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- 1. Noncommutative Quantum Field Theory: Problems of nonlocal time Tapio Salminen University of Helsinki Noncommutative Quantum Field Theory: A Confrontation of Symmetries M. Chaichian, K. Nishijima, TS and A. Tureanu On Noncommutative Time in Quantum Field Theory TS and A. Tureanu
- 2. Part 1Introduction
- 3. Quantizing space-time MotivationBlack hole formation in the process of measurement at smalldistances (∼ λP ) ⇒ additional uncertainty relations forcoordinates Doplicher, Fredenhagen and Roberts (1994)
- 4. Quantizing space-time MotivationBlack hole formation in the process of measurement at smalldistances (∼ λP ) ⇒ additional uncertainty relations forcoordinates Doplicher, Fredenhagen and Roberts (1994)Open string + D-brane theory with an antisymmetric Bij ﬁeldbackground ⇒ noncommutative coordinates in low-energylimit Seiberg and Witten (1999)
- 5. Quantizing space-time MotivationBlack hole formation in the process of measurement at smalldistances (∼ λP ) ⇒ additional uncertainty relations forcoordinates Doplicher, Fredenhagen and Roberts (1994)Open string + D-brane theory with an antisymmetric Bij ﬁeldbackground ⇒ noncommutative coordinates in low-energylimit Seiberg and Witten (1999) A possible approach to Planck scale physics is QFT in NC space-time
- 6. Quantizing space-time ImplementationWe 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 ImplementationWe 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 noncommutativity also between the coordinate operators [ˆµ , x ν ] = 0 x ˆ Snyder (1947); Heisenberg (1954); Golfand (1962)
- 8. Quantizing space-time ImplementationWe 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 ImplementationWe 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 transformations.
- 10. Does this meanLorentz invarianceis lost?
- 11. Quantizing space-time ImplementationWe 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 -productIn noncommuting space-time the analogue of the action 1 µ 1 λS (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 λS 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 symmetryThe action of NC QFT written with the -product, though itviolates Lorentz symmetry, is invariant under the twistedPoincar´ algebra e Chaichian, Kulish, Nishijima and Tureanu (2004) Chaichian, Preˇnajder and Tureanu (2004) s
- 17. Eﬀects of noncommutativity The actual symmetryThe action of NC QFT written with the -product, though itviolates Lorentz symmetry, is invariant under the twistedPoincar´ algebra e Chaichian, Kulish, Nishijima and Tureanu (2004) Chaichian, Preˇnajder and Tureanu (2004) sThis is achieved by deforming the universal enveloping of thePoincar´ algebra U(P) as a Hopf algebra with the Abelian etwist 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 happensto representations,causality etc?
- 22. Eﬀects of noncommutativity Twisted Poincar´ algebra eThe representation content is identical to the correspondingcommutative theory with usual Poincar´ symmetry =⇒ erepresentations (ﬁelds) are classiﬁed according to theirMASS and SPIN
- 23. Eﬀects of noncommutativity Twisted Poincar´ algebra eThe representation content is identical to the correspondingcommutative theory with usual Poincar´ symmetry =⇒ erepresentations (ﬁelds) are classiﬁed according to theirMASS and SPINBut 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 CausalitySO(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. Part 2Noncommutative time and unitarity
- 27. Noncommutative time String theory limitsUntil now we have had all coordinates noncommutative 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0
- 28. Noncommutative time String theory limits The low-energy limit of string theory with a background Bij ﬁeld gives 0 0 0 0 0 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0This is referred to as space-like noncommutativity.
- 29. Noncommutative time String theory limitsThis string theory is S-dual to another string theory with an Eij background. There we would have 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 0 0 0 0 0 The so called time-like noncommutativity.
- 30. Noncommutative time String theory limitsThis string theory is S-dual to another string theory with an Eij background. There we would have 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 0 0 0 0 0 The so called time-like noncommutativity.However, it has been shown that the low-energy limit does not exist for these theories. Seiberg and Witten (1999)
- 31. Could you please stop talking about strings?
- 32. Noncommutative time UnitarityWe may still consider quantum ﬁeld theories with 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0
- 33. Noncommutative time UnitarityWe may still consider quantum ﬁeld theories with 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 But in the interaction picture it has been shown that perturbative unitarity requires 2 2 2 2 θ (p0 − p1 ) + θ(p2 + p3 ) > 0
- 34. Noncommutative time Unitarity We may still consider quantum ﬁeld theories with 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 But in the interaction picture it has been shown that perturbative unitarity requires 2 2 2 2 θ (p0 − p1 ) + θ(p2 + p3 ) > 0Time-like noncommutativity → violation of unitarity Gomis and Mehen (2000)
- 35. Noncommutative time UnitarityWe may still consider quantum ﬁeld theories with 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 → Forget about the interaction picture and go to the Heisenberg picture.
- 36. Noncommutative time Unitarity We may still consider quantum ﬁeld theories with 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 → Forget about the interaction picture and go to the Heisenberg picture.However, using the Yang-Feldman approach one can show: S † ψin (x)S = ψout (x) + g 4 (· · · ) = ψout (x) Salminen and Tureanu (2010)
- 37. Noncommutative time Unitarity We may still consider quantum ﬁeld theories with 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 → Forget about the interaction picture and go to the Heisenberg picture.However, using the Yang-Feldman approach one can show: There is no unitary S-matrix. Salminen and Tureanu (2010)
- 38. Part 3Tomonaga-Schwinger equation & causality
- 39. 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
- 40. 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 )
- 41. 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) 1K(x; y , z) = exp[−2i(xθ−1 y + y θ−1 z + zθ−1 x)] πD det θ
- 42. 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 θIn 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
- 43. Tomonaga-Schwinger equation In commutative theoryGeneralizing 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)
- 44. Tomonaga-Schwinger equation In commutative theoryGeneralizing 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.
- 45. 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
- 46. 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
- 47. 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
- 48. Tomonaga-Schwinger equation The causality conditionThe commutators of products of ﬁelds decompose into factors like φ(˜, a1 ) . . . φ(˜, an−1 )φ(˜ , b1 ) . . . φ(˜ , bn−1 ) φ(˜, an ), φ(˜ , bn ) x x y y x y
- 49. Tomonaga-Schwinger equation The causality conditionThe commutators of products of ﬁelds decompose into factors like φ(˜, a1 ) . . . φ(˜, an−1 )φ(˜ , b1 ) . . . φ(˜ , bn−1 ) φ(˜, an ), φ(˜ , bn ) x x y y x y All products of ﬁelds being independent, the necessary condition is φ(˜, ai ), φ(˜ , bj ) = 0 x y
- 50. Tomonaga-Schwinger equation The causality conditionThe commutators of products of ﬁelds decompose into factors like φ(˜, a1 ) . . . φ(˜, an−1 )φ(˜ , b1 ) . . . φ(˜ , bn−1 ) φ(˜, an ), φ(˜ , bn ) x x y y x y All products of ﬁelds being independent, the necessary condition is φ(˜, ai ), φ(˜ , bj ) = 0 x ySince 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
- 51. All the hard work andwe end up withthe light-cone?
- 52. Tomonaga-Schwinger equation The causality conditionHowever, since a and b are integration variables in the range 0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞ the causality condition is not in fact
- 53. Tomonaga-Schwinger equation The causality conditionHowever, 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
- 54. Tomonaga-Schwinger equation The causality conditionHowever, since a and b are integration variables in the range 0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞ the necessary condition becomes
- 55. Tomonaga-Schwinger equation The causality conditionHowever, 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
- 56. 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 < 0This is the light-wedge causality condition, invariant under the stability group of θµν ,O(1, 1) × SO(2). Chaichian, Nishijima, Salminen and Tureanu (2008)
- 57. Tomonaga-Schwinger equation The causality conditionThis is the light-wedge causality condition, invariant under the stability group of θµν ,O(1, 1) × SO(2). Chaichian, Nishijima, Salminen and Tureanu (2008)
- 58. Tomonaga-Schwinger equation The causality condition If we had taken 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0
- 59. Tomonaga-Schwinger equation The causality condition If we had taken 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 we would change(x 0 − y 0 )2 − (x 1 − y 1 )2 − (ai2 − bj2 ) − (ai3 − bj3 )2 < 0 Chaichian, Nishijima, Salminen and Tureanu (2008)
- 60. Tomonaga-Schwinger equation The causality condition If we had taken 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 into(ai0 − bj0 )2 − (ai1 − bj1 )2 −(ai2 − bj2 ) − (ai3 − bj3 )2 < 0 Chaichian, Nishijima, Salminen and Tureanu (2008) Salminen and Tureanu (2010)
- 61. Tomonaga-Schwinger equation The causality condition If we had taken 0 θ 0 0 −θ 0 0 0 θµν = 0 0 0 θ 0 0 −θ 0 into(ai0 − bj0 )2 − (ai1 − bj1 )2 −(ai2 − bj2 ) − (ai3 − bj3 )2 < 0 → No solution to the Tomonaga-Schwinger equation for any x and y . Salminen and Tureanu (2010)
- 62. In Sum
- 63. In SumRequiring solutions to theTomonaga-Schwinger eq. → light-wedge causality.
- 64. In SumRequiring solutions to theTomonaga-Schwinger eq. → light-wedge causality. Unitarity & causality violated in theories with noncommutative time.
- 65. Thank you Photo credits everystockphoto.com“Meet Charlotte” @ slideshare.net
- 66. Extra material Confrontation of symmetries
- 67. Confrontation of symmetries Twisted Poincar´ algebra eWriting down the coproducts of Lorentz generators (only θ23 = 0):
- 68. Confrontation of symmetries Twisted Poincar´ algebra eWriting 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
- 69. Confrontation of symmetries Twisted Poincar´ algebra eWriting 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.
- 70. Confrontation of symmetries Hopf dual algebraThe 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
- 71. 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 νCoordinates change by coaction, but [xµ , xν ] = iθµν is preserved (x )µ = δ(x µ ) = Λµ ⊗ x α + aµ ⊗ 1 α [xµ , xν ]= iθµν
- 72. Confrontation of symmetries A simple example 0 1 cosh α sinh α 0 0 B sinh α cosh α 0 0 CΛ01 =@ B C 0 0 1 0 A 0 0 0 1 0 1 1 0 0 0 B 0 1 0 0 CΛ23 =B @ 0 C 0 cos γ sin γ A 0 0 − sin γ cos γ 0 1 1 0 0 0 B 0 cos β sin β 0 CΛ12 =@ B C 0 − sin β cos β 0 A 0 0 0 1
- 73. Confrontation of symmetries A simple example 0 1 cosh α sinh α 0 0Λ01 B sinh α =@ B cosh α 0 0 C C [aµ , aν ] = 0 0 0 1 0 A 0 0 0 1 0 1 1 0 0 0 B 0 1 0 0 C [aµ , aν ] = 0Λ23 =B @ 0 C 0 cos γ sin γ A 0 0 − sin γ cos γ 0 1 1 0 0 0 B 0 cos β sin β 0 C [a2 , a3 ] = iθ(1 − cos β)Λ12 =@ B C 0 − sin β cos β 0 A [a1 , a3 ] = −iθ sin β 0 0 0 1
- 74. By imposing a Lorentz transformationwe get accompanying noncommuting translationsshowing up as the internal mechanism by whichthe twisted Poincar´ symmetry keeps the ecommutator [xµ , xν ] = iθµν invariant
- 75. Theory of induced representations Fields in commutative spaceA commutative relativistic ﬁeld carries a Lorentzrepresentation and is a function of x µ ∈ R1,3
- 76. Theory of induced representations Fields in commutative spaceA commutative relativistic ﬁeld carries a Lorentzrepresentation and is a function of x µ ∈ R1,3It is an element of C ∞ (R1,3 ) ⊗ V , where V is aLorentz-module. The elements are deﬁned as: Φ= fi ⊗ vi , fi ∈ C ∞ (R1,3 ) , vi ∈ V i
- 77. 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)
- 78. Theory of induced representations Fields in noncommutative spaceIn 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
- 79. Theory of induced representations Fields in noncommutative spaceIn 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 ) 2If V is a Lorentz module in Φ = i fi ⊗ vi , vi ∈ V , the Pµ ofM02 cannot act on Φ
- 80. Theory of induced representations Fields in noncommutative spaceIn 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 ) 2If V is a Lorentz module in Φ = i fi ⊗ vi , vi ∈ V , the Pµ ofM02 cannot act on ΦOur proposition: Retain V as a Lorentz-module but forbid allthe transformations requiring the action of Pµ on vi Chaichian, Nishijima, Salminen and Tureanu (2008)
- 81. Theory of induced representations Fields in noncommutative spaceIn 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 ) 2If V is a Lorentz module in Φ = i fi ⊗ vi , vi ∈ V , the Pµ ofM02 cannot act on ΦOur proposition: Retain V as a Lorentz-module but forbid allthe transformations requiring the action of Pµ on vi Chaichian, Nishijima, Salminen and Tureanu (2008) ⇒ Only transformations of O(1, 1) × SO(2) allowed
- 82. 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)

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