Int J Theor Phys (2012) 51:2564–2584DOI 10.1007/s10773-012-1137-3

Electromagnetic Interaction of Composite Particlesin a Higher-Derivative Nonrelativistic Gauge Field Model

Edmundo C. Manavella · Carlos E. Repetto

Received: 9 November 2011 / Accepted: 14 March 2012 / Published online: 29 March 2012© Springer Science+Business Media, LLC 2012

Abstract A higher-derivative classical nonrelativistic U(1) × U(1) gauge field model thatdescribes the electromagnetic interaction of composite particles in 2 + 1 dimensions is pro-posed. The model contains a Chern-Simons U(1) field and the electromagnetic U(1) field,and it uses either a composite boson system or a composite fermion one. The second caseis explicitly considered. The model is obtained by adding suitable higher-derivative termsto the Lagrangian of a model previously proposed. One of them for the electromagneticfield and the other for the Chern-Simons field. By following the usual Hamiltonian methodfor singular higher-derivative systems, the canonical quantization is carried out. By extend-ing the Faddeev-Senjanovic formalism, the path integral quantization is developed. Con-sequently, the Feynman rules are established and the diagrammatic structure is discussed.In this context, only a mixed bosonic propagator can be defined. The use of the higher-derivative terms eliminates the ultraviolet divergence of the primitively divergent Feynmandiagrams where the bosonic propagator is present. The unitarity problem, related to thepossible appearance of states with negative norm, is treated. A generalization of the Becchi-Rouet-Stora-Tyutin algorithm is applied to the model. We will focus our attention on theissue from the point of view of the field theory.

Keywords Lagrangian and Hamiltonian formalisms · Higher derivatives · Chern-Simonsgauge theory

E.C. Manavella (�) · C.E. RepettoInstituto de Física Rosario (CONICET), 27 de Febrero 210 bis, S2000EZP Rosario, Argentinae-mail: manavella@ifir-conicet.gov.ar

C.E. Repettoe-mail: repetto@ifir-conicet.gov.ar

E.C. Manavella · C.E. RepettoFacultad de Ciencias Exactas, Ingeniería y Agrimensura (UNR), Av. Pellegrini 250, S2000BTP Rosario,Argentina

Int J Theor Phys (2012) 51:2564–2584 2565

1 Introduction

As it is well known, the composite particle theory has significant relevance in the under-standing of the quantum Hall effect, in its integer and fractional versions, and has been longregarded with increasing interest [1–10]. Currently, there are three typical lines of work thatuse: composite bosons (CB) [11, 12], composite fermions (CF) [13–19], and both compos-ite bosons and fermions [20–22]. Moreover, the importance of these theories is enhancedbecause not only explain the quantum Hall effect but they are also related to superconduc-tivity [23–25].

We are interested in studying the electromagnetic interaction of these composite particlesin 2 + 1 dimensions. We will focus our attention on the issue from the point of view of thefield theory. Then, in a forthcoming paper, we will consider this model in the context ofcondensed matter in order to obtain possible new results.

In this context, we start by referring to what has been done in previous papers where wehave developed other models related to the subject. Therefore, we proposed [26] a classi-cal nonrelativistic U(1) × U(1) gauge field model which contains two U(1) gauge fields, aChern-Simons (CS) field aμ [4, 27, 28] and the electromagnetic field Aμ.

Likewise, we performed the canonical quantization for the model. This was done bymeans of the Dirac Hamiltonian formalism for constrained systems [29–31].

Furthermore, we implemented the path integral quantization method through theFaddeev-Senjanovic (FS) procedure [32, 33] because the model has first and second-classconstraints. Consequently, we established the Feynman rules of the model.

Moreover, we considered a simplified version of the model related to one used within theframework of condensed matter [34].

Also, we applied the Becchi-Rouet-Stora-Tyutin (BRST) algorithm to the model [35–41].We treated explicitly the CF case.On the other hand, we proposed to extend the model by adding different terms to the

Lagrangian density.Thus, in [42], we treated the general case by considering a topological mass term for the

electromagnetic field and interaction terms between the CS and electromagnetic fields.Besides, we analyzed the obtained model following the same steps as the ones carried

out in [26] and we compared the resultant and original models.Furthermore, we found that the equations corresponding to the pure interactive case can

be obtained directly from the ones corresponding to [42], by cancelling the terms with topo-logical mass. Nevertheless, we encountered the trouble that the equations corresponding tothe pure topologically massive case cannot be found from the ones corresponding to thatreference by cancelling the interaction terms between the gauge fields. This is because theconstraint and diagrammatic structures for these models are different.

For this reason, we had to consider [43] the pure topologically massive case to which weanalyzed similarly to what has been done in [26, 42].

On the other hand, it is known that, in different models with gauge invariances, it hasbeen considered the addition of higher-derivative (HD) terms for the gauge fields to the cor-responding Lagrangians, keeping their gauge invariances. The reason for this procedure isthat, in general, these terms improve the ultraviolet behaviour of the gauge field propagators.Consequently, the divergence of the Feynman diagrams in which these propagators appearcan possibly be eliminated [44–50].

Due to this, we proposed to apply this procedure to the composite particle models of [26,42, 43].

In accordance with the above mentioned issue, we had to consider separately the puretopologically massive and general cases.

2566 Int J Theor Phys (2012) 51:2564–2584

With reference to the first case, we modified [50] the model proposed in [43] by intro-ducing a HD term for the electromagnetic field in the Lagrangian of this model, preservingits gauge invariance. Then, we repeat what has been done in previous models of [26, 42, 43]for this model.

Moreover, we studied the ultraviolet behaviour of the new electromagnetic propagatorfound, as well as the ultraviolet convergence of the primitively divergent Feynman diagramsof the original model in which this propagator appears.

We also dealt with the unitarity of the model.Besides, we considered the HD nontopologically massive case. As we saw, except for

the equations involved in the treatment of the unitarity problem, the equations for that casecan be obtained directly from the ones corresponding to the treated model by cancelling theterms with topological mass.

The second case is analyzed here. Thus, in this paper, we add in gauge invariant formHD terms for the electromagnetic and CS fields to the Lagrangian of the model of [42].

Next, we study the resultant model following the same stages as the ones performedin [50].

The article is organized as follows. In Sect. 2, we present our classical model and we carryout its canonical quantization. Later, in Sect. 3, we perform the path integral quantization,we establish the Feynman rules and we study the ultraviolet convergence. Afterwards, inSect. 4, we analyze the unitarity. In Sect. 5, we develop the BRST quantization. Finally, inSect. 6, we display our conclusions and outlook.

2 Classical Model and Canonical Quantization

In the present paper, as it was proposed in the introduction, we add HD terms for the elec-tromagnetic and CS fields to the Lagrangian of the model of [42], preserving its gaugeinvariance.

Hence, we consider a HD classical nonrelativistic field model with U(1) × U(1) gaugesymmetry for the electromagnetic interaction of composite particles in 2 + 1 dimensions.We explicitly analyze a CF system. We think that this system can be treated through thefollowing second-order singular Lagrangian density:

L = Lemcf + Ltm + Lint + Lh, (1)

where

Lemcf = iψ† D0ψ + 1

2mb

ψ†D2ψ − μψ†ψ + 1

4πφεμνρaμ∂νaρ, (2)

Ltm = 1

2σεμνρAμ∂νAρ − 1

4FμνF

μν, (3)

Lint = ζe

mb

εμνρ (aμ∂νAρ + Aμ∂νaρ) + ηe

mb

fμνFμν, (4)

Lh = κ∂ρFμν∂ρFμν + κ ′∂ρfμν∂

ρFμν. (5)

In (2)–(5), the Greek indices take the values μ,ν,ρ = 0,1,2.We employ natural units where � = c = 1. The Minkowskian metric used is gμν =

diag(1,−1,−1) and ε012 = ε12 = 1.

Int J Theor Phys (2012) 51:2564–2584 2567

In (2), the covariant derivative which involves both the CS U(1) gauge field aμ and theelectromagnetic U(1) gauge field Aμ, is given by Dμ = ∂μ−iaμ−ieAμ (we take the electroncharge as −e) and furthermore, D2 = D2

1 + D22 . The matter field ψ is a charged spinorial

field describing CF. mb and μ are the band mass and the chemical potential of the electrons,respectively. φ is the strength of the flux tube in units of the flux quantum 2π . (We fix thestrength for the fictitious charge of each particle that interacts with the fictitious gauge fieldin the unit.)

In (3), the first term on the right-hand side is the topological mass term for the electro-magnetic field. The topological mass is given by 2π/σ and so the real magnetic flux at-tached to the electrons is eσ/2π . In the second term on the right-hand side of that equation,Fμν=∂μAν − ∂νAμ is the electromagnetic field tensor.

In (4), Lint is the Lagrangian density corresponding to the interaction between the gaugefields and fμν=∂μaν − ∂νaμ is the CS field tensor. Furthermore, in that equation, the CSterm is written in such a way to obtain symmetric expressions for the canonically conjugatemomenta corresponding to the gauge fields.

In (5), Lh is the Lagrangian density corresponding to the HD terms and κ and κ ′ areconstants.

Therefore, the Lagrangian density corresponding to the original model is obtained from(1) by removing the HD terms.

By using the expression for the covariant derivative, we rewrite (2) as follows:

Lemcf = i

τ + 1

2ψ† ∂0ψ + i

τ − 1

2∂0ψ

† ψ + ψ†(a0 + eA0)ψ + 1

4πφεμνρ aμ ∂νaρ

+ 1

2mb

ψ† D2ψ − μψ† ψ. (6)

In this equation, the kinetic fermionic term is written in the general form through thearbitrary parameter τ [31].

Now we are going to develop the canonical quantization for the model.The canonical formalism for HD theories was developed by Ostrogradski [51]. Later, that

formalism was generalized to the case of singular systems [52].Consequently, we use the Hamiltonian treatment for singular HD systems developed in

this last reference. Therefore, let us consider a second-order Lagrangian density of the form

L = L(Ak, ∂μAk, ∂μ∂νAk), (7)

for a set of independent dynamical field variables Ak, k = 1, . . . , n, and where the Greekindices take the values μ,ν = 0,1,2.

Next, from the principle of least action, we obtain the Euler-Lagrange equations

δLδAk

− ∂μ

δLδ(∂μAk)

+ ∂μ∂ν

δLδ(∂μ∂νAk)

= 0, (8)

where δ denotes functional derivative.In this case the canonical dynamical field variables are introduced according to the Ostro-

gradski transformation. These are A(1)k = Ak , A

(2)k = Ak and the corresponding canonically

conjugate momenta, defined in the way

P(1)k = δL

δA(1)k

− ∂μ

δLδ(∂μA

(2)k )

, (9)

2568 Int J Theor Phys (2012) 51:2564–2584

P(2)k = δL

δA(2)k

, (10)

respectively, where we denote by Ak = ∂0Ak the time derivative.Because the Lagrangian density (1) is of the kind given by (7), to generate the phase

space of the present model, we must consider the dynamical field variables: AI = (aμ, bν =aν , Aρ, Bε = Aε, ψα, ψ

†β) and their respective canonically conjugate momenta P I =

(pμ, qν, P ρ, Qε, π†α, πβ) defined by (9) and (10). In these sets, the compound index I

runs over the components of the different field variables and the new Greek indices take thevalues α,β = 1,2.

We find that the momenta have the expressions

p0 = 2κ ′(∂iBi − ∇2B0

), (11)

pi = 1

4πφεij aj + ζ

e

mb

εijAj + 2ηe

mb

(Bi − ∂iA0

)

+ 2κ ′(∂iB0 − Bi + ∂i∇2A0 − 2∇2Bi + ∂i∂jBj

), (12)

q0 = 0 , (13)

qi = 2κ ′(Bi − ∂iB0

), (14)

P 0 = 4κ(∂iBi − ∇2B0

) + 2κ ′(∂i bi − ∇2b0), (15)

P i = 1

2σεijAj + ζ

e

mb

εij aj + 2ηe

mb

(bi − ∂ia0

) + ∂iA0 − Bi

+ 4κ(−Bi + ∂iB0 − 2∇2Bi + ∂i∇2A0 + ∂i∂jBj

)

+ 2κ ′(−bi + ∂i b0 − 2∇2bi + ∂i∇2a0 + ∂i∂j bj

), (16)

Q0 = 0, (17)

Qi = 4κ(Bi − ∂iB0

) + 2κ ′(bi − ∂ib0

), (18)

π†α = −i

τ + 1

2ψ†

α, (19)

πα = iτ − 1

2ψα, (20)

where ∇2 = ∂i∂i is the Laplacian operator and the Latin indices take the values i, j = 1,2.

The nonvanishing fundamental equal-time (x0 = y0) Bose-Fermi brackets [53, 54] aregiven by

[aμ(x),pν(y)

]− = δν

μδ(x − y), (21)[bμ(x), qν(y)

]− = δν

μδ(x − y), (22)[Aμ(x),P ν(y)

]− = δν

μδ(x − y), (23)[Bμ(x),Qν(y)

]− = δν

μδ(x − y), (24)[ψα(x),π

†β(y)

]+ = −δαβδ(x − y), (25)

[ψ†

α(x),πβ(y)]+ = −δαβδ(x − y), (26)

Int J Theor Phys (2012) 51:2564–2584 2569

where we have used the notation [ . , . ]∓ to point out brackets between bosonic and betweenfermionic Grassmann variables, respectively.

From (11)–(20), we find the primary constraints

Φ0a = q0 ≈ 0, (27)

Φ0A = Q0 ≈ 0, (28)

Ω†α = π†

α + iτ + 1

2ψ†

α ≈ 0, (29)

Ωα = πα − iτ − 1

2ψα ≈ 0. (30)

The canonical Hamiltonian density is defined as Hc = bμpμ + bμqμ +BμP μ + BμQμ +ψπ† + ψ†π − L. By using (11)–(20), we have that

Hc = bμpμ + BμP μ − κ

2κ ′2 qiqi + 1

2κ ′ qiQi + qi∂ib0 + Qi∂iB0 + μψ† ψ

− ψ†(a0 + eA0)ψ − 1

2mb

ψ† D2ψ + 1

4FijF

ij + 1

2F0iF

0i

+ 1

2σεij (−2A0∂iAj + AiBj ) + 1

4πφεij (−2a0∂iaj + aibj )

+ ζe

mb

εij(aiBj + Aibj − 2a0∂iAj − 2A0∂iaj ) − ηe

mb

(fijF

ij + 2f0iF0i)

+ κ(−∂iFjk∂

iF jk + 2∂iBj∂jBi − 2∂i∂jA0∂

i∂jA0 − 4∂iBj∂iF0j

)

+ κ ′(−∂ifjk∂iF jk + 2Gij∂jbi − 2∂if0j ∂

iF 0j), (31)

where Gij =∂iBj − ∂jBi .Furthermore, the primary Hamiltonian density is defined as

Hp = Hc + λaΦ0a + λAΦ0

A + λ†αΩα + Ω†

αλα, (32)

where λa and λA are bosonic Lagrange multipliers and λ†α and λα are fermionic ones.

Now we must impose the consistency condition on the primary constraints. Thus, weobtain the secondary constraints

Φ1a = [

Φ0a ,Hp

] = −p0 + ∂iqi ≈ 0, (33)

Φ1A = [

Φ0A,Hp

] = −P 0 + ∂iQi ≈ 0, (34)

Φ2a = −[

Φ1a ,Hp

] = ∂ipi + 1

4πφεij ∂iaj + ζe

mb

εij ∂iAj + ψ†ψ ≈ 0, (35)

Φ2A = −[

Φ1A,Hp

] = ∂iPi + ζe

mb

εij ∂iaj + 1

2σεij ∂iAj + eψ†ψ ≈ 0, (36)

where Hp = ∫d2x Hp is the primary Hamiltonian.

By using (1) and (8), it is easy to show that (35) and (36) are the equations of motioncorresponding to the time components of the aμ and Aμ, respectively.

2570 Int J Theor Phys (2012) 51:2564–2584

When the consistency on the constraints (29) and (30) is implemented, the Lagrangemultipliers λ†

α and λα , appearing in (32), remain determined, respectively. Besides, whenthe consistency on the constraints (35) and (36) is imposed, the obtained equations are au-tomatically satisfied.

Next, we find that the constraints (27), (28), (33) and (34) are first-class whereas theconstraints (29), (30), (35) and (36) are second-class. However, the latter do not constitutea minimal set of second-class constraints. This is because the determinant of the matrixconstructed with the Bose-Fermi brackets between these constraints vanishes.

Hence, there must be at least two linear combinations of second-class constraints whichare independent of the above first-class constraints and they must also be first-class. It iseasy to see that there are only two of such combinations, which are

Σ1 = eΦ2a − Φ2

A

= e∂ipi − ∂iP

i + e

(1

4πφ− ζ

mb

)εij ∂iaj +

(ζe2

mb

− 1

2σ

)εij ∂iAj ≈ 0, (37)

Σ2 = ψ†Ω − ψΩ† − i

eΦ2

A

= ψ†π − ψπ† − i

e

(∂iP

i + ζe

mb

εij ∂iaj + 1

2σεij ∂iAj

)≈ 0. (38)

Therefore, two second-class constraints can be eliminated and so the final set of con-straints is the following:

i. The first-class constraints defined by the functions Σ1,Σ2,Σ3 = Φ0a , Σ4 = Φ0

A, Σ5 = Φ1a

and Σ6 = Φ1A.

ii. The second-class constraints defined by Ω†α and Ωα .

The extended Hamiltonian is defined by

He =∫

d2x(

Hc + ρaΣa

) −∫

d2x d2y ΓI (x)F−1IJ (x,y)

[ΓJ (y),Hc

], (39)

where ρa, a = 1, . . . ,6, are bosonic Lagrange multipliers, F−1 is the inverse of the matrix F

whose elements are [ΓI ,ΓJ ], I, J = 1, . . . ,4, and Γ1 = Ω†1 ,Γ2 = Ω

†2 ,Γ3 = Ω1 and Γ4 = Ω2

are the second-class constraints.The matrix F reads

F =

⎛

⎜⎜⎝

0 0 −i 00 0 0 −i

−i 0 0 00 −i 0 0

⎞

⎟⎟⎠ δ(x − y). (40)

Its determinant is

detF = δ(x − y) (41)

and its inverse is given by

F−1 =

⎛

⎜⎜⎝

0 0 i 00 0 0 i

i 0 0 00 i 0 0

⎞

⎟⎟⎠ δ(x − y). (42)

Int J Theor Phys (2012) 51:2564–2584 2571

Now we calculate the Dirac brackets D(F) with regard to the matrix F . The D(F)

bracket between the functions R(x) and S(y) is defined as follows:

[R(x), S(y)

]D(F) = [R(x), S(y)

] −∫

d2ud2v[R(x),ΓI (u)

]F−1

IJ (u,v)[ΓJ (v), S(y)

].

(43)From this equation, we obtain the following nonvanishing D(F) brackets:

• Field-field:[ψ†

α(x),ψβ(y)]D(F)

+ = −iδαβδ(x − y), (44)

• Field-momentum:

[aμ(x),pν(y)

]D(F)

− = δνμδ(x − y), (45)

[bμ(x), qν(y)

]D(F)

− = δνμδ(x − y), (46)

[Aμ(x),P ν(y)

]D(F)

− = δνμδ(x − y), (47)

[Bμ(x),Qν(y)

]D(F)

− = δνμδ(x − y). (48)

When we impose the D(F) brackets, we must take the second-class constraints as equa-tions strongly equal to zero. Thus, the following field variables remain determined:

π†α = −i

τ + 1

2ψ†

α, (49)

πα = iτ − 1

2ψα. (50)

Moreover, the last term on the right-hand side of (39) vanishes and so the extended Hamil-tonian remains

He =∫

d2x(

Hc + ρaΣa

). (51)

Now we calculate the final Dirac brackets or simply, Dirac brackets. The Dirac bracketbetween the functions R(x) and S(y) is defined as follows:

[R(x), S(y)

]D = [R(x), S(y)

]D(F)

−∫

d2ud2v[R(x),ΔA(u)

]D(F)G−1

AB(u,v)[ΔB(v), S(y)

]D(F). (52)

In this equation, G−1 is the inverse of the matrix G whose elements are [ΔA,ΔB ], A,B =1, . . . ,12,Δa = Σa and Δ6+a = Θa , where Θa ≈ 0 are admissible gauge-fixing conditions,one for each first-class constraint. The gauge-fixing conditions must verify that detG �≈ 0and must be compatible with the equations of motion.

We choose the following gauge-fixing conditions:

Θ1 = ∂iai ≈ 0, (53)

Θ2 = ∂iAi ≈ 0, (54)

Θ3 = b0 ≈ 0, (55)

2572 Int J Theor Phys (2012) 51:2564–2584

Θ4 = B0 ≈ 0, (56)

Θ5 = 2

(− ηe

mb

+ κ ′∇2

)∇2a0 + (

1 + 4κ∇2)∇2A0

+ εij

(ζe

mb

∂iaj + 1

2σ∂iAj

)− ∂iP

i ≈ 0, (57)

Θ6 = 2

(− ηe

mb

+ κ ′∇2

)∇2A0 − ∂ip

i

+ εij

(1

4πφ∂iaj + ζe

mb

∂iAj

)≈ 0, (58)

which satisfy the above requirements.We find that the matrix G is given by

G =(

O P

−P T Q

)δ(x − y), (59)

where

P =

⎛

⎜⎜⎜⎜⎜⎜⎝

e∇2 −∇2 0 0 0 00 − i

e∇2 0 0 0 0

0 0 −1 0 0 00 0 0 −1 0 00 0 0 0 t 00 0 0 0 s t

⎞

⎟⎟⎟⎟⎟⎟⎠

, (60)

Q =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 ∇2

0 0 0 0 ∇2 00 0 0 0 0 00 0 0 0 0 00 −∇2 0 0 0 0

−∇2 0 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎠

, (61)

where s = (1 + 4κ∇2)∇2, t = 2(− ηe

mb+ κ ′∇2)∇2, O is the 6 × 6 null matrix and “T ”

indicates transposition.The determinant of G holds

detG = −t4(∇2

)4δ(x − y) �≈ 0 (62)

and its inverse reads

G−1 =(

R S

−ST O

)δ(x − y), (63)

where

R =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 −e−1tw2 e−1w

0 0 0 0 −i(tw − e)w iw

0 0 0 0 0 00 0 0 0 0 0

e−1tw2 i(tw − e)w 0 0 0 0−e−1w −iw 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎠

, (64)

Int J Theor Phys (2012) 51:2564–2584 2573

S =

⎛

⎜⎜⎜⎜⎜⎜⎝

−e−1u 0 0 0 0 0−iu −ieu 0 0 0 0

0 0 1 0 0 00 0 0 1 0 00 0 0 0 −w tw2

0 0 0 0 0 −w

⎞

⎟⎟⎟⎟⎟⎟⎠

, (65)

where uδ(x − y) = −(2π)−1 ln |x − y| and

w δ(x − y) = (2π)−1 ln |x − y|8κ(|x − y| ln |x − y|)−2 − 1

. (66)

Therefore, from (52), we find the following nonvanishing Dirac brackets:

• Field-field:[ψ†

α(x),ψβ(y)]D

+ = −iδαβδ(x − y), (67)

• Field-momentum:

[ai(x),pj (y)

]D

− = [Ai(x),P j (y)

]D

−

= δj

i δ(x − y) + 1

2π∂x

i ∂xj ln |x − y|, (68)

[bi(x),pj (y)

]D

− = 2εkj ∂xi ∂x

k

(1

4πφtw − ζe

mb

)wδ(x − y), (69)

[bi(x), qj (y)

]D

− = [Bi(x),Qj (y)

]D

− = δj

i δ(x − y), (70)

[Bi(x),P j (y)

]D

− = −2ζe

mb

εkj ∂xi ∂x

k wδ(x − y), (71)

• Momentum-momentum:

[pi(x),pj (y)

]D

− = − 1

4πφεij δ(x − y), (72)

[pi(x),P j (y)

]D

− = − ζe

mb

εij δ(x − y), (73)

[P i(x),P j (y)

]D

− = − 1

2σεij δ(x − y). (74)

When we impose the Dirac brackets, we must take the first-class constraints and thegauge-fixing conditions as equations strongly equal to zero. Hence, the following field vari-ables remain determined:

b0 = 0, (75)

B0 = 0, (76)

p0 = ∂iqi, (77)

q0 = 0, (78)

2574 Int J Theor Phys (2012) 51:2564–2584

P 0 = ∂iQi, (79)

Q0 = 0, (80)

and a0 is obtained through (57) and (58) and A0 through (58).Furthermore, the second term on the right-hand side of (51) vanishes and so the extended

Hamiltonian coincides with the canonical one.Now we must go to the independent dynamical field variables aμ,Aν,ψα and ψ

†β , and

their respective canonically conjugate momenta pμ,P ν,π†α and πβ . In this way, some results

found are the following:

(i) As regards the constraint structure:The constraints (27) and (28) disappear and the ones given by (33) and (34) turn out

to be

Φ1′a = −p0 + 2κ ′(∂iAi − ∇2A0

) = 0, (81)

Φ1′A = −P 0 + 4κ

(∂iAi − ∇2A0

) + 2κ ′(∂i ai − ∇2a0

) = 0, (82)

without no other change in the set of constraints.The first-class constraints related to the symmetries of the gauge group U(1) × U(1)

are the ones defined by the functions Σ1,Σ2,Φ1′a and Φ1′

A .The admissible gauge-fixing conditions are the ones defined by the functions

Θ1,Θ2,Θ5 and Θ6.Consequently, in this case, instead of the matrix G, given by (59), we must consider

the matrix G′ whose elements are [Δ′A,Δ′

B ],A,B = 1, . . . ,8,Δ′a = Σa,a = 1,2,Δ′

3 =Φ1′

a ,Δ′4 = Φ1′

A ,Δ′5 = Θ1,Δ

′6 = Θ2,Δ

′7 = Θ5 and Δ′

8 = Θ6. This is written as

G′ =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 e∇2 −∇2 0 00 0 0 0 0 − i

e∇2 0 0

0 0 0 0 0 0 t 00 0 0 0 0 0 s t

−e∇2 0 0 0 0 0 0 ∇2

∇2 ie∇2 0 0 0 0 ∇2 0

0 0 −t −s 0 −∇2 0 00 0 0 −t −∇2 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

δ(x − y) (83)

and its nonvanishing determinant coincides with the one corresponding to the matrix G.(ii) As regards the final results corresponding to the canonical quantization:

The nonvanishing Dirac brackets are given by (67), (68), (72), (73) and (74).The field variables that remain determined are given by (49), (50), a0 through (57)

and (58), A0 through (58) and by the following equations:

p0 = 2κ ′(∂iAi − ∇2A0

), (84)

P 0 = 4κ(∂iAi − ∇2A0

) + 2κ ′(∂i ai − ∇2a0). (85)

In this way, the dynamics of the classical model remains completely specified.Finally, the canonical quantization is made by replacing the Dirac brackets between

field variables by the equal-time commutators or anticommutators between field opera-tors according to the usual rule [30].

Int J Theor Phys (2012) 51:2564–2584 2575

As it can be shown, by taking κ = κ ′ = 0 in the results denoted in the items (i) and (ii),we obtain the ones corresponding to the original model, as must be expected. In particular,we find

a0(x) = mb

4πηe

∫d2y

[mb

2ηe∂ip

i + ∂iPi −

(ζe

mb

+ mb

8πφηe

)εij ∂iaj

− 1

2

(1

σ+ ζ

η

)εij ∂iAj

](y) ln |x − y|, (86)

A0(x) = mb

4πηe

∫d2y

[∂ip

i − εij

(1

4πφ∂iaj + ζe

mb

∂iAj

)](y) ln |x − y|. (87)

On the other hand, let us note that a CB system can be studied in the same way. In thiscase, the matter field is a charged scalar field. Thus, the fermionic second-class constraints(29) and (30) turn into bosonic second-class ones. Moreover, the fermionic brackets (25),(26), (44) and (67) become into bosonic ones.

3 Path Integral Quantization, Feynman Rules and Diagrammatic Structure

Now we develop the Feynman path integral quantization method. For this purpose, we usethe FS formalism because the model has first and second-class constraints. Therefore, wewrite the generating functional of the model as follows [32, 33]:

Z =∫

DaμDpμDbνDqν

DAρDP ρDBεDQε

DψαDπ†αDψ

†βDπβ(detF)

12(detG)

12 δ(ΓI )

× δ(ΔA) exp

[i

∫d3x

(bμpμ + bμqμ + BμP μ + BμQμ + ψπ† + ψ†π − He

)], (88)

where δ(ΓI ) and δ(ΔA) stand for products of Dirac delta functions and the Hamiltoniandensity He was given in (39).

We must write the generating functional in terms of the independent dynamical fieldvariables, aμ,Aρ,ψα and ψ

†β .

The determinant of the matrix F does not depend on the canonical dynamical field vari-ables (see (41)). Thus, we include (detF)

12 in the path integral normalization factor. The

same occurs with the other determinant appearing in (88) (see (62)).Besides, by means of the Dirac deltas δ(ΔA),A = 3, . . . ,6,9,10, δ(ΓI ), I = 1, . . . ,4,

we calculate the path integrals over q0,Q0,p0,P 0, b0,B0,π†1 ,π

†2 ,π1 and π2, respectively.

By means of the Dirac deltas δ(Δ11) and δ(Δ12), we calculate the path integrals over a0

and A0.In addition, we make the integrations over bi and Bi .Moreover, by means of the Fourier integral corresponding to the Dirac delta we have that

δ(Δn) = ∫DΛn exp(i

∫d3x ΛnΔn),n = 1,2. Due to the arbitrariness of the Lagrange mul-

tipliers Λn it is possible to make a scale change in the corresponding integration variablesin such a way that the generating functional remains [31]

Z =∫

DaμDAρDψαDψ†βδ

(∂lal

)δ(∂mAm

)exp

(i

∫d3xL

), (89)

where L is the starting Lagrangian density given by (1).

2576 Int J Theor Phys (2012) 51:2564–2584

This result was expected. Nevertheless, we consider necessary to start formally from thecanonical path integral (88) and to prove that it is possible to arrive at the Lagrangian pathintegral (89). This is because, as it is well known, there are many field theories in whichthe simple Lagrangian path integral cannot be obtained from the canonical one (see, forinstance, [55] and references therein).

Finally, we use the Faddeev-Popov trick. Hence, we write the gauge-fixing condi-tions in the form ∂μaμ(x) = ca(x) and ∂μAμ(x) = cA(x) (generalized Lorentz gauge).By considering the first of these conditions we write δ[∂μaμ(x) − ca(x)] = ∫

Dca(x) ×exp{i λa

2

∫d3x[∂μaμ(x)]2}, with a Gaussian weight independent of ca(x). Therefore, the in-

tegrand in (89) does not depend on ca(x) and the integration over this quantity can be per-formed, appearing exp{i λa

2

∫d3x[∂μaμ(x)]2} instead of δ[∂μaμ(x) − ca(x)]. We proceed in

an analogous way for the second condition.In this way, the generating functional remains

Z =∫

DaμDAρDψαDψ†β exp

(i

∫d3x Leff

), (90)

where the Lagrangian density Leff is expressed in terms of the independent dynamical fieldvariables and so constitutes the effective Lagrangian density of the model. This is given by

Leff = L + Lgf , (91)

where

Lgf = λa

2

(∂μaμ

)2 + λA

2

(∂μAμ

)2(92)

is the gauge-fixing Lagrangian density.Now we are going to establish the Feynman rules of the model [56].Since in (4) and (5) appear quadratic terms that mix aμ and Aμ, these must contribute to

the propagators. Therefore, the only possibility is to consider a unique auxiliary extendedquantity XΛ = (aμ,Aν), where the compound index Λ runs over the components of the fieldvariables [47]. In this way, the Lagrangian density (91), written in terms of this quantity, canbe partitioned as follows:

Leff = Leff (XΛ) + Leff

(ψ,ψ†

) + Linteff

(XΛ,ψ,ψ†

), (93)

where

Leff (XΛ) = 1

2XΛ

(D−1

)ΛΠXΠ, (94)

Leff

(ψ,ψ†

) = ψ†G−1ψ, (95)

Linteff

(XΛ,ψ,ψ†

) = ψ†V ΛXΛψ + ψ†XΛWΛΠXΠψ. (96)

Consequently, from (94)–(96), we calculate the propagators and vertices of the model in themomentum space.

The matrix D−1 appearing in (94) is nondegenerate. As a result, the propagator DΛΠ ofthe gauge field XΛ is written as

DΛΠ(k) =(

Mμν(k) Lμν(k)

Lμν(k) Nμν(k)

), (97)

Int J Theor Phys (2012) 51:2564–2584 2577

where

Mμν(k) = μ1(k2

)gμν + μ2

(k2

)kμkν − iμ3

(k2

)εμνρk

ρ, (98)

Lμν(k) = λ1

(k2

)gμν + λ2

(k2

)kμkν − iλ3

(k2

)εμνρk

ρ, (99)

Nμν(k) = ν1(k2

)gμν + ν2

(k2

)kμkν − iν3

(k2

)εμνρk

ρ. (100)

In (98)–(100), the coefficients are given by

μ1

(k2

) = −4acϕ − θ(c2 + k2ϕ2)

A, (101)

μ2

(k2

) = − Bλak4 A

, (102)

μ3(k2

) = b(4a2 − k2θ2) − 4[ck2θϕ + a(c2 + k2ϕ2)]2k2 A

, (103)

λ1(k2

) = b(cθ + 2aϕ) + 2ϕ(c2 − k2ϕ2)

A, (104)

λ2

(k2

) = −λ1

k2, (105)

λ3

(k2

) = 2c(c2 − k2ϕ2) − b(2ac + k2θϕ)

k2 A, (106)

ν1

(k2

) = − b

A(bθ + 4cϕ) , (107)

ν2

(k2

) = − CλAk4 A

, (108)

ν3

(k2

) = 2b

k2 A(ab − c2 − k2ϕ2

), (109)

where

θ = θ(k2

) = 1 − 4κk2 , (110)

ϕ = ϕ(k2

) = d + κ ′k2 , (111)

A = A(k2

) = 8b[a(c2 + k2ϕ2

) + ck2θϕ] − b2

(4a2− k2θ2

) − 4(c2− k2ϕ2

)2, (112)

B = B(k2

) = −A − λak2[4acϕ + θ

(c2 + k2ϕ2

)], (113)

C = C(k2

) = −A − λAk2b(bθ + 4cϕ) , (114)

with a = (2σ)−1, b = (4πφ)−1, c = ζe/mb , d = ηe/mb and k2 = kμkμ.In (95), G is the propagator of the matter field. This is given by

G(p,E) =(

E−μ − p2

2mb

)−1

, (115)

where E is the particle energy, p its ordinary momentum and p2 = p21 + p2

2 .

2578 Int J Theor Phys (2012) 51:2564–2584

In (96), the vector V Λ gives the 3-point vertex of the model. This is expressed as

V Λ =(

1,pi

mb

, e, epj

mb

), (116)

with i, j = 1,2.Finally, in (96), the matrix WΛΠ gives the 4-point vertex. This reads

WΛΠ = − 1

2mb

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 00 1 0 0 e 00 0 1 0 0 e

0 0 0 0 0 00 e 0 0 e2 00 0 e 0 0 e2

⎞

⎟⎟⎟⎟⎟⎟⎠

. (117)

Consequently, the Feynman rules of the model are:

(i) Propagators. We associate with the propagator of the gauge field XΛ a wavy line

and with the propagator of the matter field ψ a straight line

(ii) Vertices. Hence, the 3 and 4-point vertices of the model remain represented respectivelyby

, .

The remaining Feynman rules are the usual ones.Now we are going to analyze, in the HD context, the ultraviolet behaviour of the primi-

tively divergent Feynman diagrams of the original model that contain the propagator DΛΠ .As it can be seen, for large momenta the bosonic propagator corresponding to the orig-

inal model behaves like k−2. From (97), we find that for the present model this propagatorbehaves like k−4, regardless the chosen gauge. Consequently, we gain two powers of k withregard to the propagator obtained without the HD term and so its ultraviolet behaviour isimproved.

As it can be shown, the primitively divergent diagrams of the original model that containthe propagator DΛΠ are the following:

Int J Theor Phys (2012) 51:2564–2584 2579

(i) The CF self-energy diagram

.

By using the Feynman rules given above, its Feynman integral is written as

Σ(p) ∝∫

d3k

(2π)3V ΛG(p − k,Ep−k)V

ΠDΛΠ(k). (118)

(ii) The correction to the 3-point vertex

,

whose Feynman integral reads

ΛΠ(p,q,p + q) ∝∫

d3k

(2π)3V ΘDΘΩ(p + k)V ΩG(k − q,Ek−q)V

ΠG(k,Ek). (119)

For the present model, we find that the above diagrams are now convergent. This canbe seen putting the expressions for the propagators DΛΠ and G, and the vertices V Λ andWΛΠ , given by (97), (115), (116) and (117), respectively, in the Feynman integrals (118)and (119), and carrying out the summations over the repeated indices. For large momenta,the Feynman integral (118) behaves like k−1 and the one given by (119) like k−2.

In this way, we see that the new bosonic propagator has a better ultraviolet behaviourthan the one obtained when the HD terms are not present. Furthermore, by means of a powercounting analysis, we find that the primitively divergent diagrams of the original model withsuch propagator turn out to be convergent.

Nevertheless, in order to know the true ultraviolet behaviour of the above diagrams withand without the HD terms, we must evaluate the Feynman integrals (118) and (119).

In this case, as it is known, the dimensional regularization procedure cannot be safelyused due to the presence of the volume form εμνρ in the electromagnetic field propaga-tor. Therefore, another gauge-invariant regularization method, for instance, the Pauli-Villarsone [57], must be used.

The renormalization procedure is implemented as in the usual quantum electrodynamics.We will discuss these issues in a future paper.

2580 Int J Theor Phys (2012) 51:2564–2584

4 Unitarity

In this section, we will devote our attention to studying the problem of unitarity of the model.As it is well known, in quantum field theories with HD terms unitarity can be violated at

the quantum level [58–61]. This happens when there are states with negative norm. Thesestates are called ghost states.

To investigate the unitarity of the model, we must carefully analyze the bosonic propa-gator considered.

In this way, we write the propagator (97) as follows:

DΛΠ = KΛΠ

Δ. (120)

In this equation Δ = det(D−1), which contains the poles of this propagator, is written as

Δ(α) = −4λaλAα4 A, (121)

where A = A(α) was given in (112) and α = k2.The matrix KΛΠ , called matrix residue, is Hermitian. Therefore, this is diagonalizable

and it has real eigenvalues ξ (p),p = 1, . . . ,6, and eigenstates mutually orthogonal J(p)

Λ .In consequence, we can define the currents for emission of a particle corresponding to

incoming particles of the S matrix as the real eigenstates of KΛΠ , one for every nonzeroeigenvalue of this matrix, normalized in the form

J(p)T

Λ (k)KΛΠ(k)J(p)

Π (k) = 1. (122)

As it can be proved [56], in this situation, if all the eigenvalues at every pole of thepropagator (120) are positive, the model is unitary.

The case of a negative eigenvalue at a pole corresponds to a nonreal current. In such acase, we must consider an alternative real current, for instance, J (p)

Λ (k) = iJ(p)

Λ (k). Unfortu-nately, such current does not satisfy (122) because it has negative norm, and so the unitarityof the model is lost. The prescription to restore the unitarity, usually known as indefinitemetric prescription, establishes that for such currents the normalization must be done with aminus sign attached on the right-hand side of (122).

Therefore, we must know the sign of all the eigenvalues at every pole of the propagator(120).

To achieve this goal, we have to consider the secular equation corresponding to the matrixresidue. The expression is extremely complicated, so we do not write it here explicitly.Nevertheless, we can assert that at the pole α = 0 the eigenvalue is ξ = 0.

Then, in order to deal with simpler expressions, we are going to work in the Landaugauge in which λa → ∞ and λA → ∞. Therefore, we must rewrite (120) and (121) takinginto account these conditions:

D′ΛΠ = K ′

ΛΠ

Δ′ , (123)

with

Δ′(α) = 2αA. (124)

In this case, the secular equation is more compact and it is written as below

ξ 2(ξ 4 + p3ξ

3 + p2ξ2 + p1ξ + p0

) = 0, (125)

Int J Theor Phys (2012) 51:2564–2584 2581

where

p3 = −4k20

[(b2 + c2

)θ + 4(a + b)cϕ + αθϕ2

],

p2 = −{(α − 2k2

0

)[4(a2 + b2 + 2c2

) − αθ2] − 8

(α2 − 2αk2

0 + 2k40

)ϕ2

}A,

p1 = −4k20

(α − 2k2

0

)θ A2,

p0 = −4(α − 2k2

0

)2 A3. (126)

On the other hand, from (124) we see that the propagator (123) has the pole α1 = 0 andother six poles that verify A = 0.

In this way, from (125) we find that for α1 = 0 the eigenvalues verify

ξ 2{ξ 4 − 4k2

0

[b2 + c2 + 4cd(a + b)

]ξ 3

+ 8k20

(a2 + b2 + 2c2 + 2k2

0d2)

A|0ξ 2 + 8k40 A|20ξ − 16k4

0 A|30} = 0, (127)

in which A|0 = A(α = 0) = −4(c2 − ab)2. Consequently, the eigenvalues are either zero orobtained as solutions of a fourth degree equation.

For the other poles, we have that

ξ 5(ξ + p3) = 0. (128)

Thus, ξ = 0 or ξ = −p3.

5 BRST Formalism

Now we apply a generalization to the HD case of the BRST algorithm to the model.As it can be shown, in this case, the mathematical developments used and the found

results are similar to the ones corresponding to the model of [50]. The only changes appearin the quantities listed below:

(i) The canonical Hamiltonian density, given by (31), written in terms of the first-classHamiltonian density H0 and the first-class constraints

Hc = H0 − b0Σ5 − B0Σ6. (129)

(ii) The dynamical field variables of the model in the BRST context

AF = (aμ, bν,Aρ,Bε,ψα,ψ

†β, ρa

), (130)

where ρa are Lagrange multipliers, with a = 1, . . . ,6. The respective canonically con-jugate momenta are

P F = (pμ,qν,P ρ,Qε,π†

α,πβ, ξa). (131)

(iii) The expressions for the Bose-Fermi brackets between the gauge-fixing conditions andthe first-class constraints

[Θa(x),Σb(y)

]− = [

f ab + ga

b∇2 + hab

(∇2)2]

δ(x − y), (132)

2582 Int J Theor Phys (2012) 51:2564–2584

where

f ab =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 00 0 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 0 00 0 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎠

, (133)

gab =

⎛

⎜⎜⎜⎜⎜⎜⎝

−e 0 0 0 0 01 i

e0 0 0 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0 2ηe

mb−1

0 0 0 0 0 2ηe

mb

⎞

⎟⎟⎟⎟⎟⎟⎠

, (134)

hab =

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 −2κ ′ −4κ

0 0 0 0 0 −2κ ′

⎞

⎟⎟⎟⎟⎟⎟⎠

, (135)

with a, b = 1, . . . ,6.

6 Conclusions and Outlook

A HD classical nonrelativistic U(1) × U(1) gauge field model that describes the electro-magnetic interaction of composite particles in 2 + 1 dimensions has been constructed. Thiswas done by adding two HD terms to the Lagrangian of the model proposed by us in [42],preserving its gauge invariance.

Later, by means of the usual Hamiltonian treatment for singular HD systems, the canon-ical quantization was performed.

Next, by extending the FS procedure, the path integral quantization was performed.Hence, the Feynman rules of the model were established. It was found that the inclusionof the HD terms improves the behaviour of the propagator DΛΠ of the gauge field XΛ, forlarge momenta. Consequently, by using the diagrammatic technique, it was seen that theprimitively divergent diagrams of the original model in which such propagator takes placeturn out to be convergent. We must note that in [50], by considering only the electromagneticfield propagator, this improvement is achieved only in the Landau gauge, in which λA → ∞.

Afterwards, the unitarity problem was analyzed.Finally, the application of the BRST algorithm to the model was briefly described.The paper was developed considering explicitly the CF case.In a future paper, as we said in Sect. 1, we will consider the present model in the frame-

work of condensed matter and, as we said in Sect. 3, we will study the regularization andthe renormalization of this model.

Acknowledgements The authors would like to thank the Facultad de Ciencias Exactas, Ingeniería y Agri-mensura and the Consejo Nacional de Investigaciones Científicas y Técnicas, for financial support.

Int J Theor Phys (2012) 51:2564–2584 2583

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