Post on 25-Aug-2020
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.)
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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
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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.