Quantum Binding in Newton Potential: a Source for Dark Energy? T. - - PowerPoint PPT Presentation

▶
quantum binding in newton potential a source for dark
SMART_READER_LITE
LIVE PREVIEW

Quantum Binding in Newton Potential: a Source for Dark Energy? T. - - PowerPoint PPT Presentation

May 27, 2023 259 likes •692 views

Nonrelativistic: Madelung fluid and Fisher entropy Spec. Rel.: off mass-shell non-plane-wave Gen. Rel.: energy-momentum with quantum binding Consequences in Approximate Numbers Quantum Binding in Newton Potential: a Source for Dark Energy? T.

slide-1
SLIDE 1

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Quantum Binding in Newton Potential: a Source for Dark Energy?

  • T. S. Biró, P

. Ván: Foundations of Phys. 45 (2015) 1465

T.S. Biró, P . Ván1

1Heavy Ion Research Group

MTA Research Centre for Physics, Budapest

August 27, 2018

Talk given by T.S.Biró at Mátraháza, Sept. 5. 2018. Biro, Ván QGR in Madelung Variables 1 / 39

slide-2
SLIDE 2

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Dark Energy

75% of large scale universe

Can it be related to dark matter? Can it be quantum gravity? Can it be vacuum polarization, Casimir energy? Can it be a more subtle quantum effect? Present talk: couple Einstein to Schrödinger/Madelung eqs

Biro, Ván QGR in Madelung Variables 2 / 39

slide-3
SLIDE 3

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Outline

1

Nonrelativistic: Madelung fluid and Fisher entropy

2

  • Spec. Rel.: off mass-shell non-plane-wave

3

  • Gen. Rel.: energy-momentum with quantum binding

4

Consequences in Approximate Numbers

Biro, Ván QGR in Madelung Variables 3 / 39

slide-4
SLIDE 4

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Outline

1

Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

2

  • Spec. Rel.: off mass-shell non-plane-wave

3

  • Gen. Rel.: energy-momentum with quantum binding

4

Consequences in Approximate Numbers

Biro, Ván QGR in Madelung Variables 4 / 39

slide-5
SLIDE 5

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Schrödinger with Madelung

− 2 2m∇2ϕ + V(x)ϕ = i ∂ ∂t ϕ (1) Ansatz ϕ = R e

i α

Classical action and momentum ∂α ∂t = −E, ∇α = P (2)

Biro, Ván QGR in Madelung Variables 4 / 39

slide-6
SLIDE 6

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Logarithmic Derivatives

∂ ∂t ϕ = 1 R ∂R ∂t − i E

  • ϕ,

∇ϕ = 1 R ∇R + i P

  • ϕ

(3) Laplacian ∇2ϕ =

  • ∇

∇R R + i P

  • +

∇R R + i P 2 ϕ (4)

Biro, Ván QGR in Madelung Variables 5 / 39

slide-7
SLIDE 7

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Real and Imaginary Part

E = V − 2 2m

  • ∇∇R

R + ∇R R 2 − P2 2

  • (5)

i R ∂R ∂t = − 2 2m i

  • ∇P + 2

R P · ∇R

  • (6)

Biro, Ván QGR in Madelung Variables 6 / 39

slide-8
SLIDE 8

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Interpretation

Energy = Classical + Quantum contributions E =

P2 2m + V − 2 2m ∇2R R

(7) Mass density continuity m∂R2 ∂t + ∇

  • R2 P
  • = 0

(8)

Biro, Ván QGR in Madelung Variables 7 / 39

slide-9
SLIDE 9

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Action Principle

Variational Principle behind the Schrödinger equation S = ∂S ∂t + |∇S|2 2m + V

  • |ϕ|2 d3x dt

(9) ”Boltzmannian” ansatz: ϕ = e

i S

Using this ansatz: S =

  • i ϕ∗ ∂ϕ

∂t + 2 2m∇ϕ∗ · ∇ϕ + Vϕ∗ϕ

  • d3x dt

(10) Variation against ϕ∗ delivers δS δϕ∗ = i ∂ϕ ∂t − 2 2m∇2ϕ + Vϕ = 0 (11)

Biro, Ván QGR in Madelung Variables 8 / 39

slide-10
SLIDE 10

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Action with Madelung Variables

Split to Quantum + Classical parts S = 2 2m (∇R)2 + R2 (∇α)2 2m + V + ∂α ∂t

  • d3x dt

(12) Structure of Quantum Principle: S = 2( quantum kinetic ) + R2( classical Hamilton-Jakobi ) Path integral, tunneling: S = α − i ln R

Biro, Ván QGR in Madelung Variables 9 / 39

slide-11
SLIDE 11

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Madelung fluid density

Canonical momenta from S =

  • L d3xdt:

ΠR = ∂L ∂∇R = 2 m ∇R, Πα = ∂L ∂∇α = R2 m ∇α, PR = ∂L ∂ ∂R

∂t

= 0, Pα = ∂L ∂ ∂α

∂t

= R2 (13) Continuity eq: ∂Pα ∂t + ∇Πα = 0 fluid density ρ = Pα = R2 .

Biro, Ván QGR in Madelung Variables 10 / 39

slide-12
SLIDE 12

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Madelung current

The ”classical” momentum defines a velocity as P = m v The continuity equation reads as ∂ρ ∂t + ∇(ρ v) = 0. (14)

Biro, Ván QGR in Madelung Variables 11 / 39

slide-13
SLIDE 13

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Bohm potential

The quantum correction to the energy can be expressed as − 2 2m ∇2R R = − 2 2m ∇2ρ 2ρ − (∇ρ)2 4ρ2

  • (15)

Biro, Ván QGR in Madelung Variables 12 / 39

slide-14
SLIDE 14

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid

Fisher entropy

The action expressed by ρ becomes S =

P2 2m + V − E

  • ρ +
  • 2i

∂ρ ∂t + 2 2m (∇ρ)2 4ρ

d3x dt (16) The last term in the quantum part looks like Fisher entropy.

(The other is a total time derivative, can safely be neglected.)

Biro, Ván QGR in Madelung Variables 13 / 39

slide-15
SLIDE 15

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Outline

1

Nonrelativistic: Madelung fluid and Fisher entropy

2

  • Spec. Rel.: off mass-shell non-plane-wave

Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

3

  • Gen. Rel.: energy-momentum with quantum binding

4

Consequences in Approximate Numbers

Biro, Ván QGR in Madelung Variables 14 / 39

slide-16
SLIDE 16

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Quantum Lagrangian

Lagrange density L = 1 2∂iψ∗ ∂iψ − 1 2 mc

  • 2

ψ∗ ψ. (17) Action and other conventions S =

  • L d4x

(18) with dxi = (cdt, d r). Physical units [L] = energy density / c = [mc/L3].

Biro, Ván QGR in Madelung Variables 14 / 39

slide-17
SLIDE 17

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Madelung ansatz

Complex scalar field (related to wave function) ψ =

  • √mc R eiα/

(19) Here α is only the (real) classical action. Units of R from mc

  • 2

ψ∗ψ = mcR2 (20) is part of L, it follows that R2 is number density . Compare this with Maupertuis action for a classical mass point: −1 2 mc2R2d3x

  • dt = −
  • mc2dτ

(21) It follows a normalization

  • R2 d3x = 2.

Biro, Ván QGR in Madelung Variables 15 / 39

slide-18
SLIDE 18

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Derivative

First derivative of ψ: ∂iψ = ∂iR R + i ∂iα

  • ψ

(22) The derivative of the classical action is a classical momentum: Pi = ∂iα, ui = Pi/(mc). (23)

Biro, Ván QGR in Madelung Variables 16 / 39

slide-19
SLIDE 19

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Quantum Action

S = 2 2mc ∂iR ∂iR + R2 2

  • ∂iα ∂iα − (mc)2

d4x (24) Rewritten as a sum of a classical and a quantum part: S = 2 2mc ∂iR ∂iR + R2 2mc

  • Pi Pi − (mc)2

d4x (25)

Biro, Ván QGR in Madelung Variables 17 / 39

slide-20
SLIDE 20

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

U(1) Noether charge

Jk = i 2

  • ψ∂kψ∗ − ψ∗∂kψ
  • =

1 mc R2Pk = R2uk (26) is a number density 4-current R2uk = ρuk. Conserved by variation of S wrsp. α: δS δα = −∂k 1 mc R2∂kα

  • = 0

(27) Facit: δS δα = δSquantum δSclassical = −∂kJk = −∂k(ρuk) = 0. (28)

Biro, Ván QGR in Madelung Variables 18 / 39

slide-21
SLIDE 21

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Off Mass Shell

Variation against R delivers δS δR = − 2 mc R + R mc

  • PiPi − (mc)2

= 0. (29) Off-mass shell dispersion relation for the classical 4-momentum

PiPi − (mc)2 = 2 R

R

(30) Metric view: gijuiuj = 1 + mc 2 R R (31) Note: this is a Compton wavelength scaled, locally Lorentzian spacetime metric.

Biro, Ván QGR in Madelung Variables 19 / 39

slide-22
SLIDE 22

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Energy-Momentum Tensor I

Derivation I: using ψ and ψ∗. Πk = δL δ∂kψ = 1 2∂kψ∗ (32) Tij = Πi∂jψ + Π∗

i ∂jψ∗ − gijL.

(33) in terms of R and α: Tij = mcR2wij + 2 mc Uij, wij = uiuj − 1 2gij(ukuk − 1), Uij = ∂iR ∂jR − 1 2gij∂kR ∂kR. (34)

Biro, Ván QGR in Madelung Variables 20 / 39

slide-23
SLIDE 23

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Energy-Momentum Tensor I

Using the off-mass-shell relation leads to: Tij = mcR2uiuj + 2 mc

  • ∂iR ∂jR − 1

2gij

  • ∂kR∂kR + RR
  • (35)

Here the O(2) part is the quantum contribution, the rest is just dust .

Biro, Ván QGR in Madelung Variables 21 / 39

slide-24
SLIDE 24

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Energy-Momentum Tensor II

Take the derivative of the off-mass-shell eqation (31): R2 2 ∂i

  • ukuk − 1 −

2 (mc)2 R R

  • = 0

(36) Use Compton wavelength LC = /mc and expand: R2uk∂iuk − 1 2L2

CR2∂i

R R

  • = 0

(37)

Biro, Ván QGR in Madelung Variables 22 / 39

slide-25
SLIDE 25

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Energy-Momentum Tensor II

The Madelung fluid is irrotational: ∂iuk = 1 mc ∂i∂kα = 1 mc ∂k∂iα = ∂kui (38) Therefore R2uk∂k(ui) = ∂k(R2ukui) − ui∂k(R2uk) (39) and due to continuity the last term vanishes.

Biro, Ván QGR in Madelung Variables 23 / 39

slide-26
SLIDE 26

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Energy-Momentum Tensor II

By these manipulations we obtain ∂k

  • R2ukui
  • = 1

2L2

CR2∂i

R R

  • (40)

Further use of the Leibniz rule leads to R2∂i R R

  • = R∂iR − ∂iR R

= ∂k (R∂k∂iR − ∂kR ∂iR) (41)

Biro, Ván QGR in Madelung Variables 24 / 39

slide-27
SLIDE 27

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Energy-Momentum Tensor II

This reveals a vanishing divergence of the Bohm-Takabayashi tensor Tij = mcR2uiuj − 2 2mc

  • R∂i∂jR − ∂iR ∂jR
  • (42)

It differs from the Klein-Gordon one (35) by ∆ij = Tij−Tij = 2 2mc

  • ∂iR∂jR + R∂i∂jR − gij(∂kR∂kR + RR)
  • (43)

Biro, Ván QGR in Madelung Variables 25 / 39

slide-28
SLIDE 28

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Difference

We note that

  • gij − ∂i∂j

R2 2 = gij(∂kR∂kR+RR)−∂iR∂jR−R∂i∂jR (44) One realizes that ∆ij = 2 4mc

  • ∂i∂j − gij
  • R2

(45) has a vanishing divergence.

Biro, Ván QGR in Madelung Variables 26 / 39

slide-29
SLIDE 29

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Full Quantum Energy-Momentum Tensor III

It can be derived in two ways: from Klein-Gordon Lagrangian and from Madelung fluid hydrodynamics. They differ in a tensor part with vanishing divergence. The general tensor contains parameter µ multiplying the divergenceless part: Tij = mcR2uiuj + Uij + µ∆ij (46) with the general Bohm-Takabayashi term Uij = 2 2mc

  • ∂iR∂jR − R∂i∂jR
  • .

(47)

Biro, Ván QGR in Madelung Variables 27 / 39

slide-30
SLIDE 30

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Trace of Energy-Momentum Tensor

Ti

i =

  • 1 + L2

C

1 − 3µ 4

  • (mcR2)

(48) For µ = 0 original Bohm potential. For µ = 1/3 only classical contribution to trace. For µ = 1 the original Klein-Gordon case.

Biro, Ván QGR in Madelung Variables 28 / 39

slide-31
SLIDE 31

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid

Energy-Momentum Tensor in scaling variables

Use R = eσ/ √ V, then Ti

j = mc

V e2σuiuj + Ui

j

(49) with Ui

j =

2 2mcV e2σ 2µ∂iσ ∂jσ + (µ − 1)∂i∂jσ − µδi

j

  • 2∂kσ∂kσ + σ
  • (50)

Biro, Ván QGR in Madelung Variables 29 / 39

slide-32
SLIDE 32

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Einstein equation Conseqences of a conformal transformation Identification of a Bohm term

Outline

1

Nonrelativistic: Madelung fluid and Fisher entropy

2

  • Spec. Rel.: off mass-shell non-plane-wave

3

  • Gen. Rel.: energy-momentum with quantum binding

Einstein equation Conseqences of a conformal transformation Identification of a Bohm term (Re-)construction of a common action

4

Consequences in Approximate Numbers

Biro, Ván QGR in Madelung Variables 30 / 39

slide-33
SLIDE 33

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Einstein equation Conseqences of a conformal transformation Identification of a Bohm term

Einstein equation

Consider the downscaled µ-Madelung-Bohm-Takabayashi energy-momentum as a source of gravity: Gi

j − Λδi j = 8πG

c3 e−2σTi

j.

(51) Use the Schwarzschild length (half-radius) LS = Gm

c2 , to achieve

Gi

j − Λδi j

= 8πLS V uiuj + 4πLSL2

C

V Qi

j

Qi

j

= 2µ∂iσ∂jσ + (µ − 1)∂i∂jσ − µδi

j(2∂kσ∂kσ + σ).

Biro, Ván QGR in Madelung Variables 30 / 39

slide-34
SLIDE 34

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Einstein equation Conseqences of a conformal transformation Identification of a Bohm term

Conformal transformation by Ω(x)

From flat spacetime to curved one by scaling with Ω(x): ωa = Ω(x)dxa = esdxa (52) (induced) Christoffel symbol Γj

ik = ∂is δj k + ∂ks δj i − ∂js ηik

(53) (induced) Ricci tensor Rik = 2 (∂is ∂ks − ∂i∂ks) −

  • s + 2 ∂js ∂js
  • ηik

(54) (induced) Ricci scalar R = gikRik = −6e−2s s + ∂js ∂js

  • (55)

Biro, Ván QGR in Madelung Variables 31 / 39

slide-35
SLIDE 35

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Einstein equation Conseqences of a conformal transformation Identification of a Bohm term

Transformed Einstein equation

Assume modified Einstein equation: G

i j − Λδi j = 8πG

c3 e−2σ Ti

j.

(56) After conformal transformation one obtains (Delphenic)

  • Gi

j +

  • 2s + ∂ks∂ks − Λ
  • δi

j − 2∂i∂js + 2∂is ∂js

  • = 8πG

c3 e−2σ Ti

j

(57) We want to connect s and σ. As a source we insert our µ-Madelung Energy-Momentum-Tensor as obtained above.

Biro, Ván QGR in Madelung Variables 32 / 39

slide-36
SLIDE 36

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Einstein equation Conseqences of a conformal transformation Identification of a Bohm term

Term by term identification

Gi

j

= 8πLS V uiuj

  • 2s + ∂ks∂ks − Λ
  • =

−µ4πLSL2

C

V

  • 2∂kσ∂kσ + σ
  • −2∂i∂js

= (µ − 1)4πLSL2

C

V ∂i∂jσ 2∂is∂js = 2µ4πLSL2

C

V ∂iσ∂jσ

Biro, Ván QGR in Madelung Variables 33 / 39

slide-37
SLIDE 37

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers Einstein equation Conseqences of a conformal transformation Identification of a Bohm term

Resolution

Obviously s = σ, and furthermore Gi

j = 8πLS

V uiuj, (58) µ = 1/3 neither Klein-Gordon nor Madelung V = 4π 3 LSL2

C

(59) is a scaled Planck volume (by MP/m). Cosmolgical term Λ = 3

  • σ + ∂kσ∂kσ
  • = 3R

R (60)

Biro, Ván QGR in Madelung Variables 34 / 39

slide-38
SLIDE 38

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Outline

1

Nonrelativistic: Madelung fluid and Fisher entropy

2

  • Spec. Rel.: off mass-shell non-plane-wave

3

  • Gen. Rel.: energy-momentum with quantum binding

4

Consequences in Approximate Numbers

Biro, Ván QGR in Madelung Variables 35 / 39

slide-39
SLIDE 39

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Average Cosmological Term

For a mass m particle - bound in −α/r potential Λ = −3 ∇2R R = 3 a2 (61) with a = LC/α being the Bohr radius . Observed cosmological effect: Λ ≈ 10−60M2

P ≈ 3α2m2

− → source mass: m = 10−30 α √ 3 MP ≈ 100 α √ 3 meV. (62)

Biro, Ván QGR in Madelung Variables 35 / 39

slide-40
SLIDE 40

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Neutrinonium

Speculation: weak interaction coupled Majorana νν-neutrinonium. reduced mass m = mν/2, αweak ≈ 1/30 gives mν ≈ 2 100 30 √ 3 ≈ 2 √ 3 meV In this way neutrinos may be responsible for both dark matter and dark energy.

Biro, Ván QGR in Madelung Variables 36 / 39

slide-41
SLIDE 41

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Gravitonium

c = 1 units

Speculation: bound in Newton potential neutral GG-gravitonium . reduced mass m = mG/2, αNewton ≈ Gm2/ = (m/MP)2 Bohr radius aB = /mα = LP(MP/mG)3. Λ = 3 a2 = 3 4L2

P

mG MP 6 . (63) This explains a small value, L2

PΛ = 2.56 × 10−122 with

mG = 68 MeV

Biro, Ván QGR in Madelung Variables 37 / 39

slide-42
SLIDE 42

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Physical Objects on LC − LS

Characteristic mass m and size L compared to LC and LS.

Biro, Ván QGR in Madelung Variables 38 / 39

slide-43
SLIDE 43

Nonrelativistic: Madelung fluid and Fisher entropy

  • Spec. Rel.: off mass-shell non-plane-wave
  • Gen. Rel.: energy-momentum with quantum binding

Consequences in Approximate Numbers

Summary

QM is an off-mass-shell effect (2R/R). Via QGR assumption it relates to a cosmological term (Λ = 3R/R). Natural quantization volume: V = 4π

3 LSL2 C.

µ = 1/3 neither Bohm-Takabayashi nor Klein-Gordon, but has classical trace. Outlook

Madelung ansatz for Pauli and Dirac. Madelung as a conformal transformation Nonabelian external fields. Dilaton type actions may deliver our assumption?

Biro, Ván QGR in Madelung Variables 39 / 39