A How Does U(1) Double-Cover SO(2) for a Specified Angle?

  • A
  • Thread starter Thread starter pellis
  • Start date Start date
  • Tags Tags
    Complex Numbers
AI Thread Summary
The discussion focuses on demonstrating how the U(1) circle group of complex numbers serves as a double cover for the rotation group SO(2) through specific examples. The user seeks clarity on the relationship between unit complex numbers and 2D rotations, noting that while some sources suggest a two-fold cover, others imply a one-to-one mapping. The conversation highlights the confusion surrounding the mapping of rotations to points in U(1) and the implications of clockwise versus anticlockwise rotations reaching the same point. The user plans to provide a proof based on a Clifford algebra approach, using a table to illustrate the double cover concept. A clear example is requested to solidify understanding of the U(1) to SO(2) relationship.
pellis
Messages
80
Reaction score
19
TL;DR Summary
Can someone please provide an explicit example of two complex numbers for double cover U(1) of SO(2) for a specified angle R(θ)?
I'm trying to find an explicit example showing exactly how the U(1) “circle group” of complex numbers double-covers 2D planar rotations R(θ) that form the rotation group SO(2).

There are various explanations available online, some of which are clear but seem to be at variance with other explanations. (I leave aside other more technical explanations suited only to graduate students in mathematics - I'm only a chemist - such as those in https://ncatlab.org/nlab/, math.stackexchange, or even https://en.wikipedia.org/wiki/Covering_space.)

Like some other texts, even the usually clear (late) Pertti Lounesto wrote, in his “Clifford Algebras and Spinors”: “The fact that two opposite elements of the spin group Spin(2) represent the same rotation in SO(2) is expressed by saying that Spin(2) is a two-fold cover of SO(2), and written as Spin(2)/{±1} is isomorphic to SO(2).” … without giving any example.

1. In what seems like a trivial sense, the unit complex numbers exp(i(θ+2nπ)) for integers n>±1 appear to provide not just a two-fold but rather an n-fold cover of R(θ); but the relevant points in U(1) would then be identical for a given R(θ), which to me looks like 1:1 rather than 2:1 (?)

2. Another account, if I read it correctly, appears to suggest that a rotation R(θ) maps to the two points exp(±iθ) in U(1) – but that results in a reflection in the real plane z = (cos(θ) + isin(θ)) and z = (cos(θ) - isin(θ)), which doesn’t seem to agree with what someone wrote elsewhere, that "one rotation in SO(2) maps to two rotations in U(1)".

3. There’s also the case where an anticlockwise rotation by θ in the 2D plane reaches the same point as a clockwise rotation (i.e. in the alternative direction) by (θ-2π), and which yields the same final position, as (1.) above, for an anticlockwise rotation by (θ + 2π). And this is just a special case of how the same point is reached by π rotations in opposite directions.

The answer I seek is a clear and unambiguous example of the U(1) double cover of SO(2), identifying the angle θ of a single rotation R(θ) and the corresponding two (distinct?) resulting points exp(i….) and a different exp(I,,,,).

Clarification or other advice will be much appreciated.

Thank you for reading the whole question.
 
Mathematics news on Phys.org
I think I've resolved the question and will post the proof in a day or so.
 
pellis said:
I think I've resolved the question and will post the proof in a day or so.
Tabulated data towards a solution for the question, based on Cl2 Clifford algebra approach that uses the factorised rotor exp() in a generalisable "sandwich product" form exp(-iθ/2)rexp(+iθ/2) to rotate a vector.

More detailed explanation to follow, once I'm satisfied that the table leads to a correct illustration of the double cover.
 

Attachments

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Thread 'Imaginary Pythagorus'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top