The gravity-related decoherence master equation from hybrid dynamics - PowerPoint PPT Presentation

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

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

Naive hybrid: Dirac+Poisson Liouville equation for ρ ( q , p ) with Hamilton func. H ( q , p ) : � ∂ H ∂ρ − ∂ρ ∂ H � � ρ = { H , ρ } P ≡ ˙ ∂ q n ∂ p n ∂ q n ∂ p n n ρ with Hamiltonian ˆ von Neumann equation for ˆ H : ρ = − i ρ ] ≡ − i ˙ � [ˆ � ˆ ρ ˆ � ˆ H , ˆ H ˆ ρ − ˆ H . � ρ ( q , p ) with ˆ H ( q , p )= ˆ H Q + H C ( q , p )+ˆ Naiv hybrid equation for ˆ H QC ( q , p ) : ρ = − i ˙ � [ˆ ρ ] + Herm { ˆ ( Aleksandrov 1981 ) ˆ H , ˆ H , ˆ ρ } P ρ ( q , p ) ≥ 0. ( Boucher , Traschen 1988 ) It may not preserve ˆ

Blurring Dirac+Poisson In total Hamiltonian ˆ H = ˆ H Q + H C + ˆ H QC , the interacting part: ˆ � ˆ f r φ r H QC = r Blurring the ’interacting’ currents by classical noises δ f , δφ : f r + δ f r ( t ) φ r + δφ r ( t ) � �� � ˆ � ˆ H noise = QC r � � Q δ ( t ′ − t ) δ f r ( t ′ ) δ f s ( t ) noise = D rs � � C δ ( t ′ − t ) δφ r ( t ′ ) δφ s ( t ) noise = D rs H noise = ˆ Total Hamilton becomes noisy: ˆ H Q + H C + ˆ H noise QC .

Hybrid master equation ρ = − i ˙ � [ˆ ρ ] + Herm { ˆ ˆ H , ˆ H , ˆ ρ } P Recall, this naive hybrid dynamics may not preserve ˆ ρ ( q , p ) ≥ 0. We H noise and take the average of the naive dynamics. replace ˆ H by ˆ H noise = ˆ f r δφ r + φ r δ f r ) ˆ � (ˆ H + r Noise terms add − [ˆ f , [ˆ f , ˆ ρ ]] and { φ, { φ, ˆ ρ } P } P to the naive hybrid eq.: ρ = − i ˙ � [ˆ ρ ] + Herm { ˆ ˆ H , ˆ H , ˆ ρ } P − − 1 ρ ]] + 1 � C [ˆ f r , [ˆ � D rs f s , ˆ D rs Q { φ r , { φ s , ˆ ρ } P } P 2 � 2 2 r , s r , s ρ ( q , p ) ≥ 0 if D Q D C ≥ � 2 / 4. ( D . 1995 ) . This hybrid master eq. preserves ˆ

Hybrid master equation for matter plus gravity Quantized matter Hamiltonian ˆ H Q , its mass density ˆ f ( r ) , coupled to weak classical gravitational field φ ≡ 1 2 c 2 ( g 00 − 1 ) . Conjugate variables q n ⇒ φ ( r ) and p n → ξ ( r ) . Hybrid state: ˆ ρ ( φ, ξ ) . The total Hamiltonian: H ( φ, ξ ) = ˆ ˆ H Q + H C ( φ, ξ ) + ˆ H QC ( φ ) 2 π Gc 2 ξ 2 + |∇ φ | 2 � � � � ˆ ˆ H C ( φ, ξ ) = ; H QC ( φ ) = f ( r ) φ ( r ) 8 π G r r f r ⇒ ˆ f ( r ) and φ r ⇒ φ ( r ) , and � As ˆ � r ⇒ r , the hybrid master eq. reads: ρ = − i ˙ � [ˆ ρ ] + Herm { ˆ ˆ H , ˆ H , ˆ ρ } P − − 1 ρ ]] + 1 � � D C ( r , s )[ˆ f ( r ) , [ˆ f ( s ) , ˆ D Q ( r , s ) { φ ( r ) , { φ ( s ) , ˆ ρ } P } P 2 � 2 2 r , s r , s Recall D Q D C ≥ � 2 / 4. � We concentrate on the reduced dynamics of ˆ ρ Q = ρ ( φ, ξ ) D φ D ξ . Most ˆ terms on the r.h.s. cancel but we are left with:

Reduced quantum master equation ρ Q = − i ρ Q ] − i � ρ ( φ )] D φ − 1 � ˙ � [ˆ φ ( r )[ˆ D C ( r , s )[ˆ f ( r ) , [ˆ ˆ H Q , ˆ f ( r ) , ˆ f ( s ) , ˆ ρ Q ]] � 2 � 2 r r , s � where ˆ ρ ( φ ) = ρ ( φ, ξ ) D ξ . Suppose the post-mean-field Ansatz: ˆ � ρ ( φ ) D φ = Herm ˆ φ ( r )ˆ φ ( r )ˆ ρ Q . ˆ � f ( s ) ˆ φ ( r ) =: − G | r − s | s We obtain the following result: ρ Q = − i 1 � � ˙ � [ˆ H Q + ˆ D C ( r , s )[ˆ f ( r ) , [ˆ ˆ H G , ˆ ρ Q ] − f ( s ) , ˆ ρ Q ] 2 � 2 r s where ˆ H G is the well-known Newtonian potential energy: ˆ f ( r )ˆ H G = − G � � f ( s ) ˆ | r − s | . 2 r s

The gravity-related decoherence matrix We determine the decoherence matrix D C ( r , s ) . Recall: � � noise = D Q ( r ′ , r ) δ ( t ′ − t ) δ f ( r ′ , t ′ ) δ f ( r , t ) noise = D C ( r ′ , r ) δ ( t ′ − t ) � � δφ ( r ′ , t ′ ) δφ ( r , t ) The mean-fields satisfy: ∆ � φ ( r ) � = 4 π G � ˆ f ( r ) � . If we requested the same equation for the fluctuations, the above two correlations would lead to: ∆∆ ′ D C ( r , r ′ ) = ( 4 π G ) 2 D Q ( r , r ′ ) With minimum blurring D C D Q = � 2 / 4, the unique translation invariant solution: ( cf . D ., Lukacs 1987 ) D C ( r , r ′ ) = ( G � / 2 ) | r − r ′ | − 1 The reduced master eq. of quantized matter follows: ( D . 1987 ; cf . Penrose 1994 ) ρ Q ] − 1 G / � ρ Q = − i � � | r − r ′ | [ˆ f ( r ) , [ˆ ˙ � [ˆ H Q + ˆ ˆ H G , ˆ f ( r ′ ) , ˆ ρ Q ] . 4 r r ′

Appendix: Full expansion of the hybrid master equation ρ = − i ρ ] − i � φ ( r )[ˆ ˙ � [ˆ ˆ H Q , ˆ f ( r ) , ˆ ρ ] � r � 4 π Gc 2 ξ ( r ) δ ˆ ρ 4 π G ∆ φ ( r ) δ ˆ 1 ρ � f ( r ) δ ˆ ρ � � ˆ − δφ ( r ) + + Herm δξ ( r ) δξ ( r ) r r − 1 ρ ] ] + 1 δ 2 ˆ ρ � � � � r ′ D C ( r , r ′ )[ˆ f ( r ) , [ˆ f ( r ′ ) , ˆ r ′ D Q ( r , r ′ ) δξ ( r ) δξ ( r ′ ) . 2 � 2 2 r r The status of the post-mean-field Ansatz � ρ ( φ ) D φ = Herm ˆ φ ( r )ˆ φ ( r )ˆ ρ Q , ˆ � f ( s ) ˆ φ ( r ) =: − G | r − s | 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.

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); Gen.Rel.Grav. 28 , 581 (1996) 581.