Lorentz invariance of scalar field action on $κ$-Minkowski space-time

0.15 MB·English

by  Sebastian Nowak

Lorentz invariance of scalar field action on
κ
-Minkowski space-time
5 Sebastian Nowak∗
0
0
2 Institute for Theoretical Physics
University of Wroc law
n
a Pl. Maxa Borna 9
J
Pl–50-204 Wroc law, Poland
4

February 1, 2008
1
v
7
1
0 Abstract
1
0 We construct field theory on noncommutative κ-Minkowski space-
5 time. Having the Lorentz action on the noncommutative space-time co-
0 ordinates we show that the field lagrangian is invariant. We show that
/ noncommutativityrequiresreplacingtheLeibnitzrulewiththecoproduct
h
one.
t
-
p
e 1 Introduction
h
:
v
κ-Minkowski space-time with no-trivial commutator between time and space
i
X being
i
r [x ,x ]=− x (1)
a 0 i κ i
has been first constructed in [1], [2] as a Hopf algebra dual to the translational
partofκ-Poincar´ealgebra[3]. After Doubly SpecialRelativity (DSR) has been
formulatedin[4],[5],[6],itwassoonrealized[7]thatκ-Minkowskispace-timeis
anaturalcandidateforspace-timeofDSR1.Thisspace-timearisesalsonaturally
in investigations of 2+1 (quantum) gravity [10], [11], giving rise to the claim
that it is the space-time emerging from semiclassical, weak field approximation
of 3+1 gravity as well.
It seems natural to investigate properties of classical and quantum fields
on κ-Minkowski space-time. Indeed such investigations has been undertaken,
among others in [12], [13], [14], [15], [17], [16]. It has been shown [19] that the
invarianceofactiononscalarfieldsonκ-Minkowskileadstodeformedalgebraof
∗e-mail: [email protected]
1ForuptodatereviewsofDSRprogramsee[8],[9].
1
symmetries and some nontrivial co-algebrastructure. In this paper we proceed
in different manner, we consider Lorentz action on generators of κ-Minkowski
spaceandcomputeLorentzactiononnormallyorderedplanewaveandthenwe
show Lorentz invariance of κ-deformed Klein-Gordon action.
2 κ-deformed algebra of symmetries
In this section we review the construction of κ-Poincar´e algebra [3] which can
be seen as deformed algebra of symmetries. Lorentz sector of this algebra is
standard but the action of boost generators on momenta is deformed. Here we
write down this algebra in so-called bicrossproduct basis [2]
[M ,M ]=iǫ M , [M ,N ]=iǫ N ,
i j ijk k i j ijk k
[N ,N ]=−iǫ M . (2)
i j ijk k
[M ,p ]=iǫ p , [M ,p ]=0
i j ijk k i 0
κ p2 i
[N ,p ]=iδ 1−e−2p0/κ + − p p , [N ,p ]=ip . (3)
i j ij(cid:18)2 (cid:16) (cid:17) 2κ(cid:19) κ i j i 0 i
Thisalgebratogetherwithcoalgebrastructurewrittenbelowformsnoncommu-
tative, noncocommutative Hopf algebra
△(M )=M ⊗1+1⊗M
i i i
1
△(N )=N ⊗1+e−p0/κ⊗N + ǫ p ⊗M
i i i κ ijk j k
1
S(N )=−ep0/κ(N − ǫ p M ), S(M )=−M (4)
i i κ ijk j k i i
△(p )=p ⊗1+e−p0/κ⊗p
i i i
△(p )=p ⊗1+1⊗p
0 0 0
S(p )=−p ep0/κ, S(p )=−p (5)
i i 0 0
Followingthe generalscheme wecanintroduce κ-Minkowskispace T∗ asa dual
Hopf algebra to algebra of translations (momenta). With the use of pairing
<p ,x >=−iη (6)
µ ν µν
together with axioms of a Hopf algebra duality
<t,xy >=<t ,x><t ,y >,
(1) (2)
<ts,x>=<t,x ><s,x >,
(1) (2)
∀t,s∈T, x,y ∈T∗ (7)
we get
i
[x ,x ]=0, [x ,x ]=− x
i j 0 i κ i
2
△x =x ⊗1+1⊗x . (8)
µ µ µ
where we use Sweedler notation for coproduct
△t= t ⊗t .
(1) (2)
X
Having the Hopf algebra of space-time coordinates we can write down the co-
variant action of T on it:
t⊲x=<x ,t>x , ∀x∈T∗, t∈T. (9)
(1) (2)
In our case it reads
1 ∂
p ⊲:ψ(x ,x ):= : ψ(x ,x ):,
i i 0 i ∂x i 0
i
1 ∂
p ⊲:ψ(x ,x ):=− : ψ(x ,x ): (10)
0 i 0 i ∂x i 0
0
which is like in classical case but remembering normal ordering.
The actionof U(so(1,3))generatorsonmomenta,defined informula (3) (we
denote itherefor simplicity by⊳), canbe translatedbyduality intogenerators
of κ-Minkowski space. We have
<t,h⊲x>=<t⊳h,x>
∀t∈T, h∈U(so(1,3)), x∈T∗. (11)
Using duality pairing written in (6) we get
M ⊲x =0, M ⊲x =iǫ x ,
i 0 i j ijk k
N ⊲x =ix , N ⊲x =iδ x . (12)
i 0 i i j ij 0
We see that the action is the same as in classical case. The difference occurs
while acting on product of coordinates. In noncommutative case we must use
thecoproductformulainsteadofusualLeibnitzrulesoasnocontradictionwith
relation
i
[x ,x ]=− x
0 i κ i
arises. We have
h⊲(xy)=(h ⊲x)(h ⊲y) (13)
(1) (2)
Moreoverfromthe structure ofbicrossproductconstructionit appearsthat one
can also define the action of space-time algebra on Lorentz algebra and then
define the cross relations between them [2]. In our case it has following form
[M ,x ]=0, [M ,x ]=iǫ x ,
i 0 i j ijk k
i i
[N ,x ]=ix − N , [N ,x ]=iδ x − ǫ M . (14)
i 0 i κ i i j ij 0 κ ijk k
Ithasbeenshownthatso(1,3)algebraandspace-timealgebra(7)togetherwith
above cross relations form so(1,4) algebra. Moreover the action on momenta
written in equation (3) may be derived from the action of so(1,4) algebra on
four dimensional de Sitter space of momenta [23].
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3 Integration and Fourier transform
InordertodefineFouriertransformonκ-Minkowskiwehavetodefinenormally
orderedplane wave,which is a solutionof κ-deformedKlein-Gordonfield equa-
tion [17]. Normal ordering is necessary because space-time coordinates do not
commute, here we choose ”time to the right” ordering
:eipx :=eipxe−ip0x0 (15)
which simply means all x shifted to the right. Having normally ordered plane
0
waveone candefine κ-deformedFourier transformdescribing fields onnoncom-
mutative space:
1
Φ(x ,x)= dµΦ(p ,p)eipxe−ip0x0, (16)
0 (2π)4 Z 0
where
dµ=d4pe3p0/κ (17)
isthemeasureinvariantundertheactionofU(so(1,3))algebra. Thefieldwritten
above is an element of Hopf algebra described in equations (8). One can equip
this algebra with hermitian conjugation. This implies the conjugation of field:
1
Φ+(x ,x)= dµΦ⋆(p ,p)eip0x0e−ipx, (18)
0 (2π)4 Z 0
which can be rewritten in the following form:
1
Φ+(x ,x)= dµΦ⋆(p ,p)eiS(p)xe−iS(p0)x0, (19)
0 (2π)4 Z 0
where we used equality
eip0x0e−ipx =e−ipep0/κxeip0x0 (20)
together with definition of antipode S (equation (5)).
In order to define the field theory on κ-Minkowski we need to define κ-
deformedintegration. Toendthiswemustintroduce,inanalogytotheclassical
case, the following formula
1
d4x:eipx :=e3p0/κδ4(p) (21)
(2π)4 Z Z
This formula is invariantunder the action of Lorentz generatorson momentum
space defined in equations (3) . Let us check it for boost transformation; we
have
1 d4x:eip′x :=e3p′0/κδ4(p′)=e3p′0/κJ(∂p′µ)δ4(p)=e3p0/κδ4(p) (22)
(2π)4 Z Z ∂p
ν
and the same result holds for rotation generators.
4
Using the above definition of delta function one can find
d4xΨ(x)Φ(x)=
Z
d4x dµdµ′ Ψ˜(p)Φ˜(p′) : eipx : : eip′x :=
Z Z
dµdµ′Ψ˜(p)Φ˜(p′)δ(p+˙ p′) (23)
Z
To derive this formula one uses the fact that
: eipx : : eip′x :=: ei(p+˙ p′)x : (24)
where
p+˙ p′ =(p +p′;p+e−p0/κp′) (25)
0 0
represents the “co-productsummation rule”, related to the Hopf algebra struc-
ture of κ-Poincare algebra.
We can define the scalar product of fields on κ-Minkowski as
(Φ(x),Ψ(x))= d4xΦ+(x)Ψ(x)= d4pe3p0Φ∗(p)Ψ(p) (26)
Z Z
NowitisagoodpointtointroduceLorentzactiononfieldsonnoncommuta-
tivespace-time. We startwithnormallyorderedplane wave. Usingthe Lorentz
actiononnoncommuting coordinates(12)together withcoproductrule (13)we
find
M ⊲(eipjxje−ip0x0)=ǫ x p eipjxje−ip0x0
i ijk j k
N ⊲(eipjxje−ip0x0)=
i
eipjxj(cid:20)(cid:18)δij(κ2(1−e−2Pκ0)+ 21κp~2)− κ1pipj(cid:19)xj +(−pix0)(cid:21)e−ip0x0 (27)
and together with action of translation generators defined in equation (10) we
have the action of the κ-Poincar´e algebra on normal ordered plane wave. One
can easily show that the generators of this action are hermitian with respect
to scalar product (26). One can also find differential realization of the above
action [17], which turns out to be nonlinear.
NowtakinginfinitesimalparameterεwecanintroduceinfinitesimalLorentz
transformations on normal ordered plane waves.
(1+iεMi)⊲(eipjxje−ip0x0)=eip′jxje−ip0x0
where
p′ =p −iε[M ,p ] (28)
j j i j
and
(1+iεNi)⊲(eipjxje−ip0x0)=eip′jxje−ip′0x0
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p′ =p −iε[N ,p ], p′ =p −iε[N ,p ] (29)
j j i j 0 0 i 0
We see that our normally orderedplane wave is an analog of classicalplane
wave eipx. The action on space-time coordinates interchanges with the action
on energy-momentum space. Moreover in both cases one can see the phase
space as the Lee algebra of space-time coordinates and the energy-momentum
Hopf algebra as the algebra of functions on group generated by the algebra of
space-timecoordinates. Thesignificantdifference isthattheclassicalalgebrais
commutative while the quantum is not.
KnowingtheactionofLorentzalgebraonplanewavewecanwritedownthe
transformations of Fourier transform (16)
(1+iεNi)⊲Φ(x,t)= dµΦ(p0,~p)eip′jxje−ip′0x0. (30)
Z
Now we can change variables under the integral and since the measure dµ is
invariant under the action of U(so(1,3)) algebra we get
(1+iεN )⊲Φ(x,t)= dµΦ(p′,p~′)eipjxje−ip0x0 (31)
i Z 0
where
p′ =p +iε[N ,p ], p′ =p +iε[N ,p ] (32)
j j i j 0 0 i 0
but on the other side we can write
(1+iεN )⊲Φ(x,x )=Φ′(x,x )= dµΦ′(p ,p~)eipjxje−ip0x0. (33)
i 0 0 0
Z
Compering equations (31) and (33) we see that
Φ′(p′,p~′)=Φ(p ,~p)
0 0
which means that fields in energy-momentum space are scalar fields. Here we
consider only the action of algebra but one can also consider the action of κ-
Poincar´egrouponfields. Thiswasdoneinpaper[14],withthehelpofκ-Wigner
construction.
Using the first order Taylor expansion we can write equation (31) in the
following form
(1+iεN )⊲Φ(x,t)= dµ(1+iεN )Φ(p ,p~)eipjxje−ip0x0, (34)
i i 0
Z
where the boost generator on the right hand side stands for differential oper-
ator in momentum space. Having the above transformation rules we see that
variables p ,p may be indeed identified with energy and momentum in bi-
0 i
crossproduct basis.
Another important thing to notice is the action of κ-Poincar´e algebra on
product of fields. To end this we must apply the coalgebra structure, which in
6
classical case, where the coproducts are trivial, is just the Leibnitz rule. The
action of Lorentz algebra is due to the covariance condition (13). We get
N ⊲(Φ(x)Ψ(x))= d4pe3p0/κd4ke3k0/κ [(N Φ(p))Ψ(k)+
i i
Z Z
e−p0/κΦ(p)(N Ψ(k))+ǫ p Φ(p)(M Ψ(k)) :eipx ::eikx :. (35)
i ijk j k
i
Fromtheaboveequationitisseenthatnoncommutativityofspace-timeenforces
applying the coproduct rule on fields in momentum space [18]. The action of
translation generators is due to the equation (10) and it reads
p ⊲(Φ(x)Ψ(x)) = d4pe3p0/κd4ke3k0/κ(p +k )Φ(p)Ψ(k):eipx ::eikx :
0 0 0
Z Z
p ⊲(Φ(x)Ψ(x))= d4pe3p0/κd4ke3k0/κ(p +e−p0/κk )Φ(p)Ψ(k):eipx ::eikx :
i i i
Z Z
(36)
which is nothing but the coproduct summation rule. In the next section we’ll
see that we can also define invariant action and field equation.
4 Actions and field equations
Weconsiderhereonlythefreeκ-deformedKG(Klein-Gordon)actionwhichhas
the following form [13]:
1
S = d4x ηµνΦ(x)∂ˆ ∂ˆ Φ(x)−M2Φ(x)2 . (37)
µ ν
2Z (cid:16) (cid:17)
where∂ˆ meanscovariantdifferentiationonnoncommutingspace-time[20],[21],
ν
[22]. We consider only the real fields (Φ+(x) = Φ(x)). By construction the
action should be invariant under action of deformed κ-Poincar´e algebra. Let
us show this invariance explicitly. To do that we write the action in somewhat
convenient form
S = d4x d4pe3p0/κd4kΦ(p)Φ(S(k)):eipx ::eiS(k)x :M(k) (38)
Z Z Z
where M(k) is the mass shell condition and will be explicitly written below.
Using formula (35) we have
(1+iεN )⊲S = d4x d4pe3p0/κd4k
i
Z Z Z
Φ(p)Φ(S(k))+(iεN Φ(p))Φ(S(k))+e−p0/κΦ(p)(−iεS(N )Φ(S(k)))
i i
h
+ǫ p Φ(p)(−iεS(M )Φ(S(k)))]M(k):eipx ::eiS(k)x :. (39)
ijk j k
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Now using the following formulas for antipodes S
1
S(M )=−M , S(N )=−ep0/κ(N − ǫ p M ) (40)
i i i i κ ijk j k
together with definition of delta function (21) we get
(1+iεN )⊲S = d4pe3p0/κ(1+εN )(Φ(p)Φ(S(p)))M(k)=S (41)
i i
Z
where we made use of the invariance of the measure. We get the same result if
we repeat the above calculations for generators of rotations.
Letusnowturntotheremainingpartofthe(deformed)Poincar´esymmetry,
namely the symmetries with respect to the space time translations. We have
εP ⊲S = d4x d4pe3p0/κd4ke3k0/κ
µ
Z Z Z
ε(p +˙ k )Φ(p)Ψ(k)M(k):eipx ::eikx :=0 (42)
µ µ
whereweuseddefinitionofdeltafunction. followingthesameprocedurewecan
show invariance of the scalar product (26).
In terms of the Fourier transformed fields the action reads
1
S = dp d3pΦ(p ,p)Φ(−p ,−ep0/κp)
0 0 0
2Z
p 1 p4
κ2sinh2 0 − p2 e2p0/κ+1 + e2p0/κ−M2 (43)
(cid:20) κ 2 (cid:16) (cid:17) 4κ2 (cid:21)
where we used the explicit form of M(p ,p)
0
p 1 p4
M(p ,p)≡ κ2sinh2 0 − p2 e2p0/κ+1 + e2p0/κ−M2 (44)
0 (cid:20) κ 2 (cid:16) (cid:17) 4κ2 (cid:21)
Note that the factors e3p0/κ in the integration measure cancel. This action can
bealsoexpressedinslightlymorecompactwayusingtheantipodeS(generalized
“minus”) of the κ-Poincar´ealgebra defined in equations (5)
1
S = dp d3pΦ(p ,p)Φ(S(p ),S(p))M(p ,p) (45)
0 0 0 0
2Z
Varying this action with respect to Φ(p ,p) and noting that
0
M(p ,p)=M(S(p ),S(p))
0 0
we find the on shell condition of the form
p 1 p4
κ2sinh2 0 − p2 e2p0/κ+1 + e2p0/κ−M2 =0 (46)
κ 2 4κ2
(cid:16) (cid:17)
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Now we can write down the real field on shell in the following form
Φ(x)= d3ka∗eikxe−ik0x0 + d3ka eiS(k)xe−iS(k0)x0 (47)
Z k Z k
which is very similar to the classical case. The difference is the antipode S in
exponent instead of minus. On the mass shell we have
1
ek0/κ = . (48)
1− |k|
κ
where for simplicity we consider only the massless case.
5 conclusions
We have shown that in analogy to the classical case, for scalar fields on κ-
Minkowskione canconstruct scalarproduct and Hermitianaction of κ-Poinar´e
algebragenerators. A subalgebraofthis algebracanbe interpretedas infinites-
imal Lorentz transformations, under which the field action is invariant. The
solutions of κ-deformed Klein-Gordon equation have the form similar to classi-
cal field.
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