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Information about Confrontation of symmetries

Published on May 29, 2008

Author: lxsalmin

Source: slideshare.net

A seminar talk on NC QFT

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Part 1 Introduction

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)

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)

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

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

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)

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

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.

Does this mean Lorentz invariance is lost?

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

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).

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.

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

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

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

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)

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 )

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

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

Then what happens to representations, causality etc?

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

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

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

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

“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)

Part 2 Tomonaga-Schwinger equation & causality

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

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 )

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

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

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)

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.

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

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

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

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

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

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

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

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

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

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

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

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)

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)

Part 3 Confrontation of symmetries

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

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

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.

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

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

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

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

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

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

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

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)

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

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

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)

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

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)

In Sum

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

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.

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