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The relativistic particle with curvature and torsion of world trajectory

Local symmetries of the action for a relativistic particle with curvature and torsion of its world curve in the (2+1)-dimensional space-time are studied. With the help of the method, worked out recently by the authors (Phys.Rev., D56, 1135, 1142 (1997)), first the local-symmetry transformations are obtained both in the phase and configuration space. At the classical level, the dependence of the particle mass on the parameters of curvature and torsion and the Regge trajectory are obtained. It is shown that the tachyonic sector can be removed by a proper gauge choice.


Introduction
In recent years, interest in the point relativistic particle has been revived by investigation of the action depending on the curvature and/or torsion of the world trajectory of that particle (see, e.g. [1] and refs. therein). This relates, on the one hand, to the development of the string theory: the above-mentioned object can be considered as a one-dimensional analog of the relativistic string with rigidity [2]. On the other hand, such models seem to continue realizing an old idea of modelling the particle spin without introducing additional spin variables [3]. Note also a connection of the indicated models with the problem of describing planar phenomena: the high-temperature superconductivity [4] and the fractional quantum Hall effect [5].
The present work is devoted to an investigation of the action of a massive relativistic particle with curvature and torsion in (2+1) dimensions. Here there are the first-and second-class constraints, both primary and secondary (see, e.g. [1,6]). Therefore, already the initial stage of constructing the theory needs a complete separation of constraints of the first and second class and finding gauge transformations. This can be made only with the help of the methods, recently proposed in papers [7]- [9]. An explicit form of gauge transformations is required, e.g., in eliminating unphysical degrees of freedom (i.e., a reduction of the theory), carried out before or after the quantization (the schemes of covariant and canonical quantization).
The layout of the paper is as follows. In Sec. 2, the system under investigation is considered in the framework of the generalized Hamiltonian formalism by Dirac for theories with higher derivatives [10], and by the method, suggested in Ref. [7], we separate completely the constraints of the first and second class and obtain a canonical set of constraints. In Sec. 3, we obtain gauge transformations for the considered system both in the phase and configuration space and the Noether identity. In Sec. 4, the dependence of the particle mass on the parameters of curvature and torsion and the Regge trajectory are obtained. It is shown that the tachyonic sector can be removed by a proper gauge choice.

Generalized Hamiltonian formalism
So, let us consider a relativistic particle with curvature and torsion of its world curve in the (2+1)-dimensional space-time. Given a parametrization of the world trajectory x µ (τ ) (µ = 0, 1, 2), the action can be written in the form whereẋ ≡ dx(τ )/dτ,ẋ 2 > 0, dimensionless parameters α and β characterize the contributions of curvature (k) and torsion (κ), respectively, m is a parameter of the mass dimension, ε µνρ is a completely antisymmetric unit tensor (ε 012 = 1), the Lorentz metric with signature (+, −, −) is used. In the following we shall use the notation We see that the action 1 depends on the derivatives of the particle coordinates with respect to τ up to and including the third order. Therefore, when passing to the generalized Hamiltonian formalism for theories with higher derivatives, the canonical variables are to be introduced according to the Ostrogradsky method [10] where L is the Lagrangian in 1. The canonical Hamiltonian has the form The Poisson brackets (PB) are defined in the following way From the definition of the momentum p 3 three primary constraints are derived where g = (q 2 q 3 ) 2 − q 2 2 q 2 3 . Making use of the orthonormal basis in a three-dimensional space-time and the completeness condition n µ 1 n ν 1 − n µ 2 n ν 2 − n µ 3 n ν 3 = g µν , it is convenient to decompose the vector 7 in this basis and to pass from three primary constraints 7 to the equivalent constraints (the projections of 7 onto the vectors 8) Determinant of the matrix of the transformation to the constraint set 9 equals −g. When investigating self-consistency of the theory, we apply the Dirac scheme for breeding constraints. From the conditions of the time conservation of constraintsφ 1 = 0 (α = 1, 2, 3; u α are the Lagrange multipliers; Σ 1 = means this equality to hold on the surface of the primary constraints Σ 1 ) we obtain the secondary constraints From the conditionsφ 2 α Σ 2 = 0 (α = 1, 2) two ternary constraints are obtained and the equationφ 2 3 Σ 2 = 0 serves for determining the Lagrangian multiplier u 3 Further we obtainφ from the conservation (in time) conditions of which the Lagrangian multiplier u 2 is determined. In eq.13 it is taken into account that . So, the complete set of constraints is obtained with the following PB among them: . We see that from three chains of the constraints the two ones are candidates for second-class constraints. Making use of the method developed in Ref. [7], let us separate the constraints into the first-and second-class ones and pass to an equivalent canonical set of constraints. For this purpose, we perform the equivalence transformation to obtain the desired structure of the second-class constraints of the canonical set and then completely to separate the first-class constraints: where the coefficients a 1 , a 2 and a 3 are determined to fulfil the requirements ( Σ = means this equality to hold on the surface of all constraints Σ). Now we carry out the following equivalence transformation with coefficients b 1 , b 2 and b 3 , determined to fulfil the requirements The obtained set of constraints is canonical, because the only non-vanishing (on Σ) PB among them are i.e., on Σ the maximal partition of the set of constraints is achieved: each second-class constraint of the final set has the vanishing (on the constraint surface) PB with all the constraints of the system except one, and the first-class constraints have the vanishing PB with all the constraints. Therefore, there are three constraints of the first class 20 and two chains of the second-class constraints 21, 22 (second-class constraints are denoted by the letter Ψ).
The total Hamiltonian has now the form i.e., is a function of the first-class. Note that the secondary constraints of the first class have the following PB with the primary constraint Φ 1 The second PB 25 are not of the form needed for obtaining local-symmetry transformations -the right-hand side must be expressed only through Φ 1 1 (first-class primary constraints are to be the ideal of quasi-algebra of all the first-class constraints [8]). Therefore, instead of Φ 3 1 in 20, the equivalent constraint will be used

Local-symmetry transformations
Now we can say that, according to the theorem proved in Ref. [8], the action 1 is quasiinvariant under the one-parameter quasigroup of local-symmetry transformations. We shall derive these transformations by the method developed in Ref. [8,9], where it has been proved that for the obtained canonical set of constraints the generator of local-symmetry transformations is to be the form where the coefficients ε m 1 1 (m 1 = 1, 2, 3) are the solution of the system of equationṡ with the functions g m ′ 1 m 1 (here H = H c + u 2 Ψ 1 2 + u 3 Ψ 1 3 with u 2 and u 3 given by eqs. 14 and 12). As g 3 3 1 1 = q 2 q 3 /q 2 2 , g 2 3 1 1 = −1, g 3 2 1 1 = −q 2 3 /q 2 2 , g 2 2 1 1 = −q 2 q 3 /q 2 2 , g 1 2 1 1 = −1, the system of equations 29 for determining ε m 1 1 becomesε Denoting ε 3 1 ≡ λ, we obtain We see that the quantity G in 28 depends onq 3 ; therefore, for the desired local-symmetry transformations being canonical it is necessary to extend the phase space in the following way: Define the coordinates Q i = q i (i = 1, 2, 3), Q 4 =q 3 ; then their conjugate momenta, calculated in accordance with the formula of theories with higher derivatives [10], are P i = p i (i = 1, 2, 3), P 4 = 0. The generalized momentum P 4 is an extra primary constraint of the first class. In the extended phase space the total Hamiltonian is written down as where H T is of the same form as in the initial phase space 24 andū is an arbitrary function of the time. From 33 we may conclude that there do not appear additional secondary constraints corresponding to P 4 . The set of constraints in the extended phase space remains the same as in the initial phase space, obeys the same algebra and does not depend on new coordinates and momenta as H T does. We shall seek a generator G in the extended phase space in a form analogous to that in the initial phase space. Then from the requirements of quasi-invariance of the action under the transformations generated by G, we obtain the same relations 32 for determining ε m 1 1 . As a result, we find the expression for G: Note that the obtained generator 35 satisfies the group property where the transformation G 3 35 is realized by carrying out two successive transformations G 1 and G 2 35. Now the local-symmetry transformations of the coordinates in the extended one are of the form here the functions f and h are rather cumbersome. One can verify that to within quadratic terms in δQ i and δP j i.e., the obtained infinitesimal gauge transformations are canonical in the extended (by Ostrogradsky) phase space. We see from 37 that the vector P µ 1 is gauge-invariant, and it is conserved also in the time evolution.
The corresponding transformations of local symmetry in the Lagrangian formalism are determined in the following way: We obtain δx µ = −λẋ µ .
One can see that the Lagrangian 1 is quasi-invariant under these transformations. The corresponding Euler -Lagrange equation is Under the transformations 39, 40 We obtain the Noether identity 4 Spin-mass-curvature relation Consider some physical conclusions (first on the classical level) from the obtained set of constraints and the transformations of local symmetry. We have already seen that the integral of motion P 1 is gauge-invariant. From invariance of the action 34 under translations we obtain that P 1 is the energy-momentum vector. From invariance of the action 34 under rotations we obtain that the conserved angular momentum tensor is This quantity is gauge-invariant in the initial phase space, i.e. on the surface P 4 = 0. The quantities P 1 and M µν make up the Lie algebra with respect to PB: One can construct the Casimir operators P 2 1 = M 2 and W of the Poincaré group, where W is determined as W = −S 2 = 1 2 M µν M µν P 2 1 − M µσ M µτ P σ 1 P 1τ (48) with the particle spin defined in the case d = 2 + 1 by Let us obtain the dependence of the particle mass M on parameters of the curvature and torsion. To this end, we solve the equation of the constraint Ψ 4 2 with respect to P 2 1 : where k = √ g/(q 2 2 ) 3/2 is the curvature. At a given value of curvature k, the particle has the mass either M + or M − . Since we can impose up to three gauge conditions χ α (q, p, τ ) = 0, α = 1, 2, 3 with the requirements det||{χ α , Φ m 1 }|| ≈ 0, m = 1, 2, 3, (52) (v m are determined from eqs.53 for constructing generalized Hamiltonian H E ), the appearence of the tachyonic sector (M 2 < 0) can be prevented. Indeed, choosing we obtain the determinant 52 to be equal to −4cq 0 2 q 2 2 q 2 3 . Then solving the inequality M 2 > 0, one can limit desired values c.
For the relation between the mass of the particle and its spin we obtain the same twobranched expression as in Ref. [11] i.e., for a fixed mass there are two world curves corresponding to different spin values.