Variational principle for cylindrical curves and dynamics of spinning particles in $d=3$ Minkowski space

TL;DR

This paper develops a variational framework for describing the classical trajectories of spinning particles in 3D Minkowski space, revealing their geometric nature and connection to Poincare group orbits.

Contribution

It introduces a novel variational principle for cylindrical curves representing spinning particles, including gauge invariance and explicit connection to Poincare group co-orbits.

Findings

Classical paths lie on a circular cylinder in Minkowski space.

Derived non-Lagrangian equations of motion with a variational interpretation.

Confirmed states lie on Poincare group co-orbits.

Abstract

We proceed from the fact that the classical paths of irreducible massive spinning particle lie on a circular cylinder with the time-like axis in Minkowski space. Assuming that all the classical paths on the cylinder are gauge-equivalent, we derive the equations of motion for the cylindrical curves. These equations are non-Lagrangian, but they admit interpretation in terms of the conditional extremum problem for a certain length functional in the class of paths subjected to the constant separation conditions. The unconditional variational principle is obtained after inclusion of constant separation conditions with the Lagrange multipliers into the action. We explicitly verify that the states of the obtained model lie on the co-orbit of the Poincare group. The relationship with the previously known theory is demonstrated.

Equations

$${\eta }_{\mu \nu }=\text{diag}(-1,1,1),\mu ,\nu =0,1,2.$$

$$(x-a{)}^2+(n,x{)}^2=r^2.$$

$$(n,x)=n_{\mu }{x}^{\mu },(x-a{)}^2=(x-a{)}_{\mu }(x-a{)}^{\mu }.$$

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

$$p=mn,J=m[a,n]-sn,$$

$$[n,a{]}_{\mu }={\epsilon }_{\mu \nu \rho }{n}^{\nu }{a}^{\rho },$$

$$p^2=-m^2,(p,J)=ms.$$

$$(\dot{x},x-y)=0,(x-y{)}^2=r^2.$$

$$[\dot{y},\ddot{y}]=0.$$

$$y=x-d.$$

$$(\dot{x},d)=0,d^2-r^2=0.$$

$$[\dot{x}-\dot{d},\ddot{x}-\ddot{d}]=0.$$

$$L[x(\tau ),d(\tau )]=-m\int \sqrt{-(\dot{x}-\dot{d}{)}^2}d\tau ,$$

$$S_0[x(\tau ),d(\tau )]=-m\int \sqrt{-(\dot{x}-\dot{d}+\frac{s}{m{r}^2}[d,\dot{d}]{)}^2}d\tau .$$

$$S[x(\tau ),d(\tau ),\alpha (\tau ),\beta (\tau )]=S_0[x(\tau ),d(\tau )]+\int \alpha (\dot{x},d)+\frac{\beta }{2}(d^2-r^2)d\tau .$$

$$\frac{\delta S}{\delta x}=-\frac{m}{(-(v,v){)}^{\frac{1}{2}}}\dot{v}-\frac{m(v,\dot{v})}{(-(v,v){)}^{\frac{3}{2}}}v-\alpha \dot{d}-\dot{\alpha }d=0,$$

$$\frac{\delta S}{\delta d}=\alpha \dot{x}+\beta d+\frac{m}{(-(v,v){)}^{\frac{1}{2}}}\dot{v}+\frac{m(v,\dot{v})}{(-(v,v){)}^{\frac{3}{2}}}v-\frac{s}{r^2}\frac{(v,\dot{v})}{(-(v,v){)}^{\frac{3}{2}}}[v,d]-\frac{s}{r^2}\frac{[\dot{v},d]+2[v,\dot{d}]}{(-(v,v){)}^{\frac{1}{2}}}=0,$$

$$\frac{\delta S}{\delta \alpha }=(\dot{x},d)=0,\frac{\delta S}{\delta \beta }=\frac{1}{2}(d^2-r^2)=0,$$

$$v\equiv \dot{x}-\dot{d}+\frac{s}{m{r}^2}[d,\dot{d}].$$

$$\alpha =0,\beta =\frac{2s}{r^4}\frac{(d,v,\dot{d})}{\sqrt{-(v,v)}}.$$

$$[v,\dot{v}]=0.$$

$$\delta x=[\omega ,x]+a,\delta d=[\omega ,d],\delta \alpha =0,\delta \beta =0,$$

$$p=\frac{mv}{\sqrt{-(v,v)}},J=\frac{m[x-d,v]-sv}{\sqrt{-(v,v)}}.$$

$$x(\tau )=(\kappa \tau ,r\cos \psi (\tau ),r\sin \psi (\tau )),d(\tau )=(0,r\cos \psi (\tau ),r\sin \psi (\tau )).$$

$$S({\tau }_1,{\tau }_0)=-m\kappa ({\tau }_1-{\tau }_0)-s\left(\psi ({\tau }_1)-\psi ({\tau }_0)\right).$$

$$r=\frac{s}{m}.$$

$$d=-\frac{s}{m}\frac{[n,\xi ]}{(n,\xi )},n=\frac{\dot{x}}{\sqrt{-{\dot{x}}^2}},\xi \equiv (1,\sin \phi ,\cos \phi ),$$

$$S[x(\tau),\varphi(\tau)]=-m\int\sqrt{-\dot{x}^{2}\bigg{(}1-2\frac{s}{m}\frac{\dot{\varphi}}{(\dot{x},\xi)}\bigg{)}}\,\,d\tau\,.$$

Full text

1. INTRODUCTION

The classical spinning particles are studied for decades starting from the paper of Frenkel [1]. The first seventy years of the development and corresponding bibliography are briefly summarized in [2]. The discussion of recent researches and references can be found in [3], [4].

The Kirillov-Kostant-Souriau [5], [6], [7] geometric quantization method tells us that the quantization of the classical spinning particle model corresponds to the irreducible representation of the Poincare group if the classical limit of the dynamical system is the co-adjoint orbit. The classical action functional is defined by the symplectic form on the co-orbit. The latter fact means that the momentum and total angular momentum are the only independent gauge invariant observables in the irreducible spinning particle theory. The admissible values of these quantities are restricted by the mass-shell and spin-shell constraints.

The irreducible spinning particle models are known in four-dimensional Minkowski and (A)dS spaces [8], [9], [10], [11]. For lower and higher dimensions we refer the articles [12], [13], [14]. In all these models the physical degrees of freedom involve besides particle’s position in space-time, the position in the ”internal space”. The structure of internal space depends on the model. In particular, the spinor, twistor, vector, and tensor spinning particle models are distinguished [2]. The common feature shared by the irreducible particle models is that the classical paths of free particle lie on a certain surface in space-time irrespectively to the specifics of the model.

The geometry of irreducible classical paths has been studied in [15]. In this work, it has been shown that the gauge equivalence class of particle’s paths necessarily forms a cylindrical surface in Minkowski space called the world sheet. In $d=3,4$, the world sheets are $2d$ circular cylinders with a time-like axis. The cylinder position is determined by the particle momentum and total angular momentum, while the representation defines the cylinder radius. The geometrical equations describing the motion of the particle along the world sheet are obtained for space-time dimension $d=3,4$ in the papers [15], [16]. These equations involve higher derivatives, and they are non-Lagrangian.

In the present paper, we construct the action functional for a massive spinning particle in $3d$ Minkowski space from the requirement that its paths are general cylindrical curves. We proceed as follows. First, the auxiliary variables are introduced to reduce the order of cylindrical curve equations. The obtained system of equations is non-Lagrangian, but it admits interpretation in terms of conditional extremum problem in the class of constant separation curves [17]. The non-variational dynamics is embedded into the variational one after inclusion of constant separation conditions with the Lagrange multipliers into the action. We explicitly verify the obtained action describes irreducible particle. At the final step, we establish the relationship with the previously known spinning model from the work [18].

2. CIRCULAR CYLINDERS AND CYLINDRICAL CURVES

Consider $3d$ Minkowski space with the local coordinates $x^{\mu}\,,\,\mu=0,1,2$ equipped with the metric with the mostly positive signature,

[TABLE]

Introduce the circular cylinder with time-like axis as the hypersurface defined by the equation

[TABLE]

Here, the round brackets denote the scalar products,

[TABLE]

All the tensor indices are raised and lowered by the metric. The real number $r$ is the cylinder radius. The parameters $n$, $a$ determine the cylinder position in space, and they meet the conditions

[TABLE]

The geometric meaning of $n,a$ is follows: $n$ is the normalized tangent vector to the cylinder axis, and $a$ connects the cylinder axis and the origin by the shortest path.

A cylindrical curve $x(\tau)$, with $\tau$ being the parameter on them, is a curve traced on a cylinder (2). The cylindrical curves are known to be the paths of irreducible massive spinning particles. Once the classical trajectory of the particle lies on the cylinder (2), the linear momentum $p$ and total angular momentum $J$ of the particle are determined by the formula from the paper [15],

[TABLE]

where $m,s$ are particle’s mass and spin. The square bracket denotes the cross product of two vectors,

[TABLE]

with $\varepsilon_{\mu\nu\rho}$ being the $3d$ Levi-Civita symbol. We use the convention $\varepsilon_{012}=1$ throughout the article. The quantities $p,J$ automatically satisfy the mass-shell and spin-shell constraints,

[TABLE]

This ensures the irreducibility of the particle moving the cylindrical path.

We want to describe the cylindrical curves by the ordinary differential equations. In our classification of cylindrical paths, we mostly follow [17]. We say that the curves $x(\tau)$, $y(\tau)$ have constant separation $r$ if

[TABLE]

Here, the dot denotes the derivative by $\tau$. Conditions (8) imply that $x(\tau)$ and $y(\tau)$ have a common normal of fixed length $r$ at any moment of time. The cylinder path $x(\tau)$ and its projection onto the cylinder axis $y(\tau)$ are particular examples of constant separation curves. Whenever $x(\tau)$ runs over the cylinder, the curve $y(\tau)$ follows a rectilinear path. So, we can select the cylindrical paths by imposing the differential equation on the curve $y(\tau)$:

[TABLE]

As it has been proven in [19], the differential equations (8), (9) are satisfied if and only if the curve $x(\tau)$ lies on a cylinder. The formal proof is quite long, and we do not repeat it here.

Let us now consider another form of equations (8), (9). Introduce the difference vector $d$ that connects the current particle position on the cylinder and cylinder axis by the shortest path,

[TABLE]

In terms of the vector $d$ and current particle’s position $x$, the conditions of constant separation (8) read

[TABLE]

The substitution (10) brings the differential equations (9) to the following form:

[TABLE]

We mostly work with equations (11), (12) because the dynamical variables $x$, $d$ have clear geometrical interpretation.

3. VARIATIONAL PRINCIPLE

In this section, we derive the variational principle for the equations (11), (12) of cylindrical curves.

We begin form geometric meaning of equations (11), (12). The condition (12) ensures that the projection of the cylindrical path on the cylinder axis is a straight line. The constant separation condition (11) selects the set of possible particle’s positions with one and the same projection on the axis. Being considered in itself, equation (12) comes from the least action principle for the functional

[TABLE]

while (11) are additional conditions. Here, the parameter $m$ with the mass dimension can be interpreted as the particle’s mass, but there is no room for spin. To introduce spin, we consider the more general series of action functionals parameterized by the parameter $s$,

[TABLE]

With account additional conditions (11), the extremals of the action are cylindrical curves. The sign under the root is chosen from the fact that the cylinder axis has a time-like tangent vector.

The specifics of the variational problem (14) is that the extremum of action is sought in the class paths subjected to the constant separation conditions (11). This setting is somewhat similar to the optimal control problem [19], where the certain functional is extremized in the class of trajectories that meet differential equations. To construct unconditional variational principle, the constraints (11) and the Lagrange multipliers $\alpha,\beta$ are added to the action (14),

[TABLE]

In principle, the inclusion of Lagrange multipliers with constraints into the action can lead to appearance of parasitic degrees of freedom. This does not happen in this model.

Let us see the equations (11), (12) come from the least action principle for the functional (15). The Euler-Lagrange equations read

[TABLE]

[TABLE]

[TABLE]

where the following notation is introduced:

[TABLE]

At first we note that the Lagrange multipliers $\alpha,\beta$ can be algebraically expressed from equations of motion (16), (17),

[TABLE]

This observation means that the Lagrange multipliers $\alpha,\beta$ are auxiliary variables, while all the physical dynamics is described by the variables $x,d$. Once the Lagrange multipliers are expressed, the remaining independent equations include (16), and constant separations conditions (18). From (16), it follows

[TABLE]

Equation (12) is a consequence of the definition of vector $v$ (19), and relations (18), (21). This means that the equations (11), (12) follow from the action functional (15).

Let us now discuss the interpretation of the parameters $m$, $s$ involved in the series of functionals (15). The action (15) is Poincare-invariant. The infinitesimal rotations and translations of Minkowski space induce a following transformations of dynamic variables:

[TABLE]

where $a,\omega$ are transformation parameters. The integrals of motion, which correspond to this symmetry, are the momentum $p$ and total angular momentum $J$,

[TABLE]

The mass-shell and spin-shell constraints (7) are satisfied by these quantities, as it is expected for the irreducible spinning particle with mass $m$ and spin $s$.

It is interesting to compare the action (15) for spinning and spinless particles. Consider the value of the classical action (15) in the special class of paths with $n=(1,0,0)$ and $a=(0,0,0)$. Modulo reparametrization, the general casual trajectory on such a cylinder read

[TABLE]

where $\kappa$ is a real constant having the sense of speed, and $\psi(\tau)$ is some function. The substitution of this path into action gives

[TABLE]

This expression is a sum of two terms. The first one is the length of particle’s path projection on the cylinder axis. The second one is angle of rotation of particle’s path around the cylinder. For spinless particle the length of the particle path projection is the only contribution in the action. Once spin is nonzero, the rotation angle is also included in the action functional. In $d=4$ Minkowski space, the similar dependence of the action on the mass and spin has been observed in [10].

4. COMPARISON WITH PREVIOUS RESULTS

Let us show that the previously known model of spinning particle from ref. [18] can be derived from the action principle (14). For reasons of simplicity, we consider only the case of isotropic spin vector,

[TABLE]

In this setting, the constant separation conditions (11) can be expressly solved by means of normalized isotropic vector $\xi$,

[TABLE]

where $\varphi$ is a new dynamical variable. Once the difference vector $d$ is expressed in terms of $x,\varphi$, the action functional (15) takes the previously known form,

[TABLE]

5. CONCLUSION

In the present work, the variational model of irreducible massive spinning particle is constructed from the requirement that the class of gauge equivalence of its paths forms a cylindrical surface in Minkowski space. The specifics of the problem is that both the spinning and spinless particles have one and same classical paths, while the difference between the models appears at the quantum level. To introduce spin, we propose the series of action functionals, which have one and the same Euler-Lagrange equations. The different representatives in the action functional series are not equivalent: the action of spinless particle is just the length of the projection of path onto the cylinder axis, while in the presence of spin the rotation of the trajectory around cylinder is taken into account. To our knowledge, the constructed spinning particle model is one of a few Lagrangian theories, in which different variational formulations are physically relevant. In the final part of the work, we showed that in the certain particular case our model is equivalent to the previously known one.

6. ACKNOWLEDGMENTS

We thank A.A. Sharapov for discussions and valuable comments. IAR expresses his gratitude to the organisers of the AYSS -2019 conference for their hospitality and support. This research was funded by state task of Ministry of Science and Higher Education of Russian Federation, grant number 3.9594.2017/8.9.