Why we need Lie algebras?

1. Jun 19, 2010

qiuhd

I know we need metric, curvature, what is Lie algebras used for? What is the physics meaning?

Last edited: Jun 19, 2010
2. Jun 19, 2010

czelaya

This may not be a suitable answer (well, I know it won’t). In general relativity, differential geometry is the major mathematical language. There exist mathematical structures within differential geometry called manifolds. Lie groups, in general, are continuous groups. Consequently, lie groups are continuous groups that exist on manifolds.

3. Jun 19, 2010

Naty1

4. Jun 19, 2010

bapowell

Lie groups are heavily used in particle physics, not so much in GR as far as I know. The generators of each Lie group form an algebra, called the Lie algebra, so the Lie algebra comes along for the ride. As continuous groups, Lie groups are the symmetry groups of the standard model of particle physics. The main Lie groups of interest are SO(N), U(N), SU(N) in the standard model, but more exotic groups crop up in string theory.

5. Jun 19, 2010

qiuhd

Thanks, most differential geometry and Riemann geometry textbooks contain Lie group material, I thought they have something to do with GR.

6. Jun 19, 2010

OB1

There is much more to Lie algebras than just differential geometry and Lie groups. Sure, you can study them as the tangent space to the identity of a Lie group, but they are a very complex algebraic structure with many applications in physics, and GR is arguably the branch of physics that relies on Lie theory the least. A few examples of Lie algebras being used in physics:
1. Hamiltonian mechanics: the so-called "phase space" equipped with the Poisson bracket is an example of a Lie algebra.
2. Quantum mechanics: the Heisenberg formulation of quantum mechanics. Heisenberg postulated the existence of an infinite-dimensional Lie algebra of operators. He then derived the state space and (together with Pauli) found the appropriate representations of this Lie algebra in L2 Hilbert space. That is why in quantum mechanics we have p= -i \hbar \nabla . The RHS is the Lie algebra representation of the momentum operator. Everything else in quantum mechanics can be shown to follow from properties of the Lie algebra (highest weight and irreducible representations, nondegeneracy of the Killing form, etc.). It's a very beautiful formulation and often simpler to use than Schrodinger's picture (although it is less intuitive, which is why it is not usually taught until graduate level quantum mechanics).
3. Theories of particle physics: supersymmetry, string theory, etc.. Supersymmetry uses a somewhat more general version of a Lie algebra, namely a Lie superalgebra, as the operator algebra. Bosonic string theory uses a Lie algebra called the Virasoro algebra to formulate operators (from which you can derive that Bosonic string theory requires 26 dimensions) and the state space. Superstring theory uses a Lie superalgebra to do the same.
In general, whenever you see commutation relations in physics, you can be sure there is a Lie algebra behind it. I once heard a mathematical physicist say that all of physics is just a series of Lie algebras (and superalgebras) and finding a good TOE is just about figuring out how they are all connected. I'm sure that was an oversimplification, but it gives you an idea of the importance of Lie algebras in physics.

7. Jun 19, 2010

qiuhd

Thanks, Can you give me a example of commutation relations in physics? and what is TOE?

8. Jun 19, 2010

OB1

For example, the canonical commutation relation in quantum mechanics: [x,p]=i hbar
TOE = theory of everything.

9. Jun 19, 2010

Mentz114

Lie algebras come into mechanics because the inifinitesimal contact transformations that respect the symmetries of the Hamiltonian are composed of generators of the invariance groups.

The Lie derivative in GR is an anti-commutating covariant derivative wrt a vector space which defines the inifinitesimal changes when a vector or tensor moves an infinitesimal amount in the direction of that vector space. The Killing vector equations may be written as the Lie derivative of the metric

$$\xi_{b;a}+\xi_{a;b}=L_{\xi}g_{ab}=0$$

where $L_{\xi}$ is the Lie derivative.

10. Jun 19, 2010

psholtz

Lie algebras are also important in the study of differential equations.