A Connection between Reciprocal Space and Cotangent Space

  • Thread starter Phinrich
  • Start date
31
2
Try this link;

 

mathwonk

Science Advisor
Homework Helper
10,719
894
yes, sorry for more math speak. we take it for granted that e^(it) = cos(t) + i sin(t), which is a point of the unit circle in the complex plane. so the "circle group" means the group of unit length complex numbers, i.e. those of form cos(t) + i sin(t) = e^(it), for real t.

thus t-->e^(it) is a map from the additive group of reals to the multiplicative group U of unit complex numbers. hence by definition of the dual group, it is an element of the dual group of the reals. for any real s, t-->e^(its) is another one, and apparently all such occur this way, so this would identify the dual group of maps R-->U with the original group R. but i am really not at all proficient in this topic.
 
31
2
Thanks. Dont worry about the "maths speak". The rigour of your arguments reminds me to try to be more careful in how I phrase my arguments. Yes I enjoyed the theory of complex numbers when I was at University so I understand what you say.
 

Want to reply to this thread?

"Connection between Reciprocal Space and Cotangent Space" You must log in or register to reply here.

Related Threads for: Connection between Reciprocal Space and Cotangent Space

Replies
23
Views
15K
  • Posted
Replies
9
Views
3K
  • Posted
Replies
5
Views
2K
  • Posted
Replies
8
Views
4K
  • Posted
Replies
2
Views
2K
Replies
6
Views
4K
Replies
0
Views
2K
  • Posted
Replies
3
Views
868

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top