Geometrical model of massive spinning particle in four-dimensional Minkowski space

TL;DR

This paper introduces a geometric model for a massive spinning particle in four-dimensional Minkowski space, where particle trajectories are constrained to cylinders defined by momentum and angular momentum, emphasizing a gauge-invariant, non-Lagrangian approach.

Contribution

The paper presents a novel geometrical model of a massive spinning particle that does not require additional spin variables and derives implicit equations of motion based on path curvature and helices.

Findings

Particle paths lie on cylinders determined by momentum and angular momentum.

Equations of motion are implicit, gauge-invariant, and non-Lagrangian.

Explicit equations are derived for small curvature and helical trajectories.

Abstract

We propose the model of massive spinning particle traveling in four-dimensional Minkowski space. The equations of motion of the particle follow from the fact that all the classical paths of the particle lie on a cylinder whose position in Minkowski space is determined by the particle's linear momentum and total angular momentum. All the paths on one and the same cylinder are gauge equivalent. The equations of motion are found in implicit form for general time-like paths, and they are non-Lagrangian. The explicit equations of motion are found for trajectories with small curvature and helices. The momentum and total angular momentum are expressed in terms of characteristics of the path in all the cases. The constructed model of the spinning particle has geometrical character, with no additional variables in the space of spin states being introduced.

Equations

$$(x-y{)}^2+(n,x{)}^2=R^2,(x-y,a)=\Delta ,$$

$$(n,n)=-1,(a,a)=1,(n,y)=(a,y)=0.$$

$$(u,w)=u_{\mu }{w}^{\mu },\forall u,w.$$

$$r=\sqrt{R^2-{\Delta }^2}.$$

$$0\le |\Delta |\le R.$$

$$p=mn,J=my\wedge n+s{\epsilon }_{\mu \nu \rho \sigma }{n}^{\mu }{a}^{\nu }d{x}^{\rho }\wedge d{x}^{\sigma },$$

$$p^2+m^2=0,w^2=m^2{s}^2,$$

$$S=m(y-x)\wedge n+s{\epsilon }_{\mu \nu \rho \sigma }{n}^{\mu }{a}^{\nu }d{x}^{\rho }\wedge d{x}^{\sigma }.$$

$$\frac{1}{2}{S}_{\mu \nu }{S}^{\mu \nu }=s^2-m^2{R}^2,\frac{1}{8}{\epsilon }_{\mu \nu \rho \sigma }{S}^{\mu \nu }{S}^{\rho \sigma }=-ms\Delta .$$

$$\frac{d^{k}}{d\tau^{k}}\bigg{(}(x-y)^{2}+(n,x)^{2}-R^{2}\bigg{)}=0\,,\qquad\frac{d^{k}}{d\tau^{k}}\bigg{(}(x-y,a)-\Delta\bigg{)}=0\,,\qquad k=1,2,3,\ldots\,.$$

$$n=n(x,\dot{x},\ddot{x},\dots ),y=y(x,\dot{x},\ddot{x},\dots ),a=a(x,\dot{x},\ddot{x},\dots ),$$

$$\begin{array}[]{c}\displaystyle(\dot{x},a)=0\,,\qquad(\ddot{x},a)=0\,,\qquad(\dddot{x},a)=0\,,\qquad(\ddddot{x},a)=0\,;\end{array}$$

$$\begin{array}[]{c}\displaystyle(\dot{x},d)=0\,,\qquad(\ddot{x},d)+(\dot{x},n)^{2}=0\,,\qquad(\dddot{x},d)+3(\ddot{x},n)(\dot{x},n)=0\,,\\[17.07164pt] \displaystyle(\ddddot{x},d)+4(\dddot{x},n)(\dot{x},n)+3(\ddot{x},n)^{2}=0\,,\end{array}$$

$$d\equiv x+n(n,x)-y.$$

$$(\dot{x},\dot{x})=-1.$$

$$a=\pm \frac{[\dot{x},\ddot{x},\stackrel{...}{x}]}{\sqrt{[\dot{x},\ddot{x},\stackrel{...}{x}{]}^2}},[\dot{x},\ddot{x},\stackrel{...}{x}]\equiv {\epsilon }_{\mu \nu \rho \sigma }{\dot{x}}^{\mu }{\ddot{x}}^{\mu }{\stackrel{...}{x}}^{\rho }d{x}^{\sigma },$$

$$([\dot{x},\ddot{x},\stackrel{...}{x}],\stackrel{....}{x})\equiv (\dot{x},\ddot{x},\stackrel{...}{x},\stackrel{....}{x})=0,$$

$$\begin{array}[]{c}\displaystyle e_{1}=\dot{x}\,,\qquad e_{2}=\frac{\ddot{x}}{\sqrt{(\ddot{x},\ddot{x})}}\,,\\[19.91692pt] \displaystyle e_{3}=\frac{(\ddot{x},\ddot{x})^{2}\dot{x}+(\ddot{x},\dddot{x})\ddot{x}-(\ddot{x},\ddot{x})\dddot{x}}{\sqrt{(\dddot{x},\dddot{x})(\ddot{x},\ddot{x})^{2}+(\ddot{x},\ddot{x})^{4}-(\ddot{x},\dddot{x})^{2}(\ddot{x},\ddot{x})}}\,,\qquad e_{4}=\frac{[\dot{x},\ddot{x},\dddot{x}]}{\sqrt{[\dot{x},\ddot{x},\dddot{x}]^{2}}}\,.\end{array}$$

$$\begin{array}[]{c}\displaystyle\phantom{\frac{1}{2}}(e_{1},e_{1})=-1\,,\qquad(e_{2},e_{2})=(e_{3},e_{3})=(e_{4},e_{4})=1\,,\phantom{\frac{1}{2}}\\[8.53581pt] \displaystyle\phantom{\frac{1}{2}}(e_{i},e_{j})=0\,,\qquad i\neq j\,.\phantom{\frac{1}{2}}\end{array}$$

$$\begin{array}[]{c}\displaystyle\phantom{\frac{1}{2}}k_{1}=\sqrt{(\ddot{x},\ddot{x})}\,,\qquad k_{2}=-\frac{1}{(\ddot{x},\ddot{x})}\sqrt{(\dddot{x},\dddot{x})(\ddot{x},\ddot{x})-(\ddot{x},\dddot{x})^{2}-(\ddot{x},\ddot{x})^{3}}\,,\phantom{\frac{1}{2}}\\[14.22636pt] \displaystyle\phantom{\frac{1}{2}}k_{3}=\frac{(\dot{x},\ddot{x},\dddot{x},\dddot{x})}{\sqrt{((\dddot{x},\dddot{x})-(\ddot{x},\dddot{x})^{2}(\ddot{x},\ddot{x})^{-1}-(\ddot{x},\ddot{x})^{2})[\dot{x},\ddot{x},\dddot{x}]^{2}}}\,.\phantom{\frac{1}{2}}\end{array}$$

$$\begin{array}[]{c}\displaystyle\phantom{\frac{1}{2}}\ddot{x}=e_{1}\,,\qquad\ddot{x}=k_{1}e_{2}\,,\qquad\dddot{x}=k_{1}^{2}e_{1}+\dot{k}_{1}e_{2}+k_{1}k_{2}e_{3}\,,\phantom{\frac{1}{2}}\\[14.22636pt] \displaystyle\phantom{\frac{1}{2}}\ddddot{x}=3\dot{k}_{1}k_{1}e_{1}+(\ddot{k}_{1}+k_{1}^{3}-k_{1}k_{2}^{2})e_{2}+(2\dot{k}_{1}k_{2}+k_{1}\dot{k}_{2})e_{3}+k_{1}k_{2}k_{3}e_{4}\,.\phantom{\frac{1}{2}}\end{array}$$

$$n={\beta }_1{e}_1+{\beta }_2(-{\alpha }_2{e}_2+{\alpha }_1{e}_3),d=r({\alpha }_1{e}_2+{\alpha }_2{e}_3)\pm \Delta e_4,$$

$$\begin{array}[]{c}\displaystyle\alpha_{1}^{2}+\alpha_{2}^{2}=1\,,\qquad\beta_{1}^{2}-\beta_{2}^{2}=1\,,\qquad rk_{1}\alpha_{1}+\beta_{2}^{2}=0\,,\qquad r(\dot{k}_{1}\alpha_{1}+k_{1}k_{2}\alpha_{2})+3k_{1}\alpha_{2}\beta_{1}\beta_{2}=0\,,\\[14.22636pt] \displaystyle r\bigg{[}(\ddot{k}_{1}+k_{1}^{3}-k_{1}k_{2}^{2})\alpha_{1}+(2\dot{k}_{1}k_{2}+k_{1}\dot{k}_{2})\alpha_{2}\bigg{]}+4(\dot{k}_{1}\alpha_{2}-k_{1}k_{2}\alpha_{1})\beta_{1}\beta_{2}-k_{1}^{2}((3\alpha_{1}^{2}-7)\beta_{2}^{2}-3)=0\,.\end{array}$$

$${\beta }_1=\text{ch}\theta .$$

$$\begin{array}[]{rl}\displaystyle P_{1}(\beta_{1})&=81k_{1}^{4}\beta_{1}^{16}-162k_{1}^{4}\beta_{1}^{14}+r^{2}(18k_{1}^{4}k_{2}^{2}-18\dot{k}{}_{1}^{2}k_{1}^{2}+81k_{1}^{4})\beta_{1}^{12}+r^{2}(18k_{1}^{2}k_{1}^{2}-18k_{1}^{4}k_{2}^{2})\beta_{1}^{10}+\\[14.22636pt] \displaystyle\hfil&+r^{4}(k_{1}^{4}k_{2}^{4}+\dot{k}{}_{1}^{4}+2\dot{k}{}_{1}^{2}k_{1}^{2}k_{2}^{2}-18k_{1}^{6}k_{2}^{2})\beta_{1}^{8}+r^{4}18k_{1}^{6}k_{2}^{2}\beta_{1}^{6}-r^{6}(2\dot{k}{}_{1}^{2}k_{1}^{4}k_{2}^{2}+2k_{1}^{6}k_{2}^{4})\beta_{1}^{4}+r^{8}k_{1}^{8}k_{2}^{4}=0\,,\\[14.22636pt] \displaystyle P_{2}(\beta_{1})&=81k_{1}^{4}\beta_{1}^{24}+324k_{1}^{4}\beta_{1}^{22}+\textbf{260 terms}=0\,.\end{array}$$

$${\beta }_1={\beta }_1(k_1,k_2,{\dot{k}}_1,{\dot{k}}_2,{\ddot{k}}_1).$$

$$\begin{array}[]{c}\displaystyle\alpha_{1}=\frac{1-\beta_{1}^{2}}{rk_{1}}\,,\qquad\alpha_{2}=\frac{2r\dot{k}_{1}k_{1}^{-2}k_{2}(1-\beta_{1}^{2})((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}{r^{2}k_{1}^{2}(1-\beta_{1}^{2})^{2}+k_{1}^{2}(9\beta_{1}^{2}(1-\beta_{1}^{2})-r^{2}k_{2}^{2})((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}\,,\\[19.91692pt] \displaystyle\beta_{2}=-\frac{r^{2}k_{1}^{2}(1-\beta_{1}^{2})^{2}+k_{1}^{2}(9\beta_{1}^{2}(1-\beta_{1}^{2})-r^{2}k_{2}^{2})((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}{6rk_{1}^{2}k_{2}\beta_{1}((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}\,.\end{array}$$

$$\begin{array}[]{rl}p&\displaystyle=\beta_{1}e_{1}+\frac{\dot{k}_{1}(1-\beta_{1}^{2})}{3rk_{1}^{4}\beta_{1}}e_{2}-\\[14.22636pt] &-\displaystyle\frac{r^{2}k_{1}^{2}(1-\beta_{1}^{2})^{3}+k_{1}^{2}(1-\beta_{1}^{2})(9\beta_{1}^{2}(1-\beta_{1}^{2})-r^{2}k_{2}^{2})((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}{6r^{2}k_{1}^{3}k_{2}\beta_{1}((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}e_{3}\,,\\[14.22636pt] \displaystyle J&\displaystyle=m(x-d)\wedge n\pm s\bigg{[}\beta_{1}e_{2}\wedge e_{3}-\frac{\dot{k}_{1}(1-\beta_{1}^{2})}{3rk_{1}^{4}\beta_{1}}e_{1}\wedge e_{3}-\\[14.22636pt] &-\displaystyle\frac{r^{2}k_{1}^{2}(1-\beta_{1}^{2})^{3}+k_{1}^{2}(1-\beta_{1}^{2})(9\beta_{1}^{2}(1-\beta_{1}^{2})-r^{2}k_{2}^{2})((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}{6r^{2}k_{1}^{3}k_{2}\beta_{1}((1-\beta_{1}^{2})^{2}-r^{2}k_{1}^{2})}e_{1}\wedge e_{2}\bigg{]}\,.\end{array}$$

$${\text{Res}}_{{\beta }_1}(P_1,P_2)=0.$$

$$r{k}_1\sim r{k}_2\sim r^2{\dot{k}}_1\sim r^2{\dot{k}}_2\sim r^3{\ddot{k}}_1\ll 1.$$

Full text

1. Introduction

The classical spinning particle is irreducible if its quantization corresponds to the irreducible representation of the Poincare group. The models of irreducible spinning particles are well-known [1, 2, 3, 4]. In most of these models the particle is considered as the point object in Minkowski space, while the spin positions take values in the internal space. Depending on the choice of the coordinates in internal space, vector, tensor, spinor and twistor types of spinning particle models are distinguished [5]. In the geometrical models the spin is described in terms of derivatives of the particle’s path, and no internal space is introduced. The variational geometrical models are known for massless particles [6, 7]. The variational geometrical models of massive particles are reducible [8, 9].

In the paper [10], the concept of the world sheet is proposed to describe massive particles with spin. The idea of this concept is that the classical trajectories of irreducible particle lie on a certain cylindrical surface in Minkowski space. The position of the cylinder in space is determined by the values of the particle’s linear momentum and total angular momentum, and all the paths on one and the same cylinder are gauge equivalent. The world sheet concept can be used to deduce the ordinary differential equations describing the particle’s path. These equations are non-Lagrangian and involve higher derivatives. The geometrical model of massive spinning particle traveling light-like and isotropic paths in three-dimensional Minkowski space was developed in the original work [10]. In four-dimensional Minkowski space, the world sheet is considered in [11].

In the present research, we construct the geometrical model of massive spinning particle traveling in four-dimensional Minkowski space. The content of the article is based on the paper [11], but more attention is paid to the geometry of particle’s paths. In particular, we consider the cylindrical curves with small curvature, and also the helical paths. The construction of equations of cylindrical curves uses several techniques of differential geometry, including the Frenet moving frame. For differential geometry of the curves in (non-)Euclidean space we refer to the books [12, 13]. The method of classification of cylindrical curves in Euclidean space has been proposed in [14, 15]. We use this method with small modifications accounting the signature of the space-time metric.

The rest of the article is organized as follows. In Section 2, we describe the geometry of the world sheet of massive spinning particle in four-dimensional Minkowski space. In Section 3, we derive the equations for general cylindrical path on the world sheet with time-like tangent vector. The equations are derived in implicit form. In Section 4, some particular types of cylindrical curves with explicit equations of motion are considered. The conclusion summarizes the results.

2. World sheets and world paths

The geometry of the world sheet of massive spinning particle in four-dimensional space-time has been described in ref. [11]. It has been shown that the world surface is determined by two algebraic equations

[TABLE]

where $x^{\mu}\,,\mu=\overline{0,3}$ are the coordinates in Minkowski space, and $R,\Delta$ are constants. The vector parameters $n,\,y,\,a$, are subjected to the constraints

[TABLE]

Hereinafter, the round brackets denote the scalar product in Minkowski space,

[TABLE]

We use the mostly positive convention for the signature of the metric throughout the paper. All the world indices are raised and lowered by the metric.

The geometric meaning of the equations (1) is following. The first equation determine the circular hypercylinder with time-like axis in Minkowski space. The tangent vector to the cylinder axis is $n$, and the vector y connects the cylinder axis and the origin by the shortest path. The radius of hypercilder is $R$. The second of equations (1) determines the hyperplane with the normal. The hypercylinder axis is tangent to this hyperplane. The distance between hypercylinder axis and hyperplane is $\Delta$. The intersection of the hypercylinder and hyperplane is two-dimensional cylinder. The tangent vector to the cylinder axis is $n$. The radius of the cylinder is

[TABLE]

It is a real quantity if

[TABLE]

Otherwise, the intersection of the hypercylinder and hyperplane is empty. In the special case $r=0$, the cylinder becomes a straight line. We dot not consider this possibility below.

The linear momentum $p=p_{\mu}dx^{\mu}$ and total angular momentum $J=J_{\mu\nu}dx^{\mu}\wedge dx^{\mu}$ of the particle are determined by the world sheet position in the Minkowski space,

[TABLE]

where $\wedge$ denotes the exterior product, and $\varepsilon_{\mu\nu\rho\sigma},\varepsilon_{0123}=1$ is the Levi-Civita symbol. The constants $m,s$ are the mass and spin of the particle. The details of the derivation of the formulas (6) are given in [11]. With account of relations (2), the states of the spinning particle belong to the co-orbit of the Poincare group,

[TABLE]

where $w=sa$ is the Pauli-Lyubanski pseudovector. The spin tensor is determined by the standard rule $S=J-x\wedge p$, form which we find

[TABLE]

By construction,

[TABLE]

The right hand sides of these equalities are constants because of irreducibility conditions of the Poincare group representation [11].

The classical trajectories of spinning particles are curves on the world sheet. The casuality condition $\dot{x}^{0}>0$ is imposed on the physical path. We term the path locally regular if each its small segment lies on a unique representative in set of cylinders. Almost all the cylindrical paths are locally regular. Only locally regular paths are considered in this article.

3. Classification of cylindrical curves

In this section, we obtain the system of equations describing the cylindrical curves in four-dimensional Minkowski space.

The starting point of our consideration is that each cylindrical curve $x=x(\tau)$ lies on a certain cylinder in the set (1). In this case, the equations (1) are identically satisfied for all the values of the parameter $\tau$ on the curve. The differential consequences of these identities have the form

[TABLE]

Once the differential consequences of sufficiently large order are included, the cylinder parameters can be expressed as functions of derivatives of the path,

[TABLE]

where the dots denote the derivatives by $\tau$. The existence of unique solution (11) of the equations (10) uses the regularity assumption for the cylinder path. For singular path, the solution is not necessary unique. That is why the singular cylindrical paths are excluded. The substitution of the cylinder parameters (11) into the original equations gives us the equations for the cylindrical curves. The cylinder parameters are integrals of motion of this equation, which can be treated as the equation of motion of spinning particle. The linear momentum and angular momentum of the particle are defined by the rule (6).

We can proceed with explicit derivation of the equations of cylindrical path. It is sufficient to differentiate equations (1) four times. In this case, the differential consequences (10) have the form

[TABLE]

[TABLE]

where the notation is used,

[TABLE]

The vector $d$ connects the current position of the particle and hypercylinder axis by the shortest path. To simplify the formulas, we assume that the path is time-like and the parameter $\tau$ is natural on the curve,

[TABLE]

The relations (12), (13) constitute the system of equations (10) for finding of cylinder parameters $n,y,a$.

The normalized vector $a$ can be expressed from equations (12),

[TABLE]

where the square brackets denote the vector product of three vectors in Minkowski space. The consistency condition for equations (12) reads

[TABLE]

where the round brackets denote the mixed product of four vectors in Minkowski space. From the geometrical viewpoint, the obtained condition implies zero torsion of the curve $x(\tau)$. It ensures that this curve lies in some hyperplane, with $a$ being its normal.

Now we consider the hypercylinder condition and its differential consequences. We have nine unused conditions (1), (2), (13) to express two vectors $n,y$ and find a single consistency condition. Introduce the Frenet moving frame for time-like curves [13],

[TABLE]

All the basis vectors of Frenet frame are normalized and orthogonal to each other,

[TABLE]

The curvatures $k_{1},k_{2}$, and torsion $k_{3}$ of the curve are determined by the rule

[TABLE]

The time derivatives of particle’s position can be expressed as the linear combinations of the Frenet basis vectors (18) with the coefficients depending on the curvatures and torsion of path,

[TABLE]

The unknown vectors $n,d$ admit the following represenation:

[TABLE]

where $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ are new unknowns. Substitution of the expressions (21), (22) into (1), (2), (13) brings us the following system of algebraic equations:

[TABLE]

The other conditions are automatically satisfied.

The general method of solution of systems of algebraic equations uses techniques of resultants. In this approach, all the variables except one are eliminated, while all the other unknowns are expressed as functions of this single variable from subresultants. For system (23), $\beta_{1}$ is a good candidate for the independent variable. It has the sense of the hyperbolic tangent of the angle $\theta$ between the element of cylinder and particle’s path,

[TABLE]

Eliminating the variables $\alpha_{1},\alpha_{2},\beta_{2}$ from the system (23) in two different ways, we get two algebraic equations of order $16$ and $24$ for $\beta_{1}$,

[TABLE]

The unknown $\beta_{1}$ is the positive common root of these polynomials. We can’t express it in the explicit form, but we can argue that it is a function of first and second curvatures $k_{1},\,k_{2}$ with its derivatives,

[TABLE]

As the curvatures are functions of the derivatives of the path, this solution involves derivatives of $x$ up to fourth order. The other unknowns are expressed using $\beta_{1}$,

[TABLE]

The momentum and total angular momentum of the particle are expressed from equations (6) and (27),

[TABLE]

By construction, $p$ and $J$ belong to the co-orbit (7) of the Poincare group. This ensures the irreducibility of the particle that moves the cylindrical path. The equation of hypercylindrical curves is the consistency condition of equations (25). It is given by the resultant of two polynomials,

[TABLE]

The explicit expression of this resultant is too long and not informative. As a matter of principle, we mention that the resultant involves the derivatives of the path up to fourth order. So the curves on the hypercylinder are described by the fourth-order equation.

The paths of spinning particle lie on the intersection of hypercylinder and hyperplane. Such paths meet both the conditions (17) and (29). This is a system of two equations of fourth order. The model of spinning particle, being described by these equations, has the geometrical character because no additional variables in the spin sector are introduced.

4. Some particular cases

Let us consider some particular cases of cylindrical curves such that the system (23) can be solved explicitly.

At first, suppose that the path of particle is close to the straight line. We solve equations (23) using the following assumptions. The curvatures and derivatives are small quantities of same order of magnitude:

[TABLE]

The parameters $\alpha_{1},\,\alpha_{2},\,\beta_{1},\,\beta_{2}$ are sought in the form

[TABLE]

where the parameters $\varphi,\theta$ have the same order of magnitude as (30).

Substituting the ansatz (31) into the equations (23) and accounting (30), we get the following equations:

[TABLE]

This system includes three equations for two unknowns $\varphi,\theta$. These quantities can be expressed from the linear equations,

[TABLE]

This solution exists if the regularity condition for the curve is satisfied,

[TABLE]

The consistency condition for the equations of motion (32) has the form

[TABLE]

This forth order equation for the hypercylindrical curves with small curvature.

The momentum and angular momentum of the particle moving the path with small curvature have the form

[TABLE]

where the terms labeled $o(1)$ vanish in the straight line approximation. The obtained formulas are analogue of the formulas (28) in the small curvature limit. The system of equations (17), (35) is a system of two equations of motion of the massive spinning particle. We can explicitly see that both equations have the fourth order in derivatives of the path.

The other type of cylindrical curves are helices. The helices on $2d$ cylinder are defined by the constant curvature conditions

[TABLE]

In this case, the system of equations (23) takes the form

[TABLE]

The solution to this system reads

[TABLE]

The consistency condition for this system is the relation between the curvatures of the helix and its radius,

[TABLE]

In this way, we see that the helices are particular solutions to the equations of cylindrical curves (17), (29).

The linear momentum and total angular momentum of the spinning particle traveling the helical path have the form

[TABLE]

By construction, the momentum and total angular momentum of the particle lie on the co-orbit of the Poincare group.

5. Conclusion

In this article, we considered a geometrical model of irreducible massive spinning particle. Our initial assumption was that the classical paths of spinning particle lie on a certain cylinder whose position is determined by the values of the particle’s linear momentum and total angular momentum. Preceding from the fact that all the paths on one and the same cylinder are gauge equivalent, we derived the particle’s equations of motion. The equations of motion are found only in implicit form. The structure of these equations brings us to the conclusion that the path of spinning particle subjects to the system of two equations of forth order. The explicit structure of these equations was studied in two particular cases of the paths with small curvatures and helical trajectories. The null-like paths are null-like helices, and they are possible path of irreducible spinning particle. The model of massive spinning particle traveling isotropic helical path was proposed earlier from other considerations [16]. The space-like cylindrical trajectories can be studied in the similar way. We plan to study this problem later.

Acknowledgements

This research was funded by state task of Ministry of Science and Higher Education of Russian Federation, grant number 3.9594.2017/8.9.