A quantum particle on the real line R has as configuration space this real line R, while its state space is the infinite dimensional complex Hilbert space of square integrable complex valued functions on R. In Quantum Information, a fundamental role is played by finite dimensional complex Hilbert spaces. Typically, they are the state spaces of finite sets of qubits. Question : what is the configuration space corresponding to such a finite dimensional complex Hilbert space ? Related question : what are the position and momentum operators on a finite dimensional complex Hilbert space ?
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$\begingroup$ "The" configuration space is a finite set of size the same as the dimension of the Hilbert space. There are no position or momentum operators without specifying more data. But see my blog post "The Schrödinger equation on a finite graph" (I can't link to it at the moment) for a fairly concrete example of what this data could be. $\endgroup$– Qiaochu YuanCommented Jun 14, 2012 at 9:46
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2$\begingroup$ (In a fairly strong sense there cannot be position and momentum operators on finite-dimensional Hilbert spaces because the Weyl algebra admits no finite-dimensional representations whatsoever. In that blog post you can find a suggestion for what a reasonable analogue of (the exponential of) a momentum operator ought to be on a finite graph, as well as operators describing whether or not a particle is at one of the vertices.) $\endgroup$– Qiaochu YuanCommented Jun 14, 2012 at 9:52
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$\begingroup$ @Qiaochu: Are you saying that the phrase "configuration space" is commonly used in this context, with the meaning you specified, or are you saying that if someone (like the OP) wants to use "configuration space" in this context then is should have this meaning? $\endgroup$– Andreas BlassCommented Jun 14, 2012 at 13:34
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$\begingroup$ @Andreas: I'm not familiar enough with the context to claim the former, so I suppose I'm claiming the latter. $\endgroup$– Qiaochu YuanCommented Jun 14, 2012 at 21:50
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