# B Is imaginary "i" a purely aesthetic matter?

#### sysprog

WOW... sudden realization!! This is exciting!! For the last 20 years I meandered around them mysterious quantum formulas (without understanding anything, but fascinated on how cute they look) people would talk about SO(2), SU(3), etc... and refer to Lie algebra, multiplicative groups, ... ah!!!... no way someone with under 30 trillion neurons can understand any of that!!

Now I see - SO(2) is the same thing as a 2D rotation in ℝ2, and SO(3) is a 3D rotation in ℝ3. Hooray! So SO(4) may be related to 4D rotations? Or something else.. the wording is really difficult to translate a human English. Also I guess SU(3) too must be related to some other transformation-thingie with some particular property-thingie-thingie.

So SO(2) is related (related-sy, however the proper Mathematish phrase for that) to z ∈ ℂ such that ||z|| = 1! I learned something!!!
The membership roster of the SU(3) non-Abelian homology (gauge) group was used to (as it turns out, apparently correctly), predict the existence of the top quark in QCD, which helped to bring about the marshaling of greater effort to find it -- ephemeral, because it doesn't hadronize like the others, but part of the group, so the QCD theorists assiduously sought it.

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#### TeethWhitener

Gold Member
So SO(2) is related (related-sy, however the proper Mathematish phrase for that) to z ∈ ℂ such that ||z|| = 1! I learned something!!!
Not just related. $SO(2)$ is isomorphic to the circle group, the unit circle in the complex plane. The representation that @mfb pointed out occurs because the set of all skew-symmetric matrices under standard matrix multiplication forms a Lie algebra, $o(2)$, with the commutator as the Lie bracket. The exponential map takes elements from this Lie algebra to the Lie group $SO(2)$, just as the exponential map takes elements from the Lie algebra $i\mathbb{R}$ to the Lie group called the circle group (and denoted by Wikipedia as $\mathbb{T}$).

#### stevendaryl

Staff Emeritus
This is way over the top. To cite Pauli matrices just to have a name for $i \sigma_y$ is very far fetched. It's like starting with a factor group of $SO(4)$ just to explain a complex number. Sorry, but this is ridiculous.
Well, Hestenes had a program of replacing all occurrences of "i" in physics by elements of Clifford algebra. I don't know how successful his program was.

#### Klystron

Gold Member
Not just related. $SO(2)$ is isomorphic to the circle group, the unit circle in the complex plane. The representation that @mfb pointed out occurs because the set of all skew-symmetric matrices under standard matrix multiplication forms a Lie algebra, $o(2)$, with the commutator as the Lie bracket. The exponential map takes elements from this Lie algebra to the Lie group $SO(2)$, just as the exponential map takes elements from the Lie algebra $i\mathbb{R}$ to the Lie group called the circle group (and denoted by Wikipedia as $\mathbb{T}$).
Let me also call attention to a series of Insight articles beginning with
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/

Excellent exercise for aging brains <grin>.

#### fresh_42

Mentor
2018 Award
Well, Hestenes had a program of replacing all occurrences of "i" in physics by elements of Clifford algebra. I don't know how successful his program was.
... and $\mathbb{C}$ is a real (associative) superalgebra with $\mathbb{C}_0=\mathbb{R}$ and $\mathbb{C}_1=i \cdot \mathbb{R}$. Super!

"Is imaginary "i" a purely aesthetic matter?"

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