# what is the physical importance of unitary group

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?

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$\textrm{U}(1)$ 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
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## 1 Answer

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.

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