1
$\begingroup$

I am new to Group theory and representations and I'm having trouble with this problem in an exercise:

Given the two oscillator algebra $$[a, a^†] = 1$$ $$[b, b^†] = 1$$ $$[a, b] = 0$$ show that the forgoing generators satisfy $SU(1,1)$ algebra. $$[K_+, K_−] = 2K_z$$ $$[a, K_+] = a^†b^†$$ $$[a, K_−] = ab$$

I've read that $SU(1,1)$ algebra is defined by the the commutations: $[K_1, K_2] = −iK_0$ , $ [K_0, K_1] = iK_2$ , $[K_2, K_0] = iK_1$ .

I assumed that it was just a matter of plugging in the given generators and obtaining these relations to show that they satisfy the given algebra but I'm unable to do this.

I would really appreciate it if someone could provide me with some tips on commutator manipulation and the general strategy one would use in solving such problems.

$\endgroup$

closed as off-topic by ACuriousMind Mar 10 '18 at 16:13

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation. $\endgroup$ – ACuriousMind Mar 10 '18 at 16:13