Quantum Field Theory on NC Curved Spacetimes

th.workshop2010.desy.de

Quantum Field Theory on NC Curved Spacetimes

ong>Quantumong> ong>Fieldong> ong>Theoryong> on NC Curved Spacetimes

Alexander Schenkel

(work in part with Thorsten Ohl)

Institute for Theoretical Physics and Astrophysics, University of Würzburg

ong>Quantumong> ong>Fieldong> ong>Theoryong>: Developments and Perspectives

DESY ong>Theoryong> Workshop, Hamburg

September 21-24, 2010

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


Why?

Motivation

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


Why? Motivation

◮ QFT on curved spacetimes is important for physics

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


Why? Motivation

◮ QFT on curved spacetimes is important for physics

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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, . . . ]

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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?

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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!?!?

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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]

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


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]

→ test their physical implications by using QFT on CS

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 2 / 9


Scalar field on NC CS

Scalar field theory on NC curved spacetimes

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 3 / 9


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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 3 / 9


Scalar field on NC CS Kinematics

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

⋆-product h ⋆ k = h e iλ ←−

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


Scalar field on NC CS Kinematics

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

⋆-product h ⋆ k = h e iλ ←−

2 ∂µΘ µν−→ iλ

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

F −1 = ¯f α


⊗ ¯fα = exp

2 Θab

Xa ⊗ Xb with [Xa, Xb] = 0

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 3 / 9


Scalar field on NC CS Kinematics

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

⋆-product h ⋆ k = h e iλ ←−

2 ∂µΘ µν−→ iλ

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

F −1 = ¯f α


⊗ ¯fα = exp

2 Θab

Xa ⊗ Xb with [Xa, Xb] = 0

NC geometry via twist deformation quantization: [Wess group]

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 3 / 9


Scalar field on NC CS Kinematics

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

⋆-product h ⋆ k = h e iλ ←−

2 ∂µΘ µν−→ iλ

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

F −1 = ¯f α


⊗ ¯fα = exp

2 Θab

Xa ⊗ Xb with [Xa, Xb] = 0

NC geometry via twist deformation quantization: [Wess group]

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

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 3 / 9


Scalar field on NC CS Kinematics

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

⋆-product h ⋆ k = h e iλ ←−

2 ∂µΘ µν−→ iλ

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

F −1 = ¯f α


⊗ ¯fα = exp

2 Θab

Xa ⊗ Xb with [Xa, Xb] = 0

NC geometry via twist deformation quantization: [Wess group]

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

◮ 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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 3 / 9


Scalar field on NC CS Deformed dynamics

◮ deformed action functional:

SΦ = − 1


〈〈dΦ,

−1

g⋆ 〉⋆, dΦ〉⋆ + M

2

2 Φ ⋆ Φ ⋆ vol⋆

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 4 / 9


Scalar field on NC CS Deformed dynamics

◮ deformed action functional:

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 4 / 9


Scalar field on NC CS Deformed dynamics

◮ deformed action functional:

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 4 / 9


Scalar field on NC CS Deformed Green’s operators

◮ (M, g⋆, ⋆) λ=0 time-oriented, connected, globally hyperbolic

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 5 / 9


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)

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 5 / 9


Scalar field on NC CS Deformed Green’s operators

◮ (M, g⋆, ⋆) λ=0 time-oriented, connected, globally hyperbolic

◮ deformed Klein-Gordon operator P⋆ = ∞

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

λ

n=0

nP (n)

(M) for all n > 0

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 5 / 9


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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 5 / 9


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)±

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 5 / 9


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


Scalar field on NC CS Deformed symplectic vector space

◮ define: ∆⋆ := ∆⋆+ − ∆⋆−

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 6 / 9


Scalar field on NC CS Deformed symplectic vector space

◮ define: ∆⋆ := ∆⋆+ − ∆⋆−

◮ space of “physical sources”:

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 6 / 9


Scalar field on NC CS Deformed symplectic vector space

◮ define: ∆⋆ := ∆⋆+ − ∆⋆−

◮ space of “physical sources”:

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

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 6 / 9


Scalar field on NC CS Deformed symplectic vector space

◮ define: ∆⋆ := ∆⋆+ − ∆⋆−

◮ space of “physical sources”:

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

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

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]].

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 6 / 9


Scalar field on NC CS Deformed symplectic vector space

◮ define: ∆⋆ := ∆⋆+ − ∆⋆−

◮ space of “physical sources”:

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

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

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]].

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]),.

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 6 / 9


Scalar field on NC CS Deformed symplectic vector space

◮ define: ∆⋆ := ∆⋆+ − ∆⋆−

◮ space of “physical sources”:

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

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

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]].

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.

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 6 / 9


Scalar field on NC CS ∗-algebras of field polynomials

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 7 / 9


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 .

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 7 / 9


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 .

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 7 / 9


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 .

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

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).

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 7 / 9


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 .

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

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).

◮ Consequences:

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 7 / 9


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 .

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

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).

◮ 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〉

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 7 / 9


Scalar field on NC CS

An example of a convergent deformation

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


Scalar field on NC CS 2-dimensional FRW universe

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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



0

0

k=−∞

ϕ(t, −k)

∆(t, τ, k)

cosh(3λk) ψ(τ, k)

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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



0

0

k=−∞

ϕ(t, −k)

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

S ϕ (t, k) =

∆(t, τ, k)

cosh(3λk) ψ(τ, k)

ϕ(t, k)

cosh(3λk) ⇒ ω(Sϕ, Sψ) = ω⋆(ϕ, ψ)

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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



0

0

k=−∞

ϕ(t, −k)

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

S ϕ (t, k) =

∆(t, τ, k)

cosh(3λk) ψ(τ, k)

ϕ(t, k)

cosh(3λk) ⇒ ω(Sϕ, Sψ) = ω⋆(ϕ, ψ)

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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



0

0

k=−∞

ϕ(t, −k)

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

S ϕ (t, k) =

∆(t, τ, k)

cosh(3λk) ψ(τ, k)

ϕ(t, k)

cosh(3λk) ⇒ ω(Sϕ, Sψ) = ω⋆(ϕ, ψ)

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

◮ Interpretation:

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

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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



0

0

k=−∞

ϕ(t, −k)

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

S ϕ (t, k) =

∆(t, τ, k)

cosh(3λk) ψ(τ, k)

ϕ(t, k)

cosh(3λk) ⇒ ω(Sϕ, Sψ) = ω⋆(ϕ, ψ)

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

◮ Interpretation:

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

localized observables are excluded.

A. Schenkel (Würzburg) QFT on NC CS DESY 2010 8 / 9


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]]

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


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]]

◮ Examples of convergent deformations [arXiv:1009.1090]:

◮ construction of the deformed symplectic vector space

◮ quantization in terms of Weyl algebras

◮ QFT on NC CS

injective ∗-morphism

−−−−−−−−−−−−−−−−−→ QFT on CS

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

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