probability qm

8/3/2019 Probability Qm

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On the origin of probability in quantum

mechanics

R. Buniy, S. Hsu and A. Zee

ITS, University of Oregon

May 30, 2006


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Thought experiment

N

S

+

-

D

D

observer

Does it matter ifobserver = nothing ? (Or a few atoms? Manyatoms?)


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Thought experiment

N

S

+

-

D

D

observer

Does it matter ifobserver = amoeba ?


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Thought experiment

N

S

+

-

D

D

observer

Does it matter ifobserver = cat ?(Schrodingers cat?)


7/27

Thought experiment

N

S

+

-

D

D

observer

Does it matter ifobserver = Gork the robot ?

(S. Coleman, Quantum Mechanics, In Your Face!)


8/27

Thought experiment

N

S

+

-

D

D

observer

Does it matter ifobserver = a scientist ?


9/27

Time evolution in QM

Quantum mechanics, as conventionally formulated, has twotypes of time evolution:

U: Unitary, deterministic: (t) = eiHt(0)

C: Copenhagen, Collapse: discontinuous, probabilistic

(von Neumann projection: | |a, for outcome a.)

Is C really necessary? Perhaps C is an apparent phenomena

which emerges from unitary evolution U!

H. Everett, 1958

Quantum cosmology. Quantum computers.


10/27

Many Worlds Interpretation

N

S

+

-

D

D

observer

= No collapse or Decoherent histories.

Macroscopic beings will perceive a collapse due to decoherenceof distinct branches.

DeWitt, Hartle, Gell-Mann, Feynman, Hawking, Coleman,

Weinberg, Deutsch...


11/27

What kind of quantum mechanic are you?

The unexamined life: Gee, never really thought about it!Copenhagen is OK with me.

OK, I thought about it and something IS funny...

Instrumentalist: Copenhagen OK, were not permitted tointerpret physical models beyond experimental predictions.(How do you feel about quarks?) Also, were not permitted tothink about the QM evolution of the universe as a whole.

No Collapse: Copenhagen is just obfuscation. Its not awell-defined theory, and Many Worlds is.

QM is incomplete: Many Worlds is too weird for me. QM isincomplete and well discover something beyond it someday.(Well, what are you waiting for? Why waste time on any otherproblems in physics?)


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Steve Weinberg, Einsteins Mistakes, Physics Today 2005

...Bohrs version of quantum mechanics was deeply flawed, butnot for the reason Einstein thought. The Copenhagen

interpretation describes what happens when an observermakes a measurement, but the observer and the act ofmeasurement are themselves treated classically. This is surelywrong: Physicists and their apparatus must be governed by thesame quantum mechanical rules that govern everything else in

the universe. But these rules are expressed in terms of awavefunction (or, more precisely, a state vector) that evolves ina perfectly deterministic way. So where do the probabilisticrules of the Copenhagen interpretation come from?


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Steve Weinberg, Einsteins Mistakes, Physics Today 2005

...Considerable progress has been made in recent years towardthe resolution of the problem, which I cannot go into here. It isenough to say that neither Bohr nor Einstein had focused on thereal problem with quantum mechanics. The Copenhagen rulesclearly work, so they have to be accepted. But this leaves thetask of explaining them by applying the deterministic equationfor the evolution of the wavefunction, the Schrodinger

equation, to observers and their apparatus. The difficulty is notthat quantum mechanics is probabilisticthat is something weapparently just have to live with. The real difficulty is that it isalso deterministic, or more precisely, that it combines aprobabilistic interpretation with deterministic dynamics.


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Many Worlds Interpretation

S = (+ - + + - + - +)

Ensemble state: = Ni=1

i = 1 2 N.

Each of 2Nbranches or histories S appears in :

=

{s1,...sN} cs1...sN|s1, . . . sN

S=

(s1, s2, . . . sN), si=

.


15/27

Born rule

S = (+ - + + - + - +)

How does Born rule (p = |c|2) arise in NoCollapse interpretation?

Each observer perceives a particular history S,but all branches are possible. Structure of treecompletely independent ofc!

Multiplicity of branches S dominated by

n+ n = N/2.On most branches S, physicists would not havededuced the Born rule!


16/27

Born rule

S = (+ +

+ +

+

+

)

Let n = n+, f = f+ = n/Nand p = p+ = |c+|2.Assume Born rule. Use it to compute the norm squared ofbranches based on statistical properties ofS, such as number of

+ outcomes, n.

n + outcomes

|cs1,...,sN|2 = P(n) =

N

n

pn(1 p)Nn

1) combinatorial factorN

n

peaked at n = N/2 or f = 1/2.

2) individual probability factor pn(1 p)Nn peaked at n = 0, Nor f = 0, 1.

Competition between (1) and (2): P(n) peaked at n = pNorf = p.


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Born rule

Example: p = p+ = .3

Key observation: Born rule probability is just the norm squaredfor a set of outcomes. In conventional QM, probability =

measure. Branches which are rare according to Born rule musthave small norm.

Norm of dominated by |s1, . . . , sN with f p (fraction of+values = .3).

Branches with f p are called maverick worlds (Everett).


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Born rule

For N , states with f p (maverick worlds) have measurezero. Only states with fraction of+ values given by Born rulehave non-zero norm.

Since zero norm states are unphysical, we have derived theBorn rule within the No Collapse interpretation.

(Everett, DeWitt, Hartle)

maverick

non-maverick: f=p


19/27

Our horizon is finite

Can N ?

Our causal horizon is finite. Decoherence times are fast, but notzero. Hence Nis finite.

For Nfinite, does the argument still work?

Maverick worlds have small, but non-zero, norm. We have nojustification for removing components of with small but

non-zero norm. In fact, in the absence of wavefunctioncollapse, the norm of a subcomponent |s1, . . . , sN plays no rolein quantum mechanics!

Frequentist vs Bayesian notion of probability.


20/27

Discrete quantum state space

Quantum mechanics + gravity = minimum length.

No meaning to distance less than the Planck length: lP(Calmet, Graesser and Hsu 2004.)

This suggests a discreteness to quantum state space, or Hilbertspace, in abuse of terminology. (Buniy, Hsu and Zee 2005.)


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Discrete quantum state space

Consider rotation of Stern-Gerlachdevice by small angle . If sufficientlysmall, no component is displaced by

more than a Planck length lP. Thecorresponding spin eigenstates areindistinguishable.

We are motivated to consider a minimum

norm in Hilbert space.

| | < | | 0

, N|, N 2N

p+0

d f P(f N) . (1)

where P(f N)

2Np(1

p)

1/2 exp N(fp)22p(1

p) .

Requiring that this collective norm squared is less than N2

yields

0 > N1/2

2p(1 p)|ln (N2)|

1/2. (2)


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Born rule from discrete state space

If, for finite N, an experimenter could measure all Noutcomeswhich define his branch of the wavefunction, he might find a

deviation from the predicted Born frequency f = p as large as

|ln (N2)|1/2

standard deviations (i.e., measuring the deviation in units of

N1/2).


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An experimenter is unlikely to be able to measure more than asmall fraction of the outcomes that determine his branch.

A particular branch of the wavefunction is specified by thesequence of outcomes S = (s1, s2, . . . , sN).

Nis the total number of decoherent outcomes on a branch, so it

is typically enormous at least Avogadros number if thesystem contains macroscopic objects such as an experimenter.

B l f d


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The experimental outcomes available to test Borns rule will besome much smaller number N Ncorresponding to some

subset of the si directly related to the experiment.

Any deviation from the Born rule of order N1/2 will be wellwithin the experimental statistical error of order N1/2 .

The Born rule will be observed to hold in all the brancheswhich remain after truncation due to discreteness.

S b


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Some numbers

Let 10100 a tiny discreteness scale!

Assume N 10160 H4 in Fermis.

Then N

2

1040

and|ln (N2)|1/2 10

so the predicted deviation from Born rule is 10 standarddeviations for the entire universe.

Unless experimenters can measure more than N/(10)2 of all Nbranchings, their statistical accuracy will not be enough toexclude this deviation.

S


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Summary

Original derivation of Born rule in No Collapse interpretationsis flawed: only applies when N=

(strict frequentist limit),

but our universe is finite.

Very small discreteness of quantum state space is enough torestore the result if minimum norm in Hilbert space,detectable Maverick worlds can be excluded.

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