Special unitary group
mathematics, the special unitary group of degree "n", denoted SU("n"), is the group of "n"×"n" unitary matrices with determinant1. The group operation is that of matrix multiplication. The special unitary group is a subgroupof the unitary groupU("n"), consisting of all "n"×"n" unitary matrices, which is itself a subgroup of the general linear groupGL("n", C).
The SU(n) groups find wide application in the
standard modelof physics, especially SU(2) in the electroweak interactionand SU(3) in QCD.
The simplest case, SU(1), is the
trivial group, having only a single element. The group SU(2) is isomorphicto the group of quaternions of absolute value1, and is thus diffeomorphicto the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjective homomorphismfrom SU(2) to the rotation groupSO(3) whose kernel is .
The special unitary group SU("n") is a real
matrix Lie groupof dimension "n"2 − 1. Topologically, it is compactand simply connected. Algebraically, it is a simple Lie group(meaning its Lie algebrais simple; see below). The center of SU("n") is isomorphic to the cyclic groupZ"n". Its outer automorphism group, for "n" ≥ 3, is Z2, while the outer automorphism group of SU(2) is the trivial group.
The SU(n) algebra is generated by "n"2 operators, which satisfy the commutator relationship (for i,j,k,l = 1, 2, ..., n)
Additionally, the operator
which implies that the number of "independent" generators of SU(n) is n2-1. [R.R. Puri, "Mathematical Methods of Quantum Optics", Springer, 2001. ]
In the defining or fundamental representation the generators are represented by "n"×"n" matrices where: :*:where the "f" are the "
structure constants" and are antisymmetric in all indices, whilst the "d" are symmetric in all indices. As a consequence: :*:*
We also have:*
as a normalization convention.
adjoint representationthe generators are represented by × matrices whose elements are defined by the structure constants:::*
For SU(2), the generators T, in the defining representation, are proportional to the
Pauli matrices, via:::where:
The structure constants for SU(2) are defined by the
Levi-Civita symbol::; the rest can be determined by antisymmetry.All the "d" values vanish.
The generators of SU(3), "T", in the defining representation, are:::where , the
Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2): :
These obey the relations:*:where the "f" are the "structure constants", as previously defined, and have values given by::::::
The "d" take the values:::::::
Lie algebracorresponding to is denoted by . Its standard mathematical representation consists of the traceless antihermitiancomplex matrices, with the regular commutatoras Lie bracket. A factor is often inserted by particle physicists, so that all matrices become hermitian. This is simply a different, more convenient, representation of the same real Lie algebra. Note that is a Lie algebra over .
For example, the following antihermitian matrices used in
quantum mechanicsform a basis for over :::: (where is the imaginary unit.)
This representation is often used in
quantum mechanics(see " Pauli matrices" and " Gell-Mann matrices"), to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in quantum relativity.
Note that the product of any two different generators is another generator, and that the generators
anticommute. Together with the identity matrix(times ),:these are also generators of the Lie algebra .
Here it depends of course on the problem whether one works finally, as in non-relativistic quantummechanics, with 2-
spinors; or, as in the relativistic Dirac theory, one needs an extension to 4-spinors; or in mathematics even to Clifford algebras.
"Note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra , whereas you generate the Lie algebra with commutator brackets instead."
Back to general :
If we choose an (arbitrary) particular basis, then the
subspaceof traceless diagonal matrices with imaginary entries forms an dimensional Cartan subalgebra. Complexifythe Lie algebra, so that any traceless matrix is now allowed. The weight eigenvectors are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra is only dimensional, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which should not be thought of as an element of the Lie algebra!) for the purpose of computing weights and that only. So, we have a basis where the th basis vector is the matrix with on the th diagonal entry and zero elsewhere. Weights would then be given by coordinates and the sum over all coordinates has to be zero (because the unit matrix is only auxiliary).
So, has a rank of and its
Dynkin diagramis given by , a chain of vertices.
root systemconsists of roots spanning a Euclidean space. Here, we use redundant coordinates instead of to emphasize the symmetries of the root system (the coordinates have to add up to zero). In other words, we are embedding this dimensional vector space in an -dimensional one. Then, the roots consists of all the permutations of . The construction given two paragraphs ago explains why. A choice of simple roots is:,:,:…,:.
Cartan matrixis :.
Weyl groupor Coxeter groupis the symmetric group, the symmetry groupof the - simplex.
Generalized special unitary group
For a field "F", the generalized special unitary group over F, SU("p","q";"F"), is the group of all
linear transformations of determinant1 of a vector spaceof rank "n" = "p" + "q" over "F" which leave invariant a nondegenerate, hermitian formof signature ("p", "q"). This group is often referred to as the special unitary group of signature p q over F. The field "F" can be replaced by a commutative ring, in which case the vector space is replaced by a free module.
Specifically, fix a
hermitian matrix"A" of signature "p" "q" in GL("n","R"), then all
Often one will see the notation without reference to a ring or field, in this case the ring or field being referred to is C and this gives one of the classical
Lie groups. The standard choice for "A" when "F" = C is: However there may be better choices for "A" for certain dimensions which exhibit more behaviour under restriction to subrings of C.
A very important example of this type of group is the
Picard modular groupSU(2,1;Z ["i"] ) which acts (projectively) on complex hyperbolic spaceof degree two, in the same way that SL(2,Z) acts (projectively) on real hyperbolic spaceof dimension two. In 2003 Gábor Francsicsand Peter Laxcomputed a fundamental domain for the action of this group on , see [http://www.esi.ac.at/Preprint-shadows/esi1273.html] .Another example is SU(2,1;C) which is isomorphic to SL(2,R).
In physics the special unitary group is used to represent
bosonicsymmetries. In theories of symmetry breakingit is important to be able to find the subgroups of the special unitary group. Important subgroups of SU(n) that are important in GUT physics are, for p>1, n-p>1::For completeness there are also the orthogonaland symplectic subgroups:::Since the rank of SU(n) is n-1 and of U(1) is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. SU(n) is a subgroup of various other lie groups:::: (see Spin group)::: (see Simple Lie groupsfor E6, E7, and G2)There are also the identities SU(4)=Spin(6), SU(2)=Spin(3)=USp(2) and U(1)=Spin(2)=SO(2) .
One should finally mention that SU(2) is the double
covering groupof SO(3), a relation that plays an important role in the theory of rotations of 2- spinors in non-relativistic quantum mechanics.
Representation theory of SU(2)
Projective special unitary group, "PSU(n)"
* [http://arxiv.org/abs/math/0605784v1 Maximal Subgroups of Compact Lie Groups ]
* [http://courses.washington.edu/phys55x/Physics%20558_lec1_03.htm Physics 558 - Lecture 1, Winter 2003]
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