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is well known that composition of point reflections generate pure displacements. This implies that the commutator of two point reflections will be a pure displacement. Are there similar elemental transformation operations which can generate pure displacements from their anticommutators?

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This question is not well posed. I'm not sure what exactly you intend addition/subtraction of point reflections to mean, since this depends on the objects that the point reflection operators are acting on. In general the commutator of two point reflections is not a translation. For example, consider reflections acting on one-dimensional functions. Let P be reflection about the point p and Q be the same about the point q. Then

$$ PQf(x) = f(x-(p-q)) \\ QPf(x) = f(x-(q-p)) $$ $$[P,Q]f(x) =(PQ-QP)f(x) = f(x-p+q) - f(x-q+p)$$

If you take, for example $f(x) = x$, then $[P,Q]x = x-p+q-x+q-p = 2(q-p)$. This constant is clearly not a translation of $x$.

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  • $\begingroup$ I was wondering about this myself. $\endgroup$ – David Z Jul 10 '12 at 4:58

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