What is the significance of parity in group theory?

[xylai]

Parity vs. group theory?

Parity is a special property in Quantum mechanics.
I don't know whether it relates to group thery?
Is it O(2), U(1), or others?

Thank you!

 

[Fredrik]

Parity and time reversal are properties (isometries) of Minkowski space, not of quantum mechanics, but I guess I'm nitpicking now. Everything that's relevant for Minkowski space is of course relevant in relativistic QM too. Those two symmetries are not part of any connected Lie group (such as the ones you mention). The group of proper (no parity), orthochronous (no time reversal), homogeneous (no translations) Lorentz transformations is SO(3,1). It's universal covering group is SL(2,C), so relativistic quantum theories can be realized as representations of SL(2,C) or representations up to a phase of SO(3,1). (Chapter 2 of vol.1 of Weinberg is a good place to read about these things).

 

[xylai]

Sorry, I am not family with group theory.
As far as I know the symmetry of space (f(x)=f(-x)) is relative to parity conservation.

I don't know the relation of the symmetry of space and group theory?