REGULAR STATISTICAL THEORIES AND THE STATISTICAL THEORY APPROACH

A statistical theory (S,P,p) is defined to be regular if the following conditions hold for all X e S , A,B e ? :

1 e ? (4.1a)

If A ^ B then A 'AB e ? (4.1b)

If A=$ B then (A A B )(X ) + ( A 'A B ) ( \ ) = B (X ) . (4.1c) This is equivalent to the definition in §3.4 of Appendix II (equations (17)), noting that (4.1a) is implicitly assumed in the latter definition (§3.2 of Appendix II), and that A' e 7 follows from equations (2.7f), (2.9b), (4.1a), (4.1b) o f the thesis. Comparison o f equations (2.13), (4.1) shows that the class o f classical statistical theories is contained within the class of regular statistical theories. Further, Example 3.2 o f Appendix II shows that the Hilbert space description o f a standard quantum system yields a regular statistical theory.

The class of regular statistical theories provides the basis for a "statistical theory approach" to axiomatic quantum mechanics, briefly discussed in Hall [1988; see Appendix II]. The aim o f such an approach is to derive the characteristic "quantum" features o f standard quantum mechanics from a small number of (physically motivated) axioms. A further aim is to derive the Hilbert space formalism o f standard quantum mechanics, which I shall not address here. I will now discuss the basic assumptions o f the statistical theory approach, and in the next section make comparisons with the quantum logic approach.

The Statistical Theory Approach

While regular statistical theories are the key to the statistical theory approach to axiomatic quantum mechanics, the defining conditions (4.1a), (4.1b), (4.1c) do not directly admit a simple physical interpretation and are therefore not suitable as axioms for a physical theory. In the following I shall show that some other,

physically motivated axioms can be found from which conditions (4.1) can be derived as theorems.

First, motivated by the probabilistic nature of quantum phenomena:

Axiom 1. A physical system is described by a statistical theory as characterised in §2.2. In particular, there are a set of states, S ; a set of experimental propositions, P , which may be tested on these states; and a mapping p:PxS —♦ [0,1] , where p(A,A,) is the probability that proposition A is verified on state X .

The remaining axioms are motivated by consideration of simultaneously measurable propositions. Consider an experiment E which may be performed on the class of systems described by a statistical theory (S,P,p) . Denote the possible results of this experiment by elements of a result set . For any physical experiment will be finite. There is then an associated set of propositions tested by experiment E , of the form "The result of E is contained in a subset, X , of R£ Denote the experimental proposition corresponding to X c R^ by Ax , and the set of such propositions by P£ . Thus Ax is verified by a result x e X , and falsified otherwise. E is defined to be an experiment of (S,P,p) if P_ c P . Now if a proposition A € P has zero probability of being verified on any state

Ä Cj

X € S , i.e., p(Ax ,X.) = 0 VA. e S , then a result x e X is exceptional, and may be discarded (without changing any probabilities). It follows that R„ may be

b

replaced by the set R_\X without loss of generality. I shall therefore assume that b

R_ is minimal, in the sense that b

\ = 0 IFF X = <j> , (4.2) where denotes the abstract proposition corresponding to Ax e P£ .

The following lemma gives some technical properties of the abstract propositions in 7 , corresponding to an experiment, E , of (S,P,p) .

Lemma 4.1. If E is an experiment of the statistical theory (SJP,p) , then (i) 4 ^ = 1 ; , (4.3a) (ii)

A L

=

A

, where Xc := RC\X ; (4.3b) ^ Xc E (iii)

A x n j X )

+ i

(X)

= A

( X )

; (4.3c) x cnY Y

for all X,Y c R^. , X sS .

Proof. The proof of properties (4.3) uses the fact that E simultaneously tests all propositions in P£ . Consider an ensemble of N systems described by state

X

, and suppose that experiment E is performed on each member of this ensemble. For each system define the quantity r(Ax ,^) to be 1 if experimental proposition Ax is verified, and 0 otherwise. Since, by definition, the result of E is in , and since the result is in some Y c R g if and only if it is in exactly one of X n Y , Xc n Y , then r(AR

,X

= 1 ; t(A

,X)

+ r(A

,X)

= t(A

X)

; xnY x cny y (4.4a) (4.4b)

for every U S , X,Y c . Further, if ( • ) denotes an ensemble average, in the limit N —> oo , then equations (2.1), (2.2) imply

(r(Ax ,^)) = A^ X) (4.4c)

for all X q R_ ,

X e

S . Part (i) of the lemma now follows from definition (2.5)

h

and equations (4.4a), (4.4c); part (ii) follows from definition (2.4) and equations (4.4a), (4.4b), (4.4c) (choosing Y = R^ ); and part (iii) follows from equations

(4.4b), (4.4c). □

The second axiom of the statistical theory approach is motivated by the interpretation of the proposition A ^ ,^ e P£ for an experiment E of (S,P,p) . In particular, since x e X n Y if and only if x e X , x e Y , it follows that AxnY

is verified if and only if both Ax and A y are verified. Intuitively, the probability of this simultaneous verification on state X is just the "joint probability" of Ax and Ay verified on state X . To incorporate this as a property of the probability structure of (SJ?,p) I formulate:

In document Locality, probability and quantum mechanics (Page 59-62)