nc time seminar
DESCRIPTION
Modified from previous for different audience.TRANSCRIPT
Noncommutative Quantum Field Theory:
Problems of nonlocal time
Tapio SalminenUniversity of Helsinki
Noncommutative Quantum Field Theory: A Confrontation of SymmetriesM. Chaichian, K. Nishijima, TS and A. Tureanu
On Noncommutative Time in Quantum Field TheoryTS and A. Tureanu
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 Bij fieldbackground ⇒ noncommutative coordinates in low-energylimit
Seiberg and Witten (1999)
VA possible approach to Planck scale physics isQFT in NC space-time
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 Bij fieldbackground ⇒ noncommutative coordinates in low-energylimit
Seiberg and Witten (1999)
VA possible approach to Planck scale physics isQFT in NC space-time
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 Bij fieldbackground ⇒ noncommutative coordinates in low-energylimit
Seiberg and Witten (1999)
VA possible approach to Planck scale physics 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 noncommutativity also betweenthe coordinate operators
[xµ, xν ] 6= 0
Snyder (1947); Heisenberg (1954);
Golfand (1962)
Quantizing space-timeImplementation
We generalize the commutation relations fromusual quantum mechanics
[xi , xj ] = 0 , [pi , pj ] = 0[xi , pj ] = i~δij
by imposing noncommutativity also betweenthe coordinate operators
[xµ, xν ] 6= 0
Snyder (1947); Heisenberg (1954);
Golfand (1962)
Quantizing space-timeImplementation
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 Lorentztransformations.
Quantizing space-timeImplementation
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 Lorentz
transformations.
Does this meanLorentz invarianceis lost?
Quantizing space-timeImplementation
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
Translational invariance is preserved,but the Lorentz group breaks down to SO(1, 1)xSO(2).
=⇒ No spinor, vector, tensor etc representations.
Quantizing space-timeImplementation
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
Translational invariance is preserved,
but the Lorentz group breaks down to SO(1, 1)xSO(2).
=⇒ No spinor, vector, tensor etc representations.
Quantizing space-timeImplementation
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
Translational invariance is preserved,
but the Lorentz group breaks down to SO(1, 1)xSO(2).
=⇒ No spinor, vector, tensor etc representations.
Effects of noncommutativityMoyal ?-product
In noncommuting space-time the analogue of the action
S (cl)[Φ] =
∫d4x
[1
2(∂µΦ)(∂µΦ)− 1
2m2Φ2 − λ
4!Φ4
]can be written using the Moyal ?-product
Sθ[Φ] =
∫d4x
[1
2(∂µΦ) ? (∂µΦ)− 1
2m2Φ ? Φ− λ
4!Φ ? Φ ? Φ ? Φ
]
(Φ ?Ψ) (x) ≡[
Φ(x)ei2θµν
←−∂∂xµ
−→∂∂yν Ψ(y)
]y=x
Effects of noncommutativityMoyal ?-product
In noncommuting space-time the analogue of the action
S (cl)[Φ] =
∫d4x
[1
2(∂µΦ)(∂µΦ)− 1
2m2Φ2 − λ
4!Φ4
]can be written using the Moyal ?-product
Sθ[Φ] =
∫d4x
[1
2(∂µΦ) ? (∂µΦ)− 1
2m2Φ ? Φ− λ
4!Φ ? Φ ? Φ ? Φ
]
(Φ ?Ψ) (x) ≡[
Φ(x)ei2θµν
←−∂∂xµ
−→∂∂yν Ψ(y)
]y=x
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
(i
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
(i
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
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
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
Then what happensto representations,causality etc?
Effects of noncommutativityTwisted Poincare algebra
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
− 1
2θαβ [(ηαµPν − ηανPµ)⊗ Pβ + Pα ⊗ (ηβµPν − ηβνPµ)]
Effects of noncommutativityTwisted Poincare algebra
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
− 1
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
Part 2Noncommutative time
and unitarity
Noncommutative timeString theory limits
Until now we have had all coordinates noncommutative
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
Noncommutative timeString theory limits
The low-energy limit of string theorywith a background Bij field gives
θµν =
0 0 0 00 0 0 00 0 0 θ0 0 −θ 0
This is referred to as space-like noncommutativity.
Noncommutative timeString theory limits
This string theory is S-dual to another string theorywith an Eij background. There we would have
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 00 0 0 0
The so called time-like noncommutativity.
However, it has been shown that the low-energy limitdoes not exist for these theories.
Seiberg and Witten (1999)
Noncommutative timeString theory limits
This string theory is S-dual to another string theorywith an Eij background. There we would have
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 00 0 0 0
The so called time-like noncommutativity.
However, it has been shown that the low-energy limitdoes not exist for these theories.
Seiberg and Witten (1999)
Could you pleasestop talking
about strings?
Noncommutative timeUnitarity
We may still consider quantum field theories with
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
But in the interaction picture it has beenshown that perturbative unitarity requires
θ′(p20 − p2
1) + θ(p22 + p2
3) > 0
Time-like noncommutativity → violation of unitarity
Gomis and Mehen (2000)
Noncommutative timeUnitarity
We may still consider quantum field theories with
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
But in the interaction picture it has beenshown that perturbative unitarity requires
θ′(p20 − p2
1) + θ(p22 + p2
3) > 0
Time-like noncommutativity → violation of unitarity
Gomis and Mehen (2000)
Noncommutative timeUnitarity
We may still consider quantum field theories with
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
But in the interaction picture it has beenshown that perturbative unitarity requires
θ′(p20 − p2
1) + θ(p22 + p2
3) > 0
Time-like noncommutativity → violation of unitarity
Gomis and Mehen (2000)
Noncommutative timeUnitarity
We may still consider quantum field theories with
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 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) + g4(· · · ) 6= ψout(x)
Salminen and Tureanu (2010)
Noncommutative timeUnitarity
We may still consider quantum field theories with
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 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) + g4(· · · ) 6= ψout(x)
Salminen and Tureanu (2010)
Noncommutative timeUnitarity
We may still consider quantum field theories with
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 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)
Part 3Tomonaga-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)
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)
Tomonaga-Schwinger equationConventions
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 equationConventions
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
iδ
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
iδ
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
iδ
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)
]= 0
Tomonaga-Schwinger equationIn noncommutative theory
Moving on to NC space-time we get
iδ
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)
]= 0
Tomonaga-Schwinger equationIn noncommutative theory
Moving on to NC space-time we get
iδ
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)
]= 0
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)
]= 0
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
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)
]= 0
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
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)
]= 0
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 the hard work andwe end up withthe light-cone?
Tomonaga-Schwinger equationThe causality condition
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
Tomonaga-Schwinger equationThe causality condition
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
Tomonaga-Schwinger equationThe causality condition
However, since a and b are integration variables in the range
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)
Tomonaga-Schwinger equationThe causality condition
However, since a and b are integration variables in the range
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)
Tomonaga-Schwinger equationThe causality condition
However, since a and b are integration variables in the range
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)
Tomonaga-Schwinger equationThe causality condition
This is the light-wedge causality condition, invariant under thestability group of θµν ,O(1, 1)× SO(2).
Chaichian, Nishijima, Salminen and Tureanu (2008)
Tomonaga-Schwinger equationThe causality condition
If we had taken
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
we would change
(x0 − y0)2 − (x1 − y1)2 − (a2i′ − b2
j′)− (a3
i′ − b3
j′)2 < 0
Chaichian, Nishijima, Salminen and Tureanu (2008)
Tomonaga-Schwinger equationThe causality condition
If we had taken
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
we would change
(x0 − y0)2 − (x1 − y1)2 − (a2i′ − b2
j′)− (a3
i′ − b3
j′)2 < 0
Chaichian, Nishijima, Salminen and Tureanu (2008)
Tomonaga-Schwinger equationThe causality condition
If we had taken
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
into
(a0i′ − b0
j′)2 − (a1
i′ − b1
j′)2−(a2
i′ − b2
j′)− (a3
i′ − b3
j′)2 < 0
Chaichian, Nishijima, Salminen and Tureanu (2008)
Salminen and Tureanu (2010)
Tomonaga-Schwinger equationThe causality condition
If we had taken
θµν =
0 θ′ 0 0−θ′ 0 0 0
0 0 0 θ0 0 −θ 0
into
(a0i′ − b0
j′)2 − (a1
i′ − b1
j′)2−(a2
i′ − b2
j′)− (a3
i′ − b3
j′)2 < 0
→ No solution to the Tomonaga-Schwinger equationfor any x and y .
Salminen and Tureanu (2010)
In Sum
Requiring solutions to theTomonaga-Schwinger eq.→ light-wedge causality.
Unitarity & causalityviolated in theories with
noncommutative time.
In Sum
Requiring solutions to theTomonaga-Schwinger eq.→ light-wedge causality.
Unitarity & causalityviolated in theories with
noncommutative time.
In Sum
Requiring solutions to theTomonaga-Schwinger eq.→ light-wedge causality.
Unitarity & causalityviolated in theories with
noncommutative time.
Thank you
Photo credits
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Extra materialConfrontation 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.
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.
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.
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 =
0BB@cosh α sinh α 0 0sinh α cosh α 0 0
0 0 1 00 0 0 1
1CCA
Λ23 =
0BB@1 0 0 00 1 0 00 0 cos γ sin γ0 0 − sin γ cos γ
1CCA
Λ12 =
0BB@1 0 0 00 cos β sin β 00 − sin β cos β 00 0 0 1
1CCA
[aµ, aν ] = 0
[aµ, aν ] = 0
[a2, a3] = iθ(1− cosβ)
[a1, a3] = −iθ sinβ
Confrontation of symmetriesA simple example
Λ01 =
0BB@cosh α sinh α 0 0sinh α cosh α 0 0
0 0 1 00 0 0 1
1CCA
Λ23 =
0BB@1 0 0 00 1 0 00 0 cos γ sin γ0 0 − sin γ cos γ
1CCA
Λ12 =
0BB@1 0 0 00 cos β sin β 00 − sin β cos β 00 0 0 1
1CCA
[aµ, aν ] = 0
[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 =∑
i
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)
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 =∑
i
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)
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 =∑
i
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)
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
Chaichian, Nishijima, Salminen and Tureanu (2008)
⇒ Only transformations of O(1, 1)× SO(2) allowed
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
Chaichian, Nishijima, Salminen and Tureanu (2008)
⇒ Only transformations of O(1, 1)× SO(2) allowed
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
Chaichian, Nishijima, Salminen and Tureanu (2008)
⇒ Only transformations of O(1, 1)× SO(2) allowed
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
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 ofthe stability group of θµν, i.e. O(1, 1)× SO(2)