Solitary wave solutions of the 2+1 and 3+1 dimensional nonlinear Dirac equation constrained to planar and space curves
SSolitary wave solutions of the 2+1 and 3+1 dimensional nonlinear Dirac equationconstrained to planar and space curves
Fred Cooper,
1, 2, ∗ Avinash Khare, † and Avadh Saxena ‡ Santa Fe Institute, Santa Fe, NM 87501, USA Theoretical Division and Center for Nonlinear Studies,Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Physics Department, Savitribai Phule Pune University, Pune 411007, India (Dated: September 16, 2020)We study the effect of curvature and torsion on the solitons of the nonlinear Dirac equationconsidered on planar and space curves. Since the spin connection is zero for the curves consideredhere, the arc variable provides a natural setting to understand the role of curvature and then wecan obtain the transformation for the 1+1 dimensional Dirac equation directly from the metric.Depending on the curvature, the soliton profile either narrows or expands. Our results may beapplicable to yet-to-be-synthesized curved quasi-one dimensional Bose condensates. ∗ [email protected] † [email protected] ‡ [email protected] Typeset by REVTEX a r X i v : . [ n li n . PS ] S e p I. INTRODUCTION
Recently there has been renewed interest in the nonlinear Dirac equation (NLDE) because it arises in Bose conden-sation in honeycomb optical lattices where confining potentials allow for quasi-one dimensional (Q1D) confinement[1]. Here we show that one can find exact solitary wave solutions for the Dirac equation confined to various spacecurves by using either the arc variable representation or the vierbein formalism of Weyl [2]. The latter is elucidatedin the Appendix. We find that since the curves are only in the spatial part of the metric, one can transform theNLDE on the curved surface to another flat space NLDE by a coordinate transformation. This allows us to obtain thesolutions of the Dirac equation analytically for the conic surfaces such as a hyperbola or parabola as well as helical(space) curves in 3+1 dimensions in terms of the spatial arc length parameter s . One can furthermore analyticallyobtain the connection between the parametric description of the curve and the spatial arc length variable s .To our knowledge there are no studies of NLD solitons in curved geometries either experimentally or theoretically.However, one could envision constructing curved QID Bose condensates, which serves as one of our motivations.In contrast, solitons and breathers have been studied on curves in the context of the nonlinear Schr¨odinger (NLS)equation [3, 4] with an interesting interplay of curvature and nonlinearity through the soliton/breather solutions. Thestudy of nonlinear waves on curves and closed surfaces is important in its own right [5].This paper is structured as follows. In the next section we study the NLD equation on conic sections such as ahyperbola, parabola and ellipse. In section III we show how the normalized charge density of a soliton is modified dueto the curvature. In section IV we consider Jacobi elliptic and other parameterizations of the various curves. SectionV deals with space curves (i.e. with finite torsion) such as a helix, an elliptic helix and the helix of the hyperboloidof revolution. Finally in section VI we provide our main conclusions. II. NONLINEAR DIRAC EQUATION CONSTRAINED TO SPACE CURVES IN
DIMENSIONS
First let us consider the solutions of the Nonlinear Dirac Equation (NLD equation) in 2 + 1 dimensions (often calledthe Gross-Neveu model [6]) when it is confined to a conic section which is a hyperbola, ellipse or parabola. The NLDequation including a mass term is given by: − i ( γ µ ∂ µ + m )Ψ − g ( ¯ΨΨ)Ψ = 0 , (2.1)where γ µ are Dirac matrices. It will be convenient in studying the transformation properties of this equation tointroduce the quantity σ σ ≡ − ig ¯ΨΨ , (2.2)so that we can rewrite the NLD equation in the suggestive form often used in the large-N type expansions of thequantum version of this theory − i [ γ µ ∂ µ + ( m + σ )]Ψ = 0 . (2.3)The fact that σ transforms as a scalar is what will be important in what follows. A. Hyperbola
First let us consider the hyperbola defined by y − x = a . Here η is an azimuthal angle. The parametrization y = a sec u, x = a tan u, (2.4)with − π/ < u < π/
2, fulfills this condition. We have that ds = dx + dy = (cid:2) a sec ( u ) (cid:0) tan ( u ) + sec ( u ) (cid:1)(cid:3) du . (2.5)If we introduce a new coordinate via d s = ds = a sec( u ) (cid:112) tan ( u ) + sec ( u ) du , which is the arc length along thecurve, we can solve the usual Dirac equation in terms of s and then use the relationship: s [ a, u ] = a (sec( u ) (cid:113) tan ( u ) + sec ( u ) −
1) (2.6)
FIG. 1. The curve s vs u for a = 1 for the hyperbola parametrized by circular functions. to obtain the solutions as a function of u, t , The resulting 1+1 dimensional NLD equation is( γ ∂ t + γ ∂ s + σ + m )Ψ[ s , t ] = 0 . (2.7)We see that because the spin connection is zero, one could have obtained this answer by realizing that the metric( ds ) = − ( dt ) + ( dx ) + ( dy ) = − ( dt ) + a sec ( u ) (cid:0) tan ( u ) + sec ( u ) (cid:1) du . (2.8)The 2+1 dimensional metric can be transformed to a 1 + 1 dimensional Minkowski metric by the transformation ds = a sec( u ) (cid:113) tan ( u ) + sec ( u ) du (2.9)Then we would obtain the solutions of the NLD equation as a function of s , t and then obtain the result in terms of u using s [ u ]. The 1+1 dimensional resulting NLDE is translationally invariant under s → s − s , so there are solitarywave solutions centered at any value of the arc length s .Note that if the transformation would have mixed space and time, such as in the light cone transformation performedin studying a scale invariant initial condition on the NLDE discussed in [7] this simple result would not have beenpossible. Then we need to resort to a more general formulation, namely that of vierbeins discussed in the Appendix.For a = 1 we get the curve shown in Fig. 1. B. NLD equation constrained to an ellipse
For the other conical sections the formalism is the same, just the parametric representation is changed. Since thespin connection is zero for these transformations, one can obtain the transformation to the 1+1 dimensional Diracequation directly from the metric. For the ellipse centered at the coordinate origin x = a cos θ , y = b sin θ , (2.10)so that ds = − dt + dx + dy = − dt + ( a sin θ + b cos θ ) dθ . (2.11)We see that we can reduce the metric to a 1 + 1 dimensional Minkowski form by the transformation: d s = (cid:112) a sin θ + b cos θ dθ . (2.12)Explicitly s [ θ ] = bE (cid:18) θ (cid:12)(cid:12)(cid:12)(cid:12) − a b (cid:19) , (2.13)where E ( x, m ) is the incomplete elliptic integral of the second kind with modulus m [8]. When a = 1 , b = 2 we getthe transformation shown in Fig. 2. For a circle ( a = b ) the above equation reduces to s ( θ ) = aθ . FIG. 2. The arc variable curve s vs θ for a = 1 , b = 2 for the ellipse.FIG. 3. The arc variable curve s vs η for a = 1 for the parabola. C. NLD equation constrained to a parabola
For the parabola x = 4 ay , the parametric equations are x = 2 aη , y = aη . (2.14)Then ds = − dt + dx + dy = − dt + 4 a (1 + η ) dη . (2.15)Here changing variables to d s = 2 a (cid:90) (cid:112) η dη (2.16)brings the metric back into the Minkowski form. Explicitly s [ η ] = a (cid:16) η (cid:112) η + 1 + sinh − ( η ) (cid:17) . (2.17)When a = 1 we obtain the curve shown in Fig. 3. III. NORMALIZED CHARGE DENSITY
What we have shown for the NLD equation in 2 + 1 dimensions constrained to the three conic sections, that we canreduce the 2 + 1 dimensional NLDE to the flat space 1 + 1 dimensional NLDE if we use the arc length variable. Thestatic solutions of the NLDE in 1+1 dimensions can be written in the form: ψ ( x )= (cid:18) u ( x ) i v ( x ) (cid:19) = R ( x ) (cid:18) cos θi sin θ (cid:19) , (3.1) FIG. 4. Normalized charge density vs. the arc length s for ω = 0 . ω = 1 / where u = R cos θ = 2 g ( m − ω )( m + ω ) cosh βx ( m + ω cosh 2 βx ) ,v = R sin θ = 2 g ( m − ω )( m − ω ) sinh βx ( m + ω cosh 2 βx ) , (3.2)as discussed in [9] and more recently for arbitrary nonlinearity in [10]. As a function of ω/m the shape of the solitarywave changes from single humped to double humped as one lowers ω/m . As a function of the arc length s the densityof the solitary wave is given by ρ ( s ) = R ( s ) ρ ( s ) = Ψ † Ψ = 2 β g ( m + ω ) 1 + α tanh β s (1 − α tanh β s ) sech β s . (3.3)Here (with frequency ω ) β = (cid:112) m − ω , α = m − ωm + ω , (3.4)and the total charge is Q = (cid:90) d s Ψ † Ψ = 2 βg ω . (3.5)In Fig. 4 we plot ρ/Q [ ω ] vs. s for two values, ω = 0 . ω = 1 / A. Hyperbola
We now wish to discuss the change in shape of the solitary wave as we go from the arc variable s to the parametricrepresentation of the curve variable u when the two dimensional motion is confined to a hyperbola. The arc lengthin terms of u is given by s [ a, u ] = a (sec( u ) (cid:113) tan ( u ) + sec ( u ) − . (3.6)In what follows we will always choose the mass m = 1 , and take g = 1. Later on we will use the symbol m to describethe modulus of Jacobi elliptic functions. This is not to be confused with the Dirac mass m which we set to one. InFig. 5 we plot ρ/Q [ ω ] vs. u for the same two values ω = 0 . ω = 1 / u FIG. 5. Normalized charge density for a soliton constrained to a hyperbola vs. u for , ω = 0 . ω = 1 / a = 1 , b = 2 vs. θ for ω = 0 . ω = 1 / B. Ellipse
Choosing for the parameters of the ellipse a = 1 , b = 2, we have that the arc length is given by s [ θ ] = 2 E (cid:18) θ (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . (3.7)The curve for the normalized charge densities for the curve constrained to the ellipse with b = 2 , a = 1 is shown inFig. 6 and is contracted. C. Parabola
Choosing for the parameters of the parabola a = 1, we have that the arc length is given by s [ η ] = η (cid:112) η + 1 + sinh − ( η ) . (3.8)The curve for the normalized charge densities for the curve constrained to a parabola is shown in Fig. 7. It is evenmore contracted. IV. OTHER PARAMETERIZATIONS OF THE CONIC SECTIONS
Of course there are other parameterizations of the conic sections which give different expressions for the arc lengthin terms of these parameters. For example, we can parametrize the ellipse, using doubly periodic functions–namelythe Jacobi elliptic functions and we can parametrize the hyperbola and the parabola using hyperbolic functions.
FIG. 7. Normalized charge density for the soliton constrained to a parabola vs. η for ω = 0 . ω = 1 / A. Jacobi elliptic parametrization of the ellipse
In the simplest case we can let x = a cn( θ, m ) , y = b sn( θ, m ) , (4.1)when the modulus m → x, k ) and sn( x, k ) [8] become the circular functions cos( u )and sin( u ), respectively. The spatial part of the metric is dx + dy = dn( θ | m ) (cid:0) a sn( θ | m ) + b cn( θ | m ) (cid:1) ( dθ ) , (4.2)so that the change of variables is: d s = dn( θ | m ) (cid:112) a sn( θ | m ) + b cn( θ | m ) dθ . (4.3)Integrating we get: s = bE (cid:18) am( θ | m ) (cid:12)(cid:12)(cid:12)(cid:12) − a b (cid:19) θ , (4.4)which simplifies to our previous result when b = 2 a . Here am( θ, m ) is the Jacobi amplitude function [8]. The Jacobiellipse is shown, with a = 3, b = 5, m = 1 / θ of the Jacobielliptic function we find the results shown in Fig. 9 B. A different hyperbolic function parametrization of the hyperbola
Let us consider the following parametrization of the hyperbola in terms of hyperbolic functions x = a sinh( θ ) , y = a cosh( θ ) . (4.5)For this alternative parametrization of the hyperbola we find ds = a cosh(2 θ ) dθ , (4.6)and s = − iaE ( iu | . (4.7)In terms of this parameterization we get the densities shown in Fig. 10 when we choose a = 2. FIG. 8. The Jacobi elliptic curve { x [ θ ] , y [ θ ] } for a = 3 , b = 5 , m = 1 / θ for ω = 0 . ω = 1 / b = 5 , a = 3 , m = 1 /
3. It is contracted compared to Fig. 6.FIG. 10. Normalized charge density for the soliton constrained to a hyperbola vs. θ for ω = 0 . ω = 1 / a = 2. FIG. 11. Normalized charge density for the soliton constrained to a parabola vs. η for , ω = 0 . ω = 1 / a = 1. C. Hyperbolic function parametrization of a parabola
Instead of the parametrization given above of the parabola we could have used a parametrization in terms ofhyperbolic functions: x = 2 a sinh( η ) , y = a sinh ( η ) , (4.8)so that again x = 4 ay . Note that for small η this parametrization goes over to the parametrization used earlier. Thespatial part of the metric then becomes dx + dy = 4 a cosh ( η ) dη . (4.9)Thus the appropriate change of variables to a 1 + 1 dimensional metric is captured in ds = 2 a cosh ( η ) dη . (4.10)In this way, we obtain s ( η ) = a [ η + sinh(2 η ) / . (4.11)One obtains for the normalized densities in the two cases studied before the results shown in Fig. 11. V. SOLITARY WAVE SOLUTIONS FOR THE
DIMENSIONAL NLDE CONFINED TO A SPACECURVEA. Helix
The helix is a space curve with parametric equations: x = r cos θ , y = r sin θ , z = cθ . (5.1)Here r is the radius of the helix and 2 πc gives the vertical separation of the helix’s hoops, i.e. pitch. The curvatureof the helix κ is given by κ = rr + c , (5.2)which is constant and the torsion τ is given by τ = cr + c , (5.3)which is also constant. For this problem we find the metric is given by ds = − dτ + dx + dy + dz = − dτ + ( r + c ) dθ . (5.4)0Therefore, for this problem, the simple transformation to the 1+1 Minkowski metric is again in terms of the arc length d s = (cid:112) r + c dθ . (5.5)So again in the 1 + 1 dimensional space the transformation is given by d s = √− gdθ . (5.6) B. Elliptic helix
The elliptic helix, i.e. a helix with an elliptic cross-section on projection, is defined by x = a cos θ , y = b sin θ , z = k (cid:90) (cid:112) a sin θ + b cos θdθ . (5.7)Note that dz = k ( dx + dy ) so that the 3D arc length is proportional to the 2D arc length. The curvature isvariable in this case. Performing the integral we get z = aE (cid:18) θ (cid:12)(cid:12)(cid:12)(cid:12) − b a (cid:19) . (5.8)The metric is given by ds = − dτ + dx + dy + dz = − dτ + (1 + k )( a cos ( θ ) + b sin ( θ )) dθ , (5.9)so that again one can transform this to a Minkowski metric with the transformation utilizing the arc length d s = √− gdθ = (cid:112) k (cid:113) a cos ( θ ) + b sin ( θ ) dθ . (5.10)Thus, the explicit transformation is (for a > b > s = (cid:112) k aE (cid:18) θ (cid:12)(cid:12)(cid:12)(cid:12) − b a (cid:19) . (5.11) C. Jacobi elliptic helix
The Jacobi elliptic helix can be parametrized as follows: x = a cn( θ | m ) , y = b sn( θ | m ) , z = k (cid:90) dθ dn( θ | m ) (cid:112) ( a sn( θ | m ) + b cn( θ | m ) ) . (5.12)Performing the integral we have z = kbE (cid:18) am( θ | m ) (cid:12)(cid:12)(cid:12)(cid:12) − a b (cid:19) . (5.13)The space curve is shown in Fig. 12, and dx + dy + dz = (1 + k )(dn( θ | m )) (cid:0) a sn( θ | m ) + b cn( θ | m ) (cid:1) . (5.14)So we see that we can transform this to a 1D Dirac equation by letting d s = (cid:112) k dn( θ | m ) (cid:112) ( a sn( θ | m ) + b cn( θ | m ) ) dθ . (5.15)Integrating, we get the arc variable s = (cid:112) k bE (cid:18) am( θ | m ) (cid:12)(cid:12)(cid:12)(cid:12) − a b (cid:19) . (5.16)1 FIG. 12. The Jacobi elliptic helix curve { x [ θ ] , y [ θ ] , z [ θ ] } for a = 3 , b = 5 , m = 1 / , k = 1. D. Helix of the hyperboloid of revolution
The hyperboloid of revolution is defined by x + y a − z b = 1 . (5.17)The helix of the one-sheeted hyperboloid of revolution can be parametrized as follows: x = a cosh η cos f ( η ) , y = a cosh η sin f ( η ) , z = b sinh η , (5.18)where dfdη = (cid:115)(cid:18) ba tan α (cid:19) − tanh η . (5.19)The curve f [ η ] depends on whether β = ba tan α (5.20)is less than one or greater than or equal to one. In the latter case f [ η ] is a straight line. In our plot, Fig. 13, weconsider the cases β = 1 / , ,
10 to exemplify the possibilities. Explicitly, f / ( η )= cosh( η ) (cid:18)(cid:112) − η ) sin − (sinh( η )) − (cid:112) cosh(2 η ) − − (cid:18) η ) √ cosh(2 η ) − (cid:19)(cid:19) cosh(2 η ) − ,f ( η )= cosh( η ) (cid:113) − ( η ) (cid:18) sinh − (cid:16) sinh( η ) √ (cid:17) + tan − (cid:18) √ η ) √ cosh(2 η )+3 (cid:19)(cid:19)(cid:112) cosh(2 η ) + 3 , FIG. 13. The curves f [ η ] for β = 1 / , ,
10, drawn in black, blue and red. Here we have chosen a = 1, b = 1 / β = 1 / , ,
10, are drawn in black, blue and red on a one-sheeted hyperboloid of revlution. Here wehave chosen a = 1, b = 1 / f ( η )= cosh( η ) (cid:113) − ( η ) (cid:18) − (cid:16) η ) √ (cid:17) + tan − (cid:18) √ η ) √ η )+11 (cid:19)(cid:19)(cid:112) η ) + 11 . (5.21)From Eq. (5.18) one can show that the differential arc length is d s = b csc( α ) cosh( η ) . (5.22)Since dz = b cosh( η ) , (5.23)therefore the parameter α has the meaning dzd s = sin α (5.24)and the arc length is given by s = b csc( α ) sinh( η ) . (5.25)The helices for β = 1 / , ,
10 are shown in Fig. 14.
VI. CONCLUSIONS
We have studied the behavior of the nonlinear Dirac equation [9, 10] on planar and space curves. We have shownhere how the arc length variable is the relevant choice to study the nonlinear Dirac equation on planar and space3curves. We studied different parameterizations of various curves including those involving Jacobi elliptic functions[8] and studied the charge density in the presence of a soliton. We found the change in soliton shape in terms ofnarrowing or broadening of the soliton profile in the curved region. These results illustrate an insightful interplaybetween solitons of the nonlinear Dirac equation and the curvature (and torsion) of a variety of curves. Our resultsare relevant to curved Q1D Bose condensates assuming they can be realized experimentally. It would be instructiveto (numerically) study collisions of NLD solitons on various curves to explore how curvature (and torsion) affect theirinteraction and collision dynamics including bounce windows.
VII. ACKNOWLEDGMENT
A.K. is grateful to Indian National Science Academy (INSA) for the award of INSA Senior Scientist position atSavitribai Phule Pune University. This work was supported in part by the U.S. Department of Energy.
VIII. APPENDIX: THE VIEBEIN FORMALISM APPLIED TO A HYPERBOLA
The vierbein introduced by Hermann Weyl [2] is also called a tetrad or a frame field (in general relativity). We shalluse the metric convention ( − + +) for the 2 + 1 space-time which is commonly used in the curved-space literature.In what follows, we use Greek indices for the curvilinear coordinates η or θ , and Latin indices for the Minkowskicoordinates x, y and t . To obtain the fermion evolution equations in the new coordinate system it is simplest to use acoordinate covariant action such as that used in field theory in curved spaces, even though here the actual curvatureis zero. For the constrained system, the 2 + 1 dimensional Minkowski line element( ds ) = − ( dt ) + ( dx ) + ( dy ) = − ( dt ) + a cosh 2 η ( dη ) ≡ − ( dt ) + ( d s ) , (8.1)so that the original Minkowski metric g αβ = diag {− , , } (8.2)reduces to an effective curved space 1 + 1 dimensional metric tensor: g µν = diag {− , a cosh 2 η } (8.3)with inverse g µν = diag {− , a cosh 2 η } . (8.4)Vierbeins transform the curved 1+1 dimensional space to a locally Minkowski 1+1 dimensional space via. g µν = V aµ V bν η ab , where η ab = ( − ,
1) is the flat Minkowski metric. A convenient choice for the vierbein is V aµ = diag { , a (cid:112) cosh 2 η } , (8.5)so that V µa = diag (cid:26) , a √ cosh 2 η (cid:27) . (8.6)The determinant of the metric tensor is given bydet V = √− g = a (cid:112) cosh 2 η . (8.7)The action for our model in general curvilinear coordinates is S [Ψ , σ ] = (cid:90) d x det V (cid:18) − i γ µ ∇ µ Ψ + i ∇ † µ ¯Ψ)˜ γ µ Ψ − i ( m ¯ΨΨ) − g (cid:19) . (8.8)The coordinate dependent gamma matrices ˜ γ µ are obtained from the usual Dirac gamma matrices γ a via˜ γ µ = γ a V µa ( x ) . (8.9)4The coordinate independent Dirac matrices γ a satisfy the usual gamma matrix algebra: { γ a , γ b } = 2 η ab . (8.10)From the action Eq. (8.8) we obtain the Heisenberg field equation for the fermions,(˜ γ µ ∇ µ + σ + m ) Ψ = 0 , (8.11)where it is to be understood: σ = − ig ¯ΨΨ . (8.12)This can be generalized easily to the case where the nonlinear term in the NLDE is of the form [10] − g ( ¯ΨΨ) κ Ψ , (8.13)with κ denoting arbitrary nonlinearity. This is done by letting σ = − ig ( ¯ΨΨ) κ (8.14)in the equation of motion. This again is space and time dependent mass, but a scalar as we will show below.Here ˜ γ µ = γ a V µa , ∇ µ = ∂ µ + Γ µ , and Γ µ is the spin connection given by [11, 12]Γ µ = 12 Σ ab V aν ( ∂ µ V νb + Γ νµλ V λb ) , Σ ab = 14 [ γ a , γ b ] . (8.15)We have ˜ γ = γ ; ˜ γ = 1 a √ cosh 2 η γ . (8.16)We will choose for convenience the following representation for the matrices γ and γ in the transformed localMinkowski 1+1 dimensional Dirac equation. iγ = (cid:18) − (cid:19) , (8.17) γ = (cid:18) (cid:19) . We have that the scalar field σ is unchanged in form σ = − ig (Ψ † ˜ γ Ψ) = − ig (Ψ † γ Ψ) , (8.18)if we used Eq. (8.14) to define σ , it would again be unchanged in form.The Christoffel symbols have the usual definition, Γ σµλ = g νσ ( ∂ λ g νµ + ∂ µ g νλ − ∂ ν g µλ ). Since the only non-zeroderivative is ∂g ∂x = 2 a sinh 2 η, (8.19)we find the only nonzero Christoffel symbol is Γ = a sinh 2 η (8.20)so that Γ = tanh 2 η . (8.21)We find the spin connection is zero for the following reason. If we consider F νµ,b = ∂ µ V νb + Γ νµλ V λb , (8.22)5only F could have a non-zero contribution but we find ∂ ν V = − tanh 2 ηa √ cosh 2 η ; Γ V = + tanh 2 ηa √ cosh 2 η , (8.23)so that F = 0 which implies that the spin connection is zero.From Eq. (8.11), and the fact that the spin connection is zero we find that the equation of motion for Ψ is(˜ γ µ ∂ µ + σ + m )Ψ = 0 . (8.24)Writing this out more explicitly, we obtain (cid:18) γ ∂ t + 1 a √ cosh 2 η γ ∂ η + σ + m (cid:19) Ψ[ η, t ] = 0 . (8.25) [1] See e.g., L. H. Haddad and L. D. Carr, New J. Phys. , 113011 (2015).[2] H. Weyl, Zeitschrift Physik , 330 (1929).[3] Y. B. Gaididei, S. F. Mingaleev, and P. Christiansen, Phys. Rev. E , R53 (2000).[4] M. Ibanes, J. M. Sancho, and G. P. Tsironis, Phys. Rev. E , 041902 (2002).[5] A. Ludu, Nonlinear Waves and Solitons on Contours and Closed Surfaces (Springer Series in Synergetics, Second edition,Berlin 2012).[6] D. J. Gross and A. Neveu, Phys. Rev.
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