Causal Effective Field Eqns from
the Schwinger-Keldysh
Formalism
Richard Woodard
University of Florida
Two Common Questions
1. Aren’t QFT effective field eqns nonlocal?
Yes!
2. Doesn’t this introduce acausality?
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Not unless gravity is dynamical
In-out eqns are acausal, but from in-out BC
In-in eqns are causal
Linearized Examples
∂µ[(-g)½gµν∂ν ϕ(x)] - (-g)½ [ξR+m2] ϕ(x)
-∫d4x′ M2(x;x′) ϕ(x′) = 0
(-g)½ eµaγa [∂µ-½A µbcJbc] ψ(x) – (-g)½m ψ(x)
-∫ d4x′ [Σ](x;x′) ψ(x′) = 0
∂ν [(-g)½gνρgµσ Fρσ(x)]
-∫d4x′ [µΠν](x;x′) Aν(x′) = 0
Eg. M2(x;x′) for λϕ3
Best in position space
M2(x;x′) = ∫d4p/(2π)4
Exp[ip·(x-x′)] M2(p)
No integrals at 1 loop!
[-i M2(x;x′)]1 loop =
½ (-iλ)2 [i∆(x;x′)]2
Full Nonlinear Eqns
For in-out (in-in similar)
• Γ[ϕ] = S[ϕ] - ∑Ν=2 1/N! ∫d4x1 ϕ(x1) . . .
. . .∫d4xN ϕ(xN) ΓN(x1,...,xN)
• ΓN(x1,…,xN) ≡ N-point 1PI function
• δΓ/δϕ(x) = δS/δϕ(x) - ∑Ν=11/N!∫d4x1 ϕ(x1) . .
. . . ∫d4xN ϕ(xN) ΓN+1(x,x1,…,xN) = 0
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
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
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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
Introductory QFT Teaches
Now Multiply by Conjugate
Now Sum over States at t = t2
1. ∑Ψ |Ψ Ψ | = I
2. ∑Ψ Ψ⋆[ϕ−(t2)] Ψ[ϕ+(t2)] = δ[ϕ+(t2) - ϕ−(t2)]
Schwinger-Keldysh Formalism
1. 2 fields ϕ± line endpts have ± “polarity”
2. Exp(iS[ϕ+] – S[ϕ−]) Ints all + or all –
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3.
4.
5.
6.
7.
+ Interactions same as usual
- Interactions conjugated
T(B[φ])↔B[ϕ+] T-ordered ext. lines +
Anti-time-ordered external lines are –
4 Propagators: i∆±±(x;x′)
In-out N-point func. 2N S-K N-points
Φ[ϕ±(t1)] surface ints at t = t1 if not free
Eg. –iM2(x;x′) for λϕ3
(-iλ)2
2
′
[i∆(x;x )]
1 Loop In-Out: ½
1 Loop Schwinger-Keldysh:
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-iM2++(x;x′)
-iM2+-(x;x′)
-iM2-+(x;x′)
-iM2--(x;x′)
= ½ (-iλ)2 [i∆++(x;x′)]2
= ½ (-i λ) (+iλ) [i∆+-(x;x′)]2
= ½ (+iλ) (-iλ) [i∆-+(x;x′)]2
= ½ (+iλ)2 [i∆--(x;x′)]2
S-K Effective Field Eqns
Γ[ϕ+,ϕ−] = S[ϕ+] – S[ϕ−]
-½∫d4x∫d4x′∑±±ϕ±(x)M2±±(x;x′)ϕ±(x′) + O(ϕ3)
0 = δΓ[ϕ+,ϕ−]/δϕ+(x) (then set ϕ±=ϕ)
= δS[ϕ]/δϕ(x)
-∫d4x′ [M2++(x;x′) + M2+-(x;x′)] ϕ(x′) + O(ϕ2)
• NB: M2-+(x;y) = M2+-(y;x)
Get Props with Canonical Relation
F. F. Int. × ϕ+(x)ϕ+(x′) = <Ω0|T[φ(x)φ(x′)]|Ω0>
i∆++(x;x′) = i∆(x;x′)
F. F. Int. × ϕ+(x)ϕ−(x′) = <Ω0 |φ(x′)φ(x)|Ω0>
i∆+-(x;x′) = θ(t-t′) [i∆(x;x′)]* + θ(t′-t) i∆(x;x′)
F. F. Int. × ϕ−(x)ϕ(x′) = <Ω0|φ(x)φ(x′)|Ω0>
i∆-+(x;x′) = θ(t-t′) i∆(x;x′) + θ(t′-t) [i∆(x;x′)]*
F. F. Int. × ϕ−(x)ϕ−(x′) = <Ω0|A[φ(x)φ(x′)]|Ω0>
i∆--(x;x′) = [i∆(x;x′)]*
Properties of [M2++(x;x′) + M2+-(x;x′)]
1 Loop λϕ3: [M2++ + M2+-]
= -iλ2/2 {[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′)
[M2++
Fact 2:
+
2
M +-]
i∆-+(x;x′)
= 0 for t′ > t
= [i∆+-
*
′
(x;x )]
[M2++ + M2+-] = -λ2θ(t-t′) Im{[i∆+-(x;x′)]2]}
Manifestly real (unlike in-out!)
Causality of [M2++(x;x′) + M2+-(x;x′)]
[M2++ + M2+-] = -λ2 θ(t-t′) Im{[i∆+-(x;x′)]2}
Fact 3: i∆+-(x;x′) = <Ω0|φ(x′)φ(x)|Ω0>
= ½ <Ω0|{φ(x′),φ(x)}|Ω0>
+ ½ <Ω0|[φ(x′),φ(x)]|Ω0>
Fact 4: [φ(x′),φ(x)] = 0 for spacelike sep.
[M2++ + M2+-] contributes only for x′µ on or
within past light-cone of xµ
Not even superluminal w/o derivative ints
Blurring the Light-cone
The Physics:
• Metric operator gµν = ĝµν+ hµν sets light-cone
• hµν allows propagation not possible in ĝµν
The Math:
• Derivative ints (h∂h∂h) push inf. off light-cone
• Typical ∆c/c ∼ G2Rρσµν Rρσµν
• Cf. Larry Ford’s work
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