We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell's equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.

Lie groups play an important role in physical systems both as phase spaces and as symmetry groups. Infinite-dimensional Lie groups occur in the study of dynamical systems with an infinite number of degrees of freedom such as PDEs and in field theories. For such infinite-dimensional dynamical systems, diffeomorphism groups and various extensions and variations thereof, such as gauge groups, loop groups, and groups of Fourier integral operators, occur as symmetry groups and phase spaces. Symmetries are fundamental for Hamiltonian systems. They provide conservation laws (Noether currents) and reduce the number of degrees of freedom, that is, the dimension of the phase space.

The topics selected for review aim to illustrate some of the ways infinite-dimensional geometry and global analysis can be used in mathematical problems of physical interest. The topics selected are the following.

Infinite-Dimensional Lie Groups.

Lie Groups as Symmetry Groups of Hamiltonian Systems.

Applications.

Gauge Theories, the Standard Model, and Gravity.

SUSY (supersymmetry).

A general theory of infinite-dimensional Lie groups is hardly developed. Even Bourbaki [

An infinite-dimensional

The

These definitions in infinite dimensions are identical with the definitions in finite dimensions. The big difference although is that infinite-dimensional manifolds, hence Lie groups, are

Infinite-dimensional Lie groups are

There is NO Implicit Function Theorem or Inverse Function Theorem in infinite dimensions! (except Nash-Moser-type theorems).

If

If

If

If

If

If

Some classical examples of

Let

Let

Let

Let

We generalize the Abelian example (see Section

Applications of these infinite-dimensional Lie groups are in gauge theories and quantum field theory, where they appear as groups of gauge transformations. We will discuss these in Section

As a special case of example mentioned in Section

Certain subgroups of loop groups play an important role in quantum field theory as groups of gauge transformations. We will discuss these in Section

Among the most important “classical” infinite-dimensional Lie groups are the diffeomorphism groups of manifolds. Their differential structure is not the one of a Banach Lie group as defined above. Nevertheless they have important applications.

Let

The

The exponential map on the diffeomorphism group is given as follows. For any vector field

We see that the diffeomorphism groups are not Lie groups in the classical sense, but what we call

Several subgroups of

Let

We cannot apply the finite-dimensional theorem that if

Nevertheless

Let

The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups which are not only ILH-Lie groups but actually Hilbert-Lie groups.

Let

The

Let

The Lie algebra

On the other hand, the Lie algebra of

We have three versions of gauge groups:

On the other hand, for

To topologize

There is a natural exponential map

We have the following theorem (Schmid [

For

See [

A short introduction and “crash course” to geometric mechanics can be found in the studies by Abraham and Marsden [

A

The notion of Poisson manifolds was rediscovered many times under different names, starting with Lie, Dirac, Pauli, and others. The name

For any

Poisson manifolds are a generalization of symplectic manifolds on which Hamilton’s equations have a canonical formulated.

For finite-dimensional classical mechanics we take

As a concrete example we consider the harmonic oscillator. Here

Let

As a special case in finite dimensions, if

As a concrete example we consider the wave equations. Let

The finite-dimensional examples of Poisson brackets (

All the examples above are special cases of symplectic manifolds

(A) If

(B) If the Poisson bracket

Not all Poisson brackets are of the form given in the above examples (

A concrete example of the Lie-Poisson bracket is given by the rigid body. Here

The examples we have discussed so far are all canonical examples of Poisson brackets, defined either on a symplectic manifold

If a Hamiltonian system

For a Hamiltonian action of a Lie group

The rigid body discussed above can be viewed as an example of this reduction theorem. If

See [

We now discuss some infinite-dimensional examples of reduced Hamiltonian systems.

Maxwell’s equations of electromagnetism are a reduced Hamiltonian system with the Lie group

Let

Then Hamilton’s equations in the canonical variables

Euler’s equations for an incompressible fluid

The Maxwell-Vlasov’s equations are a reduced Hamiltonian system on a more complicated reduced space. See the study by Marsden et al. in [

Maxwell-Vlasov’s equations for a plasma density

More complicated plasma models are formulated as Hamiltonian systems. For example, for the two-fluid model the phase space is a coadjoint orbit of the semidirect product (

There are many known examples of PDEs which are infinite-dimensional Hamiltonian systems, such as the Benjamin-Ono, Boussinesq, Harry Dym, KdV, KP equations, and others. In many cases the Poisson structures and Hamiltonians are given ad hoc on a formal level. We illustrate this with the KdV equation, where at least one of the three known Hamiltonian structures is well understood [

The Korteweg-deVries (KdV) equation

The question is where this Poisson bracket (

A Fourier integral operators on a compact manifold

Both groups

For the KdV equation we take the special case where

See the study by Adams et al. in [

See [

Here we will encounter various infinite-dimensional Lie groups and algebras such as diffeomorphism groups, loop groups, groups of gauge transformations, and their cohomologies.

Consider a principal

In pure Yang-Mills theory the action functional is given by

In gauge theories the symmetry group is the group of gauge transformations. The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups, as discussed in Section

The group

We only sketch here what role this infinite-dimensional gauge group

The gauge group

The action functional (the Yang-Mills functional) is

For self-dual connections

Let

In classical field theory, one considers a Lagrangian

In QED and QCD the Lagrangian is more complicated of the form

The variational principle of the Lagrangian (

In the free case, that is, when

For

The chiral symmetry is the symmetry that leads to anomalies and the BRST invariance. In QCD the chiral symmetry of the Fermi field

This conservation law breaks down after quantization; one gets

The quantization is given by the Feynman path integral:

We can write this globally as

The effective Lagrangian

Named after Becchi et al. [

Let

For

For

The chiral anomaly

We are now going to explain the previous theorem, in particular the general definition of the Chevalley-Eilenberg [

Let

The Noether current induced by the chiral symmetry (after quantization) for the free case (

Note the similarity with the Chern-Simon Lagrangian

We are going to derive a representation of the chiral anomaly

The question is “if

Let

The form

We have an explicit form of the anomaly in

The standard model is a Yang-Mills gauge theory. Recall that the free

For different choices of the gauge Lie group

For interactions, all the relevant fields involved can be considered as sections of corresponding associated vector bundles induced by representations of the gauge groups, for example, the Dirac operator on the associated spin bundle (induced by the spin representation of

Again we do not need the metric and the curvature is determined by the potential, so the potential is the fundamental object.

Stop looking for the graviton, not because it had been found but because it does not exist. The graviton is supposed to be the particle that communicates the gravitational force. But the gravitational force is not a fundamental force.

Since Einstein in the 1920s, physicists have tried to unify what are considered the four fundamental forces, namely, electromagnetism, weak and strong nuclear forces, and the gravitational force. In the 1970s, the three nongravitational forces were unified in the standard model. At high enough energy (about

Since then, with all the string theory, SUSY, branes, and extra dimensions, the gravitational force could not be incorporated into GUT that includes all 4 forces and no graviton has been found experimentally. The reason is simple: not many people, including Einstein himself, take/took the general theory of relativity seriously enough, according to which we know that the gravitational force does not exist as fundamental force but as geometry! We do not feel it. What we feel is the resistance of the solid ground on which we stand. In general relativity, free-falling objects follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics because of the mechanical resistance of matter. Newton’s apple falls downward because the spacetime in which we exist is

(“The coefficients

“

(“For physical reasons there was the conviction that the metric field was at the same time the gravitational field”).

Therefore GUT, the grand unified theory had been completed since the 1970s with the standard model. Since the gravitational force does not exist as a fundamental force, there is nothing more to unify as forces. If we want to unify all four theories, then it has to be done in a geometric way. The equations governing gravity as well as the standard model are all

Let

Or in general, locally, in terms of the stress-energy tensor

The Levi-Civita connection

The curvature tensor

First the curvature tensor

So we can express Einstein’s equations completely in terms of the connection (potential)

The free motion in spacetime is along geodesic curves

In general relativity the diffeomorphism group plays the role of a symmetry group of coordinate transformations. Then the vacuum Einstein’s field equations

The relation between the connection

Therefore all four theories, electromagnetism, weak interaction, strong interaction, and gravity, are unified as curvature equations in vector bundles over spacetime. Different interactions require different bundles.

There is no hierarchy problem because there is no fundamental gravitational force. The question why gravitational interaction is so much weaker than electroweak and strong interactions is meaningless, comparing apples with oranges. Why are so many physicists still talking about gravitational force? It is like as if we are still talking about “sun rise” and “sun set”, 500 years after Copernicus! only worse; these are serious scientists trying to unify all four “forces” to a TOE.

I am not saying that there are no open problems in physics. Of course there is still the problem of unifying quantum mechanics and general relativity on a geometric level (not as forces). The question is “how does spacetime look at the Planck scale? Do we have to modify spacetime to incorporate quantum mechanics or quantum mechanics to accommodate spacetime, or both? We need a theory of quantum gravity. There are several theories in the developing stage that promise to accomplish this.

Superstring theory by E. Witten et al.

Discrete spacetime at Planck length by R. Loll in "Causal dynamical triangulation" and by J. Ambjorn, J. Jurkiewicz, and R. Loll in "The Universe from Scratch" [arXiv: hep-th/0509010].

Spacetime quantization: loop quantum gravity by L. Smolin in “Three Roads to Quantum Gravity” (London: Weidenfeld and Nicholson, 2000) and by S. O. Bilson-Thompson, F. Markopoulou, and L. Smolin in “Quantum Gravity and the Standard Model”, preprint 2006.

Geometric formulation of quantum mechanics by A. Ashtekar and T. A. Schilling [arXiv: gr-qc/9706069].

Deterministic quantum mechanics at Planck scale by G. t’Hooft in “Quantum Gravity as a Dissipative Deterministic System.”

Branes and new dimensions: parallel universes by L. Randell et al., D. Deutsch, PS. In the brane world, gravity is again singled out as the only force not confined to one brane.

Noncommutative Geometry. A. Connes describes the standard model form general relativity.

The most recent new development is by Verlinde [

See [

Supersymmetry (SUSY) is an important idea in quantum filed theory and string theories. The BRST symmetry we described in Section

The classical Marsden-Weinstein reduction theorem is a geometrical result stating that if a Lie group

In [

Many classes of differential equations have extensions that involve odd variables. These are called

As an example, consider the Korteweg-de Vries equation

There is a different way of writing system (

One needs some concept of supermanifold even if one only works with the component formulation. For example, one has implicitly in system (

In [

There are different approaches to supermanifolds. The

The

An element is written as

For calculus we “replace” reals

The two approaches are equivalent. There is a one-to-one correspondence between isomorphism classes of

The DeWitt topology of

Smooth functions

Smooth functions map the body to the body!

Versions of the inverse function theorem and implicit function theorem are still valid. Concepts of tangent space, vector fields, flows, and Lie groups can be developed as in the ungraded case.

A Lie supergroup is a supermanifold

Let

There are examples where

A supersymplectic supermanifold

(

(

Let

A smooth vector field

If

Let

Superversion.

Let

Suppose that the momentum map is

The supermanifold

We consider the phase space

The SUSY algebra of the Bose-Fermi oscillator is given as follows. The Lie supergroup of invertible

The algebra of Bose-Fermi supersymmetry is the intersection of the Lie superalgebras of the stabilizer of

The SUSY algebra of the Bose-Fermi oscillator

The action of

Now let

Let

Then Hamilton’s equations are the following:

The well-known SUSY algebra from the Lagrangian description can be “exported” to the Hamiltonian setup.

We reduce by an Abelian subgroup of the SUSY group:

We determine the

Finally the

See [