Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0085 Acta Polytechnica 62(1):85–89, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague MAXWELL-CHERN-SIMONS-HIGGS THEORY Usha Kulshreshthaa, ∗, Daya Shankar Kulshreshthab, Bheemraj Sihagb a University of Delhi, Kirori Mal College, Department of Physics, Delhi-110007, India b University of Delhi, Department of Physics and Astrophysics, Delhi-110007, India ∗ corresponding author: ushakulsh@gmail.com Abstract. We consider the three dimensional electrodynamics described by a complex scalar field coupled with the U(1) gauge field in the presence of a Maxwell term, a Chern-Simons term and the Higgs potential. The Chern-Simons term provides a velocity dependent gauge potential and the presence of the Maxwell term makes the U(1) gauge field dynamical. We study the Hamiltonian formulation of this Maxwell-Chern-Simons-Higgs theory under the appropriate gauge fixing conditions. Keywords: Electrodynamics, Higgs theories, Chern-Simons-Higgs theories, Hamiltonian formulations, gauge-theories. 1. Introduction We study the Hamiltonian formulation [1] of the three dimensional (3D) electrodynamics [2–22], in- volving a Maxwell term [20], a Chern-Simons (CS) term [19, 21, 22], and a term that describes a cou- pling of the U(1) gauge field with a complex scalar field in the presence of a Higgs potential [22]. Such theories in two-space one-time dimension ((2+1)D) can describe particles that satisfy fractional statistics and are referred to as the reletivistic field theoretic models of anyons and of the anyonic superconductiv- ity [21, 22]. A remarkable property of the CS action [21, 22], is that it depends only on the antisymmetric tensor ϵµνλ and not on the metric tensor gµν . As a result, the CS action in the flat spacetime and in the curved spacetime remains the same [21, 22]. Hence CS action, in both the Abelian and in the non-Abelian cases represents an example of a topological field theory [21, 22]. The systems in two-space, one-time dimensions (2+1)D (i.e., the planar systems, display a variety of peculiar quantum mechanical phenomena ranging from the massive gauge fields to soluble gravity [19– 22]. These are linked to the peculiar structure of the rotation group and the Lorentz and Poincare groups in (2+1)D. The 3D electrodynamics models with a Higgs potential, namely, the Abelian Higgs models involv- ing the vector guage field Aµ with and without the topological CS term in (2+1)D have been of a wide interest [19–22]. When these models are considered without a CS term but only with a Maxwell term accounting for the kinetic energy of the vector gauge field and they repre- sent field-theoretical models which could be considered as effective theories of the Ginsburg-Landau-type [22] for superconductivity. These models in (2+1)D or in (3+1)D are known as the Nielsen-Olesen (vortex) models (NOM) [20]. These models are the relativistic generalizations of the well-known Ginsburg-Landau phenomenological field theory models of superconduc- tivity [2, 20, 22]. The effective theories with excitations, with frac- tional statistics are supposed to be described by gauge theories with CS terms in (2+1)D and a study of these gauge field theories and the models of quantum elec- trodynamics involving the CS term represent a broad important area of investigation [21, 22]. The CS term provides a velocity dependent gauge potential [21, 22], and the presence of the Maxwell term in the action makes the gauge field dynami- cal [20]. We study the Hamiltonian formulation [1] of this Maxwell-Chern-Simons-Higgs theory under the appropriate gauge fixing conditions [20, 22]. The quantization of field theory models with con- straints has always been a challenging problem [1]. Infact, any complete physical theory is a quantum theory and the only way of defining a quantum theory is to start with a classical theory and then to quan- tize it [1]. Theory presently under consideration is also a constrained system. In the present work, we quantize this theory using the Dirac’s Hamiltonian formulation [1] in the usual instant-form (IF) of dy- namics (on the hyperplanes defined by: x0 = t = constant) under appropriate gauge-fixing conditions (GFC’s) [1, 19–22]. 2. Hamiltonian formulation The Maxwell Chern-Simons Higgs Theory in two space one time is defined by the following action: S = ∫ L(Φ, Φ∗, Aµ) d3x, (1) 85 https://doi.org/10.14311/AP.2022.62.0085 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en U. Kulshreshtha, D. S. Kulshreshtha, B. Sihag Acta Polytechnica where the Lagrangian density L (with κ = θ2π2 ; θ being the CS parameter) is given by: L = [ − 1 4 Fµν F µν + (D̃µΦ∗)(DµΦ) − V (|Φ|2) + κ 2 ϵµνλAµ∂ν Aλ ] (2) V (|Φ|)2 = γ + β|Φ|2 + α|Φ|4 = λ(|Φ|2 − Φ20) 2 ; (Φ0 ̸= 0). (3) Where the covariant derivative is defined by: Dµ = (∂µ + i eAµ) D̃µ = (∂µ − i eAµ) gµν = diag(+1, −1, −1) ϵ012 = ϵ012 = +1 µ, ν = 0, 1, 2. (4) In the above Lagrangian density the first term is the kinetic energy term of the U(1) gauge field and second term represents the coupling of U(1) gauge field with the complex scalar field as well as kinetic energy for the complex scalar field. Third term describes Higgs potential and the last term is the CS term. The model without the CS term describes an Abelian Higgs model and is defined by the Lagrangian density L = L ( Φ, Φ∗, Aµ ) where L is a function of a complex scalar field and an Abelian gauge vector field Aµ(x) defined by the above Lagrangian density. In (2+1)D this theory is called as the Nielsen Olsen (vortex) model (NOM). These models possesses sta- ble, time independent (i.e., static), classical solutions (which could be called 2D solitons). In fact, the model admits the so-called topological solitons of the vortex type [4]. Further, in this model, if we choose the parameters of the Higgs potential to be such that the scalar and vector masses become equal i.e., if we set the Higgs boson and the vector boson (photon) masses to be equal i.e., if we set: mHiggs = mP hoton = eΦ0 then that implies: V (|Φ|)2 = 1 2 e2(|Φ|2 − Φ20) 2. (5) The above model then reduces to the so-called Bo- gomol’nyi model which describes a system on the boundary between type-I and type-II superconductiv- ity [4]. In component form, the above Lagrangian density can be written as: L = ( κ 2 )[ A0F12 + A1(∂2A0) − A2(∂1A0) ] + ( κ 2 )[ A2(∂0A1) − A1(∂0A2) ] − 1 2 F 212 + [ 1 2 (∂1A0 − ∂0A1) + 1 2 (∂0A2 − ∂2A0) ] + [ (∂0Φ∗)(∂0Φ) + i e(∂0Φ∗)A0Φ − i e(∂0Φ)A0Φ∗ + e2A20Φ ∗Φ ] + [ − (∂1Φ∗)(∂1Φ) − i e(∂1Φ∗)A1Φ + i e(∂1Φ)A1Φ∗ − e2A21Φ ∗Φ ] + [ − (∂2Φ∗)(∂2Φ) − i e(∂2Φ∗)A2Φ + i e(∂2Φ)A2Φ∗ − e2A22Φ ∗Φ ] − V (|Φ|2). (6) Canonical momenta obtained from the above La- grangian density are: Π = ∂L ∂(∂0Φ) = (∂0Φ∗ − i eA0Φ∗) Π∗ = ∂L ∂(∂0Φ∗) = (∂0Φ + i eA0Φ) Π0 = ∂L ∂(∂0A0) = 0 (7) E1(:= Π1) := ∂L ∂(∂0A1) = −(∂1A0 − ∂0A1) + κ 2 A2 E2(:= Π2) = ∂L ∂(∂0A2) = (∂0A2 − ∂2A0) − κ 2 A1. (8) Here Π, Π∗, Π0, E1, E2 are the momenta canonically conjugate respectively to Φ, Φ∗, A0, A1, A2. The the- ory is thus seen to possess only one primary constraint (PC): χ1 = Π0 ≈ 0. (9) The canonical Hamiltonian density of the theory is obtained using the Legendre transformation from the Lagrangian density of the theory in the usual manner. Every term in the Lagrangian density (including the CS term) is equally important. The calculational de- tails are omitted here for the sake of brevity. The total Hamiltonian density of the theory is then obtained from the canonical Hamiltonian density by including 86 vol. 62 no. 1/2022 Maxwell-Chern-Simons-Higgs Theory in it the primary constraint of the theory with the help of the Lagrange multiplier field u ≡ u(xµ) (which is dynamical) as follows: HT = Π0u + Π Π∗ − ieA0(ΠΦ − Π∗Φ∗) + 1 2 (E12 + E22) + 1 2 F 212 + [ E1(∂1A0) + E2(∂2A0) ] + 1 2 ( κ 2 )2 (A12 + A22) − ( κ 2 )[ A2E1 − A1E2 + A0F12 ] + [ (∂1Φ∗)(∂1Φ) + i e(∂1Φ∗)A1Φ − i e(∂1Φ)A1Φ∗ + e2A21Φ ∗Φ ] + [ (∂2Φ∗)(∂2Φ) + i e(∂2Φ∗)A2Φ − i e(∂2Φ)A2Φ∗ + e2A22Φ ∗Φ ] , (10) where HT = ∫ HT d2x, (11) with the total Hamiltonian density given by: HT = [ Hc + Π0u ] . (12) . It is to be noted here that in the construction of the canonical Hamiltonian density of the theory, all the fields of the theory play an equally important role through the Legendre transformation and through the Lagrangian density of the theory that defines the theory. Also, it is worth mentioning here that the Hamilton’s equations of motion of the theory (that are omitted here for the sake of brevity) obtained from the total Hamiltonian density of the theory preserve the constraints of the theory for all time. After pre- serving the Primary constraint χ1 in the course of time, one obtains a secondary constraint χ2 = [ ie(ΠΦ − Π∗Φ∗) + (∂1E1 + ∂2E2) + κ 2 (∂1A2 − ∂2A1) ] ≈ 0. (13) The matrix of Poisson Brackets (PB’s) among the constraints χi is a null matrix and thereby theory is a gauge invariant theory and is invariant under the following local vector gauge transformations: δΦ = iβΦ, δΦ∗ = −iβΦ∗, δΠ0 = 0 δA0 = −∂0β ; δA1 = −∂1β ; δA2 = −∂2β δΠ = −iβ(∂0Φ∗) − eβA0Φ∗ + i(e − 1)(∂0β)Φ∗ δΠ∗ = iβ(∂0Φ) − eβA0Φ − i(e − 1)(∂0β)Φ δE1 = −κ 2 ∂2β; δE2 = κ 2 ∂1β; δu = −∂0∂0β. (14) Here, β is the gauge parameter β ≡ β(xµ) and the vector gauge current satisfies: ∂µJ µ = 0. The components of J µ are: J 0 = J0 = (iβΦ) [ ∂0Φ∗ − i eA0Φ∗ ] − (iβΦ∗) [ ∂0Φ + i eA0Φ ] − (∂1β) F01 − (∂2β) F02 − κ 2 [ (∂1β)A2 − (∂2β)A1 ] J 1 = −J1 = (iβΦ) [ − ∂1Φ∗ + i eA1Φ∗ ] − (iβΦ∗) [ − ∂1Φ − i eA1Φ ] − (∂0β) F10 − (∂2β) F21 + κ 2 [ (∂0β)A2 − (∂2β)A0 ] J 2 = −J2 = (iβΦ) [ − ∂2Φ∗ + i eA2Φ∗ ] − (iβΦ∗) [ − ∂2Φ − i eA2Φ ] − (∂0β) F20 − (∂1β) F12 − κ 2 [ (∂0β)A1 − (∂1β)A0 ] . (15) For quantizing the theory using Dirac’s procedure we choose the following two gauge-fixing conditions (GFC’s): ξ1 = Π ≈ 0 ξ2 = A0 ≈ 0. (16) Here the gauge A0 ≈ 0 represents the time-axial or temporal gauge and the gauge Π ≈ 0 represents the coulomb gauge. These gauges are acceptable and consistent with our quantization procedure and also physically more interesting. Corresponding to this set of gauge fixing conditions the total set of constraints now becomes: χ1 = Π0 ≈ 0 χ2 = [ ie(ΠΦ − Π∗Φ∗) + (∂1E1 + ∂2E2) + κ 2 (∂1A2 − ∂2A1) ] ≈ 0 χ3 = ξ1 = Π ≈ 0 χ4 = ξ2 = A0 ≈ 0. (17) The non-vanishing matrix elements of the matrix Rαβ (:= {χ1, χ2}P ) of the equal-time Poisson brackets of the above constraints are: R14 = −R41 = − δ(x1 − y1)δ(x2 − y2) R23 = −R32 = ieΠ δ(x1 − y1)δ(x2 − y2). (18) The above matrix is nonsingular and the set of constraints χi ; i = 1, 2, 3, 4 is now second class and the theory is a gauge non-invariant theory. The non- vanishing matrix elements of the matrix R−1αβ (which 87 U. Kulshreshtha, D. S. Kulshreshtha, B. Sihag Acta Polytechnica is the inverse of the matrix Rαβ ) are given by: R−114 = −R −1 41 = δ(x 1 − y1)δ(x2 − y2) (19) (eΠ)R−123 = −(eΠ)R −1 32 = i δ(x 1 − y1)δ(x2 − y2). Following the standard Dirac quantisation proce- dure, the non-vanishing equal time Dirac Brackets (DB’s) of the theory are obtained as: (Π) {Π∗(x0, x1, x2) , Φ(x0, y1, y2)}D = (−Π∗)δ(x1 − y1)δ(x2 − y2) {Π∗(x0, x1, x2) , Φ∗(x0, y1, y2)}D = {Π∗(x0, x1, x2) , Φ∗(x0, y1, y2)}P = −δ(x1 − y1)δ(x2 − y2) (ieΠ) {E1(x0, x1, x2) , Φ(x0, y1, y2)}D = ( κ 2 ) δ(x1 − y1) ∂2δ(x2 − y2) {E1(x0, x1, x2) , A1(x0, y1, y2)}D = {E1(x0, x1, x2) , A1(x0, y1, y2)}P = − δ(x1 − y1)δ(x2 − y2) (ieΠ) {E2(x0, x1, x2) , Φ(x0, y1, y2)}D = − ( κ 2 ) ∂1δ(x1 − y1)δ(x2 − y2) {E2(x0, x1, x2) , A2(x0, y1, y2)}D = {E2(x0, x1, x2) , A2(x0, y1, y2)}P = −δ(x1 − y1)δ(x2 − y2) (Π) {Φ(x0, x1, x2) , Φ∗(x0, y1, y2)}D = (−Φ∗)δ(x1 − y1)δ(x2 − y2) (Π) {Φ(x0, x1, x2) , A0(x0, y1, y2)}D = (Φ) δ(x1 − y1)δ(x2 − y2) (ieΠ) {Φ(x0, x1, x2) , A1(x0, y1, y2)}D = ∂1δ(x1 − y1)δ(x2 − y2) (ieΠ) {Φ(x0, x1, x2) , A2(x0, y1, y2)}D = δ(x1 − y1) ∂2δ(x2 − y2). (20) Here one finds that the product of the canonical variables appear in the expressions of the constraints as well as in the expressions of the DB’s and therefore for achieving the canonical quantisation of the theory, one encounters the problem of operator ordering while going from DB’s to the commutation relations, this problem could however be resolved by demanding that all the fields and the field momenta after quantisation become Hermitian operators and that all the canonical commutation relations need to be consistent with the Hermiticity of these operators. This completes the Hamiltonian formulation of the theory under the choosen gauge fixing conditions. It may be worthwhile to mention here that our choice of GFC’s is by no means unique. In principle, one can choose any set of GFC’s that would convert the set of constraints of the theory from first-class into a set of second-class constraints. However, it is better to choose the GFC’s that are physically more meaningful and nore relevant like the ones that we have choosen. In our case the gauge A0 ≈ 0 represents a time-axial or temporal gauge and the gauge Π ≈ 0 represents a Culomb gauge and both of them are physically important GFC’s. Another important point is that one can not choose covariant GFC’s here simply because the constraints of the theory are not covariant and therefore it would not work. In path integral quantization (PIQ) [23], transition to quantum theory is made by writing the vacuum to vacuum transition amplitude for the theory, called the generating functional Z[Jk] of the theory which in the presence of the external sources Jk for the present theory is [23]: Z[Jk] = ∫ [dµ] exp [ i ∫ d3x [JkΦk + Π∂0Φ + Π∗∂0Φ∗ + Π0∂0A0 + E1∂0A1 + E2∂0A2 + Πu∂0u − HT ] ] . (21) Here Φk ≡ (Φ, Φ∗, A0, A1, A2, u) are the phase space variables of the theory with the correspond- ing respective canonical conjugate momenta: Πk ≡ (Π, Π∗, Π0, E1, E2, Πu). The functional measure [dµ] of the theory (with the above generating functional Z[Jk]) is: [dµ] = [ (ieΠ)δ(x1 − y1)δ(x2 − y2) [dΦ][dΦ∗][dA0][dA1][dA2][du][dΠ] [dΠ∗][dΠ0][dE1][dE2][dΠu]δ[(Π0) ≈ 0] δ[[ie(ΠΦ − Π∗Φ∗) + (∂1E1 + ∂2E2) + κ 2 (∂1A2 − ∂2A1)] ≈ 0] δ[Π ≈ 0]δ[A0 ≈ 0] ] . (22) Acknowledgements We thank the organizers of the International Conference on Analytic and Algebraic Methods in Physics XVIII (AAMP XVIII) – 2021 at Prague, Czechia, Prof. Vít Jakubský, Prof. Vladimir Lotoreichik, Prof. Matej Tusek, Prof. Miloslav Znojil and the entire team for a wonderful orgnization of the conference. One of us (Bheemraj Sihag) thanks the University of Delhi for the award of a Research Fellowship. 88 vol. 62 no. 1/2022 Maxwell-Chern-Simons-Higgs Theory References [1] P. A. M. Dirac. Generalized Hamiltonian dynamics. Canadian Journal of Mathematics 2:129–148, 1950. https://doi.org/10.4153/cjm-1950-012-1. [2] V. L. Ginzburg, L. D. Landau. On the theory of superconductivity (in Russian). Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 20:1064–1082, 1950. [3] A. A. Abrikosov. On the magnetic properties of superconductors of the second group. Soviet Physics JETP 5:1174–1182, 1957. [4] H. B. Nielson, P. Olsen. Vortex line models for dual strings. Nuclear Physics B 61:45–61, 1973. [5] C. Becchi, A. Rouet, R. Stora. The abelian Higgs Kibble model, unitarity of the S-operator. Physics Letters B 52(3):344–346, 1974. https://doi.org/10.1016/0370-2693(74)90058-6. [6] E. B. Bogomolnyi. The stability of classical solutions. Soviet Journal of Nuclear Physics 24(4):449–458, 1976. [7] S. Deser, R. Jackiw, S. Templeton. Three-dimensional massive gauge theories. Physical Review Letters 48:975–978, 1982. Annals of Physics, 140:372, 1982, https://doi.org/10.1103/PhysRevLett.48.975. [8] F. Wilczek. Quantum mechanics of fractional-spin particles. Physical Review Letters 49:957–959, 1982. https://doi.org/10.1103/PhysRevLett.49.957. [9] A. J. Niemi, G. W. Semenoff. Axial-anomaly-induced fermion fractionization and effective gauge-theory actions in odd-dimensional space-times. Physical Review Letters 51:2077–2080, 1983. https://doi.org/10.1103/PhysRevLett.51.2077. [10] A. N. Redlich. Gauge noninvariance and parity nonconservation of three-dimensional fermions. Physical Review Letters 52:18–21, 1984. https://doi.org/10.1103/PhysRevLett.52.18. [11] K. Ishikawa. Chiral anomaly and quantized Hall effect. Physical Review Letters 53:1615–1618, 1984. https://doi.org/10.1103/PhysRevLett.53.1615. [12] G. W. Semenoff, P. Sodano. Non-Abelian adiabatic phases and the fractional quantum Hall effect. Physical Review Letters 57:1195–1198, 1986. https://doi.org/10.1103/PhysRevLett.57.1195. [13] L. Jacobs, C. Rebbi. Interaction energy of superconducting vortices. Physical Review B 19:4486–4494, 1979. https://doi.org/10.1103/PhysRevB.19.4486. [14] I. V. Krive, A. S. Rozhavskĭı. Fractional charge in quantum field theory and solid-state physics. Soviet Physics Uspekhi 30(5):370, 1987. https://doi.org/10.1070/PU1987v030n05ABEH002884. [15] A. L. Fetter, C. B. Hanna, R. B. Laughlin. Random-phase approximation in the fractional-statistics gas. Physical Review B 39:9679–9681, 1989. https://doi.org/10.1103/PhysRevB.39.9679. [16] T. Banks, J. D. Lykken. Landau-Ginzburg description of anyonic superconductors. Nuclear Physics B 336(3):500–516, 1990. https://doi.org/10.1016/0550-3213(90)90439-K. [17] G. V. Dunne, C. A. Trugenberger. Self-duality and nonrelativistic Maxwell-Chern-Simons solitons. Physical Review D 43:1323–1331, 1991. https://doi.org/10.1103/PhysRevD.43.1323. [18] S. Forte. Quantum mechanics and field theory with fractional spin and statistics. Reviews of Modern Physics 64:193–236, 1992. https://doi.org/10.1103/RevModPhys.64.193. [19] U. Kulshreshtha. Hamiltonian and BRST formulations of the two-dimensional Abelian Higgs model. Canadian Journal of Physics 78(1):21–31, 2000. https://doi.org/10.1139/p00-002. [20] U. Kulshreshtha. Hamiltonian and BRST formulations of the Nielsen-Olesen model. International Journal of Theoretical Physics 41(2):273–291, 2002. https://doi.org/10.1023/A:1014058806710. [21] U. Kulshreshtha, D. S. Kulshreshtha. Hamiltonian, path integral, and BRST formulations of the Chern–Simons theory under appropriate gauge-fixing. Canadian Journal of Physics 86(2):401–407, 2008. https://doi.org/10.1139/p07-176. [22] U. Kulshreshtha, D. S. Kulshreshtha, H. J. W. Mueller-Kirsten, J. P. Vary. Hamiltonian, path integral and BRST formulations of the Chern–Simons–Higgs theory under appropriate gauge fixing. Physica Scripta 79(4):045001, 2009. https://doi.org/10.1088/0031-8949/79/04/045001. [23] H. J. W. Mueller-Kirsten. Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral. World Scientific, Singapore, 2006. ISBN 9789814397735. 89 https://doi.org/10.4153/cjm-1950-012-1 https://doi.org/10.1016/0370-2693(74)90058-6 https://doi.org/10.1103/PhysRevLett.48.975 https://doi.org/10.1103/PhysRevLett.49.957 https://doi.org/10.1103/PhysRevLett.51.2077 https://doi.org/10.1103/PhysRevLett.52.18 https://doi.org/10.1103/PhysRevLett.53.1615 https://doi.org/10.1103/PhysRevLett.57.1195 https://doi.org/10.1103/PhysRevB.19.4486 https://doi.org/10.1070/PU1987v030n05ABEH002884 https://doi.org/10.1103/PhysRevB.39.9679 https://doi.org/10.1016/0550-3213(90)90439-K https://doi.org/10.1103/PhysRevD.43.1323 https://doi.org/10.1103/RevModPhys.64.193 https://doi.org/10.1139/p00-002 https://doi.org/10.1023/A:1014058806710 https://doi.org/10.1139/p07-176 https://doi.org/10.1088/0031-8949/79/04/045001 Acta Polytechnica 62(1):85–89, 2022 1 Introduction 2 Hamiltonian formulation Acknowledgements References