Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What would be the physical importance of unitary group? By physical, I mean geometric intuiton and the usages in physics.

Also, how would special unitary group be used?

share|cite|improve this question
$\textrm{U}(1)$ is the gauge group of the electromagnetic force, $\textrm{SU}(2)$ is the universal cover of the rotation group $\textrm{SO}(3)$, $\textrm{SU}(3)$ is the gauge group of the strong force... – Zhen Lin Jul 8 '12 at 0:07
up vote 2 down vote accepted

In quantum mechanics, the time evolution of states is unitary. Also, many important symmetry operations are unitary as well (there are also some anti-unitary symmetry operations, most notably time reversal).

The special unitary group can be used instead for time evolution because the global phase doesn't matter (actually states are described not by vectors, but by rays in some Hilbert space; if you change the phase of your state vector, you are still in the same ray).

Also note that the gauge transformations from the gauge theories are local unitaries; you apply a different one in each spacetime point (which means that the actual group isn't one-parameter but infinitely-many parameters). That's why the $U(1)$ symmetry is relevant there: You definitely do $not$ get the same state if you do a local $U(1)$ transformation.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.