the translation groups as generators of gauge transformation in b∧f theory

15 March 2001

Physics Letters B 502 (2001) 291–299www.elsevier.nl/locate/npe

The translation groups as generators of gauge transformationin B∧F theory

Rabin Banerjee, Biswajit ChakrabortyS.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City, Calcutta 700 098, India

Received 21 November 2000; received in revised form 9 January 2001; accepted 17 January 2001Editor: M. Cvetic

Abstract

The translation groupT (2), contained in Wigner’s little group for massless particles, is shown to generate gaugetransformations in the Kalb–Ramond theory, exactly as happens in Maxwell case. For the topologically massive (B∧F) gaugetheory, bothT (2) andT (3), act as the corresponding generators. 2001 Published by Elsevier Science B.V.

Ever since Wigner introduced the concept of little group for massive and massless relativistic particles wayback in 1939 [1], it played the most important role in classifying the various particles according to their spinquantum number. Furthermore, the transformation properties of quantum states, belonging to the Hilbert space,under Poincaré transformation can only be obtained by the method of induced representation from its correspondingtransformation properties under the little group. For example, the little group for a massive particle isSO(3) exactlyas its nonrelativistic counterpart. This implies that the transformation property of a massive particle, under rotation,will be the same as that of the nonrelativistic one allowing only the usual integer or half-integer spin for massiveparticles and the whole apparatus of spherical harmonics, Clebsch–Gordon coefficients, etc., can be carried overfrom nonrelativistic to relativistic quantum mechanics [2].

The situation for massless particles is quite different on the other hand. First of all it has no nonrelativisticcounterpart. Secondly, the structure of the corresponding little groupE(2), which is a semidirect product ofT (2)(group of translations in a two-dimensional plane perpendicular to the direction of momentum) andSO(2), isnot semisimple. In fact,T (2) is the abelian invariant subgroup ofE(2). This fact gives rise to some interestingcomplications. One can show that the spin has to align itself either in the parallel or antiparallel direction of themomentum of the particle with the allowed values of the helicity, restricted to integers and half integers, just asfor its massive counterpart [2]. The question naturally arises regarding the role of the other two “translation” likegenerators ofT (2) ⊂ E(2). Again it was shown by Weinberg [2,3] and Han et al. [4] that these objects play therole of generators of gauge transformations in Maxwell theory, which is aU(1) gauge theory. As is well known,a typical gauge theory like Maxwell theory does not allow massive excitations. This is in contrast with topologicallymassive theory like Maxwell–Chern–Simons (MCS) model in 2+ 1 dimensions [5]. The B∧F model [6] is anotherexample in the usual 3+ 1 dimensions where gauge invariance coexists with mass.

E-mail addresses: [email protected] (R. Banerjee), [email protected] (B. Chakraborty).

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00159-9

292 R. Banerjee, B. Chakraborty / Physics Letters B 502 (2001) 291–299

One can therefore ask whether the translation like generators ofT (2) can also generate gauge transformationin the rest frame of a quanta in such topologically massive gauge theories? A first attempt in this direction wasmade in [7], where it was found that the little group for massless particles in 2+ 1 dimensions, albeit in a differentrepresentation, can still generate gauge transformation in the MCS theory mentioned earlier. But the Wigner’s littlegroup for massless particles in 2+ 1 dimensions involves only a single parameter (which is isomorphic toR×Z2[8]), so the construction of the desirable (nonunique) representation generating the gauge transformation in MCStheory was straightforward. The question is what happens in the more complicated(3+ 1)-dimensional case? Forthat we consider Kalb–Ramond (KR) [9] and B∧F theory [6] — both in 3+ 1 dimension and involving a 2-formgauge field. As is well known, the KR quanta are massless unlike the topologically massive B∧F theory [10].

Let us first provide a brief review of the role of the Wigner’s little group in generating gauge transformation inMaxwell theory.1 The form of the Wigner’s little group of a massless particle moving alongz-direction is givenby [2,4]

(1)W(φ,u, v) = {Wµν} =

(1+ u2+v2

2

)(ucosφ − v sinφ) (usinφ + v cosφ)

(−u2+v2

2

)u cosφ sinφ −u

v −sinφ cosφ −v(u2+v2

2

)(ucosφ − v sinφ) (usinφ + v cosφ)

(1− u2+v2

2

)

.

By definition, this preserves the 4-momentumpµ = (ω,0,0,ω)T of the massless particle of energyω,

(2)Wµνp

ν = pµ.

This matrixW(φ,u, v) can be factorized as

(3)W(φ,u, v) =W(0, u, v)R(φ) with

(4)R(φ)=

1 0 0 00 cosφ sinφ 00 −sinφ cosφ 00 0 0 1

,

representing the rotation around thez-axis — the direction of propagation of photon andW(0, u, v) is isomorphicto the group of translationsT (2) in 2-dimensional euclidean plane. As a whole, the Lie algebra ofW(φ,u, v) isisomorphic to that ofE(2) andISO(2).

Now consider a photon having the above mentioned 4-momentumpµ and polarization vectorεµ(p), so that thenegative frequency part of the Maxwell gauge fieldAµ(x) can be written as

(5)Aµ(x)= εµ(p)eip·x.For the sake of simplicity, we shall suppress the positive frequency part and work with (5) only.

Note that a gauge transformation

(6)Aµ(x)→A′µ =Aµ + ∂µf

for some functionf (x) can be written equivalently in terms of the polarization vectorεµ as

(7)εµ(p)→ ε′µ = εµ(p)+ if (p)pµ,

wheref (x) — a scalar function — has been written asf (x)= f (p)eip·x just likeAµ (5) and again suppressingthe positive frequency part. Now the free Maxwell theory has the equation

(8)∂µFµν = 0

1 In our convention, the 4-vectorsxµ = (t, x, y, z)T ≡ (x0, x1, x2, x3)T, the signature of the metricgµν = (+,−,−,−) and the totallyantisymmetric tensorε0123= +1.

R. Banerjee, B. Chakraborty / Physics Letters B 502 (2001) 291–299 293

which follows from the LagrangianL = −14F

µνFµν . In terms of the gauge fieldAµ the above equation can berewritten as

(9)(gνµ✷ − ∂ν∂µ

)Aµ = 0,

which again can be cast in terms of the polarization vector using (5) as

(10)p2εµ − pµpνεν = 0.

One can easily see at this stage that forp2 = 0, the polarization vectorεµ is proportional topµ:

(11)εµ = (pνεν)

p2 pµ,

so that using (7), one gauges it away by making an appropriate choice forf (p):

(12)f (p)= i

p2 (pνεν).

One therefore concludes that massive excitations, if any, are gauge artefacts in pure Maxwell theory. This is nottrue for massless excitationsp2 = 0. Using (10), one finds that this only implies

(13)pµεµ = 0

which is nothing but the Lorentz gauge condition∂µAµ = 0. So in the frame, where momentum 4-vector takesthe formpµ = (ω,0,0,ω)T, the correspondingεµ takes the formεµ = (ε0, ε1, ε2, ε0)T which is again gaugeequivalent to

(14)εµ = (0, ε1, ε2,0

)T,

as one can easily show using (7) [14]. The physical polarization vector is thus confined in thexy plane forthe photon moving along thez-direction and has only two transverse degrees of freedom. The other scalar andlongitudinal polarization can be gauged away.

Coming finally to the role ofT (2) in generating gauge transformations, one can easily see that the action of thelittle group elementW(0, u, v) (1) onεµ (14) generates the following transformation:

(15)εµ → ε′µ =Wµν(0, u, v)ε

ν = εµ +(uε1 + vε2

ω

)pµ.

Clearly using (7), this can be identified as a gauge transformation. This result was obtained earlier in [3,4].We next perform a similar analysis for the Kalb–Ramond theory [9] whose dynamics is governed by the

Lagrangian

(16)L= 1

12HµνλH

µνλ, where

(17)Hµνλ = ∂µBνλ + ∂νBλµ + ∂λBµν

is the rank-3 antisymmetric field strength tensor and is derived from the rank-2 antisymmetric gauge field

(18)Bµν = −Bνµ.

The corresponding equation of motion is given by

(19)∂µHµνλ = 0.

Here the model is invariant under the gauge transformation given by

(20)Bµν → B ′µν = Bµν + ∂µfν − ∂νfµ.


In contrast to the Maxwell theory, these gauge transformations are reducible, i.e., they are not all independent. Thisis connected to the fact that it is possible to choose somefµ = ∂µΛ for which the gauge variation vanishes trivially.Proceeding just as was done for the Maxwell case, the negative frequency part for the KR gauge field for a particleof 4-momentumpµ can be written as

(21)Bµν(x)= εµν(p)eip·x,where we have introduced an antisymmetric polarization tensor (εµν = −ενµ), the counterpart ofεµ in (5). Onecan again cast the gauge transformation (20) and the equation of motion (19) in terms of the polarization tensors as

(22)εµν → ε′µν = εµν + i

(pµfν(p)− pνfµ(p)

)and

(23)pµ[pµενλ + pνελµ + pλεµν

] = 0,

respectively. We can then again consider the massive(p2 = 0) and massless(p2 = 0) cases separately. For(p2 = 0), one can write,

(24)ενλ = 1

p2

[pν(pνε

µλ)− pλ(pµεµν)

].

Using (22), this can be gauged away by choosing

(25)f λ(p)= i

p2pµε

µλ.

We thus find that massive excitations, if any, are gauge artefacts just as in the Maxwell case. Forp2 = 0, on theother hand, one gets by using (23):

(26)pµεµν = 0,

which is again equivalent to the “Lorentz condition”∂µBµν = 0. Using this condition, the six independentcomponents of the antisymmetric matrixε ≡ {εµν} can be reduced further. For example, in the frame where thelight-like vectorpµ takes the formpµ = (ω,0,0,ω)T, the condition (26) reduces to

(27)ε · p =

0 ε01 ε02 ε03

−ε01 0 ε12 ε13

−ε02 −ε12 0 ε23

−ε03 −ε13 −ε23 0

ω

00ω

= 0.

Simplifying, this yields

(28)ε03 = 0, ε01 = ε13, ε02 = ε23,

so that the reduced form of theε matrix is given by

(29)ε =

0 ε01 ε02 0−ε01 0 ε12 ε01

−ε02 −ε12 0 ε02

0 −ε01 −ε02 0

.

This form of ε can be reduced further by making the gauge transformation (22), by taking,f 1 = iωε01 and

f 2 = iωε02, wherebyε01 andε02 are gauged away and one is left with just one degree of freedom represented

by ε12. The final form of theε matrix is then

(30)ε = ε12

0 0 0 00 0 1 00 −1 0 00 0 0 0

.


We are now in a position to study the role ofT (2) in generating gauge transformations. Using the fact that therank-2 contravariant tensorεµν transforms as

(31)εµν → ε′µν = ΛµρΛ

νσ ε

ρσ

under a Lorentz transformationΛ (Λµν ≡ ∂x ′µ/∂xν), one can write down the transformation property ofε matrix

under the little groupW(0, u, v) (1), which is a subgroup of the Lorentz group, in a matrix form as

(32)ε → ε′ =W(0, u, v)εWT(0, u, v)= ε12

0 −v u 0v 0 1 v

−u −1 0 −u

0 −v u 0

= ε + ε12

0 −v u 0v 0 0 v

−u 0 0 −u

0 −v u 0

.

This is clearly a gauge transformation, as this can be cast in the form of (22) withf 1 = ivωε12, f 2 = − iu

ωε12 and

f 3 = f 0.We finally take up the case of B∧F model which has massive excitations induced by the topological term. The

B∧F model is described by the Lagrangian2 [6]

(33)L = −1

4FµνF

µν + 1

12HµνλH

µνλ − m

6εµνλρHµνλAρ,

which is obtained by topologically coupling theBµν field of Kalb–Ramond theory (16) with the Maxwell fieldAµ

so that the last term in (33) does not contribute to the energy-momentum tensor. The parameterm in this term istaken to be positive. The Euler–Lagrange equations forAµ andBµν fields are given by

(34)∂µFµρ − m

6εµνλρHµνλ = 0 and

(35)∂µHµνλ = 1

2mερσνλFρσ ,

respectively. Using the forms (5) and (21), these coupled equations (34) and (35) can be cast in terms of thepolarization vectors and polarization tensors as

(36)p2ερ − pρpµεµ + i

2mεµνλρpµενλ = 0,

(37)p2εµν + ελµpλpν − ελνpλp

µ + impρεσ ερσµν = 0.

As was done in the previous section, here too we can consider the massless (p2 = 0) and massive (p2 = 0) cases,respectively. For massless case one can easily show using (36) that

(38)εραβγ pρpµε

µ = im(pαεβγ − pβεαγ + pγ εαβ).

Contracting withpβ on either side yields

(39)pαpβεβγ + pγ p

βεαβ = 0.

Using (39) and the masslessness condition (p2 = 0), one can immediately see using (37) that

(40)pρεαεραβγ = 0,

so that any general solution ofεα can now be written as

(41)εα = f (p)pα

2 Note that the interaction term differs from the conventional B∧F termεµνλρBµνFλρ by a four-divergence.


for some functionf (p). Using (7), one can thus easily see that massless excitations, if any, are gauge artefacts now.This is in contrast with the Maxwell and KR models considered earlier, where the massive excitations are gaugeartefacts. Let us consider the massive case (p2 = θ2) now. Going to the rest frame withpµ = (θ,0), one can relatethe spatial components ofεµ andεµν by making use of (36) and (37) to get the following coupled equations:

(42)εi = − im

2θε0ijkεjk,

(43)εij = − im

θε0ijkεk,

whereasε0 andε0i remain arbitrary. However, these can be trivially gauged away by making use of the gaugetransformations (7) and (22) and the above mentioned form for the four-momentumpµ = (θ,0) in the rest frame.On the other hand, the mutual compatibility of the pair of equations (42) and (43) implies that we must have

(44)θ =m,

as we have takenm> 0. This indicates that the strength ‘m’ of B∧F term in (33) can be identified as the mass ofthe quanta in B∧F model. With this, (42) and (43) simplify further and one can writeεµν andεµ in terms of thethree independent parameters:

(45)ε = {εµν} =

0 0 0 00 0 c −b

0 −c 0 a

0 b −a 0

and

(46)εµ = −i

0a

b

c

.

Before we proceed further to construct the generators of gauge transformations in B∧F model, let us try to see howthe varying number of degrees of freedoms associated with the three different models we have considered so far,can be understood through a Hamiltonian analysis at one stroke.

For this it is convenient to consider the B∧F model (33) itself, as form = 0, this model reduces to a system ofdecoupled Maxwell and KR models (16). Introduce the momenta variables,

(47)πµ = ∂L∂Aµ

= Fµ0,

(48)πµν = ∂L∂Bµν

= 1

2

(H 0µν −mε0ρµνAρ

),

conjugate toAµ andBµν , respectively. Clearly,π0 andπ0i vanish,

(49)π0 ≈ 0, π0i ≈ 0,

and correspond to the primary constraints of the model. To check for any secondary constraints, we have to gethold of the Legendre transformed Hamiltonian, which is given by

(50)H = πµAµ + π 〈µν〉B〈µν〉 + 1

4FµνF

µν − 1

12HµνλH

µνλ + m

6εµνλρHµνλAρ,

where the symbol〈〉 means that the indices within it have to be ordered either in an increasing or decreasing orderto avoid double counting. Upon simplification, this takes the final form as,

(51)H = 1

2

[πiπi + F〈ij〉F〈ij〉 + (H123)

2] + 1

2m2A2 +m(A1π23 +A2π31 +A3π12)−A0G−B0iGi


with

(52)G≡ ∂iπi +mH123≈ 0 and

(53)Gi ≡ ∂jπji ≈ 0

being the secondary (Gauss) constraints. They are first class and generate appropriate gauge transformations. Onecan easily check that there are no tertiary constraint in the model. At this stage one can note the following points:

(i) The first-class constraintsGi are reducible as they satisfy

(54)∂iGi = 0

so that the number of such independent constraints is only 2.(ii) the pairπ0 (49),G (52) (except themH123 term) are two first-class constraints in the free Maxwell theory and

the other pairπ0i (49) andGi (53), along with the reducibility property (54), are the first-class constraints ofthe KR field.

It is thus easy to see the number of degrees of freedom in B∧F theory can be obtained by just putting togetherthe number of degrees of freedom in Maxwell and KR theories. To that end consider free Maxwell theory whichhas 4× 2 = 8 variables(Aµ,π

ν) in phase space to begin with. The two constraints(π0,G) along with two gaugefixing conditions will reduce the dimension of the physical phase space to four which is equivalent to two degreesof freedom in the configuration space. Correspondingly the polarization vector takes the form (14) with onlytransverse degrees of freedom surviving.

Taking up the case of KR theory now, it has 6× 2 = 12 variables in the phase space(Bµν,πµν). The total

number of first-class constraintsπ0i (49) andGi (52) is 3+ 2 = 5 now. Along with gauge fixing conditions,the dimension of phase space will reduce to 12− (5 × 2) = 2 so that there is only one independent variable inthe configuration space. Correspondingly the polarization matrixε involves a single parameter (30). The numberof independent configuration space variables in B∧F theory is therefore just 3. Correspondingly the polarizationvector and polarization tensor take the forms (45) and (46).

Another way of understanding the degree of freedom count is to recall that the B∧F Lagrangian (33) can beregarded either as a massive Maxwell (i.e., Proca) theory or a massive KR theory [11,12]. This can be achievedby eliminating once the KR field or, alternatively, the vector field from the coupled set of equations (34) and (35).Both these theories have three massive degrees of freedom. It is intriguing to note that this dual structure appearsto be manifested in (45) and (46), by the orthogonality relation

(55)εµνεν = 0.

Returning to the issue of gauge transformations in the B∧F theory, observe thatW(0, u, v) fails to be a generator.Does this mean thatT (2), in contrast to the Maxwell and KR examples, is not a generator of gauge transformationin the B∧F theory? Before discussing this, consider the matrix

(56)D(α,β, γ )=

1 α β γ

0 1 0 00 0 1 00 0 0 1

involving three real parametersα,β, γ . This generates gauge transformation acting on the polarization vector andpolarization tensor (45) and (46):

(57)εµ → ε′µ =Dµν(α,β, γ )ε

ν = εµ − i

m(αa + βb+ γ c)pµ,


(58)ε → ε′ =D(α,β, γ )εDT(α,β, γ )= ε +

0 (γ b− βc) (αc − γ a) (βa − αb)

−(γ b− βc) 0 0 0−(αc− γ a) 0 0 0−(βa − αb) 0 0 0

,

as both (57) and (58) can be easily cast into the form (7) and (22) with appropriate choices off (p) andf i(p).Also it preserves the 4-momentum of a massive particle at rest. We can now identify the group to whichD(α,β, γ )

belongs. One can easily show that

(59)D(α,β, γ ) ·D(α′, β ′, γ ′)=D(α + α′, β + β ′, γ + γ ′) and

(60)[P1,P2] = [P1,P3] = [P2,P3] = 0,

where

(61)P1 = ∂D(α,0,0)

∂α, P2 = ∂D(0, β,0)

∂β, P3 = ∂D(0,0, γ )

∂γ

can be thought of as three mutually commuting “translational” generators. The group can therefore be identifiedwith T (3) — the invariant subgroup ofE(3) [13] or ISO(3).

Clearly three different canonical embeddings ofT (2) within T (3) can be obtained by successively setting oneof the parametersα,β, γ to be zero in (56). Not only that these differentT (2)’s generate gauge transformations, ascan be seen trivially from (57) and (58), they also preserve the 4-momenta of massless particles moving inx, y andz directions, respectively. The only distinguishing features of these differentT (2)’s are their representations, whichthey inherit either from the Wigner’s little group in (1) or from the representation ofT (3) in (56). In the formercase, it acts as a generator in Maxwell or KR theory involving only massless excitations while in the latter case,it acts as a generator in topologically massive B∧F theory. The groupT (3), on the other hand, acts as a generatoronly in the B∧F theory.

To conclude, we have found that in B∧F theory the generator of gauge transformations can be identified toT (3) — the group of translations inR3. ClearlyT (2) ⊂ T (3) also does the job and since it is isomorphic to thegroup defined byW(0, u, v) (1) one can say that the abelian invariant subgroupT (2) of Wigner’s little group actsalso as a generator in massive B∧F theory. The B∧F theory therefore manifests both aspects of gauge invariance,conventional or massive. This does not happen in the usual gauge theories like Maxwell or KR, where only Wigner’slittle groupT (2) acts as the generator.

Our results may be compared with a dynamical (hamiltonian) approach where the Gauss operator generates thegauge transformations. The structure of this operator differs from theory to theory. On the contrary, our analysisrevealed a universal structure for the generators; namely, the translational group. We feel this is related to the factthat the abelian gauge transformations are translational in nature. It would be worthwhile to extend this analysiswhere the gauge transformations are curvilinear, as in non-abelian theories or gravity [15].

References

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D. Han, Y.S. Kim, D. Son, Phys. Rev. D 26 (1982) 3717;D. Han, Y.S. Kim, D. Son, Phys. Rev. D 31 (1985) 328.

[5] S. Deser, R. Jackiw, S. Templeton, Ann. Phys. 140 (1982) 372.[6] E. Cremmer, J. Scherk, Nucl. Phys. B 72 (1971) 117.


[7] R. Banerjee, B. Chakraborty, T. Scaria, hep-th/0011011.[8] B. Binegar, J. Math. Phys. 23 (1982).[9] M. Kalb, P. Ramond, Phys. Rev. D 9 (1974) 2273.

[10] A. Aurilia, Y. Takahashi, Prog. Theor. Phys. 66 (1981) 693;A. Aurilia, Y. Takahashi, Phys. Rev. D 23 (1981) 752.

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the translation groups as generators of gauge transformation in b∧f theory

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