7. Random variables as observables. Definition 7. A random variable (function) X on a probability measure space is called an observable

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1 7. Random variables as observables Definition 7. A random variable (function) X on a probability measure space is called an observable Note all functions are assumed to be measurable henceforth.

2 Note that if ÐHY ß ß Ñ is a probability space, then the family of (essentially) bounded random variables (functions on H ) \Ð= Ñ has the structure of an integration algebra ÐAßIÑ, with IÐ\Ñ œ ( \Ð= Ñ. Ð= ÑÞ In probability we interpret H as the space of all possible outcomes, and for each outcome =, we define \Ð= Ñ to be the value of the observable \ given the outcome = HÞ We can call \Ð= Ñ an observable.

3 We may study the integration algebra directly, since it determines the structure of the probability space up to isomorphisms (measure-preserving transformations). Note that the algebra A of observables is commutative, since \Ð= Ñ]Ð= Ñ œ ]Ð= Ñ\Ð= Ñ.

4 Generalization: Consider that in quantum mechanics observables also form an algebra which is more general, since it is noncommutative. But each observable still has an expected value. Example 3. Consider the motion of a particle on " # the line. Let [ œpð Ñ. If < ÐBÑ [ and m< m œ " then < represents a probability amplitude that particle is located at B. Also # l< ÐBÑl represents the probability that particle is located at B.

5 The operator FœQB (multiplication by B) on PÐ # Ñis the position operator. If particle is in state <ÐBÑ, then its expected position is Ø< ßF< Ù œ (.B< ÐBÑF< ÐBÑ œ (.B< ÐBÑQB< ÐBÑ œ ( < ÐBÑB< ÐBÑ.B œ ( Bl< ÐBÑl.B #

6 The operator 3`B(differentiation with respect to B) on P # Ð Ñ is the momentum operator. The expected momentum is Gœ " ` Ø< ßG< Ù œ (.B< ÐBÑ " ` < ÐBÑ 3`B œ 3 (.B < ÐBÑ< ÐBÑÞ w

7 Generally any observable quantity corresponds to a self-adjoint operator E on [, and its expectation value (in a given state < [ Ñ is IÐEÑ œ Ø< ßE< Ù.

8 8. Random variables as observables, II We now view a random variable \Ð= Ñ, = Has having a value for each outcome in the universe H of outcomes. In other words, \ has a value for each of the possible 'worlds' (possible = representing our reality) that we are in. We have seen that just know expectation values of an algebra of random variables on H is sufficient to determine the structure of H.

9 Henceforth we will focus on the algebra structure of the (bounded) random variables A œö\ð= ÑÀ\ is a RV on H.

10 Now consider [ œp ÐH ) ÞIf < Ð= Ñ [, then # " 3= Ð Ñ œ l< Ð= Ñl P ÐHÑ, and represents a probability distribution on the possible outcomes = H. If \ A, then in probability # IÐ\Ñ œ ( \Ð=3= Ñ Ð Ñ. = H œ ( H <= Ð Ñ\Ð=<= Ñ Ð Ñ.TÐ= Ñ œ Ø<= Ð Ñß\Ð=<= Ñ Ð ÑÙ œ Ø< ß\ < ÙÞ

11 Thus we can interpret a state <= Ð Ñ to be a probability distribution 3= Ð Ñ œ l<= Ð Ñl # on H. It represents our current knowledge of the world as a probability distribution 3= Ð Ñ on H.

12 9. An alternative interpretation of < Ð= Ñ Note that if we have a state <, it forms a linear functional 9 on the set of observables \ A. Specifically, define 9 Ð\Ñ œ I Ð\Ñ œ \Ð= Ñl< l ÐBÑ.T Ð= Ñ œ Ø< ß \ < ÙÞ < ( Thus a state < can be interpreted as a linear functional 9 À A Ä. #

13 Definition 8. More generally, we will define a state to be any bounded linear functional 9 on A.

14 10. Summary of commuting observables So far we have showed that studying an algebra A of random variables on a probability space H is entirely equivalent to knowing the algebra structure of A, i.e. knowing \] and +\,] (for +ß, Ñ if we know \ and ]. We call the algebra A of observables because \] œ ] \ if \ß] A. commuting

15 We have seen that if <= Ð Ñ P ÐHÑ, then # " l< l Ð= Ñ P ÐHÑ and represents a new probability distribution on H. Note that the expectation with respect to this new distribution is # I Ð\Ñ œ < ( \Ð= Ñl ðóóóóñóóóóò < l Ð= Ñ.TÐ= ÑÞ H # T Ð= Ñ < Note that this is equivalent to definining a new measure on H

16 # T< œ l< l Ð= Ñ.TÐ= Ñß i.e., Radon-Nikodym derivative.t.t < # œl< lð= Ñ. We have defined the state < through its expectation I < above as a linear functional on \ A. More generally, we have now defined a any linear functional 9 À A Ä. state as

17 But note that this I < is an expectation on the algebra A of random variables (i.e. satisfies the 4 properties defined earlier). Thus we have a Proposition. Given a probability space H, an algebra A of random variables on H, and # <= Ð Ñ P ÐH Ñ with m< m œ ", there is a unique expectation defined on A given by I < I Ð\Ñ œ < ( \Ð= Ñl< l Ð= Ñ.TÐ= ÑÞ H #

18 Note that this expectation can be interpreted as giving the average value of any observable \ given the underlying state < on H.

19 11. Non-commuting observables Recall from above we have an algebra A of observables \ together with an expectation I on the algebra (we denote IœI < for now). These form a integration algebra defined above. We define the total measure of the integration algebra to be IÐ"Ñ (i.e. expectation of the function \œ").

20 If IÐ"Ñ œ " (i.e. total measure is 1) we will call A a probability algebra. Thus a probability algebra A is just a generalization of the collection of random variables on a probability space H. Let's generalize the above ideas to the case where observables do not commute, i.e. we have an algebra A of observables where \] Á ] \.

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The Hahn-Banach and Radon-Nikodym Theorems 1. Signed measures Definition 1. Given a set Hand a 5-field of sets Y, we define a set function. À Y Ä to be a signed measure if it has all the properties of