of bounded bras equipped with the scalar product
( , ) : 1/.* x 1/.* -+ C, ((1/11, (rfJl) = (11/1), IrfJ)) = (1/IIrfJ)· (B.5J) is a pre-Hilbert space. proposition 39. The closure in the norm (given by eq.(B.SO» of the pre-Hilbert space of all bounded bras, with the scalar product given by eq.(B.SI) is a Hilbert space, which is called the conjugate (or normed) dual of 1/. and denoted 1/.t. observation 40. Let 8 be a differential manifold3, e.g., 8 may be the Newtonian spacetime(2), or Minkowski spacetime[3,4), or the configuration space of a many particle system (see, e.gYI of Appendix A.). Let (8, VR, Jl.) be a measurable space and consider the realization of the abstract Hilbert space 1/. (of infinite dimension) as £2(8, VR, Jl.) (see definition 89 of Appendix A). The elements of £2 (8, VR, Jl.) are mappings usually denoted by 1/1, but here also denoted conveniently by 11/1) such that
(B.52) and the scalar product and the norm are given VI1/I), IrfJ) E £2(8, VR, Jl.),by
(1/1, rfJ) = (1/IIrfJ) = J1/1* (x)rfJ(x)dJl., 111/1 11=1111/1) II = J (1/1,1/1) = J (1/JI1/J)· (B.53)
definition 41. The Dirac measure is the functional (xl E (£2(8, VR, Jl.))* such that VI1/J) E £2(8,VR,Jl.)
(xl1/J) = 1/J(x). (B.S4)
It is clear that (xl E (£2(8, VR, Jl.))* is not a bounded bra, and since VI1/J) E £2(8, VR, Jl.) there is no Ix) E £2(8, VR, Jl.) such that (xl1/J) = (Ix), 11/J) ).4 APPENDIX B: HILBERT SPACES 209
B.4 TENSOR PRODUCTS definition 42. Let Vi, i = 1,2, ... , n and V be vector spaces over a commutative field K. A mapping
T : VI x V2 X ... X Vn -+ V, (B.55) is called multilinear if
(i)T(VI, ... ,Vi +v~,,,,,vn) = T(VI,,,,,Vi,, ... ,Vn) +T(VI, ... ,V~, ... ,vn), (ii) T(VI, ... , kVi, ... , vn) = kT(VI, ... , Vi, ... , Vn), k E K. (B.56)
proposition 43. Let L(VI , V2 , ... , Vn; V) be the set of all multilinear forms from VI x V2 X ... X Vn -+ V. L(VI , V2 , ... , Vn; V) has a natural linear (vector) space structure over K if we define for \fT, Q E L(VI , V2 , ... , Vn; V) and k E K, (T + Q)(VI, ... , Vi, ... , Vn) (kT)(VI, ... , Vi, ... , Vn) k[T(VI, ... , Vi, ... , Vn)]. (B.57) When all the Vi = W we denote L(W, W, ... , W; V) by Ln(w; V) and T E Ln(W;K) is called a multilinearform on W. In particular, LI(W;K) = L(W;K) = W* is the dual space of W. definition 44. A tensor product of the linear spaces VI, V2 , ... , Vn is a linear space V together with a multilinear mapping denoted 181,
181 VI X V2 X ... X Vn -+ V = VI 181 V2 181 ... 181 Vn, (VI, V2, ... , Vn) f---7 VI 181 V2 181 ... 181 Vn, (B.58) which for any multilinear mapping T : VI x V2 X ... X Vn -+ K is such that there exists a unique mapping
for which T = T 0 181. proposition 45. A tensor product of the linear spaces VI, V2 , ... , Vn always exists and any two tensor products are naturally isomorphic. In view of proposition 45, we talk from now on of the tensor product. proposition 46. The elements of VI 181 V2 181 ... 181 Vn are sums of scalar multiples of tensor products of vectors (like in eq.(B.58), i.e.,
VI 181 ... 181 Vn 3 T = a(vi 181 ... 181 vn) + a'(v~ 181 ... 181 v~) + ( ... ), (B.59) , # v finite number of telms where Vi"" V; E Vi, i = 1,2, ... , n and a, a' E K, and the following properties hold
VI 181 ... 181 (Vi + vD 181 ... 181 Vn VI 181 ... 181 Vi 181 ... 181 Vn +vII8I ... l8lv~I8I ... l8Ivn, (avdl8l .. ·l8lv;I8I ... l8Ivn ... = VI 181 ... 181 (avD 181 ... 181 Vn = ...
a(vi 181 ... 181 v~ 181 ... 181 v n ). (B.60) 210 NONLOCALITY IN QUANTUM PHYSICS
proposition47. The tensor product oJ Hilbert spaces 'Hi, i = 1,2, ... , n, is a Hilbert space denoted 'HI 0 'H2 0 ... 0 'Hn =® 'H if the multilinear mapping
o : 1-£1 x 'H2 X ... X 'Hn -+ 11.1 011.2 0 ... 011.n, (i'l/J1), 17fJ2), ... , l7fJn)) t-+ l7fJl) 017fJ2) 0 ... 017fJn), (B.61) is such that: (i) the set of all vectors {1 IIJI) = 2: 7fJi l ... inl = 2: 7fJil '''in cfJit ... jn ( = 2: 7fJit, .. in cfJil ... in; (B.63) il ... i n (iii) the norm of IIJI) E® 11. is given by IIIIJI) 11= y!(\lII\lI) ~ O. (B.64) The equality sign in eq.(B.64) holds only if IIJI) = 10). The proof of proposition 47 is the simple verification that ®'H is indeed separable and complete. Moreover, the notation for the scalar product in the second line of eq.(B.63) implies that (IJII E® 'Ht , where ®1I.t is the conjugate dual of ®11.. 8.5 LINEAR OPERATORS definition 48. Let 11.1, 'H2 be Hilbert spaces. The linear space 11.1 Q911.~ is the set of all linear mappings (operators) from 'H2 to 11.1 ,i.e., 1I.1011.~ 3 T: 'H2 -+ 'HI. (B.65) definition 49. A bounded linear operator on 'H is a mapping 'H 011.t 3 ° 'H: ;2 Do -+ Ro ~ 11. (B.66) where Do (the domain of 0) and Ro (the range or codomain of 0) are linear manifolds, not necessarily subspaces of 'H, and such that O(lcfJ) + 17fJ)) = 0lcfJ) + 017fJ), 'v'lcfJ) , 17fJ) E Do, O(elcfJ)) = eOlcfJ)), 'v'lcfJ) E Do, 'v'e E C, II 0lcfJ) II = m IIlcfJ) II, 0 :::; m < 00, 'v'lcfJ) E Do· (B.67) APPENDIX B: HILBERT SPACES 211 The greatest lower bound of the numbers m is called the norm of °and is denoted by II °II. If an m as above does not exist, then °is said to be unbounded. Unbounded operators which exist only when H is infinite dimensional are fundamental in functional analysis and quantum theory, as, e.g., the so-called creation and annihilation operators of quantum field theory (see[7] of Appendix A). For an unbounded operator K E 1{ ® 1{t there always exists a 1<1» E H such that for any finite real number m we have II KI (B.69) The limit in eq.(B.69) always exists since 0l (B.70) With definition 53 we obtain an extension of the linear bounded operator °from Do to the closure Do, which means that every bounded linear operator can be assumed as acting on subspaces of H. If it happens that Do = H (i.e., Do is dense on H), then the bounded linear operator acts on allH. It is worthwhile observing that even if Do f:. H, °can be extended to all1i. This can be done first by extending °to Do using definition 53 and then defining °in DB, as an arbitrarily bounded linear operator, e.g., putting 01'I/J) = 10 ), 'v'1'I/J) E DB· Since 1i = Do E9 DB the action of °on an arbitrary ket 1<1» E H is given by linearity. Unless otherwise stated, all linear bounded operators used in what follows and in the main text are supposed to act on allH. definition 54. A sesquilinearJunctional is an element of Ht ®1it whose domain is 1i ® H. definition 55. Any bounded linear operator °E H ® Ht defines a sesquilinear functional G E Ht®Ht by G(I G(I definition 57. A bounded linear operator 0 such that 0 = ot is said to be self-adjoint or symmetrical or Hermitian. proposition 58. If 0, 0 1 , O2 E H @ Ht, if ot, ot, O~ E H @ Ht are respec• tively their adjoints and if c E C we have, (01 + 02)t = Or +ot (ot)t = 0, (cO)t = C*01' (0102)t = o~ot· (B.73) Besides the self-adjoint operators, which are the representatives of observables in the formalism of quantum theory, there are other important kinds of linear bounded operators which are used in that theory, e.g.: (i) the projection operators P E H @ Ht with domain H and range Rp which is always a subspace of H defined by (B.74) Also, given any subspace HI ~ H there is a unique projection PI such that RPI = HI. If H = (HI + Ht) ~ I¢} = I¢I} + 1¢2} then for a projection PI such that Rpl = HI we have (B.75) (ii) the unitary operators U E H @ Ht which satisfy UUt= UtU = 1, (B.76) where 1 is the identity operator in H. definition 59. The matrix representation of a linear bounded operator 0 E 1-£ @ 1-£t is given by the set of numbers (Oij), such that j (B.77) where {1'Pi)} is an orthonormal basis of 1-£. definition 60. A linear operator 0 ~ ott ~ ot is said to be closed if for every sequence {1'Pi)}, l'Pi)EDo ~ H such that we have I'l/J)EDO, and 0lcp) -1 I'l/J)· (B.78) observation 61. To be closed and to be continuous are different properties that a given 0 E H ® Ht mayor may not possess. Indeed, it is easy to find examples APPENDIX B: HILBERT SPACES 213 of closed operators that are not continuous and also it is easy to prove that every continuous operator is closed. Also it can be proved that every non closed operator can be extended to a closed one which is unique. proposition 62. If K E ti o tit is closed then K is necessarily bounded. proposition 63. A closed and unbounded operator cannot be defined in allti. i.e., if 0 E ti o tit is closed and unbounded, then Do C ti. definition 64. Let 0 be a closed and unbounded operator and suppose that Do is dense in ti and let 11/1) EDo. The adjoint ot of 0 is defined if there exists Ie/» EDO! such that (le/», 011/1)) = (otle/», 11/1))· (B.79) definition 65. Let 0 be an unbounded operator and suppose that Do is dense in ti. 0 is said to be symmetrical (or Hermitian) if ot ;2 0,0 is said to be self-adjoint (and thus symmetrical) if ot = o. proposition 66. For unbounded symmetrical operators. the following properties hold: (i) 0 C ott ~ ot, (ii) 0 C ott and 0 tt cot ::} 0 is closed and is not self-adjoint, (iii) 0 C ott = ot ::} 0 is not closed. ott, its smallest closed extension is self·adjoint (it is called essentially self-adjoint), (iv) 0 C ott cot ::} 0 is neither closed nor essentially self-adjoint, (v) 0 = ott = ot ::} 0 is self-adjoint and closed. We end this section with the remark that since for closed unbounded operators 0 1 , O 2 E 11. 011.t in general 11. ;2 DOl i' R02 ~ ti (and also in general DOl i' D02 i' Rol ) the composition (given by the product) of 0 1 and O 2 and also the composition of O 2 and 0 1, may be not defined. Finally, when the composition of linear operators is well defined the linear space of the set of operators close an algebra. the so called operator algebra. which is extensively used in quantum theory. There. we have to use Lie algebras[6] where the product of two of its elements which are linear operators on 11.. let us say A and B is defined by the commutator product [A,B] = AB - BA (B.80) B.6 PROJECTION OPERATORS AND LATTICE STRUCTURE We have already defined projection operators as being bounded linear operators P E 11. ® tit. Dp = ti such that ppt = P, an equation that implies that p2 = P and p = pt. i.e .• projection operators are idempotent and Hermitian. Let til ~ ti be a one dimensional subspace generated by Icp) E ti. i.e .• til = {clcp), c E C}. The projection on til is given by (B.81) 214 NONLOCALITY IN QUANTUM PHYSICS This equation means that there is a one-to-one correspondence between P 1 and 11.1. More generally, we can show without difficulties that given an indexed family {P",} the ranges P '" are Rp Q = 11.", ~ 11. where {11.",} is an indexed family of subspaces. Conversely, to any arbitrary indexed family of subspaces {11.{3}, there corresponds an indexed family of projections {11.{3} such that Rp/3 = 11.{3 ~ 11.. As a consequence of this one-to-one correspondence the lattice structure 5!,(11.) of the subspaces of 11. (see section B.l of this Appendix) can be transfered to the set of all projection operators. Let us see how this can be done. proposition 67. The set of all projection operators is a poset. Proof: If {11.",} is an indexed family of subspaces and {P",} the associated family of projection operators we can define the order relation by writing (B.82) which proves the proposition. Observe also that: (i) since 11.", = Rp" ~ Rp/3 = 11.{3, it follows that P",P{3 = P"" (B.83) (ii) If P'" is the projection on 1-1."" then 1 - P '" is clearly the projection on 11.~. In section B.l of this Appendix, we identified in 5!,(H) the lattice operations 1\ and V respectively with the set theoretical intersection of subspaces (denoted n) and the algebraic union of subspaces (denoted I±J). We now present how the lattice operations can be expressed in terms of algebraic operations involving projection operators. proposition 68. Let {1-1."'} be an indexed family of subspaces and {P Q} the associated family of projection operators. Then: (i) if[P"" P {3] = 0, and denoting P ",1\{3 = P '" 1\ P (3 the projection corresponding to the intersection H", n H{3, we have (B.84) (ii) If [P "" P fJ] ::j: 0, then (B.85) Proof: (i) is trivial. We now give a proof of (ii) for the particular case when 11.", and 11.{3 are one dimensional nonorthogonal subspaces. To see that eq.(B.85) is correct for this case, consider the subspaces 11.", and 11.{3 generated respectively by 11/1",), 11/1(3) E 11.. Then, (B.86) and we get immediately that (B.87) APPENDIX B: HILBERT SPACES 215 Also, if [P ",PfJl = °we have (p"PfJ)n = P"PfJ and we have eq.(B.84) again. The lattice operation V for the set of all projection operators will be denoted by the same symbol. Consider then the subspace 1l" l±J 1lfJ' the corresponding projection on this subspace denoted by P" VfJ is a well defined operation and we can verify that the following rule is the appropriate one (B.88) The algebraic expression for P" V P fJ can, in general, be calculated without diffi• culties. For the important case when [P", P fJl = 0, it is easy to verify that (B.89) For the algebraic union of an indexed family of subspaces eq.(B.87) is generalized in an obvious way. With the above identifications, it follows that the set of all projection operators (which is a vector space, since it is a subset of 1l ® 1lt) has the same lattice structure (an isomorphism) as the set of subspaces of a given Hilbert space 1l. In what follows, we denote the lattice generated by the set of all projection operators by £P (1l) and £P(1l) "'£(1l). proposition 69. Let £~ (1l) ~ £P (1l) be an indexed family {P;} of commuting projection operators which are: (i) closed with respect to countable intersections (!\) and unions (V), i.e., !\iPi E £~(1l) and Vi Pi E £~(1l), (ii)'v'Pi E £~(1l) => (I-Pi) E £~(1l), (iii) 0 E £~(1l), 1 E £~(1l). Then £~ (1l) is a Boolean algebra. These Boolean algebras are fundamental in quantum theory, since they correspond to the compatible propositions of the so called proposition calculus of quantum theory which is discussed in Chapter 7. Before concluding this Appendix we shall show how to construct Boolean algebras for sets of commuting projections for the case where the Hilbert space 1l is identified with L2 (S, 9J1, f-L) as in observation 40 above. Let A E 9J1, XA the characteristicfunction of A and consider any 1/1 E L2 (S, 9J1, f-L). To the measurable set A we associate a projection operator P A by (P A1/1)(X) = XA(X)1/1(X), 'v'x E S. (B.90) proposition 70. The projections P A are a-additive. This follows immediately from our discussion in Appendix A, since 9J1 is a-additive. Then, if {Ai} is a sequence of sets such that Ai n Aj = 0, we have 00 L P Ai = PUi,;lAi , i=l = PAiUAj, PAiPAj = PAinAj' 0, P Ai = 1. (B.91) 216 NONLOCALITY IN QUANTUM PHYSICS definition 71. A mapping P : m -t .c~(1-l), A t-+ P A, (B.92) satisfying the properties in eq.(B.90) is said to be a spectral measure over (8, m, p). Given a spectral measure, it is possible to associate to it several different numerical measures. Particularly important for applications in quantum theory are the ones given by (B.93) Notes I. In this book, when V is a Hilbert space (definition I), the vectors are denoted by kets like la), If), 14», etc. In this case the null vector is denoted by 10). 2. In the case of a non commutative field, like e.g., the quatemionic field used in qUatemionic quantum theory, it is necessary to distinguish between a left and a right scalar multiplication. See[l] for details. 3. A good reference for a first study on differential manifolds is [6] of Appendix A. 4. To give more details on this topic it is necessary to introduce the concept of rigged Hilbel1 spaces, which is fundamental for certain problems of analysis and the study of quantum field theory. This concept will not be given here since it is not necessary for the main text. The interested reader is invited to consult [7) of Appendix A. References [1] S. De Leo and W. A. Rodrigues, Jr, Quantum mechanics: from complex to complexified quaternions, Int. 1. Theor. Phys. 36(12),2725-2575 (1997). [2] W. A. Rodrigues, Jr., Q. A. G. de Souza and Y. Bozhkov, The mathematical structure of Newtonian spacetime: classical dynamics and gravitation, Found. Phys. 25(6),871-924 (1995). [3] W. A. Rodrigues, Jr. and M. A. F. Rosa, The meaning of time in the theory of relativity and Einstein's later view of the twin paradox, Found. Phys. 19(6), 705-727 (1989). [4] R. K. Sachs and H. Wu, General Relativity for Mathematicians (Springer-Verlag, Berlin, 1977). [5] C. T. J. Dodson and T. C. Post, Tensor Geometry (Springer-Verlag, Berlin, 1991). Index absolute ego, 53 beable(s), 37,44,91,98,115, 124, 134 actual infinity, 62 Beauregard, O. C. de, 51, 57,95,128, 130 actualization, 154 Belinfante, F. J., 98, 99, 101, 113 Adler, S., 7 Bell, J S., 35, 41, 42, 44,55,60,64, 144, 148, Aharanov, y, 41,181,184 183 Aharanov-Bohm effect, 118, 181, 182 inequalities, I, 2, 35, 40, 46, 52, 83, 91, Alexandrov, A. D., 141 94,95, 160, 168, 183 algebra breakdown of, 91, 92, 95 Clifford, 112 d'Espagnat form, 91, 135, 139 a-Boolean, 72 original form of, 40 algebraic union, 204 original proof of, 39 and, see conjunction version of MWI, 59 antisymmetric wave function, 182 Bennett, C. H., 168, 173, 179 anyons, 182 Bernoulli theorem, 25 approximation Bessel's inequality, 203 Born-Oppenheimer, 16 Big Bang, 1,53, 165, 183 WKB,19 Birkhoff, G., 67, 80,81,199 Aristotle, 67 birth of time, 76, 79 Aspect, A., 4, 41,101,112,113,180 bit commitment, 169 experiment, 183 Bilbol, M., 61, 64, 135, 141 atomistic, 190 Blokhintsev, D. I., 134 automaton Bogolubov, N. N., 199 nonOliented, 94 Bohm, D., 1,3,36,41,99, 112-119, 121, 124, normalized, 86, 88 128,129, 181, 184 averages, 120 Bohr, A., 2, 28, 47-49, 56 Avogadro, 144 Bohr,N.,I,2,25.43.47,55.112,164,179 awareness, 51 Boole.67 axiom Boolean algebra. 67, 191 von Neumann's, 104 Boolean logic, see logic Boolean mind. 79,128 Badurek, G., 104, 114 Boolean observer. 67. 79 Ballentine, L. E., 25, 28, 54, 55, 141 Borel field, 195 Barashenkov, v., 55, 56 Borel function, see function barriers, 178 Borel sets, 194, 195 Barrow, J., 57 Born, M., I. 50 Barut, A. 0.,121,122,131 postulate, 12,63 Bastin, E., 55 quantum mechanical rule, 60 Bauer, F., 50, 57, 63 rule, 76 Baylis, W. E., 112, 114, 130 Boschi, D .. 178-180 BBT, see theory, de Brog1ie-Bohm HVT bosons, 182 217 218 INDEX Bourbaki, N., 79 contextual dependence phenomena, 105 Bouwmeester, D., 180 coordinate chaIt, 123 Branca, S., 180 Copenhagen interpretation, 2, see interpretation Branning, D., 178, 179 correlation function, see function Brassard, G., 168, 170, 179, 180 Crepeau, C., 170, 180 Braunstein, S. L., 180 Cramer, J., 95, 96 Breidbart, S., 168, 179 cryptography, 159, 168 Broglie, L. de, I, 3, 100, 112, 113, 115, 122, 129,130 d'Espagnat, B., 4, 5, 10,25,27,29,33,35,41, waves, 21 44,54,55,57,63,64, 157 Brown, H. R., 114 version of Bell's inequalities, 36 Brune, M., 141 Dalibard, 1.,41,180 Bub, 1., 79, 81, 112 data, 154 bunch,89 datum, ISS bundle De Leo, S., 7, 216 Clifford, 121 De MUttini, F, 173, 178-180 cotangent, 129 decoherence, 7, 14, 20, 22 spin-Clifford, 121 delayed choice experiment, 164 density matrix, 9 Deutsch, D., 27,168, 179 Caldeira, A. 0.,21,22,28 Dewdney, c., 104, 114 canonical mapping, 189 DeWitt, B. S., 25, 28, 54, 59, 60, 64, 65 Cmtesian product, 188 DeWitt, c., 64 cascade, SPS, 38 diagram Casimir effect, 181, 182 Hasse, 84, 88, 89 Casimir invariants, 48 sagittal, 84 Casimir, H. B. G., 181, 184 difference, 188 Cauchy symmetric, 188 sequence in measure, 197 DiGiuseppe, G., 178, 179 Chau, H. F, 170, 180 Diosi, L., 146, 149, ISS, 157 Chiao, R. Y., 178 Dirac current, 117 Choquet-Bruhat, Y., 199 Dirac measure, 208 dassicallimit, 14 Dirac spinor field, 125 classicity parameter, 18 disjoint, 188 Clauser, 1. F., 35,40,41 disjoint subs paces, 205 collapse disjunction, 72 of wave function, 30 dispersion, 112, 134 of wave packet, 29, 50 dissipation, 16 collective system, 16 dissipation effect, 21 Collins, G., 173, 180 Dodson, C. T. J., 216 commutator product, 213 Doran, C., 126, 130 compatible observables, 8 Dorofeev, V. Yu., 27 compatible subs paces, 205 Dowker, F, 155, 157 complementarity, 14, 64 Dowling, J. P., 121, 131 whole-pUtts, 92 dragging effect, 146 configuration space, 127, 128, 182 Dreyer, J., 141 conjugate dual, 208 Dunne, B. J., 55, 56 conjugate dual space, 7 conjunction, 72, 152 Eccles, J. c., 55, 57, 182, 184 consciousness, 3, 5 I, 97 Ehrenfest, P. Boolean, 79 theorem, 15,20 world - idea, 53 Eibl, M., 180 consistency conditions, 152 Einstein, A., 1,37,41,55,56,95,112,133 context Einstein-Podolsky-Rosen, see EPR, see EPR independence of, 109 Ekelt, A. K., 168, 179 context dependent Elitzur, A. c., 4, 163, 179 HVT of the first kind, 100 Elitzur-Vaidman proposal, 163 contextual dependence, 98, 105 Ellis, J., 146, 149 INDEX 219 Emch, G., 26 least element, 189 empty set, 187 Fock, V.A., 1,2,49,50,55,56, 134 empty wave, 119 Fourier transform, 17 Enders, A., 178 Freidstadt, H., 131 energy Frenkel, A., 146, 148 non conservation, 117, 145 frequency energy-momentum tensor, 124 prediction of, 46 Englelt, B.-G., 165, 179 frequency interpretation, see interpretation, fre- ensemble, 32, 102, 112, 133, 134 quency homogeneous, 97 FUNC, see principle, functional composition environment, 16, 144 FUNC rule, 109 environmental approach, 146 function EPR,2,35,36,43,46,51,61, 155, 168, 181-183 Borel,71 equation(s) characteristic, 71, 188 boson, 123 correlation, 39 continuity, 20 frame, 105, 108 Dirac, 46, 117, 121, 125, 126, 147 Liouville, 16 BBT interpretation, 125 nonexistence of truth value, 107 Hamilton-Jacobi, 20,116,117,119,124 Wigner,17 Klein-Gordon-Fock (KGFE),46, 123, 147 Maxwell,46 Giihler, R, 147, 148 n-patticle Dirac, 127 Gell-Mann, M., 134, 143, 148, 151-153, 156, SchrOdinger, 29, 30, 44, 46, 50, 60-64, 157 116-118,143-145,151 George, C., 26, 27 master, 60 Ghirardi, G. c., 143, 144, 148 nonlinear, 147 Ghirardi-Rimini-Weber localization processes, super, 125 183 stochastic, 143 Gibbs pat'adox, 181 Weyl,46 Gillespie, D. T., 135, 141 equivalence relation, 189 Glauber, R. 1., 26, 28 erasers, 165 Gleason, A. M., 3, 101, 105, 113 Esposito, S., 55, 56 GMSZ relations, 159 EV (Elitzur-Vaidman), 163 Golub, R., 147, 148 Everett, H., 23, 25, 27, 54, 59, 64,156 Graham, N., 64, 65 evolution Graham, R. D., 28 deterministic, II Grungier, P., 41,113,180 graph,83 indeterministic, II Grassi, R, 143, 148 Schrodinger, 60, 63 Greenberger, D. M., 4, 35, 41, 159, 178 Grib, A. A., 27, 48, 56, 64, 67, 77. 80, 81, 83, facts, 154 86,91,95,96, 101, 114, 149, 184, factual, 154 185 Farris, w., 155-157 Griffiths, R. B., 152, 157 Felber, 1., 147, 148 GRW, see Ghirardi, Rimini, Weber Feller, w., 27 GRWeffect, 147 fermions, 182 Guenin, M., 26 Feynman, R. P., 21, 28,151,157 Gueret, Ph., 113,130,131 field guidance formula, 116 randomly fluctuating, 144 guidance law, 124, 127 field(s) Gull, S., 130 boson, 115 Gutler, R., 122 fields, 45 filter, 68 Hormander, L., 28 Finkelstein, D., 2, 23, 26, 27, 48,56,67,80,83, Haag, R., 79,81 86,96 Hagley, E., 141 Finkelstein, S. R., 26, 73, 83, 86,96 Hamiltonian, 8 first element super, 125 smallest element Hardy,L.,4,35,41, 161, 178-180 220 INDEX Haroche, S., 141 frequency. 7. II, 23. 50 Hartle. J. B., 23, 27. 143. 148. 151-153. 156. histories, 183 157 many worlds. 25. 59-62, 64. 183 Hasse diagram. 74. 77 objective potentialities existence (OPE), Hawking, S. w.. 64. 146. 149, 157 77 Heisenberg. w.. 1.2.44,49,55,56. 112 of objectively existing potentialities. 49 uncertainty principle. 118 ontological. liS. 128 uncertainty relation. 134 quantum logic. 54 uncertainty relations, 83. 89. 90, 178 realistic. 112, 128. 183 Heisenberg. w.. 49 statistical - of quantum theory (SIQT). Heitmann. W.• 178 133-138 helicity, 61 intersection. 188 Hepp. K.• 23. 28 meet Hestenes. D.• 117. 121. 122 conjunction, 190 hidden parameter. 39 Isham, C. 1.. 157 hidden variable. 116 Israeli. 1..72 hidden vruiables context -dependent. 39 Jabs, A.• 184 local. 39 Jahn, R. G., 55, 56 Hilbert space. 7. 16. 18.24, 30, 72. 74, 76. 79. Jammer, M.• 4. 5. 67, 79, 80.112, 141, 143, 148. 83. 102. 104, 107. 110. 129, 135. 199 154,155.198 Jauch. J. M.• 26, 67. 68, 79, 80. 101, 109. 114. rigged. 216 199 tensor product. 210 Jordan, P., 23. 28. 79 Hilley, B. 1.. 99, 112, 113, 115-119. 121, 124. Josephson junctions. 1 128.129 Jozsa, R., 170. 180 histories Jung, C. G., 50 complete family of. 152 consistent. 152, 153. ISS Karolyhazy. A., 146, 148 inconsistent, 153. ISS Kafatos. M., 178 histories approach. 144. 151. 183 Kant. I.. 3 history. 143. lSI, 152 Kastler, D., 79,81 larger. 152 Keller, 1..42 Holland, P. R.• 99, 104, 112-114. 117, 121. 124, Kent. A.• 155. 157 126, 129. 130 Killing vectors. 47 holomovement. 119 Kochen, S., 3, 101. 105. 109. Ill. 113 Holt. R. A.• 41 Kochen-Specker horizontal sum. 89 paradox. 11 0 Koimogorov,I2 Hom. M. A.• 159 Krypianidis, A., 104, 114 Home. M. A.• 35. 40, 41.178 Kwiat, P. G.• 178 Huttner. B., 168. 179 HIT. theory. hidden vruiables see Lagrangian formalism, 126 Landau measurement, see measurement. Landau improper mixtures. 10 Landau. L. D.• 30, 31, 33 inclusion. 187 Langlois, D., 170, 180 indexed family. 188 Laplace determinism, 64 inequalities, see Bell Lasenby, A., 126. 130 Bell·s.44 lattice Clauser-Home, 40 atomic. 190 Infeld. L.. 146. 149 Boolean, 71 infimum complemented, 191 greatest lower bound. 190 complete orthocomplemented, 69 internal degree of freedom. 48 correspondence with graphs, 83 interpretation definition, 190 absolute W. 54 Hasse diagram of, 74 Copenhagen (CI). 37, 43. 54. 59. 60, 62, modular. 191 74. ISS. 182. 183 non Boolean, 74, 79 INDEX 221 non distributive, 55 Hermitian, 102 of propositions, 70, 87 Mattie, K., 180 orthocomplemented, 72 maximal element, 190 quantum, 92 Maximov, Yu. M., 146, 149 uniquely complemented, 191 Mayers, D., 180 Laurikainen, K. v., 57 measure Leggett, A. J., 21, 22, 28, 149 probability, 105 Leibniz, G. W. von, 77 measurement, 8, 14 Leinaas, J., 181, 182, 184 Hamiltonian, 62 lemma ideal,8 Gleason's, lOS, 107, 108 Landau,29 Lifshitz, E. M., 30, 31, 33 non ideal, 9, 30 Linden, N., 157 of the first kind, 8 Liouville density, 17 of the second kind, 9 Lo, H-K., 170, 180 von Neumann, 8, 62 locality, 39 measures, 194 logic Mermin, N. D., 108, 114, 161, 179 Aristotelian, 74 example, 109 Boolean, 36, 72, 74, 76, 80, 97, 182 micro local analysis, 17 Boolean distributive, 76 Miller Jr., w., 26, 27, 114 formal,67 minimal element, 190 non Boolean, 74, 76, 182 Minkowski quantum, 2, 74, 77, 80, 97 vacuum, 48 macroscopic realizations of, 83 Minkowski inequality, 202 quantum (QL), 67 Minkowski spacetime, 47, 48, 79, 123, 128, 165 Logunov, A. A., 199 miracle, 162, 163 London, E, 3, 50, 57, 63 mixture, 145 Lorentz chart, 126 mixture of states, 9 Lorentz contraction, 45, 55 Mjakishev, G. 1.,141 Lorentz, H. A., 45 modular boost, 48 lattice of propositions, 87 invariance, 127, 128 modular law, 206 breakdown of, 128 Mohanty, S., 146, 149 invariant processes, 128 Mohrhoff, U., 167, 179 Lounesto, P., 121, 125, 130 momentum lower bound, 190 non conservation, 117 Lu, J. Y., 42, 129, 130 Monroe, c., 134, 141 Ludwig, G., 2, 45,55 MOll'is, M. S., 181, 184 Lukas, B., 146 Mostepanenko, V. M., 48, 56, 149 Lukacs, B., 148 Mouken, C. H., 178, 179 Moyal, J. E., 17,27 MWI (many worlds interpretation), see interpre• Maitre, X., 141 tation Maali, A., 141 Myrheim, R., 181, 182, 184 Mach-Zender interferometer, 164 Mackey, G. w., 199 Nanoupolos, D., 146, 149 Mackinon, L., 129, 130 Nauenberg, G. M., 28 macro-observable, 16 negative experiments, 163 commuting, 15 Nimtz, G., 55, 56,178 macroscopic quantum computer, 101 non-polarized particles, 10 Maiorino, 1. E., 42 nonchaotic system, 18 Mamayev, S. G., 48,56, 149 nonlocal, 119 Mandel, L., 178, 179 nonlocal connections, 128 Mann, A., 180 nonlocal influence, 140 many pasts, 63 nonlocality, 2, 2, 39, 44, 46, 51, 61, 77, 95,155 Marlow, A. R., 57 active, 128 matrix passive, 128 Dirac, 125 nonlocality in time, 181 222 INDEX non separability. 155 pertinence. 187 nontrivial topology. 181 phase space. IS normed dual. 208 phenomena. 154 photon. 124 objective potentialities. 77 picture observable Heisenberg, 154 collective. IS. 16 two worlds. 23 collective position. 16 Piron. C, 67, 80. 81, 109. 114 preferable. 22 Planck. M. observable algebra. 8 constant n. 83 observer constant n. 45. 47 Boolean. 91 mass, 48 consciousness of the. 51 Podolsky. B., 41 ultimate. 50 Poincare. H. Ocklo.55 group. 47 Odzijewicz. A .• 146 Popescu, S .. 180 Omnes. R .. 16-18.21.27. 143. 144. 148. 151- Porteous. 1. R .• 199 154. 156. 157 poset. 189. 190 ontological status of physical field. 98 definition, 189 OPE. see interpretation Post, T.. 216 operationalistic philosophy. 45 postulate operator statistical. 99 adjoint. 211 potential advanced. 95 algebra. 213 quantum, 119 annihilation. 211 potential barrier. 126 bounded. 210 potentialities. 63 commuting potentiality complete set of. 109 objective existing, 51 maximal complete set of. 109 pre-Hilbert space. 208 continuous. 211 preferable basis. 62, 144 creation. 211 pre felTed Lorentz frame, 127 projection. 212 Prigogine, 1..26.27, 143. 144, 147-149 unbounded. 211 principle or (inclusive). see disjunction functional composition (FUNC). 109. liD order relation. 189 of superposition (PS). 7 orthocomplement. 92 strong anthropic, 54 011hogonal complement. 73 probability amplitude, 75 Oziewicz. Z .. 42 proper mixtures. 10 proper subspace. 205 Palma. G. M .• 168. 179 proposition calculus. 67, 70 Pan. J. W.. 180 pseudo-differential calculus, 17 Papaliolios. C. 99. 113 psychokinetic effect. 128 paradox. see under specific name. Putnam. H .• 67. 73.81 relativistic. 79 particle(s) QL. see logic. quantum fermion, liS QT. see theory. quantum partition. 188 quantum parts, 12, 14,31 beginning of the universe. 165 Pauli. w.. 50. 55. 117 channel. 168 exclusion principle. 47 computers. 3 matrices. 102 cosmology. 59, 181 principle, 146 cryptography. 168 Pavsic, M., 112, 114. 121. 130 ensemble. 134 Pearle, P., 143, 148 indeterminism. 47. 79 Penrose. R., 56. 146. 149. 157. 183. 184 leap. IS theory. 48 miracie(s). 159. 160. 162 Peres. A., 42.54,55. 168. 179. 180 object. 45 INDEX 223 physics Schriidinger, E., I, 112 fundamentals, 7 cat, 144 potential, 124 cat paradox, 3 theory cat paradox, 15, 20, 64, 144, 182 orthodox, 137 cat state, 134, 147 theory of motion, 115 current, 117 topology, 182 equation, 8, 11, 14 quantum bit commitment evolution operator, 151 QBC, 170 Schwartz inequality, 202 quantum cryptography, I Scully, M. 0.,165,179 quantum key distribution, 169 Segal, I. E., 79,81 quasi-projectors, 18 Selleri, F., 4,5,41,99,112-114,121,122,184 quatemionic spaces, 7 sequence, 188 quotient set, 189 sesquilinear functional, 211 set, 187 random classical fields, 124 Shimony, A., 35, 41, 159 random field, 147 Simmons, G. F., 199 Rao, K. Ramakrishna, 80, 81 SIQT, see interpretation, statistical Rapoport, D., 143 Souza, Q. A. G. de, 121, 125, 130,216 Rapoport, D. L., 148 Specker, E. P., 3,101,105,109, 111,113 Rauch, H., 104, 114 spectrum, 152 Raymond, 1. P., 141 spin projection, 38, 94 Recami, E., 130 Spliser, D., 26 Redhead, M., 4, 5 SQUID,147 reduction, see wave packet reduction Squires, E. J., 53, 57, 63, 64, 128, 130, 183, 184 of state vector, 8 standard deviation, 134 Reivelt, K., 41, 96,131 Stapp, H. P., 4, 5, 35, 41, 49, 50, 53-57, 128, relation, 189 130, 151, 157, 183, 184 relative complement, 188 formulation of Bell's inequalities, 37,46 relativity breakdown of, 39 special - dogma, 36 relaxation time, 99 state Renninger, M., 163, 179 antisymmetric, 92, 94 representation coherent, 19, 20 Heisenberg, 151, 152 mixed,133 Schriidinger, 151 mixture of, 22, 29,31,33 resu[t(s), 154, 155 pure, 14, 133 Revzen, M., 180 stationary, 117 Rimini, A., 143,144,148 states, 7 Rindler vacuum, 48 statistical operator, 9 RIQT (realistic interpretation of quantum the• Steinberg, A. M., 178 ory), see interpretation, realistic Stem-Gerlach Rodrigues Jr., W. A., 7, 42, 55, 56, 112, 114, apparatus, 104, 160 121,122,125,129,130,216 experiment, 63 Roger, G., 41, 180 stochasticity, 145 Roman, P., 27,199 Stroud, c., 28 Rosa, M. A. F., 55, 56, 216 Stuckelberg, E. C. G., 26 Rosen, N., 41 subconsciousness, 51 Rosenfeld, L., 26, 27 subensemble, 134 Rucker, R., 199 sum of subs paces, 204 super quantum potential, 125 Saari, P., 41, 96, 131 superluminal Sachs, R. K., 129,216 processes, 128 Sallesi, G., 130 signal(s), 36, 95, 127 Schiller, R., 114 waves, 95 Schilpp, P. A., 56, 164 superselection rule, 79 Schiminovich, S., 26 supremum Schmidt, H., 55, 56, 128, 130 least upper bound, 190 224 INDEX symbol,I7 evolution, 30 symmetric wave function, 182 Unruh, W. G., 56 system radiation, 48 chaotic, 19 upper bound, 190 tangent bundle, 26 vacuum fluctuations, 118 Tarozzi, G., 114 Vaidman, L., 4,163,179 Taylor, A. E., 199 van der Merwe, A., 114 Taylor, M., 27 Varadarajan, V. S., 81,199 teleportation, I, 173 variable tensor product, 209 hidden, 98, 134 theorem variable{s) Gleason's, 105, 109 collective, 144 Kochen-Specker, 108, 109 Vaz Jr., 1., 42, 112, 114, 121, 125, 130 non-go, 101 vectors, 7 von Neumann's, 101, 102, 104 Vernon Jr., F. L., 21, 28,151,157 theory Vigier, J. P., 113, 129-131 continuous spontaneous localization, 144 von Mises, 12 de Broglie-Bohm, 98 von Neumann, 1., I, 3, II, 23, 27-29, 33, 49, Dirac, 126 50,53,56,63,67,79-81,101,112, GRW,I46 113, 182 hidden variables, 97-101, 109, III, 112, axiom, 103, 104 143 measurement, see measurement, von Neu• contextuality in, 108 mann de Broglie-Bohm, 100, 101, 183 von Weizsacker, C. F., 43, 55, 164 de Broglie-Bohm non relativistic, 115, 117-120, 126 de Broglie-Bohm relativistic, 123 Walther, H., 165, 179 de Broglie-Bohm relativistic many• wave function fermion, 126 Dirac, 126 de Broglie-Bohm relativistic n-particle, Klein-Gordon-Fock, 123 127 localized, 15 de Broglie-Bohm relativistic n-particle quasi-classical, 20 , 128 wave function collapse, 8 of the first kind, 99 wave guides, 178 of the second kind, 101 wave packet collapse, 147 of the zeroth kind, 10 I wave packet reduction, 97 realistic, 100 wave(s) realistic, 183 quantum, 115 Thome, K. S., 181, 184 wavelength time machine, 181 Compton, 15 Tipler, 1., 57 de Broglie, 15 Tittel, w., 173, 180 weak modularity postulate Todorov, I. T., 199 WMP, 70 Torgerson, J. R., 178, 179 Weber, T., 143, 144, 148 tunneling effect, 178 Weinfurter, U., 180 tunneling time, 126 Wesley, J. P., 41, 42, 112, 114 Tuppinger, D., 104, 114 Wheeler, J. A., 25, 53, 54, 57, 59, 64, 164, 165, 178,179 Ultbeck, 0., 25, 28, 47,56 whole, 12, 14,31 ultimate Ego, 182 whole, the - and its parts, 7 ultimate Observer, 183 wholeness, 14,97 Onal, N., 121, 131 Wiesner, S., 168, 178, 179 undistorted progressive waves (UPW), 129 Wigner union, 188 functions, 17 join Wigner, E. P., 3, 16,23,25,27-29,31,33,47, disjunction, 190 50,56,57,79, 143, 148 unitary formula, 18 INDEX 225 formula for probabilities, 152 Yeazell, J., 28 friend,3 yes-no experiment friend paradox, 53 YNE,67 index function, see function Wooters, w., 180 world Zapatrin, R. R., 67,80,83,86,95,96,101, 114, non Boolean, 76 184 Wu, H., 129,216 Zeilinger, A., 35, 41, 159, 173, 178, 180 Wunderlich, C., 141 Zeno effect, quantum, 1 zitterbewegung, 126 Yao, A., 170, 180 Zurek, W. H., 21, 28,144, 148, 151, 157