Entropic Dynamics: from Entropy and Information Geometry to Quantum Mechanics
Ariel Caticha, Daniel Bartolomeo Physics Dept. University at Albany – SUNY, USA
Marcel Reginatto Physicalische-Technische Bundesanstalt, Germany
MaxEnt 2014, Amboise
J. A. Wheeler (1983): Law without Law: “The only thing harder to understand than a law of statistical origin would be a law that is not of statistical origin, for then there would be no way for it—or its progenitor principles—to come into being.” Two tests: “No test of these views looks like being someday doable, nor more interesting and more instructive, than a derivation of the structure of quantum theory...” “No prediction lends itself to a more critical test than this, that every law of physics, pushed to the extreme, will be found statistical and approximate, not mathematically perfect and precise.”
The subject matter: The goal is to predict the positions of particles x. (Or other configurational variables, such as fields.)
Positions have definite but unknown values.
Dynamics:
X
Change happens.
x
x′ Configuration space
The first goal: find P ( x′ | x) . (Config. space + probability → non-locality.)
4
Entropic Dynamics Maximize the entropy
P ( x′ | x ) S [ P, Q ] = − ∫ dx′ P( x′ | x) log Q ( x′ | x ) Constraints: short steps: some directionality:
uniform
〈 ∆x ⋅ ∆x 〉 = κ 〈 ∆x 〉 ⋅ ∇ φ = κ ′ 5
The result:
2 1 1 P( x′ | x) = exp [− α ∆x + α ∆x′∆⋅x∇⋅φ∇]φ ] ζ 2 Displacement: Expected drift : Fluctuations:
∆x = 〈 ∆x 〉 + ∆w 1 〈 ∆x 〉 = ∇φ α
← O(α −1 )
〈∆w a 〉 = 0
1 ab 〈 ∆w ∆w 〉 = δ α a
b
← O(α −1/ 2 ) 6
Entropic Time (1) Introduce the notion of an instant,
ρ ( x′, t ′) = ∫ dx P ( x′ | x) ρ ( x, t ) (2) Instants are ordered: the Arrow of Entropic Time (3) Define duration so that the dynamics looks simple. Fluctuations:
Cn mn α n ( x, t ) = = ∆t η∆t
1 ab η 〈 ∆w ∆w 〉 = δ = ∆tδ ab αn mn a n
b n
clocks mass 7
Entropic dynamics:
ρ ( x′, t ′) = ∫ dx P ( x′ | x) ρ ( x, t )
Fokker-Planck equation: ∂ t ρ = −∇ ⋅ ( ρ v ) mv = ∇Φ Problem:
Φ = φ − log ρ 1/ 2 ( x, t ) η
this is just standard diffusion, not QM! 8
allow ϕ to be dynamic.
Solution:
φ ( x, t )
wave X
x particle
ϕ dynamics?
9
ϕ dynamics? Define H [ ρ , Φ ] so that
(
δH 1 = ∂ t ρ = − ∇ ⋅ ρ ∇Φ δΦ m
)
1 H [ ρ , Φ] = ∫ d x ρ ∇Φ ⋅ ∇Φ + F [ ρ ] 2m 3
10
Impose conservation of H :
dH [ ρ , Φ ] = ∫ dx dt
δH δH δΦ ∂ t Φ + δρ ∂ t ρ = 0 = ∂t ρ
⇒
dH = ∫ dx dt
δH ∂ t Φ + δρ ∂ t ρ = 0
δH ∂t ρ = δΦ
δH ∂tΦ = − δρ
FP eq.
and
HJ eq. 11
Choosing F [ ρ ] : information geometry g ab = ∫ dx ρ ( x | θ )
∂ log ρ ( x | θ ) ∂ log ρ ( x | θ ) ∂θ a ∂θ b
Two tensors: P ( x′ | x )
ρ (x)
⇒ ⇒
Natural choice:
γ ab = mδ ab 1 ∂ρ ( x) ∂ρ ( x) I ab = ∫ dx ρ ( x) ∂x a ∂x b
(Jacobi) (Fisher)
F [ ρ ] = ξ γ ab I ab + ∫ dx ρ ( x) V ( x) 12
Ψk = ρ
Combine ρ and Φ into
1/ 2
Φ exp(ik ) η
2 2 ∇ Ψk η η η 2 i ∂ t Ψk = − 2 ∇ Ψk + VΨk + 2 − 4ξ Ψk k 2k m 2k m Ψk 2
Natural choice:
⇒
to get QM:
⇒
1/ 2
η ˆ k = 8ξ 2
and set
η = kˆ
2 2 i ∂ t Ψ = − ∇ Ψ + VΨ . 2m
Φ 1/ 2 Ψ = ρ exp(i ) ,
2 ξ= 8 13
Conclusions: • ED derives Laws of Physics from entropic inference. • The ED of non-dissipative diffusion leads to Hamiltonians and to Quantum Theory. • Information geometry is crucial: it explains the configuration space metric and the quantum potential. • Position is “real”. Other observables are not. • The t in the Laws of Physics is “entropic” time. 14
