Quantum Field Theory on NC Curved Spacetimes

<strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> on NC Curved Spacetimes 

Alexander Schenkel 

(work in part with Thorsten Ohl) 

Institute for Theoretical Physics and Astrophysics, University of Würzburg 

<strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong>: Developments and Perspectives 

DESY <strong>Theory</strong> Workshop, Hamburg 

September 21-24, 2010 

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 1 / 9

Why? 

Motivation 


Why? Motivation 

◮ QFT on curved spacetimes is important for physics 




→ cosmology (CMB fluctuations) and black holes (Hawking radiation) 





◮ precise formulation via algebraic approach [Wald, Hamburg AQFT group, . . . ] 






◮ But why should we make all of this noncommutative? 







◮ NC geometry from quantum gravity!?!? 








→ include some quantum gravity effects in QFT on CS 









◮ NC in cosmology and black hole physics is of physical interest 










→ provide formal background for phenomenology 











◮ ∃ NC gravity solutions [Schupp, Solodukhin; Ohl, AS; Aschieri, Castellani] 











◮ ∃ NC gravity solutions [Schupp, Solodukhin; Ohl, AS; Aschieri, Castellani] 

→ test their physical implications by using QFT on CS 


Scalar field on NC CS 

Scalar field theory on NC curved spacetimes 


Scalar field on NC CS Kinematics 

◮ Simple example of a twist: (Moyal product/twist) 

⋆-product h ⋆ k = h e iλ ←− 

2 ∂µΘ µν−→ ∂ν k 





2 ∂µΘ µν−→ iλ 

∂ν −1 k twist F = e 2 Θµν∂µ⊗∂ν A. Schenkel (Würzburg) QFT on NC CS DESY 2010 3 / 9





∂ν −1 k twist F = e 2 Θµν∂µ⊗∂ν ◮ More general class of twists: (abelian twists) 

F −1 = ¯f α 

iλ 

⊗ ¯fα = exp 

2 Θab 

Xa ⊗ Xb with [Xa, Xb] = 0 







F −1 = ¯f α 

iλ 


2 Θab 


◮ NC geometry via twist deformation quantization: [Wess group] 

◮ algebra of functions (C ∞ (M)[[λ]], ⋆), where h ⋆ k := ¯f α (h) · ¯fα(k) 







F −1 = ¯f α 

iλ 


2 Θab 




◮ exterior algebra (Ω • [[λ]], ∧⋆, d), where ω ∧⋆ ω ′ := ¯f α (ω) ∧ ¯fα(ω ′ ) 







F −1 = ¯f α 

iλ 


2 Θab 




◮ exterior algebra (Ω • [[λ]], ∧⋆, d), where ω ∧⋆ ω ′ := ¯f α (ω) ∧ ¯fα(ω ′ ) 

◮ pairing 〈v, ω〉⋆ := 〈 ¯f α (v), ¯fα(ω)〉 among vector fields and 1-forms 


Scalar field on NC CS Deformed dynamics 

◮ deformed action functional: 

SΦ = − 1 

 

〈〈dΦ, 

−1 

g⋆ 〉⋆, dΦ〉⋆ + M 

2 

2 Φ ⋆ Φ ⋆ vol⋆ 




SΦ = − 1 

 

〈〈dΦ, 

−1 

g⋆ 〉⋆, dΦ〉⋆ + M 

2 

2 Φ ⋆ Φ ⋆ vol⋆ 

◮ deformed equation of motion (top-form valued): 

0 = 1 

 

⋆[Φ] ⋆ vol⋆ + vol⋆ ⋆ 

2 

⋆[Φ ∗ ] ∗ 2 

− M Φ ⋆ vol⋆ − M 2 

vol⋆ ⋆ Φ 




SΦ = − 1 

 

〈〈dΦ, 

−1 

g⋆ 〉⋆, dΦ〉⋆ + M 

2 

2 Φ ⋆ Φ ⋆ vol⋆ 

◮ deformed equation of motion (top-form valued): 

0 = 1 

 

⋆[Φ] ⋆ vol⋆ + vol⋆ ⋆ 

2 

⋆[Φ ∗ ] ∗ 2 

− M Φ ⋆ vol⋆ − M 2 

vol⋆ ⋆ Φ 

=: P⋆[Φ] ⋆ vol⋆ 

NB: P⋆ : C ∞ (M)[[λ]] → C ∞ (M)[[λ]] scalar valued 


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 

λ 

n=0 

nP (n) 

(M) for all n > 0 





λ 

n=0 

nP (n) 

◮ technical assumption: P (n) : C∞ (M) → C∞ 0 

◮ fulfilled for all twists of compact support 


◮ or g⋆ asymptotically (outside compact region) symmetric under F 





λ 

n=0 

nP (n) 





◮ based on strong results for the commutative case we find: 

there exist unique deformed Green’s operators ∆⋆± := ∞ 

λ 

n=0 

n∆ (n)± 





λ 

n=0 

nP (n) 





◮ 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 ) 


Scalar field on NC CS Deformed symplectic vector space 

◮ define: ∆⋆ := ∆⋆+ − ∆⋆− 



◮ define: ∆⋆ := ∆⋆+ − ∆⋆− 

◮ space of “physical sources”: 

H := ϕ ∈ C ∞ 0 (M)[[λ]] : ∆⋆±(ϕ) ∗ = ∆⋆±(ϕ) 



◮ define: ∆⋆ := ∆⋆+ − ∆⋆− 


H := ϕ ∈ C ∞ 0 (M)[[λ]] : ∆⋆±(ϕ) ∗ = ∆⋆±(ϕ) 

◮ we obtain: ∆⋆[H] = Sol R P⋆ , i.e. H/Ker(∆⋆) Sol R P⋆ 



◮ define: ∆⋆ := ∆⋆+ − ∆⋆− 


H := ϕ ∈ C ∞ 0 (M)[[λ]] : ∆⋆±(ϕ) ∗ = ∆⋆±(ϕ) 


Proposition (T. Ohl, AS) 

(V⋆, ω⋆) with V⋆ := H/Ker(∆⋆) and 

 

ω⋆([ϕ], [ψ]) := ϕ ∗ ⋆ ∆⋆(ψ) ⋆ vol⋆ 

is a symplectic vector space over R[[λ]]. 



◮ define: ∆⋆ := ∆⋆+ − ∆⋆− 


H := ϕ ∈ C ∞ 0 (M)[[λ]] : ∆⋆±(ϕ) ∗ = ∆⋆±(ϕ) 




 

ω⋆([ϕ], [ψ]) := ϕ ∗ ⋆ ∆⋆(ψ) ⋆ vol⋆ 


Proposition 

There is a symplectic isomorphism S : (V⋆, ω⋆) → (V[[λ]], ω) between the NC 

and com. field theory[[λ]], i.e. ω(S[ϕ], S[ψ]) = ω⋆([ϕ], [ψ]),. 



◮ define: ∆⋆ := ∆⋆+ − ∆⋆− 


H := ϕ ∈ C ∞ 0 (M)[[λ]] : ∆⋆±(ϕ) ∗ = ∆⋆±(ϕ) 




 

ω⋆([ϕ], [ψ]) := ϕ ∗ ⋆ ∆⋆(ψ) ⋆ vol⋆ 


Proposition 

There is a symplectic isomorphism S : (V⋆, ω⋆) → (V[[λ]], ω) between the NC 

and com. field theory[[λ]], i.e. ω(S[ϕ], S[ψ]) = ω⋆([ϕ], [ψ]),. 

Proof by constructing S order-by-order in λ. 


Scalar field on NC CS ∗-algebras of field polynomials 

◮ Consider unital ∗-algebras over C[[λ]] [math/0408217 (Waldmann), . . . ] 




Def: Let (V, ρ) be a symplectic vector space over R[[λ]]. Then A (V,ρ) is 

∗-algebra of field polynomials if it is generated by Φ(ϕ), ϕ ∈ V, such that 

Φ(α ϕ + β ψ) = α Φ(ϕ) + β Φ(ψ) , 

Φ(ϕ) ∗ = Φ(ϕ) , 

Φ(ϕ), Φ(ψ) = i ρ(ϕ, ψ) 1 . 






Φ(α ϕ + β ψ) = α Φ(ϕ) + β Φ(ψ) , 

Φ(ϕ) ∗ = Φ(ϕ) , 

Φ(ϕ), Φ(ψ) = i ρ(ϕ, ψ) 1 . 

◮ A (V⋆,ω⋆) ˆ= NC QFT , A (V[[λ]],ω) ˆ= com. QFT[[λ]] 






Φ(α ϕ + β ψ) = α Φ(ϕ) + β Φ(ψ) , 

Φ(ϕ) ∗ = Φ(ϕ) , 

Φ(ϕ), Φ(ψ) = i ρ(ϕ, ψ) 1 . 


Corollary 

The symplectic isomorphism S : (V⋆, ω⋆) → (V[[λ]], ω) induces a ∗-algebra 

isomorphism S : A (V⋆,ω⋆) → A (V[[λ]],ω) between NC and com. QFT[[λ]], 

via S Φ⋆(ϕ) := Φ(Sϕ). 






Φ(α ϕ + β ψ) = α Φ(ϕ) + β Φ(ψ) , 

Φ(ϕ) ∗ = Φ(ϕ) , 

Φ(ϕ), Φ(ψ) = i ρ(ϕ, ψ) 1 . 


Corollary 




◮ Consequences: 

◮ Induction of states Ω⋆ := Ω ◦ S : A(V⋆,ω⋆) → C[[λ]] 






Φ(α ϕ + β ψ) = α Φ(ϕ) + β Φ(ψ) , 

Φ(ϕ) ∗ = Φ(ϕ) , 

Φ(ϕ), Φ(ψ) = i ρ(ϕ, ψ) 1 . 


Corollary 




◮ Consequences: 

◮ Induction of states Ω⋆ := Ω ◦ S : A(V⋆,ω⋆) → C[[λ]] 

◮ NC correlation functions from commutative correlation functions 

〈0|Φ⋆(ϕ1) · · · Φ⋆(ϕn)|0〉 = 〈0|Φ(Sϕ1) · · · Φ(Sϕn)|0〉 


Scalar field on NC CS 

An example of a convergent deformation 


Scalar field on NC CS 2-dimensional FRW universe 

◮ spacetime: M = (0, ∞) × S1 with g = −dt ⊗ dt + t 2 dφ ⊗ dφ 




◮ deformation: F −1 = e iλ(t∂t⊗∂φ−∂φ⊗t∂t) → t ⋆ e iφ = e −2λ e iφ ⋆ t 





◮ in Fourier space (φ → k ∈ Z) we have 

P⋆ = cosh(3λk) ◦ , ∆±⋆ = ∆± ◦ cosh(3λk) −1 







◮ symplectic VS (V⋆, ω⋆) = (V, ω⋆) with 

∞ ∞ 

ω⋆(ϕ, ψ) = − dt t dτ τ 1 

∞ 

2π 

0 

0 

k=−∞ 

ϕ(t, −k) 

∆(t, τ, k) 

cosh(3λk) ψ(τ, k) 








∞ ∞ 

ω⋆(ϕ, ψ) = − dt t dτ τ 1 

∞ 

2π 

0 

0 

k=−∞ 

ϕ(t, −k) 

◮ symplectic map S : (V⋆, ω⋆) → (V, ω) in Fourier space 

S ϕ (t, k) = 

∆(t, τ, k) 


ϕ(t, k) 

cosh(3λk) ⇒ ω(Sϕ, Sψ) = ω⋆(ϕ, ψ) 








∞ ∞ 

ω⋆(ϕ, ψ) = − dt t dτ τ 1 

∞ 

2π 

0 

0 

k=−∞ 

ϕ(t, −k) 


S ϕ (t, k) = 

∆(t, τ, k) 


ϕ(t, k) 


◮ S is injective but not surjective (all S ϕ fall of faster than e −3λ|k|/2 for |k| ≫ 1) 








∞ ∞ 

ω⋆(ϕ, ψ) = − dt t dτ τ 1 

∞ 

2π 

0 

0 

k=−∞ 

ϕ(t, −k) 


S ϕ (t, k) = 

∆(t, τ, k) 


ϕ(t, k) 



◮ Interpretation: 

QFT on NC CS is ∗-isomorphic to a reduced QFT on CS, 








∞ ∞ 

ω⋆(ϕ, ψ) = − dt t dτ τ 1 

∞ 

2π 

0 

0 

k=−∞ 

ϕ(t, −k) 


S ϕ (t, k) = 

∆(t, τ, k) 


ϕ(t, k) 



◮ Interpretation: 

QFT on NC CS is ∗-isomorphic to a reduced QFT on CS, where strongly 

localized observables are excluded. 


Summary and references 

◮ QFT on NC curved spacetimes [arXiv:0912.2252]: 

◮ formally self adjoint EOM operators P⋆ 

◮ existence, uniqueness and construction of the deformed Green’s operators 

◮ symplectic structure on the space of real solutions of P⋆ 

◮ quantization via ∗-algebras of field polynomials 

◮ QFT on NC CS 

∗-algebra isomorphism 

←−−−−−−−−−−−−−−−→ QFT on CS[[λ]] 


Summary and references 

◮ QFT on NC curved spacetimes [arXiv:0912.2252]: 

◮ formally self adjoint EOM operators P⋆ 

◮ existence, uniqueness and construction of the deformed Green’s operators 

◮ symplectic structure on the space of real solutions of P⋆ 

◮ quantization via ∗-algebras of field polynomials 


∗-algebra isomorphism 

←−−−−−−−−−−−−−−−→ QFT on CS[[λ]] 

◮ Examples of convergent deformations [arXiv:1009.1090]: 

◮ construction of the deformed symplectic vector space 

◮ quantization in terms of Weyl algebras 


injective ∗-morphism 

−−−−−−−−−−−−−−−−−→ QFT on CS

Quantum Field Theory on NC Curved Spacetimes

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