Combinatorics of the Star Product in AQFT Eli Hawkins Kasia Rejzner - - PowerPoint PPT Presentation

Combinatorics of the Star Product in AQFT

Eli Hawkins Kasia Rejzner

The University of York

July 2, 2019

Classical Field Theory

Classical Field Theory

• M spacetime

Classical Field Theory

• M spacetime
• E vector bundle

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

• Peierls bracket of F, G : E → C:

{F, G}S(ϕ) . = ∆S[ϕ], F (1)(ϕ) ⊗ G (1)(ϕ)

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

• Peierls bracket of F, G : E → C:

{F, G}S(ϕ) . = ∆S[ϕ], F (1)(ϕ) ⊗ G (1)(ϕ)

Quantization of Free Theory

S0 quadratic = ⇒ ∆R

S0(ϕ; x, y) = ∆R S0(x, y) .

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

• Peierls bracket of F, G : E → C:

{F, G}S(ϕ) . = ∆S[ϕ], F (1)(ϕ) ⊗ G (1)(ϕ)

Quantization of Free Theory

S0 quadratic = ⇒ ∆R

S0(ϕ; x, y) = ∆R S0(x, y) .

Denote m(F, G)(ϕ) . = F(ϕ)G(ϕ)

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

• Peierls bracket of F, G : E → C:

{F, G}S(ϕ) . = ∆S[ϕ], F (1)(ϕ) ⊗ G (1)(ϕ)

Quantization of Free Theory

S0 quadratic = ⇒ ∆R

S0(ϕ; x, y) = ∆R S0(x, y) .

Denote m(F, G)(ϕ) . = F(ϕ)G(ϕ) A distribution K on M × M defines DK(F⊗G)(ϕ1, ϕ2) . = K, F (1)(ϕ1), G (1)(ϕ2)

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

• Peierls bracket of F, G : E → C:

{F, G}S(ϕ) . = ∆S[ϕ], F (1)(ϕ) ⊗ G (1)(ϕ)

Quantization of Free Theory

S0 quadratic = ⇒ ∆R

S0(ϕ; x, y) = ∆R S0(x, y) .

Denote m(F, G)(ϕ) . = F(ϕ)G(ϕ) A distribution K on M × M defines DK(F⊗G)(ϕ1, ϕ2) . = K, F (1)(ϕ1), G (1)(ϕ2) and an exponential star product F ⋆K G . = m ◦ e¯

hDK (F ⊗ G) .

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

• Peierls bracket of F, G : E → C:

{F, G}S(ϕ) . = ∆S[ϕ], F (1)(ϕ) ⊗ G (1)(ϕ)

Quantization of Free Theory

S0 quadratic = ⇒ ∆R

S0(ϕ; x, y) = ∆R S0(x, y) .

Denote m(F, G)(ϕ) . = F(ϕ)G(ϕ) A distribution K on M × M defines DK(F⊗G)(ϕ1, ϕ2) . = K, F (1)(ϕ1), G (1)(ϕ2) and an exponential star product F ⋆K G . = m ◦ e¯

hDK (F ⊗ G) .

If K(x, y) − K(y, x) = i∆S0(x, y), then this is a quantization for S0.

Classical Field Theory

• M spacetime
• E vector bundle
• E .

= Γ(M, E) smooth sections

• S action

S′′(φ) is linearized equation of motion

• perator about ϕ ∈ E (off shell).
• ∆R

S retarded Green’s function

• ∆A

S (ϕ; x, y) .

= ∆R

S (ϕ; y, x) advanced

• ∆S .

= ∆R

S − ∆A S

• Peierls bracket of F, G : E → C:

{F, G}S(ϕ) . = ∆S[ϕ], F (1)(ϕ) ⊗ G (1)(ϕ)

Quantization of Free Theory

S0 quadratic = ⇒ ∆R

S0(ϕ; x, y) = ∆R S0(x, y) .

Denote m(F, G)(ϕ) . = F(ϕ)G(ϕ) A distribution K on M × M defines DK(F⊗G)(ϕ1, ϕ2) . = K, F (1)(ϕ1), G (1)(ϕ2) and an exponential star product F ⋆K G . = m ◦ e¯

hDK (F ⊗ G) .

If K(x, y) − K(y, x) = i∆S0(x, y), then this is a quantization for S0. Changing K by a smooth, symmetric function gives an equivalent star product.

Quantization Maps

Quantization Maps

A quantization map Q : {Classical observables} → {Quantum observables} is a “choice of operator ordering”.

Quantization Maps

A quantization map Q : {Classical observables} → {Quantum observables} is a “choice of operator ordering”. It induces a star product by Q(F ⋆ G) = Q(F)Q(G) .

Quantization Maps

A quantization map Q : {Classical observables} → {Quantum observables} is a “choice of operator ordering”. It induces a star product by Q(F ⋆ G) = Q(F)Q(G) . Equivalent star products come from different choices of Q.

Quantization Maps

A quantization map Q : {Classical observables} → {Quantum observables} is a “choice of operator ordering”. It induces a star product by Q(F ⋆ G) = Q(F)Q(G) . Equivalent star products come from different choices of Q. Ordering K Product Symmetric

i 2∆S0 = i 2

• ∆R

S0 − ∆A S0

• Moyal-Weyl

Quantization Maps

A quantization map Q : {Classical observables} → {Quantum observables} is a “choice of operator ordering”. It induces a star product by Q(F ⋆ G) = Q(F)Q(G) . Equivalent star products come from different choices of Q. Ordering K Product Symmetric

i 2∆S0 = i 2

• ∆R

S0 − ∆A S0

• Moyal-Weyl

Normal ∆+

S0 = i 2∆S0 + H

Wick

Quantization Maps

A quantization map Q : {Classical observables} → {Quantum observables} is a “choice of operator ordering”. It induces a star product by Q(F ⋆ G) = Q(F)Q(G) . Equivalent star products come from different choices of Q. Ordering K Product Symmetric

i 2∆S0 = i 2

• ∆R

S0 − ∆A S0

• Moyal-Weyl

Normal ∆+

S0 = i 2∆S0 + H

Wick Time-ordered −i∆A

S0

⋆T

Quantization Maps

A quantization map Q : {Classical observables} → {Quantum observables} is a “choice of operator ordering”. It induces a star product by Q(F ⋆ G) = Q(F)Q(G) . Equivalent star products come from different choices of Q. Ordering K Product Symmetric

i 2∆S0 = i 2

• ∆R

S0 − ∆A S0

• Moyal-Weyl

Normal ∆+

S0 = i 2∆S0 + H

Wick Time-ordered −i∆A

S0

⋆T The product ⋆T will be the most useful here.

Star Product by Graphs

Star Product by Graphs

A graph describes a multidifferential operator.

Star Product by Graphs

A graph describes a multidifferential operator.

• The vertex j represents (a derivative of) the j’th argument.

Star Product by Graphs

A graph describes a multidifferential operator.

• The vertex j represents (a derivative of) the j’th argument.
• = ∆A

S0.

Star Product by Graphs

A graph describes a multidifferential operator.

• The vertex j represents (a derivative of) the j’th argument.
• = ∆A

S0.

E.g.,

1 2 = m, i.e., m(F, G) = F · G.

Star Product by Graphs

A graph describes a multidifferential operator.

• The vertex j represents (a derivative of) the j’th argument.
• = ∆A

S0.

E.g.,

1 2 = m, i.e., m(F, G) = F · G.

⋆T = 1

2 − i¯

h 1

2 + (−i¯

h)2 2

1 2 + (−i¯

h)3 6

1 2 + . . .

Star Product by Graphs

A graph describes a multidifferential operator.

• The vertex j represents (a derivative of) the j’th argument.
• = ∆A

S0.

E.g.,

1 2 = m, i.e., m(F, G) = F · G.

⋆T = 1

2 − i¯

h 1

2 + (−i¯

h)2 2

1 2 + (−i¯

h)3 6

1 2 + . . .

=

• γ

(−i¯ h)e(γ) |Aut γ| γ

• e = # edges;
• the sum is over γ with vertices 1 and 2 and edges from 2 to 1.

Time-ordered Product

For x, y ∈ M, x y = ⇒ ∆A

S0(x, y) = 0 .

Time-ordered Product

For x, y ∈ M, x y = ⇒ ∆A

S0(x, y) = 0 .

This implies that supp F supp G = ⇒ F ⋆T G = F · G .

Time-ordered Product

For x, y ∈ M, x y = ⇒ ∆A

S0(x, y) = 0 .

This implies that supp F supp G = ⇒ F ⋆T G = F · G . So, the trivial pointwise product is the time-ordered product for ⋆T .

Time-ordered Product

For x, y ∈ M, x y = ⇒ ∆A

S0(x, y) = 0 .

This implies that supp F supp G = ⇒ F ⋆T G = F · G . So, the trivial pointwise product is the time-ordered product for ⋆T . This simplifies calculations.

Time-ordered Product

For x, y ∈ M, x y = ⇒ ∆A

S0(x, y) = 0 .

This implies that supp F supp G = ⇒ F ⋆T G = F · G . So, the trivial pointwise product is the time-ordered product for ⋆T . This simplifies calculations.

Møller Operator

Now consider the action S = S0 + λV .

Time-ordered Product

For x, y ∈ M, x y = ⇒ ∆A

S0(x, y) = 0 .

This implies that supp F supp G = ⇒ F ⋆T G = F · G . So, the trivial pointwise product is the time-ordered product for ⋆T . This simplifies calculations.

Møller Operator

Now consider the action S = S0 + λV . The quantum Møller operator simplifies to RT ,λV (F) =

• eiλV /¯

h⋆T −1

⋆T

• eiλV /¯

h · F

• .

Time-ordered Product

For x, y ∈ M, x y = ⇒ ∆A

S0(x, y) = 0 .

This implies that supp F supp G = ⇒ F ⋆T G = F · G . So, the trivial pointwise product is the time-ordered product for ⋆T . This simplifies calculations.

Møller Operator

Now consider the action S = S0 + λV . The quantum Møller operator simplifies to RT ,λV (F) =

• eiλV /¯

h⋆T −1

⋆T

• eiλV /¯

h · F

• .

Its inverse is just R−1

T ,λV (G) = e−iλV /¯ h ·

• eiλV /¯

h ⋆T G

• .

Møller by Graphs

Møller by Graphs

An unlabelled vertex represents V .

Møller by Graphs

An unlabelled vertex represents V . eiλV /¯

h = 1 + iλ¯

h−1 + (iλ)2 2¯ h2 + (iλ)3 6¯ h3 + . . .

Møller by Graphs

An unlabelled vertex represents V . eiλV /¯

h = 1 + iλ¯

h−1 + (iλ)2 2¯ h2 + (iλ)3 6¯ h3 + . . . =

• γ

(−i¯ h)−v(γ)λv(γ) |Aut(γ)| γ

• v = #unlabelled vertices;
• the sum is over γ with only unlabelled

vertices and no edges.

Møller by Graphs

An unlabelled vertex represents V . eiλV /¯

h = 1 + iλ¯

h−1 + (iλ)2 2¯ h2 + (iλ)3 6¯ h3 + . . . =

• γ

(−i¯ h)−v(γ)λv(γ) |Aut(γ)| γ

• v = #unlabelled vertices;
• the sum is over γ with only unlabelled

vertices and no edges. Recall, F ⋆T G =

• γ

(−i¯ h)e(γ) |Aut γ| γ(F, G) sum over γ with edges from 2 to 1.

Møller by Graphs

An unlabelled vertex represents V . eiλV /¯

h = 1 + iλ¯

h−1 + (iλ)2 2¯ h2 + (iλ)3 6¯ h3 + . . . =

• γ

(−i¯ h)−v(γ)λv(γ) |Aut(γ)| γ

• v = #unlabelled vertices;
• the sum is over γ with only unlabelled

vertices and no edges. Recall, F ⋆T G =

• γ

(−i¯ h)e(γ) |Aut γ| γ(F, G) sum over γ with edges from 2 to 1. eiλV /¯

h · R−1 T ,λV (G) = eiλV /¯ h ⋆T G

Møller by Graphs

An unlabelled vertex represents V . eiλV /¯

h = 1 + iλ¯

h−1 + (iλ)2 2¯ h2 + (iλ)3 6¯ h3 + . . . =

• γ

(−i¯ h)−v(γ)λv(γ) |Aut(γ)| γ

• v = #unlabelled vertices;
• the sum is over γ with only unlabelled

vertices and no edges. Recall, F ⋆T G =

• γ

(−i¯ h)e(γ) |Aut γ| γ(F, G) sum over γ with edges from 2 to 1. eiλV /¯

h · R−1 T ,λV (G) = eiλV /¯ h ⋆T G

=

• γ

(−i¯ h)e(γ)−v(γ)λv(γ) |Aut(γ)| γ(G) sum over γ with edges from 1 to unlabelled vertices.

Møller by Graphs

An unlabelled vertex represents V . eiλV /¯

h = 1 + iλ¯

h−1 + (iλ)2 2¯ h2 + (iλ)3 6¯ h3 + . . . =

• γ

(−i¯ h)−v(γ)λv(γ) |Aut(γ)| γ

• v = #unlabelled vertices;
• the sum is over γ with only unlabelled

vertices and no edges. Recall, F ⋆T G =

• γ

(−i¯ h)e(γ) |Aut γ| γ(F, G) sum over γ with edges from 2 to 1. eiλV /¯

h · R−1 T ,λV (G) = eiλV /¯ h ⋆T G

=

• γ

(−i¯ h)e(γ)−v(γ)λv(γ) |Aut(γ)| γ(G) sum over γ with edges from 1 to unlabelled vertices. R−1

T ,λV (G) =

• γ

(−i¯ h)e(γ)−v(γ)λv(γ) |Aut(γ)| γ(G) sum over connected γ with edges from 1 to unlabelled vertices.

Møller by Graphs

An unlabelled vertex represents V . eiλV /¯

h = 1 + iλ¯

h−1 + (iλ)2 2¯ h2 + (iλ)3 6¯ h3 + . . . =

• γ

(−i¯ h)−v(γ)λv(γ) |Aut(γ)| γ

• v = #unlabelled vertices;
• the sum is over γ with only unlabelled

vertices and no edges. Recall, F ⋆T G =

• γ

(−i¯ h)e(γ) |Aut γ| γ(F, G) sum over γ with edges from 2 to 1. eiλV /¯

h · R−1 T ,λV (G) = eiλV /¯ h ⋆T G

=

• γ

(−i¯ h)e(γ)−v(γ)λv(γ) |Aut(γ)| γ(G) sum over γ with edges from 1 to unlabelled vertices. R−1

T ,λV (G) =

• γ

(−i¯ h)e(γ)−v(γ)λv(γ) |Aut(γ)| γ(G) sum over connected γ with edges from 1 to unlabelled vertices.

1

Interacting Product

The Møller operator intertwines the interacting and non-interacting products. F ⋆T ,int G . = R−1

T ,λV (RT ,λV F ⋆T RT ,λV G)

Interacting Product

The Møller operator intertwines the interacting and non-interacting products. F ⋆T ,int G . = R−1

T ,λV (RT ,λV F ⋆T RT ,λV G)

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−λ)v(γ) |Aut(γ)| γ

Interacting Product

The Møller operator intertwines the interacting and non-interacting products. F ⋆T ,int G . = R−1

T ,λV (RT ,λV F ⋆T RT ,λV G)

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−λ)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;

Interacting Product

The Møller operator intertwines the interacting and non-interacting products. F ⋆T ,int G . = R−1

T ,λV (RT ,λV F ⋆T RT ,λV G)

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−λ)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;
• every unlabelled vertex has at least one ingoing edge and one outgoing edge;

Interacting Product

The Møller operator intertwines the interacting and non-interacting products. F ⋆T ,int G . = R−1

T ,λV (RT ,λV F ⋆T RT ,λV G)

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−λ)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;
• every unlabelled vertex has at least one ingoing edge and one outgoing edge;
• the vertices can be ordered with 1 lowest, 2 highest, and edges pointing down.

Resummation

These include arbitrary chains of bivalent vertices.

Resummation

These include arbitrary chains of bivalent vertices.

Resummation

These include arbitrary chains of bivalent vertices. − λ

Resummation

These include arbitrary chains of bivalent vertices. − λ + λ2 − . . .

Resummation

These include arbitrary chains of bivalent vertices. − λ + λ2 − . . . = ∆A

S

Resummation

These include arbitrary chains of bivalent vertices. − λ + λ2 − . . . = ∆A

S

• This accounts for all bivalent vertices in ⋆T ,int.

Resummation

These include arbitrary chains of bivalent vertices. − λ + λ2 − . . . = ∆A

S

• This accounts for all bivalent vertices in ⋆T ,int.
• A valency r > 2 vertex represents λV (r) = S(r).

Resummation

These include arbitrary chains of bivalent vertices. − λ + λ2 − . . . = ∆A

S

• This accounts for all bivalent vertices in ⋆T ,int.
• A valency r > 2 vertex represents λV (r) = S(r).

Now let:

• Unlabelled vertices represent derivatives of S;

Resummation

These include arbitrary chains of bivalent vertices. − λ + λ2 − . . . = ∆A

S

• This accounts for all bivalent vertices in ⋆T ,int.
• A valency r > 2 vertex represents λV (r) = S(r).

Now let:

• Unlabelled vertices represent derivatives of S;
• = ∆A

S .

Nonperturbative Expression

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−1)v(γ) |Aut(γ)| γ

Nonperturbative Expression

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−1)v(γ) |Aut(γ)| γ

Nonperturbative Expression

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−1)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;

Nonperturbative Expression

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−1)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;
• every unlabelled vertex has at least one ingoing edge and one outgoing edge;

Nonperturbative Expression

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−1)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;
• every unlabelled vertex has at least one ingoing edge and one outgoing edge;
• unlabelled vertices have valency at least 3;

Nonperturbative Expression

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−1)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;
• every unlabelled vertex has at least one ingoing edge and one outgoing edge;
• unlabelled vertices have valency at least 3;
• the vertices can be ordered with 1 lowest, 2 highest, and edges pointing down.

Nonperturbative Expression

⋆T ,int =

• γ

(−i¯ h)e(γ)−v(γ)(−1)v(γ) |Aut(γ)| γ

• γ has labelled vertices 1 and 2;
• every unlabelled vertex has at least one ingoing edge and one outgoing edge;
• unlabelled vertices have valency at least 3;
• the vertices can be ordered with 1 lowest, 2 highest, and edges pointing down.

This is a star product quantizing S.

⋆T ,int = m + ¯ hB1 + ¯ h2B2 + ¯ h3B3 + . . .

⋆T ,int = m + ¯ hB1 + ¯ h2B2 + ¯ h3B3 + . . . B1 = −i

1 2 ,

⋆T ,int = m + ¯ hB1 + ¯ h2B2 + ¯ h3B3 + . . . B1 = −i

1 2 ,

B2 = −1 2

1 2 + 1

2

1 2 + 1

2

1 2 − 1

2

1 2 ,

⋆T ,int = m + ¯ hB1 + ¯ h2B2 + ¯ h3B3 + . . . B1 = −i

1 2 ,

B2 = −1 2

1 2 + 1

2

1 2 + 1

2

1 2 − 1

2

1 2 ,

B3 = i 6

1 2 − i

2

1 2 − i

2

1 2 + i

2

1 2

+ i 1

2 − i

4

1 2 + i

4

1 2 − . . .

· · · − i 6

1 2 + i

2

1 2 + i

2

1 2 − i

2

1 2

− i 1

2 + i

4

1 2 − i

4

1 2

− i 6

1 2 + i

2

1 2 + i

2

1 2 − i

2

1 2

− i 1

2 + i

4

1 2 − i

4

1 2 + . . .

· · · + i 6

1 2 − i

2

1 2 − i

2

1 2

+ i 2

1 2 + i 1 2

− i 4

1 2 + i

4

2 1

· · · + i 6

1 2 − i

2

1 2 − i

2

1 2

+ i 2

1 2 + i 1 2

− i 4

1 2 + i

4

2 1

• E. Hawkins, K. Rejzner:

The Star Product in Interacting Quantum Field Theory. arXiv:1612.09157 [math-ph]