NC time seminar

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

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)

Motivation

coordinates
Open string + D-brane theory with an antisymmetric Bij ﬁeld
background ⇒ noncommutative coordinates in low-energy
limit
Seiberg and Witten (1999)

Motivation

coordinates
Open string + D-brane theory with an antisymmetric Bij ﬁeld
background ⇒ noncommutative coordinates in low-energy
limit
A possible approach to Planck scale physics is
QFT in NC space-time

Implementation

We generalize the commutation relations from
usual quantum mechanics

[î , xj ] = 0 , [î , pj ] = 0
x ˆ p ˆ
[î , pj ] = i δij
x ˆ

Implementation

We generalize the commutation relations from
usual quantum mechanics

[î , xj ] = 0 , [î , pj ] = 0
x ˆ p ˆ
[î , 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)

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

Implementation

 
0 θ 0 0
 −θ 0 0 0 
θµν =   0

0 0 θ 
0 0 −θ 0

θµν does not transform under Lorentz
transformations.

Does this mean
Lorentz invariance
is lost?

Implementation

 
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).

Implementation

 
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

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

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

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)

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 )

e


[Pµ , Pν ] = 0

But we change the coproduct (Leibniz rule)

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

e


[Pµ , Pν ] = 0

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?

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

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

Causality

SO(1, 3)

Minkowski 1908

Causality

=⇒

SO(1, 3) O(1, 1)xSO(2)

Minkowski 1908 ´
Alvarez-Gaum´ et al. 2000
e

Part 2
Noncommutative time
and unitarity

Noncommutative time
String theory limits

Until now we have had all coordinates noncommutative

 
0 θ 0 0
 −θ 0 0 0 
θµν =
 0

0 0 θ 
0 0 −θ 0

Noncommutative time

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 0

This is referred to as space-like noncommutativity.

Noncommutative time

This 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.

Noncommutative time

This 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.


Could you please
stop talking
about strings?

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

Noncommutative time
Unitarity

 
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

Noncommutative time
Unitarity

 
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

Time-like noncommutativity → violation of unitarity
Gomis and Mehen (2000)

Noncommutative time
Unitarity

 
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.

Noncommutative time
Unitarity

 
0 θ 0 0
 −θ 0 0 0 
θµν =
 0

0 0 θ 
0 0 −θ 0


However, using the Yang-Feldman approach one can show:

S † ψin (x)S = ψout (x) + g 4 (· · · ) = ψout (x)

Salminen and Tureanu (2010)

Noncommutative time
Unitarity

 
0 θ 0 0
 −θ 0 0 0 
θµν =
 0

0 0 θ 
0 0 −θ 0


However, using the Yang-Feldman approach one can show:

There is no unitary S-matrix.


Part 3
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

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 )

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

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

In commutative theory

Generalizing the Schr¨dinger equation in the interaction picture to
o
incorporate arbitrary Cauchy surfaces, we get the

δ
i Ψ[σ] = Hint (x)Ψ[σ]
δσ(x)

In commutative theory

Generalizing the Schr¨dinger equation in the interaction picture to
o
incorporate arbitrary Cauchy surfaces, we get the

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.

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


δ
δC

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

which can be written as


δ
δC

[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

The causality condition

The 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



x x y y x y

All products of ﬁelds being independent,
the necessary condition is

φ(˜, ai ), φ(˜ , bj ) = 0
x y



x x y 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?


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



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



0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞

the necessary condition becomes



0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞


(x 0 − y 0 )2 − (x 1 − y 1 )2 < 0



0 ≤ (ai2 − bj2 )2 + (ai3 − bj3 )2 < ∞


(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)


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



If we had taken
 
0 θ 0 0
 −θ 0 0 0 
θµν =
 0

0 0 θ 
0 0 −θ 0


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



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



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 .

In Sum

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

In Sum

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

Unitarity & causality
violated in theories with
noncommutative time.

Thank you

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Extra material
Confrontation of
symmetries

Confrontation of symmetries
e

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

e

∆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

e

∆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.

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

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

A simple example

0 1
cosh α sinh α 0 0
B sinh α cosh α 0 0 C
Λ01 =@
B C
0 0 1 0 A
0 0 0 1

0 1
1 0 0 0
B 0 1 0 0 C
Λ23 =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
Λ12 =@
B C
0 − sin β cos β 0 A
0 0 0 1

A simple example

0 1
cosh α sinh α 0 0
Λ01
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 ] = 0
Λ23 =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

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


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


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)

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


∆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


∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2
Our proposition: Retain V as a Lorentz-module but forbid all
the transformations requiring the action of Pµ on vi



∆t (M01 ) = ∆0 (M01 ) = M01 ⊗ 1 + 1 ⊗ M01
θ
∆t (M02 ) = ∆0 (M02 ) + (P0 ⊗ P3 − P3 ⊗ P0 )
2
Our proposition: Retain V as a Lorentz-module but forbid all
the transformations requiring the action of Pµ on vi


⇒ 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)

NC time seminar

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