I am trying to do some self-study and plow through Ballentine's book on Quantum Mechanics. I thought I was following the majority of it until I got to Sec. 3.3, in particular the derivation of the multiples of identity for the commutators of these generators.(adsbygoogle = window.adsbygoogle || []).push({});

For example, the commutator of two rotations is given as

[tex][J_\alpha,J_\beta]=i\epsilon_{\alpha\beta\gamma}J_\gamma + i\epsilon_{\alpha\beta\gamma}b_\gamma I[/tex]

He then states that one can redefine the phase of certain vectors to transform the phase term to 0 and does so with the substitutions

[tex]J_\alpha + b_\alpha I \rightarrow J_\alpha[/tex]

leading to

[tex][J_\alpha,J_\beta]=i\epsilon_{\alpha\beta\gamma}J_\gamma[/tex]

I feel I am missing something key in what is going on here because I am not understanding why one can't just arbitrarily say the extra phase piece in the first equation is just 0 by setting [tex]b_\gamma = 0[/tex]. Where are these phase differences coming from that one needs to change your generator as Ballentine does?

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# Some confusion about Ballentine Sec 3.3 Generators of Gallilei Groups

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