SOME REMARKS ON RICCI FLOW AND THE QUANTUM POTENTIAL
arXiv:math-ph/0703065v2 1 Apr 2007
ROBERT CARROLL UNIVERSITY OF ILLINOIS, URBANA, IL 61801 Abstract. We indicate some formulas connecting Ricci flow and Perelman entropy to Fisher information, differential entropy, and the quantum potential.
1. FORMULAS INVOLVING RICCI FLOW Certain aspects of Perelman’s work on the Poincar´e conjecture have applications in physics and we want to suggest a few formulas in this direction; a fuller exposition will appear in a book in preparation [8]. We go first to [13, 24, 25, 26, 27, 28, 33, 39] and simply write down a few formulas from [28, 39] here with minimal explanation. Thus one has Perelman’s functional (R˙ is the Riemannian Ricci curvature) Z (R˙ + |∇f |2 )exp(−f )dV (1.1) F= M R and a so-called Nash entropy (1A) N (u) = M ulog(u)dV where u = log(−f ). One considers Ricci flows g = gt with δg ∼ ∂t g = h and for 2 ˙ ˙ (1B) 2∗ u = −∂ R t u−∆u+ Ru = 0 (or equivalently ∂t f +∆f −|∇f | + R = 0) it follows that M exp(−f )dV = 1 is preserved and ∂t N = F. Note the Ricci flow equation is ∂t g = −2Ric. Extremizing F via δF ∼ ∂t F = 0 involves Ric + Hess(f ) = 0 or Rij + ∇i ∇j f = 0 and one knows also that Z ˙ (|∇f |2 + R)exp(−f )dV = F; (1.2) ∂t N = M Z |Ric + Hess(f )|2 exp(−f )dV ∂t F = 2 M
¨ 2. THE SCHRODINGER EQUATION AND WDW Now referring to [3, 4, 5, 7, 8, 9, 10, 11, 12, 15, 16, 18, 19, 20, 21, 22, 23, 29, 30, 31, 32, 35, 36, 37, 38, 40] for details we note first the important observation in [39] that F is in fact a Fisher information functional. Fisher Date: March, 2007. email:
[email protected].
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ROBERT CARROLL UNIVERSITY OF ILLINOIS, URBANA, IL 61801
information has come up repeatedly in studies of the Schr¨ odinger equation (SE) and the Wheeler-deWitt equation (WDW) and is connected to a differential entropy correspondingto the Nash entropy above (cf. [4, 7, 18, 19]). The basic Rideas involve (using 1-D for simplicity) a quantum potential Q such that M P Qdx ∼ F arising from a wave function ψ = Rexp(iS/~) where Q = −(~2 /2m)(∆R/R) and P ∼ |ψ|2 is a probability density. In a WDW context for example one can develop a framework Z Z −1/2 1/2 (2.1) Q = cP ∂(GP ); QP = c P 1/2 ∂(GP 1/2 )Dhdx → → −c
Z
∂P 1/2 G∂P 1/2 Dhdx
where G is an expression involving the deWitt metric Gijkℓ (h). In a more simple minded context consider a SE in 1-D i~∂t ψ = −(~2 /2m)∂x2 ψ + V ψ where ψ = Rexp(iS/~) leads to the equations 1 ~2 Rxx 1 2 Sx + Q + V = 0; ∂t R2 + (R2 Sx )x = 0 : Q = − 2m m 2m R In terms of the exact uncertainty principle of Hall and Reginatto (see [21, 23, 34] and cf. also R[4, 6, 7, 31, 32]) the quantum Hamiltonian has a Fisher information term c dx(∇P ·∇P/2mP ) added to the classical Hamiltonian (where P = R2 ∼ |ψ|2 ) and a simple calculation gives Z Z Z 1 1 ~2 ~2 2 3 3 |∇P |2 d3 x 2∆P − |∇P | d x = (2.3) P Qd x ∼ − 8m P 8m P (2.2) St +
In the situation of (2.1) the analogues to Section 1 involve (∂ ∼ ∂x ) (2.4) P ∼ e−f ; P ′ ∼ Px ∼ −f ′ e−f ; Q ∼ ef /2 ∂(G∂e−f /2 ); P Q ∼ e−f /2 ∂(G∂e−f /2 ); Z Z Z −f /2 −f /2 P Q → − ∂e G∂e ∼ − ∂P 1/2 G∂P 1/2 In the context of the SE in Weyl space developed in [1, 2, 4, 7, 10, 11, 12, √ 35, 36, 40] one has a situation |ψ|2 ∼ R2 ∼ P ∼ ρˆ = ρ/ g with a Weyl ~ = −∇log(ˆ vector φ ρ) and a quantum potential (2.5) √ p 8 1 8 p ~2 ~2 ik ˙ ˙ R + √ √ ∂i R + √ ∆ ρˆ gg ∂k ρˆ = − Q∼− 16m 16m ρˆ g ρˆ √ √ mn (recall divgrad(U ) = ∆U = (1/ g)∂m ( gg ∂n U ). Here the Weyl-Ricci curvature is (2A) R = R˙ + Rw where √ ∆ ρˆ 2 ~ ~ (2.6) Rw = 2|φ| − 4∇ · φ = 8 √ ρˆ
SOME REMARKS ON RICCI FLOW AND THE QUANTUM POTENTIAL
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and Q = −(~2 /16m)R. Note that
ρ|2 ∆ˆ ρ |∇ˆ ~ ∼ −∆log(ˆ + −∇·φ ρ) ∼ − ρˆ ρˆ2 and for exp(−f ) − ρˆ Z Z |∇ˆ ρ|2 ~ (2.8) ρˆ∇ · φdV = −∆ˆ ρ+ dV ρˆ with the first term in the last integral vanishing and the second providing Fisher information again. Comparing with Section 1 we have analogues ~ 2 ) with φ ~ = −∇log(ˆ (2B) G ∼ (R + |φ| ρR) ∼ ∇f to go with (2.4). Clearly ρˆ is basically a probability concept with ρˆdV = 1 and quantum mechanics (QM) (or rather perhaps Bohmian mechanics) seems to enter the picture through the second equation in (2.2), namely (2C) ∂t ρˆ+ (1/m)div(ˆ ρ ∇S) = 0 with p = mv = ∇S, which must be reconciled with (1B). In any event ~ 2 can be written as (2D) R˙ + Rw + (|φ| ~ 2 − Rw ) = the term G = R˙ + |φ| R R ~2 ~ − |φ| ~ 2 ) which leads to (2E) F ∼ α αQ + (4∇ · φ M QP dV + β |φ| P dV putting Q directly into the picture and suggesting some sort of quantum mechanical connection. (2.7)
REMARK 2.1. We mention also that Q appears in a fascinating geometrical role in the relativistic Bohmian format following [3, 15, 37, 38] (cf. also [4, 7] for survey material). Thus e.g. one can define a quantum mass field via √ α −~2 2( ρ) 2 2 2 ∼ Rw (2.9) M = m exp(Q) ∼ m (1 + Q); Q ∼ 2 2 √ c m ρ 6 where ρ refers to an appropriate mass density and M is in fact the Dirac field β in a Weyl-Dirac formulation of Bohmian quantum gravity. Further one can change the 4-D Lorentzian metric via a conformal factor Ω2 = M2 /m2 in the form g˜µν = Ω2 gµν and this suggests possible interest in Ricci flows etc. in conformal Lorentzian spaces (cf. here also [14]). We refer to [3, 15] for another fascinating form of the quantum potential as a mass generating term and intrinsic self energy. NOTE. Publication information for items below listed by archive numbers can often be found on the net listing. References [1] J. Audretsch, Phys. Rev. D, 27 (1983), 2872-2884 [2] J. Audretsch, f. G¨ ahler, and N. Straumann, Comm. Math. Phys., 95 (1984), 41-51 [3] G. Bertoldi, A. Faraggi, and M. Matone, Class. Quant. Grav., 17 (2000), 3965 (hep-th 9909201) [4] R. Carroll, Fluctuations, information, gravity, and the quantum potential, Springer, 2006
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ROBERT CARROLL UNIVERSITY OF ILLINOIS, URBANA, IL 61801
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