Quantum Field Theory on NC Curved Spacetimes
Quantum Field Theory on NC Curved Spacetimes
Quantum Field Theory on NC Curved Spacetimes
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Scalar field <strong>on</strong> <strong>NC</strong> CS ∗-algebras of field polynomials<br />
◮ C<strong>on</strong>sider unital ∗-algebras over C[[λ]] [math/0408217 (Waldmann), . . . ]<br />
Def: Let (V, ρ) be a symplectic vector space over R[[λ]]. Then A (V,ρ) is<br />
∗-algebra of field polynomials if it is generated by Φ(ϕ), ϕ ∈ V, such that<br />
Φ(α ϕ + β ψ) = α Φ(ϕ) + β Φ(ψ) ,<br />
Φ(ϕ) ∗ = Φ(ϕ) ,<br />
Φ(ϕ), Φ(ψ) = i ρ(ϕ, ψ) 1 .<br />
◮ A (V⋆,ω⋆) ˆ= <strong>NC</strong> QFT , A (V[[λ]],ω) ˆ= com. QFT[[λ]]<br />
Corollary<br />
The symplectic isomorphism S : (V⋆, ω⋆) → (V[[λ]], ω) induces a ∗-algebra<br />
isomorphism S : A (V⋆,ω⋆) → A (V[[λ]],ω) between <strong>NC</strong> and com. QFT[[λ]],<br />
via S Φ⋆(ϕ) := Φ(Sϕ).<br />
A. Schenkel (Würzburg) QFT <strong>on</strong> <strong>NC</strong> CS DESY 2010 7 / 9
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