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The gravity-related decoherence master equation from hybrid dynamics Lajos Disi KFKI Research Institute for Particle and Nuclear Physics H-1525 Budapest 114, POB 49, Hungary Hybrid dynamics Poisson, Dirac, Aleksandrov brackets Blurring

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The gravity-related decoherence master equation from hybrid dynamics

Lajos Diósi KFKI Research Institute for Particle and Nuclear Physics H-1525 Budapest 114, POB 49, Hungary

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Hybrid dynamics Poisson, Dirac, Aleksandrov brackets Blurring Dirac+Poisson Hybrid master equation Gravity-related decoherence Hybrid master equation for matter plus gravity Reduced quantum master equation The gravity-related decoherence matrix Appendix: Full expansion of the hybrid master equation References

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Naive hybrid: Dirac+Poisson

Liouville equation for ρ(q, p) with Hamilton func. H(q, p): ˙ ρ = {H, ρ}P ≡

∂H ∂qn ∂ρ ∂pn − ∂ρ ∂qn ∂H ∂pn

von Neumann equation for ˆ

ρ with Hamiltonian ˆ H: ˙ ˆ ρ = − i [ˆ H, ˆ ρ] ≡ − i

H ˆ ρ − ˆ ρˆ H

Naiv hybrid equation for ˆ ρ(q, p) with ˆ H(q, p)= ˆ HQ+HC(q, p)+ˆ HQC(q, p): ˙ ˆ ρ = − i [ˆ H, ˆ ρ] + Herm{ˆ H, ˆ ρ}P

(Aleksandrov1981)

It may not preserve ˆ ρ(q, p) ≥ 0.(Boucher,Traschen1988)

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Blurring Dirac+Poisson

In total Hamiltonian ˆ H = ˆ HQ + HC + ˆ HQC, the interacting part: ˆ HQC =

ˆ f rφr Blurring the ’interacting’ currents by classical noises δf , δφ: ˆ Hnoise

=

f r + δf r(t)

φr + δφr(t)
δf r(t′)δf s(t)
noise = Drs

Q δ(t′ − t)

δφr(t′)δφs(t)
noise = Drs

C δ(t′ − t)

Total Hamilton becomes noisy: ˆ Hnoise = ˆ HQ + HC + ˆ Hnoise

QC .

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Hybrid master equation

˙ ˆ ρ = − i [ˆ H, ˆ ρ] + Herm{ˆ H, ˆ ρ}P Recall, this naive hybrid dynamics may not preserve ˆ ρ(q, p) ≥ 0. We replace ˆ H by ˆ Hnoise and take the average of the naive dynamics. ˆ Hnoise = ˆ H +

(ˆ f rδφr + φrδf r) Noise terms add −[ˆ f , [ˆ f , ˆ ρ]] and {φ, {φ, ˆ ρ}P}P to the naive hybrid eq.: ˙ ˆ ρ = − i [ˆ H, ˆ ρ] + Herm{ˆ H, ˆ ρ}P − − 1 22

Drs

C [ˆ

f r, [ˆ f s, ˆ ρ]] + 1 2

Drs

Q {φr, {φs, ˆ

ρ}P}P This hybrid master eq. preserves ˆ ρ(q, p) ≥ 0 if DQDC ≥ 2/4.(D.1995).

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Hybrid master equation for matter plus gravity

Quantized matter Hamiltonian ˆ HQ, its mass density ˆ f (r), coupled to weak classical gravitational field φ ≡ 1

2c2(g00 − 1). Conjugate variables

qn ⇒ φ(r) and pn → ξ(r). Hybrid state: ˆ ρ(φ, ξ). The total Hamiltonian: ˆ H(φ, ξ) = ˆ HQ + HC(φ, ξ) + ˆ HQC(φ) HC(φ, ξ) =

r
2πGc2 ξ2 + |∇φ|2

8πG

;

ˆ HQC(φ) =

ˆ f (r)φ(r) As ˆ f r ⇒ ˆ f (r) and φr ⇒ φ(r), and

r ⇒

r, the hybrid master eq. reads:

˙ ˆ ρ = − i [ˆ H, ˆ ρ] + Herm{ˆ H, ˆ ρ}P − − 1 22

DC(r, s)[ˆ f (r), [ˆ f (s), ˆ ρ]] + 1 2

DQ(r, s){φ(r), {φ(s), ˆ ρ}P}P Recall DQDC ≥ 2/4. We concentrate on the reduced dynamics of ˆ ρQ =

ρ(φ, ξ)DφDξ. Most terms on the r.h.s. cancel but we are left with:

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Reduced quantum master equation

˙ ˆ ρQ = − i [ˆ HQ, ˆ ρQ]− i

φ(r)[ˆ f (r), ˆ ρ(φ)]Dφ− 1 22

DC(r, s)[ˆ f (r), [ˆ f (s), ˆ ρQ]] where ˆ ρ(φ) =

ρ(φ, ξ)Dξ. Suppose the post-mean-field Ansatz:

φ(r)ˆ

ρ(φ)Dφ = Hermˆ φ(r)ˆ ρQ. ˆ φ(r) =: −G

ˆ f (s) |r − s| We obtain the following result: ˙ ˆ ρQ = − i [ˆ HQ + ˆ HG, ˆ ρQ] − 1 22

DC(r, s)[ˆ f (r), [ˆ f (s), ˆ ρQ] where ˆ HG is the well-known Newtonian potential energy: ˆ HG = −G 2

ˆ f (r)ˆ f (s) |r − s| .

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The gravity-related decoherence matrix

We determine the decoherence matrix DC(r, s). Recall:

δf (r ′, t′)δf (r, t)
noise = DQ(r ′, r)δ(t′ − t)
δφ(r ′, t′)δφ(r, t)
noise = DC(r ′, r)δ(t′ − t)

The mean-fields satisfy: ∆φ(r) = 4πGˆ f (r). If we requested the same equation for the fluctuations, the above two correlations would lead to: ∆∆′DC(r, r ′) = (4πG)2DQ(r, r ′) With minimum blurring DCDQ = 2/4, the unique translation invariant solution:(cf .D.,Lukacs1987) DC(r, r ′) = (G/2)|r − r ′|−1 The reduced master eq. of quantized matter follows:(D.1987;cf .Penrose1994) ˙ ˆ ρQ = − i [ˆ HQ + ˆ HG, ˆ ρQ] − 1 4

r
r ′

G/ |r − r ′|[ˆ f (r), [ˆ f (r ′), ˆ ρQ].

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Appendix: Full expansion of the hybrid master equation

˙ ˆ ρ = − i [ˆ HQ, ˆ ρ] − i

φ(r)[ˆ f (r), ˆ ρ] −

r
4πGc2ξ(r) δˆ

ρ δφ(r) + 1 4πG ∆φ(r) δˆ ρ δξ(r)

+ Herm
r

ˆ f (r) δˆ ρ δξ(r) − 1 22

r
r ′ DC(r, r ′)[ˆ

f (r), [ˆ f (r ′), ˆ ρ] ] + 1 2

r
r ′ DQ(r, r ′)

δ2ˆ ρ δξ(r)δξ(r ′).

The status of the post-mean-field Ansatz

φ(r)ˆ

ρ(φ)Dφ = Hermˆ φ(r)ˆ ρQ , ˆ φ(r) =: −G

ˆ f (s) |r − s| is yet to be clarified. It may follow from the c → ∞ non-relativistic limit, or may at least be consistent with it. But it may, in the worst case, contradict to the hybrid master equation.

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References

I.V. Aleksandrov, Z.Naturforsch. 36A, 902 (1981).

W. Boucher and J. Traschen, Phys.Rev. D37, 3522 (1988).

L.Diósi and B.Lukács, Annln.Phys. 44, 488 (1987). L.Diósi, Phys.Lett. 120A, 377 (1987); Phys.Rev. A40, 1165 (1989); Braz.J.Phys. 35 260 (2005).

R. Penrose, Shadows of the mind (Oxford University Press, 1994);