Born’s Rule and “Value” of an Observable before Measurement Akio Hosoya

1. 竹原 ６/６,’11 Born’s Rule and “Value” of an Observable beforeMeasurement Akio Hosoya • Introduction • A formal theory of (Weak) Value • Born’s rule • 3. Main theorem • 4. Summary • arXiv:1104.1873 • with Minoru Koga

2. １. Introduction It seems that the right quantity to discuss in quantum cosmology is not the probability for anything but the contextual value of observables. Example: Suppose the initial state of the universe is the one given by Hawking and that we know that the present state of the universe is such and such (anthropic? ) . What would be the value of physical observables in-between?

3. Fundamental Problem: The value of observable appears only after measurement not before in the Copenhagen interpretation. However, there is a revisionist like me Carl Friedrich von Weizsäcker denied that the Copenhagen interpretation asserted: "What cannot be observed does not exist". He suggested instead that the Copenhagen interpretation follows the principle: "What is observed certainly exists; about what is not observed we are still free to make suitable assumptions.

4. “Non-contextual values” of observables are not possible by the Kochen-Specker theorem (‘67) We cannot assign a value of physical quantity independently of how we measure it for dim(H)≥3. Example by Mermin (4 dim) cannot assign the eigenvalues ±1 consistently Eigenvalues are only non-contextual values σx1 1σx σxσx 1σz σz1 σzσz σxσz σzσz σyσy 1 1 1 1 1 -1 Mermin ‘90

5. Therefore the “value” should depend on context, i.e., how to measure it. The context is specified by the maximal set of commuting observables Vmax as explained later in detail. The non-commutativity of observables V(N) in quantum mechanics makes different choices of Vmax V’max∈V(N) non-commutable and therefore of different context. Ipropose the weak value advocated by Aharonov as a candidate of “contex dependent value of A “.

6. History of Born’s Rule: P(ω)=|<ω|Ψ>|2 The Born rule was formulated by Born in a 1926 paper. In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Einstein's work on the photoelectric effect, concluded in a footnote, that the Born rule gives the only possible interpretation of the solution. In 1954, together with Walter Bothe, Born was awarded the Nobel Prize in Physics for this and other work. 1) Anmerkung bei tier Korrektur: Genauere Uberlegung zeigt, dab die Wahrscheinlichkeit dem Quadrat der Φ proportional ist.

7. However, what is “probability”? There have been many debates over the meaning of probability. (1) frequency of events [coin tossing] (2) expectation [rain forcast,Laplace] subjective interpretation [Beysian] ……. But we do not have a consensus yet. It seems that at present we are content with the axiomatic theory of probability theory by Kolmogorov without talking about its meaning. We believe the combination of quantum mechanics and axiomatic probability theory reveals the meaning of probability on the basis of measurement.

8. Note that the context Ω is fixed once and for all in classical theory. Ex(A) :=  dP(ω) hA(ω) ω∈Ω: event (<ω|∈Vmax∈V(N) ) dP(ω): probability measure (independent of A) (P(<ω|) ) hA(ω): a random variable (real) (λω (A): complex) according to Kolmogorov.

9. ２. Formal theory of value of an observable 2.1 Quantum context (finite dimension) Let V(N) be a set of Abelian sub-algebras of all observables N .There may be many choices of the sub-algebra V1,V2,V3 ….. ∈ V(N).　 Choose Vmax∈V(N). We call Vmax as a context. The idea is that the mutually commutable set of observables {P,Q,R,….} define a set of simultaneous eigenvectors of P,Q,R,….{<ω|}, which corresponds to the resultant states after the projective measurements of P,Q,R,…. . The way of description (context) of experiments is characterized by the choice of Vmax. We are going to define the value of an observable A in the state |ψ> in the context Vmax, i.e.,. .{<ω|}.

10. Corresponding to the choice of the Abelian sub-algebra V1,V2,V3 …. ∈V(N), we have a collection of orthonormal basis {<ω|}1, {<ω|}2, {<ω|}3, …… We can think of the collection of the values of an observable A in the state |ψ> in the context V1, V2,V3…..i.e., {<ω|}1, {<ω|}2, {<ω|}3… We fix a maximal Abelian subalgebra Vmax∈V(N) for the moment of discussion and therefore the context Ω:={<ω|} ω. We shall find an expression for the value of an observable A- λ(A) ∈C, complex number.

11. 2.2　Main Theorem • We demand that the “value” λ(A) ∈ Cof an observable A∈N • satisfies the following properties: • Linearity: • λ(A+B)=λ(A)+λ(B) c.f. von Neumann, Bell….. • (2) Product rule when restricted to the Abelian subalgebra: • λ(ST)=λ(S) λ(T) close to classical theory • for all S, T∈Vmax • (3) Specification of which state we are definitely living in |Ψ> • λ(|Ψ⊥><Ψ⊥|)=０　, for all　|Ψ⊥>　s.t. <Ψ⊥|Ψ>=0 • The above reqirements lead to • λ(A)=Tr[WA]/Tr[W] ---(1) λ(1)=1 --- (2) • with • W=a|Ψ><ω|+b|ω><Ψ| ---(2)(3) • where <ω| is a simultaneous eigenvector of Vmax.

12. Note that for S, T∈Vmax <ω|S=<ω|s, <ω|T=<ω|t so that λ(ST)=λ(S) λ(T) holds. The product rule (2) implies W=|α><ω|+|ω><β|+ ΣωCnm|ωn><ωm| (♯) where <ωm|ω>=0, while the condition (3) implies W=|Ψ><q|+|r><Ψ| (♭) Putting ♯ and ♭together we arrive at W=a|Ψ><ω|+b|ω><Ψ| ( ♮)

13. The formal classical probability theory a la Kolmogorov presupposes the probability measure P(ω) and λω(A) thevalue of a physical quantity A for an event ω. The expectation value Ex[A] and the variance Var[A] aregiven by Ex[A]=ΣωP(ω)λω(A) Var[A]=ΣωP(ω)|λω(A)|2 We adopt these expressions also in quantum mechanics. （４）We demand the expectation value Ex[A] and the variance Var[A] be independent of the choice of CONS Ω={<ω|} ω,i.e., Vmax∈V(N).

14. According to the central limit theorem, the distribution of values of observable A approaches the normal (Gaussian ) distribution characterized by its mean Ex[A] and the variance Var[A]. The requirement (４) demands that the distribution should be independent of how we measure A.

15. The above requirement uniquely determines W=|Ψ><ω| and therefore the “value” λω(A)=Tr[WA]/Tr[W] = <ω|A|Ψ>/<ω|Ψ>, ( i.e., b=0) and the measure, P(ω)=|<ω|Ψ>|2 and therefore we have “derived” the Born formula for the expectation value and the variance Ex[A]=<Ψ|A|Ψ> Var[A]=<Ψ|A2|Ψ>.

16. Idea of the proof: if P(ω)=|<Ψ|ω>|2 Ex[A]=ΣωP(ω) λω (A) =Σω|<Ψ|ω>|2[<ω|A|Ψ>/<ω|Ψ>] = Σω<Ψ|ω><ω|A|Ψ> =<Ψ|A|Ψ> Var[A]=ΣωP(ω) |λω (A)|2 =Σω|<Ψ|ω>|2|<ω|A|Ψ>/<ω|Ψ>) |2 = Σω<Ψ|A|ω><ω|A|Ψ> =<Ψ|A2|Ψ> do not depend on {<ω|} i.e., the choice of Vmax∈V(N). Note thatP(ω)=|<Ψ|ω>|4 would not work! The key is the completeness relation Σω|ω><ω|=1. c.f. J.Phys. A;43 025304 (2010) with Shikano

17. Introducing the lagrange multiplier μ to ensure the completeness relation, we demand the variation of the “action” L[<ω|,μ] w.r.t. <ω| and μ vanish for all observable A L[<ω|,μ]=Ex[A]-μ(Σω<Ψ|ω><ω|A|Ψ>-<Ψ|A|Ψ>) =ΣωP(ω) λω (A)-μ(Σω<Ψ|ω><ω|A|Ψ>-<Ψ|A|Ψ>) where λω(A)=Tr[WA]/Tr[W] and W=a|Ψ><ω|+b|ω><Ψ| δL/δ<ω| =∂P(ω)/∂<ω|λω (A) +P(ω) ∂λω (A)/∂<ω|-μA|Ψ>=0 and a similar equation for Var[A] lead to both P(ω)=|<ω|Ψ>|2 W=|Ψ><ω|

18. ５. Summary Combining quantum mechanics and the formal probability theory we have shown that the context dependent value of observable A isthe weak value λω(A) := <ω|A|Ψ>/<ω|Ψ> and the probability measure is given by Born’s rule: P(ω)=|<ω|Ψ>|2, where |Ψ> is the initial state and <ω| is the post selected state, which is inferred by the values of all the elements of Vmax∈V(N) i.e., the context of how we intend to measure A. .

19. λω(A) is experimentally accessible at any time by the weak measurements. The probability measure P(ω)=|<ω|Ψ>|2 is not an axiom any more but a consequence of quantum mechanics and the probability theory. λω (A) is interpreted as a value of A in the context of the pre-selected state |Ψ> and the post-selected states {<ω|} of the intended projective measurements of a maximal set of commuting observables Vmax∈V(N).

20. Going back to the original motivation of the value of an observable before measurement we just show an example: ξ(t):=<x|X(t)|Ψ>/<x|Ψ>, where X(t) ,0≤t≤T is the position operator of a particle. <x| is the eigen state of X(T) with the eigen value x. t=T ξ(t)  x

21. We can ask the following counter-factual question. We are in a certain initial state and know the value of X as x by measuring X of  at t=T. What the value of X(t) would be before T ? We can answer in an experimentally verifiable way.

22. Two remarks: ☆ Counter-factual statement: If A were true, B would hold. A☐B Caution: the transitive law does not hold. ☆ Recently Englert and Spindel applied the weak value to the back action problem of the Hawking radiation. (2010 Arxiv) by analyzing G  [g] = 8πG[<out| T |in>/<out|in>]