Entropic Time Ariel Caticha Department of Physics University at Albany - SUNY MaxEnt 2010 Chamonix
Question: Do the laws of Physics reflect Laws of Nature?
Or...
Are they rules for processing information about Nature? Our objective: To derive Quantum Theory as Entropic Dynamics and focus on the implications for the theory of time. 2
Step 1: The Statistical Model
Y
hidden variables
y p( y | x)
M
x
X
x
statistical manifold
configuration space with metric γ = δ ab ab σ2 3
Step 1: The Statistical Model
p( y | x)
M
x
X
x
statistical manifold
configuration space with metric γ = δ ab ab σ2 4
Step 2: Entropic Dynamics Maximize
P( x′, y′ | x) S J [ P, Q ] = − ∫ dx′dy′ P ( x′, y′ | x) log Q( x′, y′ | x) uniform
P ( x′ | x ) P p( y′ | x′), x)
∈M Changes happen gradually. Short steps:
〈 ∆2 〉 = 〈γ ab ∆x a ∆x b 〉 = κ 5
The transition probability:
1 1 P( x′ | x) = exp [ S ( x′) − α ( x) ∆2 ] ζ 2 where
p ( y′ | x′) S ( x′) = − ∫ dy′ p( y′ | x′) log q( y′)
6
For short steps:
α P( x′ | x) ∝ exp [− 2 δ ab (∆x a − ∆x a )(∆x b − ∆x b )] 2σ Displacement: Expected drift : Fluctuations:
∆x = ∆x + ∆w 2 σ ∆x a = δ ab ∂ b S ( x) α 2 σ 〈 ∆wa ∆wb 〉 = δ ab α
7
Step 3: Entropic Time The foundation of any notion of time is dynamics. Time is introduced to keep track of the accumulation of many small changes.
P ( x′) = ∫ dx P ( x′, x) = ∫ dx P ( x′ | x) P ( x) (1) Introduce the notion of an instant
ρ ( x′, t ′) = ∫ dx P ( x′ | x) ρ ( x, t ) 8
The Arrow of Entropic Time A time-reversed evolution:
ρ ( x, t ) = ∫ dx′ P ( x | x′) ρ ( x′, t ′) Bayes' theorem
P ( x | x′) =
P( x) P ( x′ | x ) P ( x′)
There is no symmetry between prior and posterior. Entropic time only goes forward. Entropic time vs. physical time? 9
(2) Introduce the notion of interval between instants
For large α the dynamics is all in the fluctuations:
σ ab 〈 ∆w ∆w 〉 = δ α a
b
2
σ 2 ab = ∆t δ τ
Define duration so that motion looks simple:
10
τ α= ∆t
a Fokker-Planck equation: ∂ t ρ = −∂ a ( ρ v ) 2 σ va = ∂ aφ τ
φ ( x, t ) = S ( x) − log ρ 1/ 2 ( x, t )
two components:
va = ba + u a
drift velocity:
2 σ ba = ∂aS τ
osmotic velocity:
2 σ ua = − ∂ a logρ 1/ 2 τ
11
But this is just diffusion, not quantum mechanics! A wave function requires two degrees of freedom. ρ ( x, t )
M
X
φ ( x, t ) = S ( x, t ) − logρ 1/ 2 ( x, t )
x
x 12
Step 4: Manifold dynamics? "Energy" conservation
[Nelson (1979), Smolin (2006)]
1 1 2 2 E = ∫ d x ρ m v + m u + V ( x) 2 2 3
where
τ m = 2 σ
13
The result: two coupled equations
1) Fokker-Planck/diffusion equation
ρ = −
a ∂ ( ρ ∂ aφ ) m
2) energy conservation + diffusion 2 2 2 1/ 2 ∇ ρ 2 φ + (∂ aφ ) + V − =0 1/ 2 2m 2m ρ
14
Combine ρ and φ into
Ψ = ρ 1 / 2 e iφ
∂Ψ 2 2 ∇ Ψ + VΨ to get Quantum Mechanics, i = − ∂t 2m F = ma
and also Classical Mechanics Let
S HJ = φ then
S + 1 (∂ S ) 2 + V = 0 HJ a HJ 2m
ab 〈 ∆w ∆w 〉 = ∆t δ → 0 m a
b
15
More on Entropic Time A clock follows a classical trajectory.
x3c
t3
x
t2
c 2
"physical" space x1c
t1
16
For the composite system of particle and clock:
x3c
ρ ( x | x3c )
x
ρ ( x | x2c )
c 2
"physical" space x1c
ρ ( x | x1c )
17
Entropic time vs "physical" time? We observe correlations at an instant. We do not observe the "absolute" order of the instants. x3c
ρ ( x | x3c )
x
ρ ( x | x2c )
c 2
"physical" space x1c
ρ ( x | x1c )
Entropic time is all we need. 18
Conclusion: Entropic inference leads to Laws of Physics. It provides an alternative to Action Principles. The t in Laws of Physics is entropic time. The natural order/arrow of inference is entropic time... ...and this is the only time we need.
19
20
The Big Picture:
inference
society
probability entropy geometry...
physics
life
21
An analogy from physics:
initial state of motion.
final state of motion.
Force
dp Force is whatever induces a change of motion: F = dt 22
Inference is dynamics too!
old beliefs
new beliefs
information
Information is what induces the change in rational beliefs. 23
The Big Picture:
ethics
sociology economics anthropology linguistics... mathematics philosophy humanities art religion
society
probability entropy geometry...
inference logic inductive logic decision th. computer sci. informatics complexity...
life psychology sociobiology
biology: cell, organ, organism, ecology...
physics
??
astronomy, chemistry, earth sci. ...
organic chemistry, evolution...
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The Big Picture:
ethics
sociology economics anthropology linguistics... mathematics philosophy humanities art religion
society
probability entropy geometry...
inference logic inductive logic decision th. computer sci. informatics complexity...
life psychology sociobiology
biology: cell, organ, organism, ecology...
physics
astronomy, chemistry, earth sci. ...
organic chemistry, evolution...
25
The Big Picture:
inference
society
physics
life
26
Three ingredients: E. Jaynes
E. Nelson
J. Barbour
entropy
diffusion
time
Entropic Dynamics 27
Summary: 1) Hidden variables: no dynamics; just p ( y | x) and S ( x). 2) Changes happen gradually. 3) Entropic time: instants, interval, ordering. 4) The statistical manifold is dynamical: energy conservation.
28
Overview 1) The statistical model: Hidden variables. 2) The dynamical principle: Changes happen gradually. 3) A book-keeping device: Entropic time. 4) Statistical manifold dynamics: Conservative diffusion. 5) Entropic vs. "physical" time.
29
Further reading and references to the literature: go to arxiv.org and search under A.C.
30
The Gravitational Equivalence Principle: We accept the equivalence of gravitational with the "fictitious" forces that arise in accelerated frames because it explains the equality of inertial and gravitational masses and allows a geometrical explanation of gravity.
A Quantum Equivalence Principle? We accept the equivalence of quantum and "epistemic" probabilities because it explains the equality of inertial and osmotic masses it explains linearity, superposition, complex numbers, and allows an inferential explanation of quantum theory. 31
P( x′, y′ | x) S J [ P, Q ] = − ∫ dx′dy′ P ( x′, y′ | x) log Q( x′, y′ | x) Prior:
Q ( x′, y′ | x) ∝ q( x′) × q( y′)
∝ γ 1/ 2 × q( y′) uniform
First constraint:
P ( x′, y′ | x) = P ( x′ | x) P ( y′ | x′, x) = P( x′ | x) p ( y′ | x′)
∈M
∈M
Second constraint: 2 a b 2 Short steps: 〈 ∆ 〉 = 〈γ ab ∆x ∆x 〉 = λ ( x)
32
Step 4: Manifold dynamics? Energy conservation
[Nelson (1979), Smolin (2006)]
[
E = ∫ d 3 x ρ Aγ ab v a v b + Bγ ab u a u b + V ( x)
]
1 1 2 2 E = ∫ d x ρ m v + µ u + V ( x) 2 2 3
where
2 A τ m= 2 = 2 σ σ mass
2B µ= 2 σ
≈m
osmotic mass 33
But we can always regraduate to a more convenient description.
τ′ τ= , κ
η = κη ′ ,
new units:
New wave function:
φ′ φ= κ
Ψ ′ = ρ 1/ 2 exp iφ '
Schrödinger equation:
η′ 2 η ′ µκ 1 − iη ′ Ψ ′ = − ∇ Ψ ′ + VΨ ′ + 2m 2m m 2
2
2
∇ ( Ψ ′Ψ ′ ) Ψ′ 1 / 2 * ( Ψ ′Ψ ′ ) 2
34
* 1/ 2
