confrontation of symmetries

Noncommutative Quantum Field Theory:

A Confrontation of symmetries

Tapio SalminenUniversity of Helsinki

Based on work done in collaboration withM. Chaichian, K. Nishijima and A. Tureanu

JHEP 06 (2008) 078, arXiv: 0805.3500

Part 1Introduction

Quantizing space-timeMotivation

Black 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 tensorbackground (NOT induced!)

Ardalan, Arfaei and Sheikh-Jabbari (1998)

Seiberg and Witten (1999)

VA possible approach to physics at short distances isQFT in NC space-time

Quantizing space-timeImplementation

We generalize the commutation relations fromusual quantum mechanics

[xi , xj ] = 0 , [pi , pj ] = 0[xi , pj ] = i~δij

by imposing noncommuttativity also betweenthe coordinate operators

[xµ, xν ] 6= 0

Snyder (1947); Heisenberg (1954);

Golfand (1962)

We generalize the commutation relations fromusual quantum mechanics

[xi , xj ] = 0 , [pi , pj ] = 0[xi , pj ] = i~δij

by imposing noncommuttativity also betweenthe coordinate operators

[xµ, xν ] 6= 0

Snyder (1947); Heisenberg (1954);

Golfand (1962)

We take [xµ, xν ] = iθµν and choose the frame where

θµν =

0 θ′ 0 0−θ′ 0 0 0

0 0 0 θ0 0 −θ 0

θµν does not transform under Lorentztranformations.

θµν =

0 θ′ 0 0−θ′ 0 0 0

0 0 0 θ0 0 −θ 0

θµν does not transform under Lorentz

tranformations.

Does this meanLorentz invarianceis lost?

θµν =

0 θ′ 0 0−θ′ 0 0 0

0 0 0 θ0 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.

θµν =

0 θ′ 0 0−θ′ 0 0 0

0 0 0 θ0 0 −θ 0

Translational invariance is preserved,

but the Lorentz group breaks down to SO(1, 1)xSO(2).

θµν =

0 θ′ 0 0−θ′ 0 0 0

0 0 0 θ0 0 −θ 0

Translational invariance is preserved,

but the Lorentz group breaks down to SO(1, 1)xSO(2).

Effects of noncommutativityMoyal ?-product

In noncommuting space-time the analogue of the action

S (cl)[Φ] =

∫d4x

2(∂µΦ)(∂µΦ)− 1

2m2Φ2 − λ

]can be written using the Moyal ?-product

Sθ[Φ] =

∫d4x

2(∂µΦ) ? (∂µΦ)− 1

2m2Φ ? Φ− λ

4!Φ ? Φ ? Φ ? Φ

(Φ ?Ψ) (x) ≡[

Φ(x)ei2θµν

←−∂∂xµ

−→∂∂yν Ψ(y)

Effects of noncommutativityMoyal ?-product

In noncommuting space-time the analogue of the action

S (cl)[Φ] =

∫d4x

2(∂µΦ)(∂µΦ)− 1

2m2Φ2 − λ

]can be written using the Moyal ?-product

Sθ[Φ] =

∫d4x

2(∂µΦ) ? (∂µΦ)− 1

2m2Φ ? Φ− λ

4!Φ ? Φ ? Φ ? Φ

(Φ ?Ψ) (x) ≡[

Φ(x)ei2θµν

←−∂∂xµ

−→∂∂yν Ψ(y)

Effects of noncommutativityThe actual symmetry

The action of NC QFT written with the ?-product, though itviolates Lorentz symmetry, is invariant under the twistedPoincare algebra

Chaichian, Kulish, Nishijima and Tureanu (2004)

Chaichian, Presnajder and Tureanu (2004)

This is achieved by deforming the universal enveloping of thePoincare algebra U(P) as a Hopf algebra with the Abeliantwist element F ∈ U(P)⊗ U(P)

F = exp

2θµνPµ ⊗ Pν

)Drinfeld (1983)

Reshetikhin (1990)

Effects of noncommutativityThe actual symmetry

The action of NC QFT written with the ?-product, though itviolates Lorentz symmetry, is invariant under the twistedPoincare algebra

Chaichian, Kulish, Nishijima and Tureanu (2004)

Chaichian, Presnajder and Tureanu (2004)

This is achieved by deforming the universal enveloping of thePoincare algebra U(P) as a Hopf algebra with the Abeliantwist element F ∈ U(P)⊗ U(P)

F = exp

2θµνPµ ⊗ Pν

)Drinfeld (1983)

Reshetikhin (1990)

Effects of noncommutativityTwisted Poincare algebra

Effectively, the commutation relations are unchanged

[Pµ,Pν ] = 0[Mµν ,Pα] = −i(ηµαPν − ηναPµ)

[Mµν ,Mαβ] = −i(ηµαMνβ − ηµβMνα − ηναMµβ + ηνβMµα)

But we change the coproduct (Leibniz rule)

∆0(Y ) = Y ⊗ 1 + 1⊗ Y ,Y ∈ P∆0(Y ) 7→∆t(Y ) = F∆0(Y )F−1

and deform the multiplication

m ◦ (φ⊗ ψ) = φψ → m ◦ F−1(φ⊗ ψ) ≡ φ ? ψ

∆0(Y ) = Y ⊗ 1 + 1⊗ Y ,Y ∈ P∆0(Y ) 7→∆t(Y ) = F∆0(Y )F−1

m ◦ (φ⊗ ψ) = φψ → m ◦ F−1(φ⊗ ψ) ≡ φ ? ψ

∆0(Y ) = Y ⊗ 1 + 1⊗ Y ,Y ∈ P∆0(Y ) 7→∆t(Y ) = F∆0(Y )F−1

m ◦ (φ⊗ ψ) = φψ → m ◦ F−1(φ⊗ ψ) ≡ φ ? ψ

Then what happensto representations,causality etc?

The representation content is identical to the correspondingcommutative theory with usual Poincare symmetry =⇒representations (fields) are classified according to theirMASS and SPIN

But the coproducts of Lorentz algebra generators change:

∆t(Pµ) = ∆0(Pµ) = Pµ ⊗ 1 + 1⊗ Pµ

∆t(Mµν) = Mµν ⊗ 1 + 1⊗Mµν

2θαβ [(ηαµPν − ηανPµ)⊗ Pβ + Pα ⊗ (ηβµPν − ηβνPµ)]

The representation content is identical to the correspondingcommutative theory with usual Poincare symmetry =⇒representations (fields) are classified according to theirMASS and SPIN

But the coproducts of Lorentz algebra generators change:

∆t(Pµ) = ∆0(Pµ) = Pµ ⊗ 1 + 1⊗ Pµ

∆t(Mµν) = Mµν ⊗ 1 + 1⊗Mµν

2θαβ [(ηαµPν − ηανPµ)⊗ Pβ + Pα ⊗ (ηβµPν − ηβνPµ)]

Effects of noncommutativityCausality

SO(1, 3)

Minkowski 1908

O(1, 1)xSO(2)

Alvarez-Gaume et al. 2000

Effects of noncommutativityCausality

SO(1, 3)

Minkowski 1908

O(1, 1)xSO(2)

Alvarez-Gaume et al. 2000

“In commutative theories relativisticinvariance means symmetry under Poincaretranformations whereas in the noncommutativecase symmetry under the twisted Poincaretransformations is needed”

— Chaichian, Presnajder and Tureanu (2004)

Part 2Tomonaga-Schwingerequation & causality

Tomonaga-Schwinger equationConventions

We consider space-like noncommutativity

θµν =

0 0 0 00 0 0 00 0 0 θ0 0 −θ 0

and use the notation

xµ = (x , a), yµ = (y ,b)

x = (x0, x1), y = (y0, y1)

a = (x2, x3), b = (y2, y3)

We consider space-like noncommutativity

θµν =

0 0 0 00 0 0 00 0 0 θ0 0 −θ 0

and use the notation

xµ = (x , a), yµ = (y ,b)

x = (x0, x1), y = (y0, y1)

a = (x2, x3), b = (y2, y3)

We use the integral representation of the ?-product

(f ? g)(x) =

∫dDy dDz K(x ; y , z)f (y)g(z)

K(x ; y , z) =1

πD det θexp[−2i(xθ−1y + yθ−1z + zθ−1x)]

In our case the invertible part of θ is the 2x2 submatrix and thus

(f1 ? f2 ? · · · ? fn)(x) =∫da1da2 · · ·danK(a; a1, · · · , an)f1(x , a1)f2(x , a2) · · · fn(x , an)

We use the integral representation of the ?-product

(f ? g)(x) =

∫dDy dDz K(x ; y , z)f (y)g(z)

K(x ; y , z) =1

πD det θexp[−2i(xθ−1y + yθ−1z + zθ−1x)]

In our case the invertible part of θ is the 2x2 submatrix and thus

(f1 ? f2 ? · · · ? fn)(x) =∫da1da2 · · ·danK(a; a1, · · · , an)f1(x , a1)f2(x , a2) · · · fn(x , an)

Tomonaga-Schwinger equationIn commutative theory

Generalizing the Schrodinger equation in the interaction picture toincorporate arbitrary Cauchy surfaces, we get the

Tomonaga-Schwinger equation

δσ(x)Ψ[σ] = Hint(x)Ψ[σ]

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 equationIn commutative theory

Generalizing the Schrodinger equation in the interaction picture toincorporate arbitrary Cauchy surfaces, we get the

Tomonaga-Schwinger equation

δσ(x)Ψ[σ] = Hint(x)Ψ[σ]

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 equationIn noncommutative theory

Moving on to NC space-time we get

δCΨ[C]= Hint(x)?Ψ[C] = λ[φ(x)]n?Ψ[C]

The existence of solutions requires

[Hint(x)?,Hint(y)?]= 0 , for x , y ∈ C ,

which can be written as[(φ ? . . . ?φ)(x , a), (φ ? . . . ? φ)(y ,b)

∫ n∏i=1

da′i K(a; a′1, · · · , a′n)

∫ n∏i=1

db′i K(b; b′1, · · · ,b′n)

×[φ(x , a′1) . . . φ(x , a′n), φ(y ,b′1) . . . φ(y ,b′n)

which can be written as

[(φ ? . . . ?φ)(x , a), (φ ? . . . ? φ)(y ,b)

∫ n∏i=1

da′i K(a; a′1, · · · , a′n)

∫ n∏i=1

db′i K(b; b′1, · · · ,b′n)

×[φ(x , a′1) . . . φ(x , a′n), φ(y ,b′1) . . . φ(y ,b′n)

which can be written as[(φ ? . . . ?φ)(x , a), (φ ? . . . ? φ)(y ,b)

∫ n∏i=1

da′i K(a; a′1, · · · , a′n)

∫ n∏i=1

db′i K(b; b′1, · · · ,b′n)

×[φ(x , a′1) . . . φ(x , a′n), φ(y ,b′1) . . . φ(y ,b′n)

Tomonaga-Schwinger equationThe causality condition

The commutators of products of fields decompose into factors like

φ(x , a′1) . . . φ(x , a′n−1)φ(y ,b′1) . . . φ(y ,b′n−1)[φ(x , a′n), φ(y ,b′n)

All products of fields being independent,the necessary condition is[φ(x , a′i ), φ(y ,b′j)

Since fields in the interaction picture satisfy free-field equations,this is satisfied outside the mutual light-cone:

(x0 − y0)2 − (x1 − y1)2 − (a2i′ − b2

j′)− (a3

i′ − b3

j′)2 < 0

]All products of fields being independent,

the necessary condition is[φ(x , a′i ), φ(y ,b′j)

(x0 − y0)2 − (x1 − y1)2 − (a2i′ − b2

j′)− (a3

i′ − b3

j′)2 < 0

]All products of fields being independent,

the necessary condition is[φ(x , a′i ), φ(y ,b′j)

(x0 − y0)2 − (x1 − y1)2 − (a2i′ − b2

j′)− (a3

i′ − b3

j′)2 < 0

All the hard work andwe end up withthe light-cone?

However, since a and b are integration variables in the range

0 ≤ (a2i′ − b2

j′)2 + (a3

i′ − b3

j′)2 <∞

the causality condition is not in fact

(x0 − y0)2 − (x1 − y1)2 − (a2i′ − b2

j′)− (a3

i′ − b3

j′)2 < 0

0 ≤ (a2i′ − b2

j′)2 + (a3

i′ − b3

j′)2 <∞

the causality condition is not in fact

(x0 − y0)2 − (x1 − y1)2 − (a2i′ − b2

j′)− (a3

i′ − b3

j′)2 < 0

0 ≤ (a2i′ − b2

j′)2 + (a3

i′ − b3

j′)2 <∞

the necessary condition becomes

(x0 − y0)2 − (x1 − y1)2 < 0

This is the light-wedge causality condition, invariant under thestability group of θµν ,O(1, 1)× SO(2).

Chaichian, Nishijima, Salminen and Tureanu (2008)

0 ≤ (a2i′ − b2

j′)2 + (a3

i′ − b3

j′)2 <∞

(x0 − y0)2 − (x1 − y1)2 < 0

0 ≤ (a2i′ − b2

j′)2 + (a3

i′ − b3

j′)2 <∞

(x0 − y0)2 − (x1 − y1)2 < 0

Part 3Confrontation of

symmetries

Confrontation of symmetriesTwisted Poincare algebra

Writing down the coproducts of Lorentz generators (only θ23 6= 0):

∆t(M01) = ∆0(M01) = M01 ⊗ 1 + 1⊗M01

∆t(M23) = ∆0(M23) = M23 ⊗ 1 + 1⊗M23

∆t(M02) = ∆0(M02) +θ

2(P0 ⊗ P3 − P3 ⊗ P0)

∆t(M03) = ∆0(M03)− θ

2(P0 ⊗ P2 − P2 ⊗ P0)

∆t(M12) = ∆0(M12) +θ

2(P1 ⊗ P3 − P3 ⊗ P1)

∆t(M13) = ∆0(M13)− θ

2(P1 ⊗ P2 − P2 ⊗ P1)

⇒ A hint of O(1, 1)xSO(2) invariance.

∆t(M01) = ∆0(M01) = M01 ⊗ 1 + 1⊗M01

∆t(M23) = ∆0(M23) = M23 ⊗ 1 + 1⊗M23

∆t(M02) = ∆0(M02) +θ

2(P0 ⊗ P3 − P3 ⊗ P0)

∆t(M03) = ∆0(M03)− θ

2(P0 ⊗ P2 − P2 ⊗ P0)

∆t(M12) = ∆0(M12) +θ

2(P1 ⊗ P3 − P3 ⊗ P1)

∆t(M13) = ∆0(M13)− θ

2(P1 ⊗ P2 − P2 ⊗ P1)

∆t(M01) = ∆0(M01) = M01 ⊗ 1 + 1⊗M01

∆t(M23) = ∆0(M23) = M23 ⊗ 1 + 1⊗M23

∆t(M02) = ∆0(M02) +θ

2(P0 ⊗ P3 − P3 ⊗ P0)

∆t(M03) = ∆0(M03)− θ

2(P0 ⊗ P2 − P2 ⊗ P0)

∆t(M12) = ∆0(M12) +θ

2(P1 ⊗ P3 − P3 ⊗ P1)

∆t(M13) = ∆0(M13)− θ

2(P1 ⊗ P2 − P2 ⊗ P1)

Confrontation of symmetriesHopf dual algebra

The coproducts induce commutation relations in thedual algebra Fθ(G ):

[aµ, aν ] = iθµν − iΛµαΛνβθαβ

[Λµν , aα] = [Λµα,Λ

νβ] = 0; Λµα, a

µ ∈ Fθ(G )

aµ(e iaαPα

)= aµ; Λµν

(e iωαβMαβ

)= (Λαβ(ω))µν

Coordinates change by coaction, but [xµ, xν ] = iθµν is preserved

(x ′)µ = δ(xµ) = Λµα ⊗ xα + aµ ⊗ 1

[x ′µ, x′ν ]= iθµν

Confrontation of symmetriesHopf dual algebra

The coproducts induce commutation relations in thedual algebra Fθ(G ):

[aµ, aν ] = iθµν − iΛµαΛνβθαβ

[Λµν , aα] = [Λµα,Λ

νβ] = 0; Λµα, a

µ ∈ Fθ(G )

aµ(e iaαPα

)= aµ; Λµν

(e iωαβMαβ

)= (Λαβ(ω))µν

Coordinates change by coaction, but [xµ, xν ] = iθµν is preserved

(x ′)µ = δ(xµ) = Λµα ⊗ xα + aµ ⊗ 1

[x ′µ, x′ν ]= iθµν

Confrontation of symmetriesA simple example

Λ01 =

cosh α sinh α 0 0sinh α cosh α 0 0

0 0 1 00 0 0 1

Λ23 =

1 0 0 00 1 0 00 0 cos γ sin γ0 0 − sin γ cos γ

Λ12 =

1 0 0 00 cos β sin β 00 − sin β cos β 00 0 0 1

[aµ, aν ] = 0

[a2, a3] = iθ(1− cosβ)

[a1, a3] = −iθ sinβ

Confrontation of symmetriesA simple example

Λ01 =

cosh α sinh α 0 0sinh α cosh α 0 0

0 0 1 00 0 0 1

Λ23 =

1 0 0 00 1 0 00 0 cos γ sin γ0 0 − sin γ cos γ

Λ12 =

1 0 0 00 cos β sin β 00 − sin β cos β 00 0 0 1

[aµ, aν ] = 0

[a2, a3] = iθ(1− cosβ)

[a1, a3] = −iθ sinβ

By imposing a Lorentz transformationwe get accompanying noncommuting translationsshowing up as the internal mechanism by whichthe twisted Poincare symmetry keeps thecommutator [xµ, xν ] = iθµν invariant

Theory of induced representationsFields in commutative space

A commutative relativistic field carries a Lorentzrepresentation and is a function of xµ ∈ R1,3

It 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

⇒ Action of Lorentz generators on a field requires the coproduct

Chaichian, Kulish, Tureanu, Zhang and Zhang (2007)

Φ =∑

fi ⊗ vi , fi ∈ C∞(R1,3) , vi ∈ V

Φ =∑

fi ⊗ vi , fi ∈ C∞(R1,3) , vi ∈ V

Theory of induced representationsFields in noncommutative space

In NC space we need the twisted coproduct, for example:

∆t(M01) = ∆0(M01) = M01 ⊗ 1 + 1⊗M01

∆t(M02) = ∆0(M02) +θ

2(P0 ⊗ P3 − P3 ⊗ P0)

If V is a Lorentz module in Φ =∑

i fi ⊗ vi , vi ∈ V , the Pµ ofM02 cannot act on Φ

Our proposition: Retain V as a Lorentz-module but forbid allthe transformations requiring the action of Pµ on vi

⇒ Only transformations of O(1, 1)× SO(2) allowed

∆t(M01) = ∆0(M01) = M01 ⊗ 1 + 1⊗M01

∆t(M02) = ∆0(M02) +θ

2(P0 ⊗ P3 − P3 ⊗ P0)

∆t(M01) = ∆0(M01) = M01 ⊗ 1 + 1⊗M01

∆t(M02) = ∆0(M02) +θ

2(P0 ⊗ P3 − P3 ⊗ P0)

∆t(M01) = ∆0(M01) = M01 ⊗ 1 + 1⊗M01

∆t(M02) = ∆0(M02) +θ

2(P0 ⊗ P3 − P3 ⊗ P0)

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 ofthe stability group of θµν, i.e. O(1, 1)× SO(2)

In Sum

Requiring solutions to theTomonaga-Schwinger eq.→ light-wedge causality.

Properties of O(1, 1)xSO(2)& twisted Poincare invariance→ field definitions compatible

with the light-wedge.

In Sum

Thank you

Photo credits

everystockphoto.com“Meet Charlotte” @ slideshare.net

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