Conservation laws and stability of higher derivative extended Chern-Simons

TL;DR

This paper investigates the stability of higher derivative extended Chern-Simons theories, showing that bounded conserved quantities can ensure classical and quantum stability, with stability linked to unitary Poincaré representations.

Contribution

It introduces a general class of gauge invariant extended Chern-Simons theories with multiple conserved energy-momentum tensors and establishes stability criteria based on boundedness and unitarity.

Findings

Existence of conserved energy-momentum tensors in extended Chern-Simons theories.

Stability requires the $00$-component of the conserved tensor to be bounded.

Unstable theories correspond to nonunitary Poincaré representations.

Abstract

The higher derivative field theories are notorious for the stability problems both at classical and quantum level. Classical instability is connected with unboundedness of the canonical energy, while the unbounded energy spectrum leads to the quantum instability. For a wide class of higher derivative theories, including the extended Chern-Simons, other bounded conserved quantities which provide the stability can exist. The most general gauge invariant extended Chern-Simons theory of arbitrary finite order $n$ admits $(n - 1)$-parameter series of conserved energy-momentum tensors. If the $00$-component of the most general representative of this series is bounded, the theory is stable. The stability condition requires from the free extended Chern-Simons theory to describe the unitary reducible representation of the Poincar\'e group. The unstable theory corresponds to nonunitary representation.

Equations

$$S[x^i(t)]=\int dtL(x^i,{\dot{x}}^i,{\ddot{x}}^i),$$

$$\displaystyle E^{can}=\ddot{x}^{i}\frac{\partial L}{\partial\ddot{x}{}^{i}}+\dot{x}^{i}\Big{(}\frac{\partial L}{\partial\dot{x}^{i}}-\frac{d}{dt}\frac{\partial L}{\partial\ddot{x}^{i}}\Big{)}-L\,.$$

$$\displaystyle S[x(t)]=\int L(x,\dot{x},\ddot{x})dt\,,\qquad L(x,\dot{x},\ddot{x})=\frac{1}{2}\big{(}-\ddot{x}^{2}+(\omega_{1}^{2}+\omega_{2}^{2})\dot{x}^{2}-\omega_{1}^{2}\omega_{2}^{2}x^{2}\big{)}\,.$$

$${\omega }_1\ne {\omega }_2,{\omega }_1^2+{\omega }_2^2\ne 0.$$

$$\frac{\delta S}{\delta x}\equiv x^{(4)}+({\omega }_1^2+{\omega }_2^2)\ddot{x}+{\omega }_1^2{\omega }_2^2{x}^2=0.$$

$$x(t)=A_1\sin ({\omega }_{1}t+{\phi }_1)+A_2\sin ({\omega }_{2}t+{\phi }_2),$$

$$|x(t)|\le |A_1|+|A_2|.$$

$${\delta }_{\beta }x={\beta }_1\dot{x}+{\beta }_2\stackrel{...}{x},$$

$$J={\beta }_1{E}^{can}+{\beta }_2{J}^2,$$

$$\begin{array}[]{rl}\displaystyle E^{can}&\displaystyle=\Big{(}\dot{x}x^{(3)}-\frac{1}{2}\ddot{x}^{2}\Big{)}+\frac{1}{2}\Big{(}(\omega_{1}^{2}+\omega_{2}^{2})\dot{x}^{2}+\omega_{1}^{2}\omega_{2}^{2}x^{2}\Big{)}\,;\\[14.22636pt] \displaystyle J^{2}&\displaystyle=\frac{1}{2}\Big{(}\dddot{x}^{2}+(\omega_{1}^{2}+\omega_{2}^{2})\ddot{x}^{2}\Big{)}+\omega_{1}^{2}\omega_{2}^{2}\Big{(}x\ddot{x}-\frac{1}{2}\dot{x}^{2}\Big{)}\,.\end{array}$$

$$\begin{array}[]{c}\phantom{\frac{1}{2}}J_{1}=J^{2}+\omega_{1}^{2}E^{can}\,,\qquad J_{2}=J^{2}+\omega_{2}^{2}E^{can}\,,\phantom{\frac{1}{2}}\\[14.22636pt] \displaystyle\phantom{\frac{1}{2}}J_{i}=\frac{1}{2}\Big{(}(x^{(3)}+\omega_{i}^{2}\dot{x})^{2}+(\omega_{1}^{2}+\omega_{2}^{2}-\omega_{i}^{2})(\ddot{x}+\omega_{i}^{2}x)^{2}\Big{)}\,,\qquad i=1,2\,.\phantom{\frac{1}{2}}\end{array}$$

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

$$\displaystyle\phantom{\frac{1}{2}}S[A(x)]=\frac{m^{2}}{2}\sum_{p=1}^{n}\bigg{(}\alpha_{p}\int A_{\mu}(x)F^{(p)\mu}(x)d^{3}x\bigg{)}\,.\phantom{\frac{1}{2}}$$

$$F^{(p)}{}_{\mu }=m^{-p}{\epsilon }_{\mu \nu \rho }{\partial }^{\nu }{F}^{(p-1)\rho },F^{(0)}{}_{\mu }\equiv A_{\mu },r=1,\dots ,n,$$

$$\frac{\delta S}{\delta A}\equiv m^2\sum _{p=1}^n{\alpha }_p{F}^{(p)}=0.$$

$${\delta }_{\xi }{A}_{\mu }={\xi }^{\nu }{\partial }_{\nu }{A}_{\mu },$$

$$\displaystyle T^{can}_{\phantom{(}\mu\nu}(\alpha)=\frac{m^{2}}{2}\sum_{p=1}^{n}\sum_{r+s=p}\alpha_{p}\Big{(}F^{(r)}{}_{\mu}F^{(s)}{}_{\nu}+F^{(r)}{}_{\nu}F^{(s)}{}_{\mu}-\eta_{\mu\nu}\eta^{\rho\sigma}F^{(r)}{}_{\rho}F^{(s)}{}_{\sigma}\Big{)}\,.$$

$$T_{00}^{can}(\alpha )=\frac{m^2}{2}\sum _{\mu =0}^2\sum _{p=1}^n\sum _{r+s=p}{\alpha }_p{F}^{(r)}{}_{\mu }{F}^{(s)}{}_{\mu }.$$

$$T_{00}^{can}(\alpha )=\frac{m^2}{2}{\alpha }_n{F}^{(n-1)}{}_0{F}^{(1)}{}_0+\dots ,$$

$${\delta }_{\xi }{A}_{\mu }=\sum _{q=1}^{n-1}{\beta }_q{\xi }^{\nu }{\partial }_{\nu }{F}^{(q-1)}{}_{\mu },$$

$$\displaystyle T_{\mu\nu}(\alpha,\beta)=\sum\limits_{r,s=1}^{n-1}C_{r,s}(\alpha,\beta)\Big{(}F^{(r)}{}_{\mu}F^{(s)}{}_{\nu}+F^{(r)}{}_{\nu}F^{(s)}{}_{\mu}-\eta_{\mu\nu}\eta^{\rho\sigma}F^{(r)}{}_{\rho}F^{(s)}{}_{\sigma}\Big{)}\,.$$

$$\displaystyle C_{r,s}(\alpha,\beta)=\frac{\partial^{r+s}}{\partial^{r}z\,\partial^{s}z^{\prime}}\Big{(}\frac{M(z)N(z^{\prime})-M(z^{\prime})N(z)}{z-z^{\prime}}\Big{)}\Big{|}_{z=z^{\prime}=0}\,,$$

$$M(z)=\sum _{p=1}^n{\alpha }_p{z}^p,N(z)=\sum _{q=1}^{n-1}{\beta }_q{z}^q.$$

$$T_{\mu \nu }^{(q)}(\alpha )=\frac{\partial T_{\mu \nu }(\alpha ,\beta )}{\partial {\beta }_q},q=1,\dots ,n-1.$$

$$T_{00}(\alpha ,\beta )=\frac{m^2}{2}\sum _{\mu =0}^2\sum _{r,s=1}^{n-1}{C}_{r,s}(\alpha ,\beta )F^{(r)}{}_{\mu }{F}^{(s)}{}_{\mu }.$$

$$C_{r,s}(\alpha ,\beta )\text{is a positive definite matrix}.$$

$$M^{\prime }(z)=\sum _{q=0}^{n-1}{\alpha }_{q+1}{z}^q$$

$$\displaystyle S[A(x)]=\frac{1}{2}\int\Big{(}\alpha_{3}m^{-1}\varepsilon_{\mu\nu\rho}\partial^{\nu}G^{\rho}+\alpha_{2}G_{\mu}+\alpha_{1}mF_{\mu}\Big{)}\,,\quad\alpha_{3}\neq 0\,,$$

$$\displaystyle\phantom{\frac{1}{2}}G_{\mu}\equiv F^{(2)}{}_{\mu}=m^{-2}\big{(}\partial_{\mu}\partial^{\nu}A_{\nu}-\partial_{\nu}\partial^{\nu}A_{\mu}\big{)}\,,\quad F_{\mu}\equiv F^{(1)}{}_{\mu}=m^{-1}\varepsilon_{\mu\rho\nu}\partial^{\rho}A^{\nu}\,.\phantom{\frac{1}{2}}$$

$${\delta }_{\xi }{A}_{\mu }={\beta }_1{\xi }^{\nu }{\partial }_{\nu }{A}_{\mu }+{\beta }_2{\xi }^{\nu }{\partial }_{\nu }{F}_{\mu },$$

Full text

1 Introduction

Theories with higher derivatives are important in modern theoretical physics. The most known examples of such models are the Pais-Uhlenbeck (PU) oscillator [1], generalized Podolsky electrodynamics [2], modified theories of gravity [3, 4, 5, 6], and conformal higher spin theories [7]. Theories with higher derivatives have higher symmetry and better convergency properties at the classical and quantum level compared to their counterparts without higher derivatives, but also they are known for classical and quantum instability.

Stability is an important characteristic of dynamics. Classical stability means that the motions of theory are finite at every moment of time. For mechanical models it can be provided by existence of the Lyapunov function. For theories without higher derivatives the canonical energy plays the role of Lyapunov function. So, if the canonical energy is bounded, the theory is classically stable. Quantum stability means that quantum system has a well-defined vacuum state with the lowest possible energy. This type of stability is provided by the bounded Hamiltonian being the phase-space equivalent of the canonical energy.

Let us consider the stability problem of higher derivative theories. The theory with second time derivatives, described by the action functional

[TABLE]

have the following canonical energy:

[TABLE]

This expression is linear in $\dddot{x}^{i}$ , and so it is unbounded. The unbounded canonical energy cannot serve as a Lyapunov function of the system, and it cannot ensure classical stability of the model. The canonical Hamiltonian is also unbounded. So, the model is not stable at quantum level.

In [8], it was shown that a wide class of theories with higher derivatives admits, except for canonical energy, another conserved quantities. If some of these conserved quantities are bounded, they can ensure classical and quantum stability of the model. First such conservation laws were introduced for the Pais-Uhlenbeck oscillator [9, 10]. Later the stability of higher derivative theories was studied in [11, 12, 13, 14]. We will apply this idea to study the stability problem for the higher derivative extended Chern-Simons theory [15].

The rest of the paper is organized as follows. In Section 2, we consider the stability of the Pais-Uhlenbeck oscillator [1] from the viewpoint of existence of conservation law, which is different from the canonical energy. In Section 3, we analyze the stability condition for the extended Chern-Simons of arbitrary finite order. In Section 4, the general construction is exemplified by the third-order extension of Chern-Simons theory. In Conclusion we summarize the results.

2 Toy model

In this section, we illustrate how the idea of stabilizing dynamics by the conserved quantities works in the simplest higher derivative theory.

The PU oscillator of fourth order is a theory of a single dynamical variable $x(t)$ with the action functional

[TABLE]

Here, the parameters $\omega_{i},\,i=1,2,$ are the frequencies of oscillations. We assume that the frequencies are different and nonzero,

[TABLE]

The Euler-Lagrange equation for the model (3) has the form,

[TABLE]

The solution to this equation is the bi-harmonic oscillation,

[TABLE]

where the amplitudes $A_{i}$, and initial phases $\varphi_{i},\,i=1,2,$ are integration constants. The motion is finite,

[TABLE]

Thus, the Pais-Uhlenbeck oscillator is a stable model.

Let us explain the stability of the PU theory from the viewpoint of the conserved quantities of the model. There are two-parameter series of symmetries of the action functional (3),

[TABLE]

where $\beta_{1},\beta_{2}$ are infinitesimal transformation parameters, being constants. The first symmetry in the set is time translation. The second transformation in (8) is the higher symmetry. The two-parameter series of conserved quantities is associated with these symmetries by the Noether theorem:

[TABLE]

where $E^{can}$ is the canonical energy of the model, and $J^{2}$ is another independent integral of motion,

[TABLE]

Both of these quantities are unbounded quadratic forms of initial data $\dot{x},\ddot{x},\dddot{x},\ddddot{x}$, but they can be joined in two bounded combinations,

[TABLE]

Any bounded combination of these bounded quantities with positive coefficients is a positive-definite quadratic form of initial data.

The positive definite conserved quantity, being constructed from the integrals of motion (11), selects the stationary bounded surface in the phase-space of the theory. This bounded conserved quantity stabilizes the dynamics of the PU theory.

3 Extended Chern-Simons

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

[TABLE]

The extended Chern-Simons is a gauge theory of vector field $A=A_{\mu}(x)dx^{\mu}$ with the action functional

[TABLE]

Here, $m$ is a parameter with dimension of mass, and dimensionless real constants $\alpha_{1},\ldots,\alpha_{n}$ are parameters of the model. Without loss of generality we assume that $\alpha_{n}\neq 0$, and the notation is used:

[TABLE]

where $\varepsilon$ denotes the $3d$ Levi-Civita symbol with $\varepsilon_{012}=1$. The Euler-Lagrange equations for the action functional (13) have the form

[TABLE]

These equations involve the $n$-th time derivatives of $A$ .

The action (13) is Poincaré-invariant. The space-time translations are symmetries of the action functional (13),

[TABLE]

where $\xi$ is the transformation parameter. The canonical energy-momentum is associated with this symmetry,

[TABLE]

The $00$-component of the energy-momentum tensor has the form

[TABLE]

This quantity is linear in $F^{(n-1)}$ for $n>2$,

[TABLE]

the dots denote the terms that do not include $F^{(n-1)}$. Thus, the energy in the extended Chern-Simons theory is unbounded whenever the higher derivatives are included in the Lagrangian.

The series of higher symmetries, which generalize (16) of the action functional (13) has the form

[TABLE]

where $\beta_{q},\,q=1,\ldots,n-1,$ are the parameters of symmetry series, being real constants. The $(n-1)$-parameter series of conserved tensors is connected with this symmetry by the Noether theorem,

[TABLE]

The Bezout matrix $C_{r,s}(\alpha,\beta)$ of two polynomials is defined by the generating relation

[TABLE]

where $z$ and $z^{\prime}$ are two independent variables, and

[TABLE]

The representatives of the series (21) are defined by the formula

[TABLE]

By construction, $T^{(1)}_{\phantom{(}\mu\nu}\equiv T^{can}_{\phantom{(}\mu\nu}$, and other conserved tensors are independent.

The $00$-component of general conserved tensor (21) has the form

[TABLE]

This quantity is a quadratic form of the variables $F^{(r)}{}_{\mu}$. So, it is bounded if

[TABLE]

This condition is a restriction on the parameters $\beta$ in the series of energy-momentum tensors (21). It is consistent, iff the polynomial

[TABLE]

has simple and real roots. From the viewpoint of the representation theory, it means that the stability condition requires from the free extended Chern-Simons theory to describe the unitary reducible representation of the Poincaré group. If the roots of $M^{\prime}(z)$ are multiple or complex, there is no bounded integral of motion that can stabilize the dynamics.

4 Extended Chern-Simons of order 3

Let us demonstrate the general construction in the case $n=3$. The action functional of the model reads

[TABLE]

where the notation is used

[TABLE]

The model (28) is invariant under the following two-parameter series of symmetries:

[TABLE]

The corresponding two-parameter series of conserved tensors reads

[TABLE]

where

[TABLE]

is the canonical energy-momentum, and

[TABLE]

is another independent conserved tensor. The canonical energy is linear in $G_{\mu}$ and unbounded,

[TABLE]

The $00$-component of the general conserved tensor (31) has the form

[TABLE]

It can be bounded. The boundedness condition for the quadratic form reads

[TABLE]

It is consistent if parameters of the model (28) satisfy the condition

[TABLE]

According to the representation theory, the stable theory corresponds to one of the two cases: theory of two self-dual massive spin 1 with different masses, or theory of massless spin 1 and massive spin 1 subject to a self-duality condition [16, 17].

5 Conclusion

We considered a class of vector field models whose wave operator is a polynomial in the Chern-Simons operator. We demonstrated that the gauge theory of order $n$ admits $(n-1)$-parameter series of conserved tensors, whose $00$-component can be bounded, while the canonical energy is always unbounded for $n>2$. The bounded conservation laws ensure the stability of dynamics at classical level. At quantum level the stability is provided by bounded Hamiltonian. The constrained Hamiltonian formulations with bounded Hamiltonian of the extended Chern-Simons were constructed in [18, 19].

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