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I've been studying the finite-dimensional representations of the lie algebra sl(2,C). I've read that these representations are related to the harmonic oscillator and the associated raising and lowering operators, but I'm not really sure how they are related. I've read you can generate a new energy state from an old one, but I'm not really sure how it works, and the energy states do not differ by 2 like the eigenspaces do for sl(2,C). I'm having trouble finding references about this, especially ones that are accessible. If anyone knows where I can find more info, or has some knowledge of the material, I would greatly appreciate their help.

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    My understanding is that the eigenvalues of the representation of $\text{sl}(2,\mathbb{C})$ are integer valued but the standard physics convention has an additional factor of $\hbar/2$ which may fix your issue.2010-12-14
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    I'm guessing you're using the basis $X_+,X_-,H$ with $[H,X_+] = 2X_+, [H,X_-] = -2X_-, [X_+,X_-] = H$? Then I think in physics they use $\frac{1}{2}H$. So if $v$ is an eigenvector of $\frac{1}{2}H$ with value $r$ then $X_+ v$ is an eigenvector of $\frac{1}{2}H$ with value $r+1$.2010-12-14
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    Err...it looks like the commutation relations in physics involve $[a_+, a_-] = 1$, so it's not quite the same since $H$ doesn't act as the identity in the rep space.2010-12-14
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    [Ryder](http://books.google.com/books?id=nnuW_kVJ500C&printsec=frontcover&dq=quantum+field+theory+ryder&hl=en&ei=0_YGTdeYO8L-8AaE85HnAg&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CCoQ6wEwAA#v=onepage&q&f=false) section 2.3. [Hall](http://books.google.com/books?id=nnuW_kVJ500C&printsec=frontcover&dq=quantum+field+theory+ryder&hl=en&ei=0_YGTdeYO8L-8AaE85HnAg&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CCoQ6wEwAA#v=onepage&q&f=false) section 1.7.2010-12-14

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