A commonly used generalization of Pauli Operators is the "clock" and "shift" operators summarized here.
Pauli Operators are generators of SU(2). Are these generalized Pauli Operators generators of SU(n)?
$\begingroup$
$\endgroup$
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
4
Yes, of course they do: J J Sylvester introduced them in 1882, p. 649, for that very purpose. They are a basis of all $n\times n$ matrices, and thus $\mathfrak{su}(n)$, the Lie Algebra of SU(n). A linear combination of them with suitable coefficients, exponentiated, provides all elements of SU(n).
The complete family of n2 independent unitary (but non-Hermitian) independent matrices $$ (\Sigma_1)^k (\Sigma_3)^j =\sum_{m=0}^{n-1} |m+k\rangle \omega^{jm} \langle m| , $$ where $\omega^n=1$, provides a trace-orthogonal basis for $\mathfrak{gl}(n,C)$, known as "nonions" $\mathfrak{gl}(3,C)$, "sedenions" $\mathfrak{gl}(4,C)$, etc...
It should be evident that $\Sigma _ 1 ^n = \Sigma _ 3 ^n = 1\!\!1$, and the braiding relation, $ \Sigma_3 \Sigma _1 = \omega \Sigma_1 \Sigma _3 = e^{2 \pi i / n} \Sigma_1 \Sigma _3$.
Since all indices are defined cyclically mod n, orthogonality follows, $$\mathrm{tr}\Sigma_1^j \Sigma_3^k\Sigma_1^m\Sigma_3^s =\omega^{km} n~\delta_{j+m,0} \delta_{k+s,0}~.$$
This basis, not very popular for its lack of hermiticity, can be systematically connected to the standard Weyl-Cartan Hermitian basis.
For instance, the powers of the basic clock matrix, $\Sigma_3$, comprising the Cartan subalgebra, map to linear combinations of the $h_k^n$ Lie Algebra elements. For n=3 ("nonions") it is easy to demonstrate their mapping to the Gell-Mann matrices and the identity.
More significantly, the n → ∞ limit is particularly transparent in this language, identifying $\mathfrak{gl}(\infty,C)$ with the Lie algebra of Poisson brackets.
answered Jan 12, 2017 at 23:53
-
$\begingroup$ The generalized Pauli matrices can be used as a basis for the Pauli subgroup of $SU(n)$ (i.e. they has group properties) and also as a basis (of non-hermitian matrices, but nevertheless a basis) for hermitian matrices, so they also have algebraic properties. See doi: dx.doi.org/10.1063/1.528006 There is a pretty cool application by Zeitlin, V. "Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure." Physica D: Nonlinear Phenomena 49.3 (1991): 353-362. $\endgroup$ Mar 28, 2017 at 22:59
-
$\begingroup$ Yes, thanks, Zeitlin actually relies on our paper, his ref 4. Neither Patera nor we, in the late 80s, had any idea about Sylvester.... It is all JJSylvester, if you read his notions, sedenions, etc... GL(N). $\endgroup$ Mar 28, 2017 at 23:11
-
$\begingroup$ which I will do after reading those other papers you recommended on reconstructing the potential from the eigenvalues. $\endgroup$ Mar 28, 2017 at 23:12
-
$\begingroup$ Oh... you can always front-load my 1989 Tahoe summary talk.... Isospectral stuff is dandy, but it can wait for a rainy day... (When Rosner, Quigg, and Thacker saw how 2 and states of the Balmer formula reproduce the singularity of the Coulomb potential, they were sold... they beat charmonia to death...). $\endgroup$ Mar 28, 2017 at 23:17
$\begingroup$
$\endgroup$
Do you mean "generator" in the group sense or the Lie algebra sense?
As a group, the Paulis don't generate SU(2) since the Pauli group is finite. The same is true for the generalized Paulis.
As a Lie algebra, the generalized Paulis generate SU(n) since $\{ \sum_{ij} \alpha_{ij} X^i Z^j : \alpha_{ij} \in \mathbb{R} \}$ consists of all Hermitian matrices.
