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 . Ván 1 T.S. Biró, P 1 Heavy 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
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
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 Nonrelativistic: Madelung fluid and Fisher entropy 1 Spec. Rel.: off mass-shell non-plane-wave 2 Gen. Rel.: energy-momentum with quantum binding 3 Consequences in Approximate Numbers 4 Biro, Ván QGR in Madelung Variables 3 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Outline Nonrelativistic: Madelung fluid and Fisher entropy 1 Schrödinger eq. with Madelung var.-s Schrödinger eq. from action principle The Madelung fluid Spec. Rel.: off mass-shell non-plane-wave 2 Gen. Rel.: energy-momentum with quantum binding 3 Consequences in Approximate Numbers 4 Biro, Ván QGR in Madelung Variables 4 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Schrödinger with Madelung − � 2 2 m ∇ 2 ϕ + V ( x ) ϕ = i � ∂ ∂ t ϕ (1) Ansatz i � α ϕ = R e Classical action and momentum ∂α ∂ t = − E , ∇ α = P (2) Biro, Ván QGR in Madelung Variables 4 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Logarithmic Derivatives � 1 � 1 ∂ ∂ R ∂ t − i � R ∇ R + i � ∂ t ϕ = � E ϕ, ∇ ϕ = � P ϕ (3) R Laplacian � � 2 � � ∇ R R + i � � ∇ R R + i ∇ 2 ϕ = ∇ � P + � P ϕ (4) Biro, Ván QGR in Madelung Variables 5 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Real and Imaginary Part � � � 2 E = V − � 2 − P 2 � ∇ R ∇∇ R R + (5) � 2 2 m R ∂ t = − � 2 i � ∂ R i � ∇ P + 2 � R P · ∇ R (6) R 2 m � Biro, Ván QGR in Madelung Variables 6 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Interpretation Energy = Classical + Quantum contributions P 2 � 2 ∇ 2 R 2 m + V − E = (7) 2 m R Mass density continuity m ∂ R 2 � R 2 P � + ∇ = 0 (8) ∂ t Biro, Ván QGR in Madelung Variables 7 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Action Principle Variational Principle behind the Schrödinger equation ∂ t + |∇ S | 2 � � ∂ S � | ϕ | 2 d 3 x dt S = + V (9) 2 m i � S ”Boltzmannian” ansatz: ϕ = e Using this ansatz: ∂ t + � 2 � � � i ϕ ∗ ∂ϕ 2 m ∇ ϕ ∗ · ∇ ϕ + V ϕ ∗ ϕ d 3 x dt S = (10) Variation against ϕ ∗ delivers ∂ t − � 2 δ S ∂ϕ δϕ ∗ = � 2 m ∇ 2 ϕ + V ϕ = 0 (11) i Biro, Ván QGR in Madelung Variables 8 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Action with Madelung Variables Split to Quantum + Classical parts � � � 2 � ( ∇ α ) 2 + V + ∂α �� 2 m ( ∇ R ) 2 + R 2 d 3 x dt S = (12) 2 m ∂ t Structure of Quantum Principle: S = � 2 ( quantum kinetic ) + R 2 ( classical Hamilton-Jakobi ) Path integral, tunneling: S = α − i � ln R Biro, Ván QGR in Madelung Variables 9 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Madelung fluid density L d 3 xdt : � Canonical momenta from S = ∂ ∇ R = � 2 ∂ ∇ α = R 2 ∂ L Π α = ∂ L Π R = m ∇ R , m ∇ α, P R = ∂ L P α = ∂ L = R 2 = 0 , (13) ∂ ∂ R ∂ ∂α ∂ t ∂ t Continuity eq: ∂ P α + ∇ Π α = 0 ∂ t fluid density ρ = P α = R 2 . Biro, Ván QGR in Madelung Variables 10 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers 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
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Bohm potential The quantum correction to the energy can be expressed as − � 2 ∇ 2 R = − � 2 � ∇ 2 ρ 2 ρ − ( ∇ ρ ) 2 � (15) 4 ρ 2 2 m R 2 m Biro, Ván QGR in Madelung Variables 12 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Schrödinger eq. with Madelung var.-s Spec. Rel.: off mass-shell non-plane-wave Schrödinger eq. from action principle Gen. Rel.: energy-momentum with quantum binding The Madelung fluid Consequences in Approximate Numbers Fisher entropy The action expressed by ρ becomes � � � � � ( ∇ ρ ) 2 � � P 2 ∂ρ ∂ t + � 2 � d 3 x dt S = 2 m + V − E ρ + 2 i 2 m 4 ρ (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
Nonrelativistic: Madelung fluid and Fisher entropy Klein-Gordon Lagrangian Spec. Rel.: off mass-shell non-plane-wave Action principle with Madelung variables Gen. Rel.: energy-momentum with quantum binding Noether currents: two energy conservations Consequences in Approximate Numbers Relativistic Madelung fluid Outline Nonrelativistic: Madelung fluid and Fisher entropy 1 Spec. Rel.: off mass-shell non-plane-wave 2 Klein-Gordon Lagrangian Action principle with Madelung variables Noether currents: two energy conservations Relativistic Madelung fluid Gen. Rel.: energy-momentum with quantum binding 3 Consequences in Approximate Numbers 4 Biro, Ván QGR in Madelung Variables 14 / 39
Nonrelativistic: Madelung fluid and Fisher entropy Klein-Gordon Lagrangian Spec. Rel.: off mass-shell non-plane-wave Action principle with Madelung variables Gen. Rel.: energy-momentum with quantum binding Noether currents: two energy conservations Consequences in Approximate Numbers Relativistic Madelung fluid Quantum Lagrangian Lagrange density � 2 L = 1 2 ∂ i ψ ∗ ∂ i ψ − 1 � mc ψ ∗ ψ. (17) 2 � Action and other conventions � L d 4 x S = (18) with dx i = ( cdt , d � r ) . Physical units [ L ] = energy density / c = [ mc / L 3 ] . Biro, Ván QGR in Madelung Variables 14 / 39
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