Is imaginary "i" a purely aesthetic matter?

[fresh_42]

"F is algebraically closed field if every non-constant polynomial has a root"... which it says it is equivalent to saying that "F has no proper extension".

This is not true. There are extensions of $\mathbb{C}$. The division algebras $\mathbb{H}$ and $\mathbb{O}$ have already be named. But you can also add a new transcendental number, say $T$, so $\mathbb{C}[T]$ which is the same as the polynomial ring over $\mathbb{C}$ first becomes an integral domain, and thus has a quotient field $\mathbb{C}(T)$, which is a strict field extension of the complex numbers. And you can repeat this process as often as you want. The correct wording is: "... has no proper algebraic field extension!" Both restrictions are necessary as the examples I mentioned show.

 

[fresh_42]

Or if you're familiar with the properties of Pauli matrices, $$\begin{eqnarray*}
\bigl( a \, \mathbb{I} + i b \, \sigma_{y} \bigr) \bigl( c \, \mathbb{I} + i d \, \sigma_{y} \bigr) &=& ac \, \mathbb{I} + i (ad + bc) \, \sigma_{y} - b d \, {\sigma_{y}}^{2} \\
&=& (ac - bd) \, \mathbb{I} + i (ad + bc) \, \sigma_{y} \,,
\end{eqnarray*}$$ since ${\sigma_{y}}^{2}$ is the identity. (It doesn't matter if you use $\sigma_{y}$ or $-\sigma_{y}$.)

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.

 

[fbs7]

This is not true. There are extensions of $\mathbb{C}$. The division algebras $\mathbb{H}$ and $\mathbb{O}$ have already be named. But you can also add a new transcendental number, say $T$, so $\mathbb{C}[T]$ which is the same as the polynomial ring over $\mathbb{C}$ first becomes an integral domain, and thus has a quotient field $\mathbb{C}(T)$, which is a strict field extension of the complex numbers. And you can repeat this process as often as you want. The correct wording is: "... has no proper algebraic field extension!" Both restrictions are necessary as the examples I mentioned show.

Oh... got it (kinda)... I always keep omitting words that end up being relevant! Thanks for the correction!

 

[fbs7]

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.

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!

 

[fresh_42]

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!

Here's a list of "what is what" resp. "what an be viewed as what": https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
The Pauli matrices just give a simple basis for regular $2\times 2$ matrices with trace zero. They are intended to deal with $SU(2)$, although dropping the factor $i$ which in QFT is noted along $\hbar$. Therefore one could write $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ as needed in post #10 as $i\cdot \sigma_y$, but as I said, this is a bit too far fetched.

 

[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.

 

[TeethWhitener]

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]

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]

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]

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!