Higher-derivative generalization of conformal mechanics

TL;DR

This paper develops higher-derivative extensions of conformal mechanics models and derives their Newton-Hooke counterparts through coordinate transformations, expanding the theoretical framework of conformal systems.

Contribution

It introduces higher-derivative generalizations of conformal particle and many-body mechanics, and constructs their Newton-Hooke analogues via coordinate transformations.

Findings

Higher-derivative conformal models are explicitly constructed.

Newton-Hooke counterparts are obtained through transformations.

The work broadens the theoretical landscape of conformal mechanics.

Abstract

Higher-derivative analogues of multidimensional conformal particle and many-body conformal mechanics are constructed. Their Newton-Hooke counterparts are derived by applying appropriate coordinate transformations.

Equations

$$S=\frac{1}{2}\int dt{\lambda }_{ij}{x}_i{x}_j^{(2l+1)},$$

$$\displaystyle\lambda_{ij}=\left\{\begin{aligned} &\delta_{ij},&&\quad i,j=1,2,..,d,\quad\mbox{for half-integer $l$;}\\[2.0pt] &\epsilon_{ij}=-\epsilon_{ji},&&\quad i,j=1,2,\qquad\quad\qquad\mbox{for integer $l$},\end{aligned}\right.$$

$$t^{\prime }=t+a+b{t}^2+ct,{x^{\prime }}_i(t^{\prime })=x_i(t)+\sum _{k=0}^{2l}{a}_i^{(k)}{t}^k+lc{x}_i(t)+2lbt{x}_i(t)-{\omega }_{ij}{x}_j(t),$$

$$t=R\tan (\tau /R),x_i(t)=\frac{{\chi }_i(\tau )}{{\cos }^{2l}(\tau /R)},$$

$$S=\frac{1}{2}\int d\tau L_{PU}=\frac{1}{2}\int d\tau {\chi }_i\prod _{k=0}^{l-1/2}\left(\frac{d^2}{d{\tau }^2}+\frac{(2k+1{)}^2}{R^2}\right){\chi }_i,$$

$$S=\frac{1}{2}\int d\tau L_{PU}=\frac{1}{2}\int d\tau {ϵ}_{ij}{\chi }_i\prod _{k=1}^l\left(\frac{d^2}{d{\tau }^2}+\frac{(2k{)}^2}{R^2}\right){\dot{\chi }}_j.$$

$$\begin{array}{rl}& {\tau }^{\prime }=\tau +a+\frac{cR}{2}\sin \frac{2\tau }{R}+b{R}^2{\sin }^2\frac{\tau }{R},\\ & {{\chi }^{\prime }}_i({\tau }^{\prime })={\chi }_i(\tau )+lc\cos \frac{2\tau }{R}{\chi }_i(\tau )+lbR\sin \frac{2\tau }{R}{\chi }_i(\tau )+\sum _{k=0}^{2l}{R}^k{\sin }^k\frac{\tau }{R}{\cos }^{2l-k}\frac{\tau }{R}{a}_i^{(k)}-{\omega }_{ij}{\chi }_j(\tau ),\end{array}$$

$$S=\frac{1}{2}\int dt\left({\dot{x}}_i{\dot{x}}_i-\frac{g}{x_i{x}_i}\right).$$

$$t^{\prime }=t+a+ct+b{t}^2,{x^{\prime }}_i(t^{\prime })=x_i(t)+\frac{c}{2}{x}_i(t)+bt{x}_i(t)-{\omega }_{ij}{x}_j(t),{\omega }_{ij}=-{\omega }_{ji},$$

$$t^{\prime }=t+a+ct+b{t}^2,{x^{\prime }}_i(t^{\prime })=x_i(t)+lc{x}_i(t)+2lbt{x}_i(t)-{\omega }_{ij}{x}_j(t),{\omega }_{ij}=-{\omega }_{ji}.$$

$$S=\frac{1}{2}\int dt\left({\lambda }_{ij}{x}_i{x}_j^{(2l+1)}-V(x)\right)$$

$$V(x)=\frac{g}{(x_i{x}_i{)}^{1/2l}}.$$

$$S=\frac{1}{2}\int dt\left({\lambda }_{ij}{x}_i{x}_j^{(2l+1)}-\frac{g}{(x_i{x}_i{)}^{1/2l}}\right)$$

$${\lambda }_{ij}{x}_j^{(2l+1)}=-\frac{g}{2l}\frac{x_i}{(x_j{x}_j{)}^{(2l+1)/2l}}.$$

$$G=G(t,z,u)=e^{itH}{e}^{izK}{e}^{iuD},[H,D]=iH,[H,K]=2iD,[D,K]=iK.$$

$$\delta t=a+b{t}^2+ct,\delta z=b(1-2tz)-cz,\delta u=c+2bt,$$

$$G^{-1}dG=i({\omega }_{H}H+{\omega }_{D}D+{\omega }_{K}K),$$

$${\omega }_H=e^{-u}dt,{\omega }_D=du-2zdt,{\omega }_K=e^u(dz+z^{2}dt).$$

$$\rho =e^{lu}.$$

$${\omega }_D=0\Rightarrow z=\frac{1}{2l}\frac{\dot{\rho }}{\rho }.$$

$$\delta t=a+b{t}^2+ct,\delta \rho =lc\rho +2lbt\rho .$$

$$\Omega =\frac{{\omega }_K}{{\omega }_H}=\frac{1}{2l}{\rho }^{\frac{2}{l}}\left(\frac{\ddot{\rho }}{\rho }-\frac{2l-1}{2l}\frac{{\dot{\rho }}^2}{{\rho }^2}\right)$$

$$D={\rho }^{\frac{1}{l}}\frac{d}{dt},$$

$$\begin{array}{rlrl}& l=1:& & D\Omega =-\frac{g}{4}\Rightarrow \stackrel{...}{\rho }=-\frac{g}{2{\rho }^2};\\ & l=\frac{3}{2}:& & D^2\Omega +3{\Omega }^2=-\frac{g}{9}\Rightarrow {\rho }^{(4)}=-\frac{g}{3{\rho }^{5/3}};\\ & l=2:& & D^3\Omega +16\Omega D\Omega =-\frac{g}{16}\Rightarrow {\rho }^{(5)}=-\frac{g}{4{\rho }^{3/2}};\\ & l=\frac{5}{2}:& & D^4\Omega +31\Omega D^2\Omega +26(D\Omega {)}^2+45{\Omega }^3=-\frac{g}{25}\Rightarrow {\rho }^{(6)}=-\frac{g}{5{\rho }^{7/5}}.\end{array}$$

$$S=\frac{1}{2}\int d\tau \left(L_{PU}-\frac{g}{({\chi }_i{\chi }_i{)}^{1/2l}}\right),$$

$$S=\frac{1}{2}\int d\tau \left({\dot{\chi }}_i{\dot{\chi }}_i-\frac{1}{R^2}{\chi }_i{\chi }_i-\frac{g}{{\chi }_i{\chi }_i}\right).$$

$$S=\frac{1}{2}\int dt\left(\sum _{\alpha =1}^N{\dot{x}}_{\alpha ,i}{\dot{x}}_{\alpha ,i}-V_{\frac{1}{2}}(x_{1,i},x_{2,i},..,x_{N,i})\right)$$

$$t^{\prime }=t+a+b{t}^2+ct,{x^{\prime }}_{\alpha ,i}(t^{\prime })=x_{\alpha ,i}(t)+a_i^{(0)}+a_i^{(1)}t+\frac{c}{2}{x}_{\alpha ,i}(t)+bt{x}_{\alpha ,i}(t)-{\omega }_{ij}{x}_{\alpha ,j}(t),$$

$$\sum _{\alpha =1}^N\frac{\partial V_{\frac{1}{2}}}{\partial x_{\alpha ,i}}=0,\sum _{\alpha =1}^N{x}_{\alpha ,i}\frac{\partial V_{\frac{1}{2}}}{\partial x_{\alpha ,i}}+2V_{\frac{1}{2}}=0,\sum _{\alpha =1}^N\left(x_{\alpha ,i}\frac{\partial V_{\frac{1}{2}}}{\partial x_{\alpha ,j}}-x_{\alpha ,j}\frac{\partial V_{\frac{1}{2}}}{\partial x_{\alpha ,i}}\right)=0.$$

$$S=\frac{1}{2}\int dt\left(\sum _{\alpha =1}^N{\lambda }_{ij}{x}_{\alpha ,i}{x}_{\alpha ,j}^{(2l+1)}-V_l(x_{1,i},x_{2,i},..,x_{N,i})\right).$$

Full text

conformal symmetry, conformal Galilei algebra, Pais-Uhlenbeck oscillator

1. Introduction

Conformal algebra in one dimension $so(1,2)$ involves three generators: the generator of time translations $H$, the generator of dilatations $D$, and the generator of special conformal transformations $K$. There are many mechanical systems which exhibit such a symmetry. These include the model of free particle Niederer_1 , conformal particle Alfaro , harmonic oscillator Niederer , the system of identical particles which interact with each other by conformal potential (e.g. the Calogero model Calogero ) and others.

There are several reasons why conformal invariant mechanical systems have attracted interest over the last four decades. First, some of these models are (super)integrable. Second, there are conformal invariant systems which are of physical interest. In particular, some of these models appear in the context of condensed matter physics Johnson , in physics of black holes Gibbons . Third, recent proposals on the nonrelativistic version of the AdS/CFT-correspondence stimulate investigations of nonrelativistic conformal algebras and their dynamical realizations.

There are two families of nonrelativistic algebras which contain $so(1,2)$ as a subalgebra. The first family involves conformal extensions of the Galilei algebra which are parameterized by a positive integer or half-integer parameter $l$ Horvathy ; Henkel ; Negro_1 ; Negro_2 ; Lukierski . This is the reason why the members of this family are called $l$-conformal Galilei algebras.

The $l$-conformal extensions of the Newton-Hooke (NH) algebra form the second family of nonrelativistic conformal algebras Negro_1 ; Galajinsky_3 . The $l$-conformal NH algebra can be viewed as an analogue of the $l$-conformal Galilei algebra in the presence of universal cosmological repulsion or attraction.

As is well known, the $l$-conformal extensions of the Galilei algebra and the NH algebra are isomorphic. Indeed, the structure relations of the $l$-conformal NH algebra can be obtained by a change of the basis $H\rightarrow H\pm\frac{1}{R^{2}}K$ in the $l$-conformal Galilei algebra, where $\Lambda=\pm\frac{1}{R^{2}}$ is the nonrelativistic cosmological constant. But when dynamical realizations are considered, this change of the basis leads to a change of the Hamiltonian and consequently alters the dynamics. For example, the Schrödinger group ($l=1/2$-conformal Galilei group) is the maximal kinematical group of the free particle Niederer_1 . At the same time, the NH counterpart of this model is the harmonic oscillator Niederer .

The free higher-derivative particle DH ; Gomis and the Pais-Uhlenbeck (PU) oscillator Pais can be viewed as higher-derivative analogues of the free particle and the harmonic oscillator, respectively. Recently, symmetries of these higher-derivative models have been extensively studied DH ; Gomis ; Galajinsky_1 ; AG ; PU ; Andrzejewski ; Andrzejewski_1 ; Masterov_2 ; AB ; Andr . In particular it has been shown that the $l$-conformal Galilei group is the maximal symmetry group of the free $(2l+1)$-order particle Gomis . Similarly, the PU oscillator accommodates the $l$-conformal NH symmetry for a special choice of its oscillation frequencies Galajinsky_1 ; PU ; Masterov_2 .

At the same time, higher-derivative analogues of other nonrelativistic conformal invariant mechanical systems are unknown. The purpose of the present work is to construct higher-derivative analogues of the conformal particle and a system of identical particles which interact with each other via a conformal-invariant potential.

The paper is organized as follows. In Sect. 2, we review the free higher-derivative particle and the PU oscillator which exhibits the $l$-conformal NH symmetry. In Section 3 and Section 4, we construct higher-derivative analogues of the conformal particle and many-body conformal mechanics, respectively. We summarize our results and discuss further possible developments in the conclusion (Section 5). Throughout the work summation over repeated spatial indices is understood. A superscript in braces as well as the number of dots over spatial coordinates designate the number of derivatives with respect to time.

2. Free higher-derivative particle and the Pais-Uhlenbeck oscillator

Let us review certain useful facts about the free higher-derivative particle and the PU oscillator which enjoys the $l$-conformal NH symmetry. The action functional of the former model reads Gomis

[TABLE]

where dimensionless parameter $l$ can take positive integer or half-integer values and

[TABLE]

with $\epsilon_{12}=1$. The action functional (1) is invariant under the transformations Gomis

[TABLE]

where $a$, $b$, $c$, $a_{i}^{(k)}$, and $\omega_{ij}$ are infinitesimal parameters. Generators of these transformations form the $l$-conformal Galilei algebra Henkel ; Negro_1 ; Negro_2 .

The NH counterpart of the model (1) can be constructed by applying the Niederer transformation Niederer

[TABLE]

where $R$ is a constant which has the dimension of time. For half-integer $l$, the implementation of this transformation to (1) results in the action functional Galajinsky_1 ; PU

[TABLE]

while for integer $l$ one has Masterov_2

[TABLE]

The actions functionals (3) and (4) describe the $(2l+1)$-order PU oscillator for a particular choice of its oscillation frequencies. This model is invariant under the transformations Galajinsky_1 ; PU ; Masterov_2

[TABLE]

whose generators form the $l$-conformal Newton-Hooke algebra Negro_1 ; Negro_2 ; Galajinsky_3 .

The NH counterparts presented in this paper all correspond to the case of negative cosmological constant. The case of positive cosmological constant can be straightforwardly reproduced by a formal change $R\rightarrow iR$.

3. Higher-derivative analogue of conformal particle

Let us consider a multidimensional conformal particle whose action functional has the form Alfaro

[TABLE]

This action is invariant under transformations

[TABLE]

whose generators form $so(1,2)\oplus so(d)$ subalgebra of the $l=1/2$-conformal Galilei algebra. For the model of a free higher-derivative particle (1), the same subalgebra is realized by the generators which produce the following transformations

[TABLE]

Taking into account (1), let us consider the action functional of the form

[TABLE]

with an arbitrary function $V=V(x)$. To obtain higher-derivative generalization of the model (6), let us require this action functional to be invariant under the transformations (7). This restriction is satisfied when the function $V=V(x)$ has the form

[TABLE]

So, the model

[TABLE]

can be viewed as a higher-derivative generalization of conformal particle (6). The dynamics of this system is governed by the following equations of motion

[TABLE]

It is interesting to note that a one-dimensional analogue of this dynamical equation can be obtained via the method of nonlinear realization Coleman_1 ; Coleman_2 ; Ivanov . Indeed, let us consider the exponential parametrization of the group $SO(1,2)$ Ivanov_1

[TABLE]

Left multiplication by a group element $G(a,b,c)$ produces the following infinitesimal coordinate transformations

[TABLE]

where $a$, $b$, and $c$ are infinitesimal parameters. Then one constructs the left-invariant Maurer-Cartan one-forms Ivanov_1

[TABLE]

where we denoted

[TABLE]

To obtain higher-derivative dynamical realization of $SO(1,2)$ group, firstly let us introduce the new variable

[TABLE]

During next step, we discard the variable $z$ from our consideration with the aid of the constraint

[TABLE]

Considering this relation, the transformations (10) may be rewritten as

[TABLE]

One may obtain $SO(1,2)$-invariant higher-derivative equations by using both the function

[TABLE]

and the differential operator

[TABLE]

which are invariant under the transformations (11). In particular, one-dimensional analogue of (9) for $l=1,\frac{3}{2},2,\frac{5}{2}$ can be reproduced as follows

[TABLE]

It should be noted that the NH counterpart of the model (8) can be obtained via a Niederer transformation (2). The action functional of this counterpart has the form

[TABLE]

where $L_{PU}$ is defined in (3) and (4). This model holds invariant under $SO(1,2)\oplus SO(d)$-subgroup of transformations (5).

The system (13) can be also viewed as a higher-derivative analogue of the conformal particle in a harmonic trap

[TABLE]

This action functional can be obtained by applying (2) to (6).

4. Higher-derivative analogue of many-body conformal mechanics

A system of $N$ identical particle whose dynamics is governed by the action functional

[TABLE]

may exhibit invariance under transformations

[TABLE]

whose generators form the Schrödinger algebra when the function $V_{\frac{1}{2}}=V_{\frac{1}{2}}(x_{1,i},x_{2,i},..,x_{N,i})$ satisfies the following equations

[TABLE]

To construct the higher-derivative analogue of this model, let us consider the following action functional:

[TABLE]

This action is invariant under the transformations

[TABLE]

when the function $V_{l}=V_{l}(x_{1,i},x_{2,i},..,x_{N,i})$ obeys the following equations

[TABLE]

It is easy to see that there is a correspondence between potentials $V_{\frac{1}{2}}$ and solutions of the system (17). Indeed, let us suppose that we have a solution $V_{\frac{1}{2}}$ of the system (15). Then we can produce a solution

[TABLE]

of the system (17) for any possible value of $l$. It should be noted that the potential $V_{\frac{1}{2}}$ should be positive-definite in order to obtain a solution $V_{l}$ for integer $l$. For example, the potential

[TABLE]

is related to the celebrated Calogero model Calogero .

To obtain NH counterpart of the model (16), it follows to apply Niederer’s coordinate transformation of the form

[TABLE]

to the action functional (16). For half-integer $l$ one finds

[TABLE]

while for integer $l$ one obtains

[TABLE]

where the function $V_{l}=V_{l}(\chi_{1,i},\chi_{2,i},..,\chi_{N,i})$ obeys the same conditions as in (17)

[TABLE]

The actions (19) and (20) describe a set of identical Pais-Uhlenbeck oscillators which interact with each other via a conformal-invariant potential. This system can be viewed as a higher-derivative generalization of many-body conformal mechanics considered in Galajinsky_8 ; Galajinsky_9 .

5. Conclusion

To summarize, in this work higher-derivative analogues of the conformal particle and a system of identical particles which interact with each other via a conformal-invariant potential were constructed. An appropriate Niederer’s coordinate transformation was applied to these higher-derivative systems so as to obtain their NH counterparts.

Turning to further possible developments, the most interesting questions are related to integrability and stability. A construction of supersymmetric extensions of higher-derivative models (8), (13), (16), (19), and (20) is worth studying as well. It would be also interesting to extend the analysis in Refs. Galajinsky_9 ; Galajinsky_5 ; Galajinsky_6 ; Galajinsky_7 , which is related to nonlocal conformal transformations, to the case of higher-derivative mechanical systems introduced in this paper.