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For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation:

$s_2=\begin{bmatrix} e^{i\alpha}\cos(\theta) & -e^{-i\beta}\sin(\theta) \\ e^{i\beta}\sin(\theta) & e^{-i\alpha}\cos(\theta) \end{bmatrix}$

For the 3-dimension $SU(3)$ and the higher-dimension $SU(N)$ matrices, what are the most general parametrization formulations? I will appreciate any references, either a book or a review paper, on this subject oriented toward the physics/engineer applications.

EDIT:

This is to clarify my question and make it more relevant to the physics. I am looking for the kind of parametrization that is generally applicable to the problems in physics, especially being intuitive and geometrically accessible. For example, for the $SU(2)$ matrices given above, $\theta$ parameterizes the coefficients of any orthonormal states, $\alpha$ and $\beta$ parameterize the phases of the orthonormal states. The orthonormal states in this kind of general expression are certainly very common and relevant to enormous problems in quantum physics.

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  • $\begingroup$ Are you asking for unique parametrizations? Otherwise, can't you just go with $e^{\theta_i T_i}$ where $T_i$ are the $N^2-1$ generators of the algebra? $\endgroup$
    – Foster Boondoggle
    Commented Feb 17, 2016 at 22:47
  • $\begingroup$ I don't think the parametrization would be unique. I guess I am looking for the kind of parametrization that will make the most physical sense. For example, in the case of the $SU(2)$, the $\theta$ parameterizes the orthonormal coefficients, while the $\alpha$ and the $\beta$ parameterize phase angles. Anything along that line would be helpful. $\endgroup$
    – user36125
    Commented Feb 18, 2016 at 0:50
  • $\begingroup$ This doesn't seem to be about physics. (That the groups $\mathrm{SU}(N)$ are used in physics doesn't make it about physics anymore than questions about addition are about physics because physicists often add things) $\endgroup$
    – ACuriousMind ♦
    Commented Feb 18, 2016 at 1:06
  • $\begingroup$ @ACuriousMind I am asking about the kind of parametrization that specifically makes physical sense, and generally applicable to the problems in physics. $\endgroup$
    – user36125
    Commented Feb 18, 2016 at 1:30
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    $\begingroup$ Your answer is here. $\endgroup$
    – Cosmas Zachos
    Commented Mar 8, 2021 at 16:00

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For SU(3) there is a parametrization with trigonometric functions similar to the one you have written for SU(2), described in http://journals.aps.org/prd/pdf/10.1103/PhysRevD.38.1994.

Matrix elements are:

However, I think it is better to work with the $exp^{i\theta_i T_i}$ formula, where $T_i$s are the generators of the group in the appropriate representation.

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  • $\begingroup$ This is behind a paywall, so many users will not be able to see the result. Despite this, link-only answers are discouraged. Maybe you can add the relevant passages to the post? $\endgroup$
    – Martin
    Commented Feb 18, 2016 at 9:22
  • $\begingroup$ J Bronzan 1994. Google scholar makes them available for free. $\endgroup$
    – Cosmas Zachos
    Commented Mar 8, 2021 at 15:59
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Seems to me that what you're asking for is the $e^{\theta_i T_i}$ representation. The $T_i$'s are the equivalent of the Pauli matrices except $N\times N$ dimensional in the fundamental rep. (I might be missing a factor of $i$.)

It's not clear what you mean by "physical sense" in your comment above. For describing rotations uniquely in 3-d. for example, your fundamental representation of $SU(2)$ is probably less useful than the adjoint one where the isomorphism to SO(3) is more apparent. For doing $SU(3)$ QCD calcs, I don't recall ever paying attention to details of how the representation was parameterized. What mattered was things like the structure constants of the commutators.

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A possible parametrization for a real transformation matrix is via the Cayley transform: an orthogonal (i.e., real unitary) matrix can be represented as $$ Q=(I-A)(I+A)^{-1}, $$ where $I$ is the identity matrix and $A$ is skew-symmetric.

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There's this article by Adam Bincer, published in Journal of Mathematical Physics, vol 31 (1990). It is titled Parametrisation of SU(n) with n-1 orthonormal vectors and it's abstract states:

A generalisation to SU(n) of a well-known relation to SU(2) is proposed. It relies on the observation that an element of SU(n) has associated with it in a natural way n-1 orthonormal vectors in $R^{n^2-1}$. The meaning of these n-1 vectors is discussed as they relate to the geometry of the adjoint representation of SU(n).

Unfortunately, although it is now published online, it's not public access, so I can't say more.

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