confrontation of symmetries
DESCRIPTION
A seminar talk on NC QFTTRANSCRIPT
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-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-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)
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)
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 Lorentztranformations.
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
tranformations.
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
“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)
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)
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.
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 =
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
[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
[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)
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
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
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.
Thank you
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