So lately I've been thinking about whether or not it'd be possible to have the commutation relation [itex] [x,p]=i \hbar [/itex] in a Hilbert Space of finite dimension d. Initially, I was trying to construct a lattice universe and a translation operator that takes a particle from one lattice point to the next and derive the commutation relation from that. Now I'm not so sure it's possible. In finite dimension d, I believe taking the trace of both sides of [itex] [x,p]=i \hbar [/itex] gives a contradiction. Where of course I use the cyclic property of the trace. Is this okay reasoning?(adsbygoogle = window.adsbygoogle || []).push({});

If so, why do the commutation relations hold in the continuum? I guess what I'm asking is, what is the analog of the trace in the continuous case and why does it not produce a contradiction in the continuous case?

Thanks!

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# Canonical Commutation Relations in finite dimensional Hilbert Space?

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