What is the physical meaning of a Lie algebra symmetry? - MathOverflow

What is the physical meaning of a Lie algebra symmetry?

2010-05-17T05:19:22Z

The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that $H$ is a representation of some Lie group $G$. So you want to understand this Lie group $G$, and generally you do it by looking at its Lie algebra. At least initially, this is understood as a way to make the problem of classifying Lie groups and their representations easier.

But there are Lie algebras which are not the Lie algebra of a Lie group, and people are still interested in them. One possible way to justify this perspective from a physical point of view is that Lie algebras might still be viewable as ("infinitesimal"?) symmetries of physical systems in some sense. However, I have never seen a precise statement of how this works (and maybe I just haven't read carefully enough). Can anyone enlighten me?

Answer by José Figueroa-O'Farrill for What is the physical meaning of a Lie algebra symmetry?

2010-05-17T08:15:11Z

I'm not sure which sort of examples of Lie algebras without the corresponding groups you have in mind, but here is a typical example from Physics.

Many physical systems can be described in a hamiltonian formalism. The geometric data is usually a symplectic manifold $(M,\omega)$ and a smooth function $H: M \to \mathbb{R}$ called the hamiltonian. If $f \in C^\infty(M)$ is any smooth function, let $X_f$ denote the vector field such that $i_{X_f}\omega = df$. If $f,g \in C^\infty(M)$ we define their Poisson bracket $$\lbrace f, g\rbrace = X_f(g).$$ It defines a Lie algebra structure on $C^\infty(M)$. (In fact, a Poisson algebra structure once we take the commutative multiplication of functions into account.)

In this context one works with the Lie algebra $C^\infty(M)$ (or particular Lie subalgebras thereof) and not with the corresponding Lie groups, should they even exist.

Symmetries in this context are functions which Poisson commute with the hamiltonian, hence the centraliser of $H$ in $C^\infty(M)$. They define a Lie subalgebra of $C^\infty(M)$.

Added

Another famous example occurs in two-dimensional conformal field theory. For example, the Lie algebra of conformal transformations of the Riemann sphere is infinite-dimensional: any holomorphic or antiholomorphic function defines an infinitesimal conformal transformation. On the other hand, the group of conformal transformations is finite-dimensional and isomorphic to $\mathrm{PSL}(2,\mathbb{C})$.

Answer by David Bar Moshe for What is the physical meaning of a Lie algebra symmetry?

2010-05-18T11:11:32Z

I would like to provide you with a non-trivial example (and a reference) of a Lie algebra of symmetries which is not a Lie algebra of a Lie group within the framework of conventional quantum mechanics. This example relies on the notion of "dynamical groups" (which you can find a lot of literature about in the net). I think that the most precise definition of a dynamical group of a quantum system would be a Lie group which the system's phase space is a coadjoint orbit of. In the cases known to me the system's Hamiltonian belongs to the Lie algebra of the dynamical group, but I don't think that this is an essential requirement (The Hamiltonian can be a member of the universal enveloping algebra). The main application of dynamical groups is to provide algebraic solution to the quantum mechanical problem. The spectrum of the systems can be obtained from the representation of the dynamical group associated with the coadjoint orbit instead of solving the Shroedinger equation.

The Lie algebra of the dynamical group generally consists of the usual space time symmetries such as the angular momentum su(2) in addition to more generators which originally required a lot of ingenuity to come up with, such as the Runge-lenz generators of the Hydrogen atom problem, sometimes referred to as the Kepler problem,which close together with the angular momentum generators to o(4) for elliptical motion and o(3,1) for hyperbolic motion.

Another known example in which a dynamical group formulation is used is the harmonic oscillator with the dynamical group SU(1,1).

Returning to the Kepler problem. The treatment described above considers only a fixed energy subspaces. In the following article , by C. Dabul, J. Dabul, P. Slodowy, a twisted Kac-Moody dynamical algebra is constructed which is a simultaneous dynamical Lie algebra of the full problem (corresponding to elliptic, parabolic and hyperbolic trajectories). This is an example of a dynamical Lie algebra of symmetries which is not a Lie algebra of a Lie group.

Regrettably, I didn't see a followup of this work in terms of algebraic solution of the full Kepler problem in terms of representations of this algebra, nor a treatment of this problem in terms of coadjoint orbits of (the non-Lie) Kac-Moody groups which correspond to this algebra and are subjects of active research. I think that these would be interesting research problems.