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

Measurement Akio Hosoya

竹原６ / ６ ,’11

1. Introduction2. A formal theory of (Weak) Value Born’s rule3. Main theorem4. Summary

arXiv:1104.1873

with Minoru Koga

１ . IntroductionIt seems that the right quantity to discuss inquantum cosmology is not the probability foranything but the contextual value of observables.Example: Suppose the initial state of the universe is theone given by Hawking and that we know thatthe present state of the universe is such and such(anthropic? ) . What would be the value ofphysical observables in-between?

The value of observable appears only aftermeasurement not before in the Copenhageninterpretation. However, there is a revisionistlike 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.

Fundamental Problem:

“Non-contextual values” of observables are not possible bythe 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)

σx1 1σx σxσx

1σz σz1 σzσz

σxσz σzσz σyσy

1

1

1

1 1 -1

cannot assignthe eigenvalues±1 consistently

Eigenvalues areonly non-contextualvalues

Mermin ‘90

I propose the weak value advocated by Aharonov as a candidate of “contex dependent value of A “.

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 ofdifferent context.

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.

However, what is “probability”?There have been many debates over the meaningof 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 theaxiomatic theory of probability theory by Kolmogorovwithout talking about its meaning.

We believe the combination of quantum mechanicsand axiomatic probability theory reveals themeaning of probability on the basis of measurement.

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.

２ . Formal theory of value of an observable2.1 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 V

1,V

2,V

3 ….. ∈ 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 resultantstates 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|}.

Corresponding to the choice of the Abelian sub-algebra V

1,V

2,V

3 …. ∈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 V

1, V

2,V

3 …..i.e.,

{<ω|}1, {<ω|}

2, {<ω|}

3…

We fix a maximal Abelian subalgebra Vmax∈V(N)for the moment of discussion and therefore thecontext Ω:={<ω|} ω

. We shall find an expression for the

value of an observable A- λ(A) C, complex number. ∈

2.2 Main Theorem We demand that the “value” λ(A) C∈ of an observable A N∈satisfies the following properties:(1) 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 V∈ max

(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.

Note that for S, T V∈ max

<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| ( ♮)

The formal classical probability theory a laKolmogorov presupposes the probabilitymeasure P(ω) and λω (A) the value of a

physical quantity A for an event ω. The expectation value Ex[A] and the variance Var[A] are given 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).

According to the central limit theorem, the distributionof values of observable A approaches the normal (Gaussian ) distribution characterized by its meanEx[A] and the variance Var[A].

The requirement ( ４ ) demands that the distributionshould be independent of how we measure A.

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 formulafor the expectation value and the variance

Ex[A]=<Ψ|A|Ψ>Var[A]=<Ψ|A2|Ψ>.

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 that P(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

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 tobothP(ω)=|<ω|Ψ>|2

W=|Ψ><ω|

Introducing the lagrange multiplier μ to ensure the completeness relation, we demand the variation ofthe “action” L[<ω|,μ] w.r.t. <ω| and μ vanish for all observable A

５ . Summary

Combining quantum mechanics and the formalprobability theory we have shown that the context dependent value of observable A is the 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 thepost selected state, which is inferred by the valuesof all the elements of Vmax∈V(N)i.e., the context of how we intend to measure A. .

λω (A) is experimentally accessible at any time by the weak measurements. The probability measure P(ω)=|<ω|Ψ>|2 isnot an axiom any more but a consequence of quantummechanics and the probability theory. λω (A) is interpreted as a value of A in the contextof the pre-selected state |Ψ> and the post-selected states{<ω|} of the intended projective measurements of amaximal set of commuting observables Vmax∈V(N).

Going back to the original motivation ofthe value of an observable before measurementwe just show an example:

ξ(t):=<x|X(t)|Ψ>/<x|Ψ>,

where X(t) ,0≤t≤T is the position operator of aparticle. <x| is the eigen state of X(T) withthe eigen value x.

x

ξ(t)

t=T

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.

☆ 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 valueto the back action problem of the Hawking radiation.(2010 Arxiv) by analyzing

G [g] = 8πG[<out| T |in>/<out|in>]

Two remarks: