A seminar talk on NC QFT

1. Noncommutative Quantum Field Theory:
A Confrontation of symmetries
Tapio Salminen
University of Helsinki
Based on work done in collaboration with
M. Chaichian, K. Nishijima and A. Tureanu
JHEP 06 (2008) 078, arXiv: 0805.3500

2. Part 1
Introduction

3. Quantizing space-time
Motivation
Black hole formation in the process of measurement at small
distances (∼ λP ) ⇒ additional uncertainty relations for
coordinates
Doplicher, Fredenhagen and Roberts (1994)

4. Quantizing space-time
Motivation
Black hole formation in the process of measurement at small
distances (∼ λP ) ⇒ additional uncertainty relations for
coordinates
Doplicher, Fredenhagen and Roberts (1994)
Open string + D-brane theory with an antisymmetric tensor
background (NOT induced!)
Ardalan, Arfaei and Sheikh-Jabbari (1998)
Seiberg and Witten (1999)

5. Quantizing space-time
Motivation
Black hole formation in the process of measurement at small
distances (∼ λP ) ⇒ additional uncertainty relations for
coordinates
Doplicher, Fredenhagen and Roberts (1994)
Open string + D-brane theory with an antisymmetric tensor
background (NOT induced!)
Ardalan, Arfaei and Sheikh-Jabbari (1998)
Seiberg and Witten (1999)
A possible approach to physics at short distances is
QFT in NC space-time

6. Quantizing space-time
Implementation
We generalize the commutation relations from
usual quantum mechanics
[ˆi , xj ] = 0 , [ˆi , pj ] = 0
x ˆ p ˆ
[ˆi , pj ] = i δij
x ˆ

7. Quantizing space-time
Implementation
We generalize the commutation relations from
usual quantum mechanics
[ˆi , xj ] = 0 , [ˆi , pj ] = 0
x ˆ p ˆ
[ˆi , pj ] = i δij
x ˆ
by imposing noncommuttativity also between
the coordinate operators
[ˆµ , x ν ] = 0
x ˆ
Snyder (1947); Heisenberg (1954);
Golfand (1962)

8. Quantizing space-time
Implementation
We take [ˆµ , x ν ] = iθµν
x ˆ and choose the frame where
 
0 θ 0 0
 −θ 0 0 0 
θµν =   0

0 0 θ 
0 0 −θ 0

9. Quantizing space-time
Implementation
We take [ˆµ , x ν ] = iθµν
x ˆ and choose the frame where
 
0 θ 0 0
 −θ 0 0 0 
θµν =   0

0 0 θ 
0 0 −θ 0
θµν does not transform under Lorentz
tranformations.

10. Does this mean
Lorentz invariance
is lost?

11. Quantizing space-time
Implementation
We take [ˆµ , x ν ] = iθµν
x ˆ and choose the frame where
 
0 θ 0 0
 −θ 0 0 0 
θµν =   0

0 0 θ 
0 0 −θ 0

12. Quantizing space-time
Implementation
We take [ˆµ , x ν ] = iθµν
x ˆ and choose the frame where
 
0 θ 0 0
 −θ 0 0 0 
θµν =   0

0 0 θ 
0 0 −θ 0
Translational invariance is preserved,
but the Lorentz group breaks down to SO(1, 1)xSO(2).

13. Quantizing space-time
Implementation
We take [ˆµ , x ν ] = iθµν
x ˆ and choose the frame where
 
0 θ 0 0
 −θ 0 0 0 
θµν =   0

0 0 θ 
0 0 −θ 0
Translational invariance is preserved,
but the Lorentz group breaks down to SO(1, 1)xSO(2).
=⇒ No spinor, vector, tensor etc representations.

14. Effects of noncommutativity
Moyal -product
In noncommuting space-time the analogue of the action
1 µ 1 λ
S (cl) [Φ] = d 4x (∂ Φ)(∂µ Φ) − m2 Φ2 − Φ4
2 2 4!
can be written using the Moyal -product

15. Effects of noncommutativity
Moyal -product
In noncommuting space-time the analogue of the action
1 µ 1 λ
S (cl) [Φ] = d 4x (∂ Φ)(∂µ Φ) − m2 Φ2 − Φ4
2 2 4!
can be written using the Moyal -product
1 µ 1 λ
S θ [Φ] = d 4x (∂ Φ) (∂µ Φ) − m2 Φ Φ − Φ Φ Φ Φ
2 2 4!
←
− →
−
i ∂ ∂
θµν ∂x
(Φ Ψ) (x) ≡ Φ(x)e 2 µ ∂yν Ψ(y )
y =x

16. Effects of noncommutativity
The actual symmetry
The action of NC QFT written with the -product, though it
violates Lorentz symmetry, is invariant under the twisted
Poincar´ algebra
e
Chaichian, Kulish, Nishijima and Tureanu (2004)
Chaichian, Preˇnajder and Tureanu (2004)
s

17. Effects of noncommutativity
The actual symmetry
The action of NC QFT written with the -product, though it
violates Lorentz symmetry, is invariant under the twisted
Poincar´ algebra
e
Chaichian, Kulish, Nishijima and Tureanu (2004)
Chaichian, Preˇnajder and Tureanu (2004)
s
This is achieved by deforming the universal enveloping of the
Poincar´ algebra U(P) as a Hopf algebra with the Abelian
e
twist element F ∈ U(P) ⊗ U(P)
i µν
F = exp θ Pµ ⊗ Pν
2
Drinfeld (1983)
Reshetikhin (1990)

18. Effects of noncommutativity
Twisted Poincar´ algebra
e
Effectively, the commutation relations are unchanged
[Pµ , Pν ] = 0
[Mµν , Pα ] = −i(ηµα Pν − ηνα Pµ )
[Mµν , Mαβ ] = −i(ηµα Mνβ − ηµβ Mνα − ηνα Mµβ + ηνβ Mµα )

19. Effects of noncommutativity
Twisted Poincar´ algebra
e
Effectively, the commutation relations are unchanged
[Pµ , Pν ] = 0
[Mµν , Pα ] = −i(ηµα Pν − ηνα Pµ )
[Mµν , Mαβ ] = −i(ηµα Mνβ − ηµβ Mνα − ηνα Mµβ + ηνβ Mµα )
But we change the coproduct (Leibniz rule)
∆0 (Y ) = Y ⊗ 1 + 1 ⊗ Y , Y ∈ P
∆0 (Y ) → ∆t (Y ) = F∆0 (Y )F −1

20. Effects of noncommutativity
Twisted Poincar´ algebra
e
Effectively, the commutation relations are unchanged
[Pµ , Pν ] = 0
[Mµν , Pα ] = −i(ηµα Pν − ηνα Pµ )
[Mµν , Mαβ ] = −i(ηµα Mνβ − ηµβ Mνα − ηνα Mµβ + ηνβ Mµα )
But we change the coproduct (Leibniz rule)
∆0 (Y ) = Y ⊗ 1 + 1 ⊗ Y , Y ∈ P
∆0 (Y ) → ∆t (Y ) = F∆0 (Y )F −1
and deform the multiplication
m ◦ (φ ⊗ ψ) = φψ → m ◦ F −1 (φ ⊗ ψ) ≡ φ ψ

21. Then what happens
to representations,
causality etc?

22. Effects of noncommutativity
Twisted Poincar´ algebra
e
The representation content is identical to the corresponding
commutative theory with usual Poincar´ symmetry =⇒
e
representations (fields) are classified according to their
MASS and SPIN

23. Effects of noncommutativity
Twisted Poincar´ algebra
e
The representation content is identical to the corresponding
commutative theory with usual Poincar´ symmetry =⇒
e
representations (fields) are classified according to their
MASS and SPIN
But the coproducts of Lorentz algebra generators change:
∆t (Pµ ) = ∆0 (Pµ ) = Pµ ⊗ 1 + 1 ⊗ Pµ
∆t (Mµν ) = Mµν ⊗ 1 + 1 ⊗ Mµν
1
− θαβ [(ηαµ Pν − ηαν Pµ ) ⊗ Pβ + Pα ⊗ (ηβµ Pν − ηβν Pµ )]
2

24. Effects of noncommutativity
Causality
SO(1, 3)
Minkowski 1908

25. Effects of noncommutativity
Causality
=⇒
SO(1, 3) O(1, 1)xSO(2)
Minkowski 1908 ´
Alvarez-Gaum´ et al. 2000
e

26. “In commutative theories relativistic
invariance means symmetry under Poincar´ e
tranformations whereas in the noncommutative
case symmetry under the twisted Poincar´e
transformations is needed”
— Chaichian, Presnajder and Tureanu (2004)

27. Part 2
Tomonaga-Schwinger
equation & causality

28. Tomonaga-Schwinger equation
Conventions
We consider space-like noncommutativity
 
0 0 0 0
 0 0 0 0 
θµν =  0 0 0 θ 

0 0 −θ 0

29. Tomonaga-Schwinger equation
Conventions
We consider space-like noncommutativity
 
0 0 0 0
 0 0 0 0 
θµν =  0 0 0 θ 

0 0 −θ 0
and use the notation
x µ = (˜, a), y µ = (˜ , b)
x y
x = (x 0 , x 1 ), y = (y 0 , y 1 )
˜ ˜
a = (x 2 , x 3 ), b = (y 2 , y 3 )

30. Tomonaga-Schwinger equation
Conventions
We use the integral representation of the -product
(f g )(x) = d D y d D z K(x; y , z)f (y )g (z)
1
K(x; y , z) = exp[−2i(xθ−1 y + y θ−1 z + zθ−1 x)]
πD det θ

31. Tomonaga-Schwinger equation
Conventions
We use the integral representation of the -product
(f g )(x) = d D y d D z K(x; y , z)f (y )g (z)
1
K(x; y , z) = exp[−2i(xθ−1 y + y θ−1 z + zθ−1 x)]
πD det θ
In our case the invertible part of θ is the 2x2 submatrix and thus
(f1 f2 · · · fn )(x) =
da1 da2 · · ·dan K(a; a1 , · · · , an )f1 (˜, a1 )f2 (˜, a2 ) · · · fn (˜, an )
x x x

32. Tomonaga-Schwinger equation
In commutative theory
Generalizing the Schr¨dinger equation in the interaction picture to
o
incorporate arbitrary Cauchy surfaces, we get the
Tomonaga-Schwinger equation
δ
i Ψ[σ] = Hint (x)Ψ[σ]
δσ(x)

33. Tomonaga-Schwinger equation
In commutative theory
Generalizing the Schr¨dinger equation in the interaction picture to
o
incorporate arbitrary Cauchy surfaces, we get the
Tomonaga-Schwinger equation
δ
i Ψ[σ] = Hint (x)Ψ[σ]
δσ(x)
A necessary condition to ensure the existence of solutions is
[Hint (x), Hint (x )] = 0 ,
with x and x on the space-like surface σ.

34. Tomonaga-Schwinger equation
In noncommutative theory
Moving on to NC space-time we get
δ
i Ψ[C]= Hint (x) Ψ[C] = λ[φ(x)]n Ψ[C]
δC
The existence of solutions requires

35. Tomonaga-Schwinger equation
In noncommutative theory
Moving on to NC space-time we get
δ
i Ψ[C]= Hint (x) Ψ[C] = λ[φ(x)]n Ψ[C]
δC
The existence of solutions requires
[Hint (x) , Hint (y ) ]= 0 , for x, y ∈ C ,
which can be written as

36. Tomonaga-Schwinger equation
In noncommutative theory
Moving on to NC space-time we get
δ
i Ψ[C]= Hint (x) Ψ[C] = λ[φ(x)]n Ψ[C]
δC
The existence of solutions requires
[Hint (x) , Hint (y ) ]= 0 , for x, y ∈ C ,
which can be written as
(φ . . . φ)(˜, a), (φ . . . φ)(˜ , b) =
x y
n n
= dai K(a; a1 , · · · , an ) dbi K(b; b1 , · · · , bn )
i=1 i=1
× φ(˜, a1 ) . . . φ(˜, an ), φ(˜ , b1 ) . . . φ(˜ , bn ) = 0
x x y y

37. Tomonaga-Schwinger equation
The causality condition
The commutators of products of fields decompose into factors like
x x y
φ(˜, a1 ) . . . φ(˜, an−1 )φ(˜ , b1 ) . . . φ(˜ , bn−1 ) φ(˜, an ), φ(˜ , bn )
y x y

38. Tomonaga-Schwinger equation
The causality condition
The commutators of products of fields decompose into factors like
x x y
φ(˜, a1 ) . . . φ(˜, an−1 )φ(˜ , b1 ) . . . φ(˜ , bn−1 ) φ(˜, an ), φ(˜ , bn )
y x y
All products of fields being independent,
the necessary condition is
φ(˜, ai ), φ(˜ , bj ) = 0
x y

39. Tomonaga-Schwinger equation
The causality condition
The commutators of products of fields decompose into factors like
x x y
φ(˜, a1 ) . . . φ(˜, an−1 )φ(˜ , b1 ) . . . φ(˜ , bn−1 ) φ(˜, an ), φ(˜ , bn )
y x y
All products of fields being independent,
the necessary condition is
φ(˜, ai ), φ(˜ , bj ) = 0
x y
Since fields in the interaction picture satisfy free-field equations,
this is satisfied outside the mutual light-cone:
(x 0 − y 0 )2 − (x 1 − y 1 )2 − (ai2 − bj2 ) − (ai3 − bj3 )2 < 0

40. All the hard work and
we end up with
the light-cone?

41. Tomonaga-Schwinger equation
The causality condition
However, since a and b are integration variables in the range
0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞
the causality condition is not in fact

42. Tomonaga-Schwinger equation
The causality condition
However, since a and b are integration variables in the range
0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞
the causality condition is not in fact
(x 0 − y 0 )2 − (x 1 − y 1 )2 − (ai2 − bj2 ) − (ai3 − bj3 )2 < 0

43. Tomonaga-Schwinger equation
The causality condition
However, since a and b are integration variables in the range
0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞
the necessary condition becomes

44. Tomonaga-Schwinger equation
The causality condition
However, since a and b are integration variables in the range
0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞
the necessary condition becomes
(x 0 − y 0 )2 − (x 1 − y 1 )2 < 0

45. Tomonaga-Schwinger equation
The causality condition
However, since a and b are integration variables in the range
0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞
the necessary condition becomes
(x 0 − y 0 )2 − (x 1 − y 1 )2 < 0
This is the light-wedge causality condition, invariant under the
stability group of θµν ,O(1, 1) × SO(2).
Chaichian, Nishijima, Salminen and Tureanu (2008)

46. Tomonaga-Schwinger equation
The causality condition
This is the light-wedge causality condition, invariant under the
stability group of θµν ,O(1, 1) × SO(2).
Chaichian, Nishijima, Salminen and Tureanu (2008)

47. Part 3
Confrontation of
symmetries

48. Confrontation of symmetries
Twisted Poincar´ algebra
e
Writing down the coproducts of Lorentz generators (only θ23 = 0):

49. Confrontation of symmetries
Twisted Poincar´ algebra
e
Writing down the coproducts of Lorentz generators (only θ23 = 0):
∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
∆t (M23 ) = ∆0 (M23 ) = M23 ⊗ 1 + 1 ⊗ M23
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2
θ
∆t (M03 ) = ∆0 (M03 ) − (P0 ⊗ P2 − P2 ⊗ P0 )
2
θ
∆t (M12 ) = ∆0 (M12 ) + (P1 ⊗ P3 − P3 ⊗ P1 )
2
θ
∆t (M13 ) = ∆0 (M13 ) − (P1 ⊗ P2 − P2 ⊗ P1 )
2

50. Confrontation of symmetries
Twisted Poincar´ algebra
e
Writing down the coproducts of Lorentz generators (only θ23 = 0):
∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
∆t (M23 ) = ∆0 (M23 ) = M23 ⊗ 1 + 1 ⊗ M23
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2
θ
∆t (M03 ) = ∆0 (M03 ) − (P0 ⊗ P2 − P2 ⊗ P0 )
2
θ
∆t (M12 ) = ∆0 (M12 ) + (P1 ⊗ P3 − P3 ⊗ P1 )
2
θ
∆t (M13 ) = ∆0 (M13 ) − (P1 ⊗ P2 − P2 ⊗ P1 )
2
⇒ A hint of O(1, 1)xSO(2) invariance.

51. Confrontation of symmetries
Hopf dual algebra
The coproducts induce commutation relations in the
dual algebra Fθ (G ):
[aµ , aν ] = iθµν − iΛµ Λν θαβ
α β
[Λµ , aα ] = [Λµ , Λν ] = 0;
ν α β Λµ , aµ ∈ Fθ (G )
α
αP αβ M
aµ e ia α
= aµ ; Λµ e iω
ν
αβ
= (Λαβ (ω))µ
ν

52. Confrontation of symmetries
Hopf dual algebra
The coproducts induce commutation relations in the
dual algebra Fθ (G ):
[aµ , aν ] = iθµν − iΛµ Λν θαβ
α β
[Λµ , aα ] = [Λµ , Λν ] = 0;
ν α β Λµ , aµ ∈ Fθ (G )
α
αP αβ M
aµ e ia α
= aµ ; Λµ e iω
ν
αβ
= (Λαβ (ω))µ
ν
Coordinates change by coaction, but [xµ , xν ] = iθµν is preserved
(x )µ = δ(x µ ) = Λµ ⊗ x α + aµ ⊗ 1
α
[xµ , xν ]= iθµν

53. Confrontation of symmetries
A simple example
 
cosh α sinh α 0 0
 sinh α cosh α 0 0 
Λ01 =
 
0 0 1 0 
0 0 0 1
 
1 0 0 0
 0 1 0 0 
Λ23 =
 0

0 cos γ sin γ 
0 0 − sin γ cos γ
 
1 0 0 0
 0 cos β sin β 0 
Λ12 =
 
0 − sin β cos β 0 
0 0 0 1

54. Confrontation of symmetries
A simple example
 
cosh α sinh α 0 0
Λ01
 sinh α
=
 cosh α 0 0 
 [aµ , aν ] = 0
0 0 1 0 
0 0 0 1
 
1 0 0 0
 0 1 0 0  [aµ , aν ] = 0
Λ23 =
 0

0 cos γ sin γ 
0 0 − sin γ cos γ
 
1 0 0 0
 0 cos β sin β 0  [a2 , a3 ] = iθ(1 − cos β)
Λ12 =
 
0 − sin β cos β 0  [a1 , a3 ] = −iθ sin β
0 0 0 1

55. By imposing a Lorentz transformation
we get accompanying noncommuting translations
showing up as the internal mechanism by which
the twisted Poincar´ symmetry keeps the
e
commutator [xµ , xν ] = iθµν invariant

56. Theory of induced representations
Fields in commutative space
A commutative relativistic field carries a Lorentz
representation and is a function of x µ ∈ R1,3

57. Theory of induced representations
Fields in commutative space
A commutative relativistic field carries a Lorentz
representation and is a function of x µ ∈ R1,3
It is an element of C ∞ (R1,3 ) ⊗ V , where V is a
Lorentz-module. The elements are defined as:
Φ= fi ⊗ vi , fi ∈ C ∞ (R1,3 ) , vi ∈ V
i

58. Theory of induced representations
Fields in commutative space
A commutative relativistic field carries a Lorentz
representation and is a function of x µ ∈ R1,3
It is an element of C ∞ (R1,3 ) ⊗ V , where V is a
Lorentz-module. The elements are defined as:
Φ= fi ⊗ vi , fi ∈ C ∞ (R1,3 ) , vi ∈ V
i
⇒ Action of Lorentz generators on a field requires the coproduct
Chaichian, Kulish, Tureanu, Zhang and Zhang (2007)

59. Theory of induced representations
Fields in noncommutative space
In NC space we need the twisted coproduct, for example:
∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2

60. Theory of induced representations
Fields in noncommutative space
In NC space we need the twisted coproduct, for example:
∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2
If V is a Lorentz module in Φ = i fi ⊗ vi , vi ∈ V , the Pµ of
M02 cannot act on Φ

61. Theory of induced representations
Fields in noncommutative space
In NC space we need the twisted coproduct, for example:
∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2
If V is a Lorentz module in Φ = i fi ⊗ vi , vi ∈ V , the Pµ of
M02 cannot act on Φ
Our proposition: Retain V as a Lorentz-module but forbid all
the transformations requiring the action of Pµ on vi
Chaichian, Nishijima, Salminen and Tureanu (2008)

62. Theory of induced representations
Fields in noncommutative space
In NC space we need the twisted coproduct, for example:
∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2
If V is a Lorentz module in Φ = i fi ⊗ vi , vi ∈ V , the Pµ of
M02 cannot act on Φ
Our proposition: Retain V as a Lorentz-module but forbid all
the transformations requiring the action of Pµ on vi
Chaichian, Nishijima, Salminen and Tureanu (2008)
⇒ Only transformations of O(1, 1) × SO(2) allowed

63. The fields on NC space-time live in C ∞ (R1,1 × R2 ) ⊗ V ,
thus carrying representations of the full Lorentz group,
but admitting only the action of the generators of
the stability group of θµν , i.e. O(1, 1) × SO(2)

64. In Sum

65. In Sum
Requiring solutions to the
Tomonaga-Schwinger eq.
→ light-wedge causality.

66. In Sum
Requiring solutions to the
Tomonaga-Schwinger eq.
→ light-wedge causality.
Properties of O(1, 1)xSO(2)
& twisted Poincar´ invariance
e
→ field definitions compatible
with the light-wedge.

67. Thank you
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