U(1) gauge field theory on κ-Minkowski space ∗)

Marija Dimitrijevic∗∗

)

Max-Planck-Institut fur Physik, Fohringer Ring 6, 80805 Munchen, Germanyand

Arnold Sommerfeld Center for Theoretical Physics, Fakultat fur Physik,Universitat Munchen, Theresienstr. 37, 80333 Munchen, Germany

Larisa Jonke

Theoretical Physics Division, Rudjer Boskovic Institute,Bijenicka 54, 10002 Zagreb, Croatia

Lutz Moller

Max-Planck-Institut fur Physik, Fohringer Ring 6, 80805 Munchen, Germany

Received 25 July 2005

This study of U(1) gauge field theory on the kappa-deformed Minkowski space-time ex-tends previous work on gauge field theories on this type of noncommutative space-time. Weconstruct the conserved gauge current, fix part of the ambiguities in the Seiberg–Wittenmap and obtain an effective U(1) action invariant under the action of the undeformedPoincare group.

PACS : 11.10.Nx, 11.15.–qKey words: gauge field theory, deformation, quantum group

1 Introduction

The combination of general relativity and quantum mechanics suggests thatspace-time might not be a differential manifold. Relying on the well-developedmathematical concept of deformation, we formulate a field theory defined on aquantum space rather than on the usual differential manifold [1]. The main idea isthe following: A differential manifold can be described by the algebra of functionson the manifold. We deform the usual algebra of functions on Minkowski space-timeto obtain the functions on the κ-Minkowski space-time. In constructing the theory,we implement the κ-Poincare algebra as a deformed symmetry. Physical fields arethose functions which are representations of the deformed symmetry algebra. Afterdefining all field-theory ingredients in an abstract algebra formalism, we use the�-product representation to establish a connection with the usual field theory. Itwas shown in Ref. [2] how the deformation concept could be applied to a gauge fieldtheory. In this paper we concentrate on U(1) gauge field theory on the κ-Minkowskispace-time in the �-product representation.

∗) Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”,Prague, 16–18 June 2005.∗∗) On leave from University of Belgrade, Faculty of Physics, Studentski trg 12, 11000 Beograd,Serbia and Montenegro

Czechoslovak Journal of Physics, Vol. 55 (2005), No. 11 1391

M. Dimitrijevic, Larisa Jonke, and Lutz Moller

In the next two sections we shortly repeat some crucial steps in constructing agauge field theory on deformed space-time [2], while in the last section we presentthe main results — an effective U(1) action invariant under the action of the un-deformed Poincare group (see Ref. [3] for more details).

2 κ-Minkowski space in the �-product representation

We work in (n+1)-dimensional κ-Minkowski space-time with the formal metricηµν = diag (1,−1, . . . ,−1). In the �-product representation the space is defined bythe following commutation relations:

[xn �, xj ] = xn � xj − xj � xn = iaxj , [xi �, xj ] = 0 ; i, j = 0, 1, . . . , n− 1 , (1)

where the constant deformation vector a ≡ an is related to the frequently usedparameter κ as a = 1/κ. The symmetric �-product for the κ-Minkowski space-time, up to first order in the deformation parameter, is given by

f(x) � g(x) = f(x)g(x) + 12 iCµν

λ xλ∂µf(x)∂νg(x) =

= f(x)g(x) + 12 iaxj

(∂nf(x)∂jg(x) − ∂jf(x)∂ng(x)

). (2)

The requirement that derivatives should be consistent with the commutationrelations (1) [4], should commute among themselves, and have a vector-like trans-formation law under κ-deformed Lorentz transformations, gives an almost [5] uniquesolution, which we call Dirac derivatives:

[D∗n

�, xj ] = −iaDj∗ , [D∗n

�, xn] =√

1 + a2Dµ∗D∗µ ,

[D∗i

�, xj ] = ηji

(−iaD∗

n +√

1 + a2Dµ∗D∗µ

), [D∗

i�, xn] = 0 .

(3)

With this particular choice of the derivatives the κ-Poincare algebra remains un-deformed

[M∗µν �, M∗ρσ] = ηµσM∗νρ + ηνρM∗µσ − ηµρM∗νσ − ηνσM∗µρ ,

[M∗µν �, D∗ρ] = δν

ρD∗µ − δµ

ρD∗ν , [D∗

ρ�, D∗

σ] = 0(4)

and the deformation is in the coalgebra sector1). In this paper we omit the rep-resentation of the generators M∗µν of κ-Lorentz transformation, because they arenot essential for the problem at hand. Using relations (3) we calculate the repre-sentation of the derivatives D∗

µ on ordinary functions,

D∗nf(x) =

(1a

sin(a∂n) − cos(a∂n) − 1ia∂2

n

∂j∂j

)f(x) ,

D∗i f(x) =

e−ia∂n − 1−ia∂n

∂if(x) .(5)

1) The κ-deformed Poincare algebra was first introduced in [6].

1392 Czech. J. Phys. 55 (2005)

U(1) gauge field theory on κ-Minkowski space

Formulae (5) deliver the representation of D∗µ derivatives in terms of the usual

partial derivatives ∂µ. The Leibniz rules for D∗µ derivatives are non-trivial as a

consequence of (3),

D∗n (f(x) � g(x)) = (D∗

nf(x)) �(e−ia∂ng(x)

)+

(eia∂nf(x)

)� (D∗

ng(x))

− ia(D∗

j eia∂nf(x))�

(Dj∗g(x)

),

D∗i (f(x) � g(x)) = (D∗

i f(x)) � (e−ia∂∗ng(x)) + f(x) � (D∗

i g(x)) .

(6)

3 Gauge theories and the Seiberg−Witten map

Gauge theories on the κ-Minkowski space-time have two new properties as aconsequence of noncommutativity, see Ref. [2] for details. A gauge field is both en-veloping algebra-valued and derivative-valued. Having an enveloping algebra-valuedgauge field leads to a theory with infinitely many degrees of freedom. The way outof this unphysical situation is provided in terms of the Seiberg–Witten map [7]. Us-ing this map one can express noncommutative variables (gauge parameter, fields)in terms of commutative ones and in this way retain the same number of degrees offreedom as in the commutative case (where the degrees of freedom are Lie algebra-valued).

Explicit solutions of the SW map for the gauge parameter and the matter fieldwere constructed in Ref. [2] from the assumption that

δαψ = iΛα(x) � ψ (7)

and that this is a gauge transformation

(δαδβ − δβδα)ψ = δ−i[α,β]ψ . (8)

From the transformation law of the covariant derivative Dµ = D∗µ − iVµ,

δα(Dµψ(x)

)= iΛα(x) �Dµψ(x) , (9)

we obtain the trasformation law for the gauge field Vµ, and from that expression wecalculate the SW map for vector fields. As a consequence of the non-trivial Leibnizrule (6) vector fields become derivative-valued. As the next step we construct thefield-strength tensor. It is defined as

Fµν = i[Dµ�, Dν ] . (10)

The tensor Fµν is also derivative-valued. Therefore, we use the same procedure asin [2], namely, we split Fµν into the curvature-like terms and torsion-like terms

Fµν = Fµν + T ρµνDρ + . . .+ T ρ1...ρl

µν : Dρ1 . . .Dρl: + . . . , (11)

and use only curvature-like terms Fµν to construct an effective action. Explicitsolutions for the SW maps for the fields we use in writing an action for U(1) gaugetheory are given in Ref. [3].

Czech. J. Phys. 55 (2005) 1393

M. Dimitrijevic, Larisa Jonke, and Lutz Moller

4 Action

In order to construct the action, we need an integral with the trace property.This is essential for both the formulation of the variational principle and the gaugeinvariance of the action. We use the usual definition of an integral of functionsof commuting variables and introduce a measure function to implement the traceproperty [1] ∫

dn+1xµ(x)(f � g) =∫

dn+1xµ(x)(g � f) . (12)

Note that µ(x) is not �-multiplied by the other functions, it is a part of the volumeelement. From (12), it follows that

∂nµ(x) = 0 , xj∂jµ(x) = −nµ(x) . (13)

The measure function is xn independent and does not depend on the deformationparameter a either. In addition, the measure function is singular at zero and is notunique [8]. However, after defining the Lagrangian density in such a way that itvanishes at zero, we can choose a positive-definite measure function. Note also thatthe explicit form of µ(x) is not required in any of the subsequent calculations. Weonly use the relations (13), and therefore the non-uniqueness of the solution forµ(x) does not affect our results2).

With this integral we define the action as follows:

S =∫

dn+1xµ(x)L ,

where L is the Lagrangian density. Since the measure function µ(x) does not vanishin the limit a→ 0, we have to define the Lagrangian density such that

lima→0

µ(x)L = L0 .

Here L is the effective Lagrangian density expanded in powers of the deformationparameter a, and L0 is the Lagrangian density of the corresponding undeformedfield theory. Although this construction may appear rather arbitrary, we find thatimposing such a “good” limit as a→ 0 is an important physical requirement.

The action for the matter fields can be written as

Sm =∫

dn+1xµ(x)(

˜ψ � (iγµD∗µ + γµVµ �−m)ψ

). (14)

The tildas on the fields indicate that after introducing the measure one has torescale fields in order to obtain the usual, undeformed theory in the a → 0 limit.Also, requirement of antihermiticity of derivatives under the integral forces us toredefine the derivatives too.

2) An alternative way of constructing the measure function is to use a map which connects theκ-Minkowski space-time coordinates and the space-time coordinates of the canonical noncommu-tative space-time [9].

1394 Czech. J. Phys. 55 (2005)

U(1) gauge field theory on κ-Minkowski space

In the action for the gauge field

Sg = − 14

∫dn+1xµ(x) Tr (X2 � Fµν � F

µν) , (15)

we have introduced the gauge covariant expression X2

δαX2 = i[Λα�, X2], (16)

to obtain the proper limit as a→ 0 of the equations of motion, see Ref. [10].Expanding the actions (14) and (15) up to first order in deformation parameter

a and inserting the solutions for the SW map we obtain the following effectiveaction:

S =∫

dn+1x

{ψ0(iγµD0

µ −m)ψ0 − 14F

0µνF

0µν

− 14C

ρσλ

(ψ0γρD0

σD0λψ0 + D0σD0λψ0γρψ

0)

− 14 (1 − 8b1)Cρσ

λ xλψ0F 0ρσ(iγµD0

µψ0 −mψ0)

− 12 i(1 − 2b2)Cρσ

λ xλψ0γµF 0µρD0

σψ0

+ ib3Cρσλ xλψ0γµF 0

ρσD0µψ

0 − 2i(b1 − 14 (b2 − 2b3))Cρσ

λ xλψ0γµ(D0σF

0µρ)ψ0

− 14 ia(n+ 8b1 − 2nb2 + 4b3 − 1)ψ0γµF 0

µnψ0

− 12 (1−2b2)Cρσ

λ xλF 0µνF 0µρF

0νσ + 1

8 (1−8b3−2b4)Cρσλ xλF 0µνF 0

µνF0ρσ

}. (17)

The fields with the superscript zero denote undeformed fields of the standard fieldtheory. All constants bi are completely undetermined, they are consequence of am-biguity of the SW map [11]. Usually, all these constants are set to zero, but wenotice another interesting possibility. There exists a particular choice of the con-stants bi such that all ambiguous, undetermined terms in the action (17) are set tozero.

For massless fermions, we choose b1 = 116 , b2 = 1

2 , b3 = 18 , b4 = 0, and for

massive fermions we choose b1 = 18 , b2 = 1

2 , b3 = 0, b4 = 12 . The effective action

for U(1) gauge theory with fermionic matter 3) up to first order in the deformationparameter a is

S =∫

dn+1x{ψ0(iγµD0

µ −m)ψ0 − 14 F

0µνF

0µν

− 14 C

ρσλ

(ψ0γρD0

σD0λψ0 + D0σD0λψ

0γρψ

0)}

. (18)

3) Of course, m = 0 in the massless case.

Czech. J. Phys. 55 (2005) 1395

M. Dimitrijevic et al.: U(1) gauge field theory on κ-Minkowski space

The corresponding equations of motion are

(iγµD0µ −m)ψ0 − 1

2 Cρσλ γρD0

σD0λψ0 − i4n

Cρσσ γµF 0

µρψ0 = 0 ,

−iD0µψ

0γµ −mψ0 − 12 C

ρσλ D0

σD0λψ0γρ +i

4nCρσ

σ ψ0γµF 0µρ = 0 ,

∂µF0ρµ = Jρ = ψ0γρψ0 − i

4

(Cαµ

λ ηλρ(D0

µψ0γαψ

0 − ψ0γαD0µψ

0)

+ Cαρλ

(D0λψ0γαψ

0 − ψ0γαD0λψ0))

,

(19)

where Jµ is the conserved current ∂µJµ = 0.

We have obtained the action (18) and the equations of motion (19) that areboth gauge invariant and invariant under the classical Poincare transformations!

L.J. would like to thank the organizers for the very interesting and pleasant workshop.This work is supported by the Ministry of Science and Technology of the Republic ofCroatia under the contract 0098003.

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