Entropic Dynamics:

an Inference Approach to

Time and Quantum Theory

Ariel Caticha

Department of Physics

University at Albany - SUNY

EmQM-13

Vienna, 10/2013

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Do the laws of Physics reflect Laws of Nature?

Question:

Or...

Are they rules for processing information about Nature?

Our objective:

To derive Quantum Theory as Entropic Dynamics

and discuss some implications for the theory of

time, quantum measurement, Hilbert spaces, etc.

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constraint

)(

)(log)(],[

xq

xpxpdxqpS

q

prior

p

posterior

(MaxEnt, Bayes' rule and Large Deviations are special cases.)

Maximize S[p,q] subject to the appropriate constraints.

dVedVqpS ],[

)Pr(

Entropic Inference:

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the world the system (particles, fields...)

other relevant

variables

The subject matter

5 5 5

The subject matter

and entropy

)(

)|(log)|()(

yq

xypxypdyxS

The goal is to predict the positions of particles, x.

Particles have definite but unknown positions.

There exist other variables y with probability p(y|x)

)(x

)(x

(y variables are not hidden variables.)

6 6

xX

M statistical manifold

Dynamics:

]|[ xyp

x

]|[ xyp

Change happens.

Configuration space

(→ non-locality)

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Entropic Dynamics

Maximize the joint entropy

)|,(

)|,(log)|,(],[

xyxQ

xyxPxyxPydxdQPSJ

uniform

M

)|()|( xypxxP

Short steps: 22 ba

ab xx

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The result:

])(2

1)([exp

1)|( 2 xxSxxP

where

)(

)|(log)|()(

yq

xypxypydxS

Displacement: wxx

)()(

1xS

xx aa

Expected drift :

Fluctuations: abba

xww

)(

1

)( 1 O

)( 2/1 O

!!!

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Entropic Time

The foundation of any notion of time is dynamics.

),()( xxPdxxP

(1) Introduce the notion of an instant

Time is introduced to keep track of the accumulation

of many small changes.

),()|(),( txxxPdxtx

)()|( xPxxPdx

10

t

Define duration so that motion looks simple:

For large the dynamics is all in the fluctuations:

(3) Duration: the interval between instants

abba ww

1 abt

m

(2) Instants are ordered: the Arrow of Entropic Time

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)( vt

Fokker-Planck equation:

),()|(),( txxxPdxtx

Entropic dynamics:

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)( vt

Fokker-Planck equation:

mv ),(log)(),( 2/1 txxStx

2/1log

mu

drift velocity: Sm

b

osmotic velocity:

ubv

mu

2diffusion current:

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xX

M

)(xS

Entropic dynamics:

Problem: this is just standard diffusion, not QM!

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xX

),( txS

M

Non-dissipative diffusion

wave

particle

The solution: allow M to be dynamic.

Quantum Mechanics

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Manifold dynamics?

Impose “energy” conservation

)(

2

1

2

1],[ 223 xVumvmxdSE

02

)(2 2/1

2/1222

2

mV

m

Quantum Hamilton-Jacobi equation:

!!!

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The result: two coupled equations

mt

1) Fokker-Planck/diffusion equation

02

)(2 2/1

2/1222

2

mV

m

2) Quantum Hamilton-Jacobi equation

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Vm

i 22

2

to get Quantum Mechanics,

Complex numbers, linearity, and Hilbert spaces are

convenient but not fundamental.

Combine into and e2/1i

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Measurement of position in ED

Particles have definite but unknown positions.

Measurement of position is “classical”:

dxx2

| dxxdxx2

)()( Born rule:

a mere ascertaining of pre-existing values,

a mere amplification from micro to macro scales.

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What about other “observables”?

source

|

A

|ˆ| AU2

|ˆ| Aii Uxp

position

detectors

potential

“complex”

detector

ia

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then ii ac ||

2| ia

iAiA aUcU |ˆ|ˆ

ii xc |

the general Born rule

If ia|

2

ii cp and

iiA xaU ||ˆ

Other “observables” and the Born rule:

The complex detector “measures” all operators of the

form

||ˆiii aaA

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Conclusions:

• The t in the Laws of Physics is entropic time.

• Entropic Dynamics provides an alternative to

Action Principles.

• Quantum measurement is just an exercise in

inference.

• Position is “real”. Other observables are not.

They are “created” by the act of measurement.

• Unitary evolution and wave function collapse

correspond to infinitesimal and discrete updating.

• No need for quantum probabilities or logic.

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Acknowledgements:

David T. Johnson

Shahid Nawaz

Carlo Cafaro

Daniel Bartolomeo

Adom Giffin

Marcel Reginatto

Kevin Knuth

Philip Goyal

Keith Earle

Carlos Rodriguez

Nestor Caticha

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Three ingredients:

entropy

E. Nelson

time

Entropic

Dynamics

E. Jaynes

diffusion

J. Barbour

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the constraints

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Entropic inference:

To update from old beliefs to new beliefs when

new information becomes available.

the prior q(x) the posterior p(x)

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Amplification from micro to macro scales

The amplifier is prepared in an unstable state.

The problem is to infer from ix .r

iiAi xaUa ||ˆ|When

)(

)|()()|(

r

iriri

P

xPxPxP

Use Bayes’ rule:

the amplifier jumps to state .r

)(

)|()()|(

2

r

iriri

P

xPxxP

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An analogy from physics:

initial state

of motion. final state

of motion.

Force

Force is whatever induces a change of motion: t

pF

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Inference is dynamics too!

old beliefs new beliefs

information

Information is what induces the change in rational beliefs.

Information is what constrains rational beliefs.

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indeterminism The sources of all quantum weirdness:

superposition

The quantum measurement problem:

Why don't we see macroscopic entanglement?

How do we get definite outcomes?

What is the source of indeterminism?

Are observed values created by measurement?

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(2) Instants are ordered: the Arrow of Entropic Time

Bayes' theorem

A time-reversed evolution:

),()|(),( txxxPxdtx

)|()(

)()|( xxP

xP

xPxxP

There is no symmetry between prior and posterior.

Entropic time only goes “forward”. Configuration space

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"physical" space

cx1

cx3

cx2

More on entropic time

A clock follows a classical trajectory.

2t

1t

3t

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"physical" space

cx1

)|( 1

cxx

cx3)|( 3

cxxcx2

)|( 2

cxx

More on entropic time

For the composite system of particle and clock:

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"physical" space

cx1

)|( 1

cxx

cx3)|( 3

cxxcx2

)|( 2

cxx

Entropic time vs "physical" time?

Entropic time is all we need.

There is an arrow of entropic time.

We observe correlations at an instant.

We do not observe the "absolute" order of the instants.