Causal Effective Field Eqns from the Schwinger-Keldysh Formalism. Richard Woodard University of Florida

Transcription

1 Causal Effective Field Eqns from the Schwinger-Keldysh Formalism Richard Woodard University of Florida

2 Two Common Questions 1. Aren t QFT effective field eqns nonlocal? Yes! 2. Doesn t this introduce acausality? Not unless gravity is dynamical In-out eqns are acausal, but from in-out BC In-in eqns are causal

3 Linearized Examples µ [(-g) ½ g µν ν ϕ(x)] - (-g) ½ [ξr+m 2 ] ϕ(x) - d 4 x M 2 (x;x ) ϕ(x ) = 0 (-g) ½ e µ aγ a [ µ -½A µbc J bc ] ψ(x) (-g) ½ m ψ(x) - d 4 x [Σ](x;x ) ψ(x ) = 0 ν [(-g) ½ g νρ g µσ F ρσ (x)] - d 4 x [ µ Π ν ](x;x ) A ν (x ) = 0

4 Eg. M 2 (x;x ) for λϕ 3 Best in position space M 2 (x;x ) = d 4 p/(2π) 4 Exp[ip (x-x )] M 2 (p) No integrals at 1 loop! [-i M 2 (x;x )] 1 loop = ½ (-iλ) 2 [i (x;x )] 2

5 Full Nonlinear Eqns For in-out (in-in similar) Γ[ϕ] = S[ϕ] - Ν=2 1/N! d 4 x 1 ϕ(x 1 ) d 4 x N ϕ(x N ) Γ N (x 1,...,x N ) Γ N (x 1,,x N ) N-point 1PI function δγ/δϕ(x) = δs/δϕ(x) - Ν=1 1/N! d 4 x 1 ϕ(x 1 )..... d 4 x N ϕ(x N ) Γ N+1 (x,x 1,,x N ) = 0

6 What Does It Mean? ϕ(x) = <Φ + φ(x) Φ > φ(x) Quantum field operator ϕ(x) C-number soln of eff. field eqn with Pos. freq. parts of ϕ and f + agree at time t + Neg. freq. parts of ϕ and f - agree at time t - Φ ± > State centered on f ± at time t ± For in-out t + + and t - - For in-in t + = t - and often finite

7 What s Wrong with In-Out? Great for scattering in flat space! Exp[iΓ[ϕ] ij ϕ] generates S-matrix Cf. on-shell finiteness But silly for cosmology or evolution Initial singularity can t have t - - Don t know state for t + + No formal S-matrix (but Fermilab still works!) In-out ϕ(x) depends on future and isn t real

8 Introductory QFT Teaches

9 Now Multiply by Conjugate

10 Now Sum over States at t = t 2 1. Ψ Ψ Ψ = I 2. Ψ Ψ [ϕ (t 2 )] Ψ[ϕ + (t 2 )] = δ[ϕ + (t 2 ) - ϕ (t 2 )]

11 Schwinger-Keldysh Formalism 1. 2 fields ϕ ± line endpts have ± polarity 2. Exp(iS[ϕ + ] S[ϕ ]) Ints all + or all + Interactions same as usual - Interactions conjugated 3. T(B[φ]) B[ϕ + ] T-ordered ext. lines + 4. Anti-time-ordered external lines are 5. 4 Propagators: i ±± (x;x ) 6. In-out N-point func. 2 N S-K N-points 7. Φ[ϕ ± (t 1 )] surface ints at t = t 1 if not free

12 Eg. im 2 (x;x ) for λϕ 3 1 Loop In-Out: ½ (-iλ) 2 [i (x;x )] 2 1 Loop Schwinger-Keldysh: -im 2 ++(x;x ) = ½ (-iλ) 2 [i ++ (x;x )] 2 -im 2 +-(x;x ) = ½ (-i λ) (+iλ) [i +- (x;x )] 2 -im 2 -+(x;x ) = ½ (+iλ) (-iλ) [i -+ (x;x )] 2 -im 2 --(x;x ) = ½ (+iλ) 2 [i -- (x;x )] 2

13 S-K Effective Field Eqns Γ[ϕ +,ϕ ] = S[ϕ + ] S[ϕ ] -½ d 4 x d 4 x ±± ϕ ± (x)m 2 ±±(x;x )ϕ ± (x ) + O(ϕ 3 ) 0 = δγ[ϕ +,ϕ ]/δϕ + (x) (then set ϕ ± =ϕ) = δs[ϕ]/δϕ(x) - d 4 x [M 2 ++(x;x ) + M 2 +-(x;x )] ϕ(x ) + O(ϕ 2 ) NB: M 2 -+(x;y) = M 2 +-(y;x)

14 Get Props with Canonical Relation F. F. Int. ϕ + (x)ϕ + (x ) = <Ω 0 T[φ(x)φ(x )] Ω 0 > i ++ (x;x ) = i (x;x ) F. F. Int. ϕ + (x)ϕ (x ) = <Ω 0 φ(x )φ(x) Ω 0 > i +- (x;x ) = θ(t-t ) [i (x;x )] * + θ(t -t) i (x;x ) F. F. Int. ϕ (x)ϕ(x ) = <Ω 0 φ(x)φ(x ) Ω 0 > i -+ (x;x ) = θ(t-t ) i (x;x ) + θ(t -t) [i (x;x )] * F. F. Int. ϕ (x)ϕ (x ) = <Ω 0 A[φ(x)φ(x )] Ω 0 > i -- (x;x ) = [i (x;x )] *

15 Properties of [M 2 ++(x;x ) + M 2 +-(x;x )] 1 Loop λϕ 3 : [M M2 +- ] = -iλ 2 /2 {[i ++ (x;x )] 2 - [i + (x;x )] 2 } Fact 1: i ++ (x;x ) = θ(t-t ) i -+ (x;x ) + θ(t -t) i +-(x;x (x;x ) [M M 2 +-] = 0 for t > t Fact 2: i -+(x;x ) = [i +-(x;x )] * [M M 2 +-] = -λ 2 θ(t-t ) Im{[i +- (x;x )] 2 ]} Manifestly real (unlike in-out!)

16 Causality of [M 2 ++(x;x ) + M 2 +-(x;x )] [M M2 +-] = -λ 2 θ(t-t ) Im{[i +- (x;x )] 2 } Fact 3: i +- (x;x ) = <Ω 0 φ(x )φ(x) Ω 0 > = ½ <Ω 0 {φ(x ),φ(x)} Ω 0 > + ½ <Ω 0 [φ(x ),φ(x)] Ω 0 > Fact 4: [φ(x ),φ(x)] = 0 for spacelike sep. [M M 2 +-] contributes only for x µ on or within past light-cone of x µ Not even superluminal w/o derivative ints

17 Blurring the Light-cone The Physics: Metric operator g µν = ĝ µν + h µν sets light-cone h µν allows propagation not possible in ĝ µν The Math: Derivative ints (h h h) push inf. off light-cone Typical c/c G 2 R ρσµν R ρσµν Cf. Larry Ford s work

18 Conclusions 1. S-K field eqns nonlocal, but good initial value problem 2. No violation of causality without quantizing gravity 3. No superluminal propagation without derivative interactions 4. Quantum gravity blurs the light-cone