On canonical coupling of classical geometry to quantum matter 1. - PDF document

On canonical coupling of classical geometry to quantum matter 1. Introduction R ab − 1 2 g ab R = 8 πG c 4 T ab 1.1 classical T , classical R — fine. 1.2 quantum T , quantum R — nobody knows. 1.3 quantum T , classical R — try it canonically! 2. Poisson, Dirac, and Alexandrov brackets 3. Coarse-grained Alexandrov equation 4. Application to gravity 5. Fundamental uncertainty of geometry, decoherence 6. Conclusion 6.1 Semiclassical evolution must be coarse-grained 6.2 Classical part, too, gets noisy. 6.3 Reasonable decoherence obtained. 6.4 Covariant coarse-graining? 1

2.1 Classical: Poisson bracket, Liouville equation of motion. � ∂A ∂B − ∂A ∂B � � { A, B } P ≡ ∂q n ∂q n ∂p n ∂p n n ρ = { H, ρ } P ˙ � ρ ( q, p ) dqdp = 1. where ρ ( q, p ) ≥ 0, 2.2 Quantum: Dirac bracket, Schrodinger (von Neumann) equation. { A, B } D ≡ − i h [ A, B ] . ¯ ρ = { H, ρ } D ˙ where ρ ≥ 0, trρ = 1. 2.3 Semiclassical: Alexandrov (1981) bracket and equation. { A, B } A = { A, B } P + 1 2 { A, B } D − 1 2 { B, A } D ρ = { H, ρ } A ˙ � ρ ( q, p ) dqdp = 1. where ρ ( q, p ) ≥ 0, tr 2

3. Coarse-grained Alexandrov equation. H = H ( q 1 , q 2 , p 1 , p 2 ) = H 1 ( q 1 , p 1 ) + H 2 ( q 2 , p 2 ) + H I ( q 1 , q 2 , p 1 , p 2 ) where ( q 1 , p 1 ) are quantum, ( q 2 , p 2 ) are classical. � J α 1 ( q 1 , p 1 ) J α ( J α 1 , J α H I ( q 1 , p 1 , q 2 , p 2 ) = 2 ( q 2 , p 2 ) , 2 � = 1) . α Noisy interaction: � �� � H noise J α 1 ( q 1 , p 1 ) + δJ α J α 2 ( q 2 , p 2 ) + δJ α ( q 1 , p 1 , q 2 , p 2 ) = 1 ( t ) 2 ( t ) I where δJ 1 , δJ 2 are classical noises. H noise = H 1 + H 2 + H noise I � � { H noise , ρ } A ρ = ˙ noise . Choose white-noices: noise = 1 n δ ( t ′ − t ) , � � δJ α n ( t ′ ) δJ β 2Λ αβ n ( t ) n = 1 , 2 . ρ = − i h [ H, ρ ] + 1 2 { H, ρ } P − 1 ˙ 2 { ρ, H } P ¯ − 1 1 , ρ ]] + 1 Λ αβ 1 , [ J β Λ αβ 2 , { J β � 2 [ J α � 1 { J α 2 , ρ } P } P 4 4 α,β α,β Lindblad’s condition (1976) for positivity: Λ 1 Λ 2 = I . 3

4. Application to gravity. ( q 1 , p 1 ) ≡ ( q, p ) quantized non − relativistic matter ( q 2 , p 2 ) ≡ ( φ, π ) weak classical gravitational field φ ≡ 1 2 c 2 ( g 00 − 1) classical Newtonian potential f = T 00 /c 2 operator of mass distribution � 1 1 � � c 2 π 2 + |∇ φ | 2 � H ( q, p, φ, π ) = H m ( q, p ) + + r φ ( r ) f ( r ) 8 πG r Coarse-grained Alexandrov equation: ρ = − i h [ H m , ρ ] − i � ˙ r φ ( r )[ f ( r ) , ρ ] ¯ ¯ h � 1 − 1 c 2 π ( r ) δρ δφ ( r ) + ∆ φ ( r ) δρ + 1 δρ � � � � � f ( r ) , 4 πG δπ ( r ) 2 δπ ( r ) + r r δ 2 ρ − 1 r ′ λ ( r, r ′ )[ f ( r ) , [ f ( r ′ ) , ρ ] ] + 1 � � � � r ′ λ − 1 ( r, r ′ ) δπ ( r ) δπ ( r ′ ) . 4 4 r r � � ρ ( φ, π ) D φ D π in Newtonian approximation. Reduced dynamics of ρ m = Drop term 1 /c 2 and assume ρ m determines Newton potential φ : G/ 2 � � � � � | r − r ′ | [ f + ( r ′ ) + f − ( r ′ )] ρ ( φ, π ) D π = δ φ ( r ) + ρ m . r ′ r Subscripts + and − assure ρ m ’s hermiticity. Integrate both sides over φ, π ! ρ m = − i h [ H m + H g , ρ m ] − 1 � � ˙ r ′ λ ( r, r ′ )[ f ( r ) , [ f ( r ′ ) , ρ m ] ¯ 4 r H g = − G f ( r ) f ( r ′ ) � � | r − r ′ | . 2 r r ′ 4

5. Fundamental uncertainty of geometry, decoherence noise = 1 2 λ − 1 ( r ′ , r ) δ ( t ′ − t ) , � � δf ( r ′ , t ′ ) δf ( r, t ) h 2 noise = ¯ 2 λ ( r ′ , r ) δ ( t ′ − t ) . � � δφ ( r ′ , t ′ ) δφ ( r, t ) In Newtonian approximation: ∆ � φ ( r ) � = 4 πG � f ( r ) � . ∆∆ ′ λ ( r, r ′ ) = (4 πG ) 2 λ − 1 ( r, r ′ ) Di´ osi and Luk´ acs (1987): h ) | r − r ′ | − 1 λ ( r, r ′ ) = ( G/ ¯ Di´ osi (1987): ρ m = − i h [ H m + H g , ρ m ] − 1 G/ ¯ h � � | r − r ′ | [ f ( r ) , [ f ( r ′ ) , ρ m ] . ˙ ¯ 4 r r ′ Ghirardi et al. (1990): cutoff 10 − 5 cm . Decoherence at characteristic time τ = ¯ h/ ∆ U G , where ∆ U G : quantum spread of the Newtonian self-energy. Pearle and Squires (1995): Verification in proton-decay experiments(!?) 5

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