THE GEOMETRY OF LIE GROUPS

Lie groups like SU(N) display a nontrivial curved geometry in the angular variable space. The geometrical properties of the group space define a local metric tensor and through that covariant operations and invariant quantities like the scalar curvature and the Laplace operator. These can be used for derive formulae for group integrals or Casimir operators and hence lattice Hamiltonians more generally. In this section we take a brief excursion to this field exploring some basic ideas and presenting various useful results.

The basis of any geometrical description is a measure of distances, surfaces and volumes which is locally governed by the metric. In order to measure distances between two elements of a group we need to know the length of their displacement vector, which requires the definition of a norm. In fact a real valued scalar product can be defined in a natural way in the matrix representation

for SU(N) or more generally for any Lie group. This scalar product has the desired properties

The elements U of the SU(N) group are vectors of unit length, so

for any matrix A (not necessarily a group element). The invariant infinitesimal distance square

defines the components of the metric tensor in a given representation

through

where are the group parameters (in SU(N) compact angular variables) and the are the basis matrices for a given representation. They satisfy the commutation relation

with the totally antisymmetric structure constants of the associated Lie algebra.

Now one point here is still to clarify: how do we differentiate a group element U? A possibility would be to follow the general definition and take the limit of the difference of two neighboring elements. To define a neighborhood, however, we need a parameterized form, therefore we would obtain a parametric derivative. Another possibility is to construct first a one-form, a combination of the coordinate differentials without making reference to a special parameter set, and then consider the length of this differential. More generally we consider the following action on a matrix X

as the covariant differential. Expanding this leads to the abstract operator equality

with

being the parallel transport or connection form. It may be familiar to the reader from Riemannian geometry or from the theory of general relativity. In fact there is a striking analogy between the gauge covariant and geometrically covariant derivatives. In the theory of Lie groups is called the Mauer-Cartan one-form. Since its square length

equals the infinitesimal distance squared (apart from the sign) it is sometimes also called the half-metric of the group. Since the vectors U have unit length it is easy to see that the Mauer-Cartan form defined above is hermitean

The following useful identity

where the index ``ad'' refers to the adjoint representation, which uses the Lie algebra structure constants as basis and therefore

for any matrix B, can be used for the evaluation of the covariant derivative. Note that the adjoint representation of is equivalent with an infinite series of using n-fold commutators

Exercise:	Prove the identity (13).
Solution:	First we define Realizing that we use the Taylor expansion to obtain

Now using the power series form of (13) and applying it for the differential operator B=d we realize that, since

the covariant derivative is

Replacing now in this result we obtain the Mauer-Cartan form

Writing this in component form

we obtain the components of the metric tensor

where a summation over the index k is understood. The normalization of the basis matrices

is representation dependent. If the group elements were represented in the fundamental representation of SU(N) the normalization is , in the adjoint representation it is . Once the metric tensor in a given representation is known, the group integral can be carried out by using the Jacobian which leads to the Haar measure

Let us demonstrate the above in the case of SU(2). In the fundamental representation the matrix has the eigenvalues and where is the length of the vector . Correspondingly the projectors onto the eigenstates with eigenvalue satisfy

Introducing the matrix the projectors are

Using the eigenvalues and projectors any function of the matrix can easily be obtained, especially

It can be rearranged into the familiar form

In order to obtain the Mauer-Cartan form and the metric tensor we need the adjoint representation as well. In this case we consider a spin 1 system and the eigenvalues of are . The corresponding projectors satisfy

leading to

with this time. The Mauer-Cartan form becomes

which can be written in the form

with

In order to have the projectors must satisfy

As a consequence two projectors have zero length

while from the trace relations for any we get

in the adjoint SU(2) representation. This finally leads to

Exercise:	Calculate the determinant of the matrix .
Solution:	The determinant of a matrix is the product of its eigenvalues. Since we have according to (30) we obtain

Finally using the radial coordinates and we obtain the following invariant infinitesimal distance

and the following integration measure

These results show that the geometry of the group SU(2) is analogous to that of a surface of the four dimensional sphere with unit radius . The essential difference is global: SU(2) can be wrapped twice over , which is obvious since here is the main asimuthal angle. It is also obvious that the curvature of this space is constant.

The Laplace operator in this space, corresponding to the electric energy in the quantized Hamiltonian lattice gauge theory, can be written according to the above metric as

with second derivatives according to spherical angular coordinates in four dimensions

The solutions of the free SU(2) Schrödinger equation,

include the spherical functions

with the coefficients being

and the associated Legendre polynomials satisfying

such that the radial dependence factorizes

The eigenvalues in this case correspond to the pure electric energy density of a flux line spanned over a lattice link.

In comparison with the familiar spherical functions known from the Schrödinger equation for radially symmetric potentials in three-dimensional space the radial part of the eigenfunctions of the SU(2) Schrödinger equation (40) is somewhat unusual. It satisfies the eigenvalue equation

The ansatz

separates some trivial factors leading to

This equation defines the ultraspherical functions so that two classes of solutions occur

The proportionality factors are obtained from the normalization of the wave function. Finally, the solution, which is regular at , is given by

with the normalization coefficient

The energy eigenvalues are those of a quantized rotator

with being an integer or a half integer.

Finally, it is interesting to note that the radial excitations of the SU(2) quantum rotator constitute the character function of the group in the representation J

Writing this result in the alternate form

one easily realizes that J=0 belongs to the vacuum, to the fundamental and J=1 to the adjoint representation of the SU(2) group. For gauge fields on lattice links the lowest excitation corresponds to J=1.

At finite temperatures on the other hand, every excitation is allowed. Since the degeneracy of a multiplet J is

the partition function for the electric field energy eigenvalues on a link of length a can be obtained as

with being the inverse temperature. In the gluon sector we sum over integer values of J only. The average electric energy can be obtained as

and the effective number of degrees of freedom can be inspected as a function of temperature.

As expected, at high temperatures the average electric energy approaches . Assuming equipartition at high temperature this yields <E>=3T including the magnetic energy. Corresponding to an ideal gas of massless relativistic SU(2) gluons. At low temperature, however, even if all links were independent, i.e. the interlink coupling through the magnetic energy were neglected, the electric gauge degrees of freedom would be frozen. This situation is analogous to the Dulong-Petit rule, known from solid state physics.