Entropic Dynamics on Curved Spaces

0.15 MB·

by  Shahid Nawaz

Entropic Dynamics on Curved Spaces
∗
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1 Shahid Nawaz†, Mohammad Abedi‡ and Ariel Caticha§
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Department of Physics, University at Albany-SUNY,
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Albany,NY 12222, USA.
n
a
J

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2 Abstract

Entropicdynamicsisaframeworkinwhichquantumtheoryisderived
]
h as an application of entropic methods of inference. Entropic dynamics
p on flat spaces has been extensively studied. The objective of this paper
- is to extend the entropic dynamics of N particles to curved spaces. The
t
n important new feature is that the displacement of a particle does not
a transformlikeavectorbecausefluctuationscanbelargeenoughtofeelthe
u effectsofcurvature. ThefinalresultisamodifiedSchr¨odingerequationin
q
which theusual Laplacian is replaced bytheLaplace-Beltrami operator.
[

Keywords: Entropic Dynamics, Quantum Theory, Riemannian Manifold, In-
2
formation Geometry
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1 1 INTRODUCTION
0
.
1 Entropicdynamics (ED) is a frameworkin which the laws of dynamics,such as
0 quantum mechanics, are derived as an example of entropic inference. In previ-
6 ous publications ED has been used to formulate the non-relativistic quantum
1
mechanics of particles moving in flat Euclidean space [1]-[5]. The objective of
:
v this paper is to extend the entropic dynamics of N particles to curved spaces.
i This is an important preliminary step toward an entropic dynamics of gravity.
X
Two other related contributions appear in [6, 7].
r
a The formalism developed here is a straightforward extension of previous
work. The main problem to be tackled is technical: in a curved space the dis-
placement of a particle does not transform like a vector. The reason is that
in entropic dynamics, just as in other forms of dynamics that involve Brow-
nian motion such as Nelson’s stochastic mechanics [8], the fluctuations in the
∗Presented at MaxEnt 2015, the 35th International Workshop on Bayesian Inference and
Maximum Entropy Methods in Science and Engineering (July 19–24, 2015, Potsdam NY,
USA).
†correspondingauther: [email protected]
‡[email protected]
§[email protected]
1
particle’s motion tend to be so large that they can feel the curvature of the
underlyingspace. Forasingleparticlethis technicalproblemwastackledin[9];
here we extend the formalism to several particles. In its early formulations ED
involvedanumberofassumptionsaboutauxiliaryvariablesandabouttheform
of the Hamiltonian and of the quantum potential that were not sufficiently jus-
tified. Laterdevelopmentsin[3]showedthattheauxiliaryvariableswereinfact
unnecessary. Furthermore, in [4] it was shown that the requirement that ED
describe a non-dissipative diffusion leads naturally to a Hamiltonian dynamics,
and that the tools of information geometry can be used to justify the metric
for the N-particle configuration space and the functional form of the quantum
potential. Thislatterdevelopmentis crucialtoourcurrentpurposes: justspec-
ifying the geometry of the curved space in which a single particle moves is not
sufficient to specify the geometry of the configuration space for N particles. In
this work all these improvements – the elimination of auxiliary variables, the
derivationofHamiltoniandynamics, andthe use of informationgeometry– are
implemented for the ED of N particles moving in a curved space. The final
result is a modified Schr¨odinger equation that takes into account the effects of
curvature.
2 ENTROPIC DYNAMICS
Just as in any framework for inference it is crucial that we start by identifying
the variables that we wish to infer. For the present paper the subject of our
inference is the positions of particles. Perhaps the central feature of ED – that
which establishes the subject matter – is the assumption that although these
positions are unknown they have definite values. This is in marked contrast
with the standardCopenhagen interpretationin which observables do not have
actual values until elicited by an act of measurement.
We consider N particles, each of which lives in a curved space X of d-
dimensions and metric h . The configuration space is X =X X... X and
ab N
× ×
has dimension n=Nd.
Inordertofindtheprobabilitydistributionfortheparticles’positionsρ(x,t)
we will first use the method of maximum entropy to determine the probability
′
P(x x) that the particles take an infinitesimally short step from an initial po-
sition| x X to a final position x′ X . Having determined the transition
N N
∈ ∈
probabilityforashortstepwewilllateriteratethisresulttofindρ(x,t). Tofind
′ ′
P(x x) we must first identify a prior Q(x x) and the appropriate constraints
| |
and then we maximize the joint entropy,
′
P (x x)
S[P,Q]= dnx′P (x′ x)log | . (1)
− | Q(x′ x)
Z |
The Prior The prior probabilityexpressesthe uncertainty aboutwhich x′ to
expect before any information is taken into account. In the state of extreme
ignorance we have no idea of where the particles will move and knowing the
2
position of one will tell us nothing about the positions of the others. The prior
is a product of the uniform priors for the individual particles. It is given by
N
′ ′
Q(x x)= Q(x x ), (2)
| i| i
i=1
Y
where, for instance, x is the initial position of ith particle. Furthermore, the
i
individualpriorsareuniformwhichmeansthatequalvolumesareassignedequal
probabilities,
Q(x′ x )=h1/2(x′) , (3)
i| i i
where h=deth . (We need not worry about normalizationbecause it has no
ab
effect on the final result.)
The Constraints The information about the motion is introduced through
the constraints. The first constraint deals with continuity of motion. This
means that the motion can be analyzed as a sequence of infinitesimally short
steps from xa to x′a = xa +∆xa (i is the particle index and a is its spatial
i i i i
coordinateindex which runs from 1 to m). This informationis incorporatedby
the following constraint
h ∆xa∆xb =κ , (4)
ab i i i
where κi is a small constant (cid:10)that is event(cid:11)ually allowed to tend to zero. These
N constraints lead to a motion in which the particles diffuse isotropically and
independently of each other. In order to introduce some directionality and also
to accountfor entanglement effects we introduce one additional constraintthat
acts on the configuration space. We impose that the expected displacement in
the direction of the gradient of a certain ”potential” φ satisfies
N
∂φ
∆xa =κ′, (5)
h ii∂xa
i=1 i
X
′
where κ is another small constant.
Finally maximize eq. (1) subject to the constraints eqs. (4, 5) and normal-
ization. We obtain
N h1/2(x′) N 1 ∂φ
P(x′ x)= i=1 i exp α h (x )∆xa∆xb α′∆xa ,
| ζ(x,αQ1,α2,··· ,αN,α′) "−i=1(cid:18)2 i ab i i i − i ∂xai (cid:19)#
X
(6)
′
where α (i=1 N) and α are Lagrangemultipliers and ζ is a normalization
i
···
constant. Thereisapotentialproblemwitheq. (6)becauseasweshallseebelow
ineq. (20) neither coordinatedifferencessuchas ∆xa northeir expectedvalues
i
<∆xa > and<∆xa∆xb > are covariantwhichmeans that the constraintseq.
i i i
(4) andeq. (5) are notcovariant. However,as we shallsee later, we require the
′
transition probability P(x x) in the limit of short steps or α . It is only
i
| →∞
in this limit that we will later achieve manifestly covariantresults.
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Entropic Time The concept of time is closely connected with motion and
change [10]. In ED motion is described by the transition probability, eq. (6),
thatdescribesinfinitesimallyshortsteps. Largerfinite changeswillbe obtained
as the accumulation of many short steps. To construct time we must specify
what we mean by an instant, how instants are ordered, and the interval or
duration between them [12]. In ED an instant is defined by the information
required to generate the next instant. In our case this information is given by
the probability density in configuration space (x,t). Thus, in ED an instant
P ′ ′
t is represented by a probability distribution. The new distribution (x,t) at
′ P
the next instant t can be constructed using the transition probability given by
eq.(6)
(x′,t′)= dnxP(x′ x) (x,t). (7)
P | P
NextweintroducethenotionofRduration–theinterval∆tbetweensuccessive
instants. For non-relativistic quantum mechanics we want to construct a time
that is Newtonian in the sense that the time intervalis independent of x and t.
This can be done through an appropriate choice of α ,
i
m
i
α = , (8)
i
η∆t
where m is a particle dependent constant that will turn out to be the mass of
i
the ith particle and η is an overallconstant that fixes the units of time relative
to those of mass and length.
TheInformationMetricofConfigurationSpace Theconfigurationspace
X is a smooth topological manifold. Although we have introduced a metric
N
for the single particle space X we have not thus far introduced a metric for
X . The natural candidate is the information distance between two neighbor-
N
′ ′
ing distributions P(x x) and P(x x+dx) which is given by the information
| |
metric
′ ′
∂logP(x x)∂logP(x x)
g =C dnx′P(x′ x) | | , (9)
AB | ∂xA ∂xB
Z
where C is an undetermined scale factor and ∂/∂xA =∂/∂xa. The capitalized
i
indices such as A=(i,a) and B =(j,b) denote both the particle index and its
spatial coordinates. Substituting eq.(6) into eq. (9) yields
Cm
i
g = δ h . (10)
AB ij ab
η∆t
Since in what follows the metric g will always appear multiplied by the
AB
combination C/η∆t, it is convenient to introduce an effective metric
η∆t
M (x)= g (x)=m δ h (x ), (11)
AB AB i ij ab i
C
which we will call the mass tensor because that is what it is in the special case
of flat spaces where M is independent of x.
AB
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The Transition Probability Substituting eq.(11) into eq.(6) the transition
probability becomes
M1/2(x′) 1 ∂φ
P(x′ x)= exp M (x)∆xA∆xB α′∆xA ,
| ζ′(x,α1,α2,··· ,αN,α′) (cid:20)−(cid:18)2η∆t AB − ∂xA(cid:19)(cid:21)
(12)
′ ′ ′
where ζ is the new normalization constant and M(x) = detM (x) . The
AB
′
detailedanalysisofα appearsin[11]. Herewewejustnotethatforourcurrent
′ ′
purposes α can be absorbed into φ which amounts to setting α =1 .
Analysis Nextwewanttoexpressagenericdisplacement∆xA intermsofan
expected drift ∆xA plus a fluctuation ∆wA. The difficulty is that due to the
determinant in the pre-exponential factor the distribution (12) is not Gaussian
(cid:10) (cid:11)
which makes it difficult to calculate expected values exactly. To solve this
problem we note that as ∆t 0, P(x′ x) probes a very localized regionof X .
N
→ |
This suggests a Taylor expansion about x. Unfortunately since we deal with a
Brownianmotiontoaccountforfluctuationsweneedtokeeptrackofquadratic
terms ∆xA∆xB which are affected by curvature. A Brownianparticle does not
follow a smooth trajectory. Its fluctuations will make it probe the curvature
effects in a local neighborhood of x.
The configuration space X is a curved space which is locally like Rn. It is
N
convenienttoexpressthetransitionprobabilityinnormalCartesiancoordinates
at x . In normal coordinates (NC) the metric tensor in the vicinity of a point
p
x is approximately that of flat Euclidean space, i.e.
p
∂h
ab
h =δ , =0. (13)
ab ab ∂xa
(cid:12)xp
(cid:12)
(cid:12)
Therefore for the configuration space metric (cid:12)we have
MA′B′(xp)=γA′B′, with γA′B′ =miδijδab, (14)
we use primed indices for NC and unprimed for generic coordinates. The first
derivative vanishes
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∂MA′B′ ∂ MA′B′
=0 but =0. (15)
∂xC′ ∂xC′∂xD′ 6
(cid:12)xp (cid:12)xp
(cid:12) (cid:12)
(cid:12) (cid:12)
In NC detM = detγ = (cid:12)N (m )d which can be abso(cid:12)rbed into a new normal-
i=1 i
ization constant. Notice that the second order term in the Taylor expansion
of MA′B′ is proportionalQto ∆xA′∆xB′ which is of order ∆t and can therefore
be neglected relative to terms of order ∆t0 and ∆t1/2 in the exponent. The
transition probability becomes Gaussian
PNC(x′ x)= 1 exp 1 γA′B′(x)(∆xA′ ∆xA′ )(∆xB′ ∆xB′ ) ,
| Z −2η∆t −h i −h i
(cid:20) (cid:21)
(16)
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whereZ isthenormalizationconstantinnormalcoordinates. Thedisplacement
∆xA′ is expressed as the expected drift plus a fluctuation
∆xA′ = ∆xA′ +∆wA′ , (17)
h i
where
∆xA′ =η∆tγA′B′ ∂φ , (18)
h i ∂xB′
and
∆wA′ =0, and ∆wA′∆wB′ =η∆tγA′B′. (19)
h i h i
Having calculated the diffusion process in the special NC coordinates we now
transform back to generic coordinates. We Taylor expand xA′(xA) about point
x . ( See e.g. [8, 9] ).
p
∆xA′ = ∂xA′∆xA+ 1∆xA∆xB ∂2xA′ + , (20)
∂xA 2 ∂xA∂xB ···
Thedisplacement∆xA′ involvesfluctuationsoftheorderof (∆t1/2),therefore
the secondorder termmust be included. This shows that ∆OxA′ does not trans-
form as a vector. To calculate the correctionterm it is convenient to introduce
a vector ∆˜xA that coincides with ∆xA′ in NC and in generic coordinates it is
given by
∆˜xA = ∂xA ∆xA′. (21)
∂xA′
Substituting eq. (20) into eq. (21) we get
1 ∂2xA′ ∂xA
∆˜xA =∆xA+ ∆xB∆xC . (22)
2 ∂xB∂xC ∂xA′
!
The second term is related to the transformation of Christoffel’s symbols
ΓABC = ∂∂xxAA′ ∂∂xxBB′ ∂∂xxCC′ΓAB′′C′ + ∂∂xxAA′ ∂∂xB2x∂Ax′C . (23)
Since ΓA′ =0 in NC, then
B′C′
1
∆xA =∆˜xA ΓA ∆xB∆xC. (24)
− 2 BC
We can rewrite this equation as
∆xA =bA(x)∆t+∆wA, (25)
where bA is the drift velocity
η
bA(x)=˜bA(x) MBCΓA , with ˜bA(x)=ηMAB∂ φ, (26)
− 2 BC B
where MAB is the inverse of M . Notice that bA does not transform like a
AB
vector but˜bA does. And the fluctuation
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∆wA =0 and ∆wA∆wB =ηMAB∆t. (27)
h i h i
Toconcludethissectionwerewritethetransitionprobabilityeq. (12)ingeneric
coordinates in the limit ∆t 0,
→
M(x′) 1
P(x′ x)= M(x′)P˜(x′ x)= exp M (∆˜xA η∆t∂Aφ)(∆˜xB η∆t∂Bφ) .
| | ζ˜ −2η∆t AB − −
p (cid:20) (cid:21)
p (28)
which ζ˜is the new normalization and P˜(x′ x) is explicitly covariant.
|
3 FOKKER-PLANCK EQUATION
The integral equation of motion eq. (7) can be rewritten using the standard
method [1] into a differential equation for the scalar ρ defined as (x,t) =
P
√Mρ(x,t). Notice that ρ is a scalarfield but is a tensor density of weight1.
P
Thentheresultofbuildingupafinitechangefrominitialtimet0uptofinaltime
′
t by iteratingmany smallchangesgivenby the transitionprobabilityP(x x) is
|
that the probability ρ evolves according a Fokker-Planck(FP) equation
∂ρ 1 η
= ∂ √M˜bAρ + ∆ ρ, (29)
A M
∂t −√M 2
(cid:16) (cid:17)
where ∆ is the Laplace-Beltramioperator
M
1
∆ = ∂ √MMAB∂ . (30)
M A B
√M
(cid:16) (cid:17)
The FP equation can be further written as a continuity equation
∂ρ 1
= ∂ √MvAρ , (31)
A
∂t −√M
(cid:16) (cid:17)
where vA is the current velocity
vA =˜bA+uA, (32)
where uA is the osmotic velocity
uA = ηMAB∂ logρ1/2. (33)
B
−
The continuity equation can also be written as
∂ρ 1
= ∂ √MρMAB∂ Φ , (34)
A B
∂t −√M
(cid:16) (cid:17)
where
vA =MAB∂ Φ with Φ(x,t)=ηφ(x,t) η logρ1/2(x,t). (35)
B
−
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Note that since ρ and φ are scalars (not densities) then vA and uA are vectors,
and Φ is a scalar. And finally, for later convenience we write the FP equation
as follows
∂ρ δH
= , (36)
∂t δΦ
where H =H(Φ,ρ). It can easily be checkedthat the appropriatefunctional H
is
1
H[ρ,Φ]= M1/2(x)dnx ρMAB∂ Φ∂ Φ+F[ρ], (37)
A B
2
Z
where F(ρ) is an integration constant to be determined below.
¨
4 THE SCHRODINGER EQUATION IN RIE-
MANNIAN MANIFOLDS
ThewavefunctionintheSchr¨odingerequationcontainstwodynamicalvariable,
the probability ρ and phase Φ. But so far we only have one dynamical variable
ρwhichevolvesaccordingtotheFokker-Planckequation(34). TheFPequation
tellshowtoupdateρforanexternallygivendriftpotentialφ. Thisisastandard
diffusion. In order to promote Φ to a fully dynamical variable we need to allow
the evolving ρ to react back and induce a change in the the potential φ. The
precise way in which this reaction is to happen is specified by requiring that
there be a conserved “energy” functional H = H(Φ,ρ) — a change in ρ must
be compensated by a corresponding change in Φ.
We require that the energy functional eq. (37) to be conserved [4],
dH δH δH
= dnx ∂ Φ+ ∂ ρ =0, (38)
t t
dt δΦ δρ
Z (cid:18) (cid:19)
where ∂ =∂/∂t. Using eq. (36) in eq. (38) we obtain
t
dH δH
= dnx ∂ Φ+ ∂ ρ=0. (39)
t t
dt δρ
Z (cid:18) (cid:19)
This must holdfor arbitraryinitial conditions, orarbitrary∂ ρ. Therefore,this
t
yields
δH
∂ Φ= , (40)
t
− δρ
for all values of t.
Substituting eq. (37) in eq. (40) we get
1 δF
∂ Φ= MAB∂ Φ∂ Φ , (41)
t A B
−2 − δρ
which is a generalized Hamilton-Jacobi equation.
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Equations (34) and (41) can be combined into a single equation by intro-
ducing a complex function Ψ
k
Ψ =ρ1/2exp(ikΦ/η), (42)
k
where k is a positive constant introduced for later convenience. This yields
iη η2 η2 ∆ Ψ δF
M k
∂ Ψ = ∆ Ψ + | |Ψ + Ψ , (43)
k t k −2k2 M k 2k2 Ψ k δρ k
k
| |
which is a non-linear Schr¨odinger equation. This can be put into standard
Schr¨odinger form by choosing the functional F(ρ) appropriately. Arguments
from information geometry suggest that the appropriate choice is [4]
F[ρ]=ξMABI + dnxρ(V +V ), (44)
AB c
Z
where ξ is a coupling constant and I is Fisher information metric
AB
1 ∂ρ(x)∂ρ(x)
I = dnx . (45)
AB ρ(x) ∂xA ∂xB
Z
The other terms in eq. (44) involve the external potential V and a possible
curvaturepotentialV thatvanishesforflatspaces. Aswecanseethecurvature
c
potential only enters through a choice of the scalar F which means that the
choice of V is arbitrary within the framework of entropic dynamics.
c
Therefore eq. (43) becomes
iη η2 η2 ∆ Ψ
M k
∂ Ψ = ∆ Ψ +( 4ξ) | |Ψ +(V +V )Ψ . (46)
k t k −2k2 M k 2k2 − Ψ k c k
k
| |
Here we take advantage of the arbitrary constant k and choose it so that ξ =
η2/8k2. Setting η/k =~ we obtain
1
i~∂ Ψ= ~2∆ Ψ+(V +V )Ψ, (47)
t M c
−2
which is the Schr¨odinger equation for N particles on the curved configuration
space. In flat space we have ∆ N 2/m , the curvature potential
M → i=1∇i i
vanishes and we recover the usual Shr¨odinger equation,
P
~2
i~∂ Ψ= 2Ψ+VΨ. (48)
t − 2m ∇i
i
i
X
5 SUMMARY
EntropicDynamicsviewsquantumtheoryasanapplicationofentropicmethods
ofinference. Inentropicdynamicsnounderlyingactionprincipleisassumed,on
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the contrary, an action principle and the corresponding Hamiltonian dynamics
are derived as a non-dissipative diffusion.
Ourgoalhasbeentoextendentropicdynamicstocurvedspaceswhichisan
important preliminary step toward an entropic dynamics of gravity. We have
derived the modified Schr¨odinger equation on a Riemannian manifold in the
frameworkof entropic dynamics. The modified equationreplacesthe Laplacian
by the Laplace-Beltrami operator.
ACKNOWLEDGMENTS
We would like to thank C. Cafaro, D. Bartolomeo, S. DiFranzo, S. Ipek, K.
Vanslette, A. Yousefi, A. Fernandes and P. Pessoa for many insightful discus-
sions.
References
[1] A.Caticha,EntropicInferenceandtheFoundationsofPhysics,(S˜aoPaulo,
Brazil, 2012);
http://www.albany.edu/physics/ACaticha-EIFP-book.pdf
[2] A. Caticha,J. Phys. A: Math. Gen. 44, 225303 (2011); arXiv:1005.2357.
[3] A. Caticha, J. Phys.: Conf. Ser. 504, 012009 (2014); arXiv:1403.3822.
[4] A.Caticha,D.Bartolomeo,andM.Reginatto, AIP Conf. Proc. 1641, 155 (2015);
arXiv:1412.5629.
[5] A. Caticha, Entropy 17, 6110 (2015); arXiv.org:1509.03222.
[6] S. Ipek and A. Caticha, AIP Conf. Proc. 1641, 345 (2015);
arXiv:1412.5637.
[7] A.Caticha,“GeometryfromInformationGeometry”,intheseproceedings;
arXiv:1512.09076.
[8] E.Nelson,“QuantumFluctuations”,(PrincetonU.Press,Princeton,1985).
[9] S. Nawaz, Momentum and Spin in Entropic Quantum Dynamics, Ph.D
Thesis, University at Albany (2014).
[10] A. Caticha, AIP Conf. Proc. 568, 72 (2001); arXiv:math-ph/0008018.
[11] D. Bartolomeo, and A. Caticha, “Entropic Dynamics: The Schr¨odinger
equationanditsBohmianlimit”,intheseProceedings; arXiv:1512.09084.
[12] A. Caticha, AIP Conf. Proc. 1305, 200 (2011); arXiv:1011.0746.
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