Quantum Field Theory on NC Curved Spacetimes

Scalar field on NC CS Deformed Green’s operators
◮ (M, g⋆, ⋆) λ=0 time-oriented, connected, globally hyperbolic
◮ deformed Klein-Gordon operator P⋆ = ∞
λ
n=0
nP (n)
◮ technical assumption: P (n) : C∞ (M) → C∞ 0
◮ fulfilled for all twists of compact support
(M) for all n > 0
◮ or g⋆ asymptotically (outside compact region) symmetric under F
◮ based on strong results for the commutative case we find:
there exist unique deformed Green’s operators ∆⋆± := ∞
λ
n=0
n∆ (n)±
∆⋆± = ∆± − λ ∆± ◦ P(1) ◦ ∆±
− λ 2
3
∆± ◦ P(2) ◦ ∆± − ∆± ◦ P(1) ◦ ∆± ◦ P(1) ◦ ∆± + O(λ )
= − λ 1 − λ 2
[higher orders in λ follow the same structure]
2 − 1 1
+ O(λ 3 )
A. Schenkel (Würzburg) QFT on NC CS DESY 2010 5 / 9