[Paper Review] N-Galilean conformal algebras and higher derivatives Lagrangians

Krzysztof Andrzejewski, Joanna Gonera|arXiv (Cornell University)|Sep 26, 2012

Black Holes and Theoretical Physics2 references3 citations

TL;DR

This paper demonstrates that the N-Galilean conformal algebra with odd N is the maximal symmetry algebra of a free Lagrangian involving $\frac{N+1}{2}$-th order time derivatives. Using Noether's theorem and infinitesimal symmetry analysis, the authors derive the full set of symmetry generators, showing they close under the N-Galilean conformal algebra, generalizing Niederer's result for the Schrödinger algebra to higher-derivative systems.

ABSTRACT

It is shown that the N-Galilean conformal algebra, with N-odd, is the maximal symmetry algebra of the free Lagrangian involving (N+1)/2-th order time derivative.

Motivation & Objective

• To determine the maximal continuous symmetry algebra of free Lagrangians with $\frac{N+1}{2}$-th order time derivatives for odd $N$.
• To generalize Niederer's characterization of the Schrödinger algebra as the maximal symmetry of free nonrelativistic dynamics to higher-derivative systems.
• To establish a dynamical realization of N-Galilean conformal algebras using higher-derivative Lagrangians and Noether symmetries.
• To clarify the role of central extensions in the symmetry algebra, showing they do not appear in the dynamical realization despite being allowed in the algebraic structure.

Proposed method

• Apply the Noether symmetry condition to the higher-derivative Lagrangian $L = \frac{m}{2} \left( \frac{d^n \vec{q}}{dt^n} \right)^2$, where $n = \frac{N+1}{2}$.
• Use infinitesimal transformations $t' = t + \epsilon \psi(t)$, $\vec{q}' = \vec{q} + \epsilon \vec{\phi}(\vec{q}, t)$ to derive the symmetry condition on the Lagrangian under time and coordinate transformations.
• Expand the transformed higher-derivative term $\frac{d^n \vec{q}'}{dt'^n}$ to first order in $\epsilon$, using the chain rule and time reparameterization.
• Derive the condition for invariance up to a total time derivative, leading to a system of differential equations for $\psi(t)$, $\vec{\phi}(\vec{q}, t)$, and the gauge function $f$.
• Solve the resulting equations by assuming polynomial forms for $\psi(t)$, $\vec{\phi}(\vec{q}, t)$, and $f$, leading to $\psi = \tau + \lambda t + c t^2$, $\vec{\phi} = \left(\frac{2n-1}{2}\right)(\lambda + 2c t)\vec{q} + \omega_{ab} q_b + \sum_{k=0}^n v_{ak} t^k$, and $f$ depending only on $\vec{q}^{(n-1)}$.
• Identify the differential operators corresponding to the symmetry generators: $H, D, K, \vec{J}, C_{ak}$, and verify they close under the N-Galilean conformal algebra with $N = 2n - 1$.

Experimental results

Research questions

• RQ1 What is the maximal continuous symmetry algebra of a free Lagrangian with $\frac{N+1}{2}$-th order time derivatives for odd $N$?
• RQ2 Can the N-Galilean conformal algebra (with odd $N$) be realized as a Noether symmetry algebra of such a higher-derivative Lagrangian?
• RQ3 How do the symmetry generators—time translations, dilatations, special conformal transformations, rotations, and boost-like symmetries—emerge from the Noether condition?
• RQ4 Why does the dynamical realization not include a central charge, despite the algebra admitting central extensions?
• RQ5 How does this result generalize Niederer's theorem for the Schrödinger algebra to higher-derivative systems?

Key findings

• The N-Galilean conformal algebra with odd $N$ is the maximal symmetry algebra of the free Lagrangian $L = \frac{m}{2} \left( \frac{d^n \vec{q}}{dt^n} \right)^2$, where $n = \frac{N+1}{2}$.
• The symmetry generators are realized as differential operators: $H = i\partial_t$, $D = -i\partial_t - i\frac{2n-1}{2} \vec{q} \cdot \partial_{\vec{q}}$, $K = i t^2 \partial_t + i(2n-1)t \vec{q} \cdot \partial_{\vec{q}}$, $\vec{J} = -i \vec{q} \times \partial_{\vec{q}}$, and $C_{ak} = i(-1)^k t^k \partial_{q_a}$.
• The symmetry algebra closes under the standard N-Galilean conformal algebra relations with $N = 2n - 1$, confirming the algebraic structure is realized dynamically.
• The central charge does not appear in the dynamical realization, even though the algebra admits central extensions, indicating it emerges only at the Hamiltonian level.
• The solution for the symmetry parameters requires $\psi(t)$ to be quadratic, $\vec{\phi}$ to be linear in $\vec{q}$ with time-dependent coefficients, and $\chi_a(t)$ to be a polynomial of degree $n$, leading to $n+1$ boost-like generators $C_{ak}$.
• The result generalizes Niederer's theorem: just as the Schrödinger algebra is the maximal symmetry of the standard free Lagrangian, the N-Galilean conformal algebra is the maximal symmetry of the higher-derivative free Lagrangian.