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The abstract Weyl Algebra $W_n$ is the *-algebra generated by a family of elements $U(u),V(v)$ with $u,v\in\mathbb{R}^n$ such that (Weyl relations) $$U(u)V(v)=V(v)U(u)e^{i u\cdot v}\ \ Commutation\ Relations$$ $$U(u+u')=U(u)U(u'),\qquad V(v+v')=V(v)V(v')$$ $$U(u)^*=U(-u),\qquad V(v)^*=V(-v)$$ It is a well known fact (Stone- von Neumann theorem) that all irreducible representations $\pi:W_n\rightarrow B(H)$ of this algebra such that $$s-\lim_{u\rightarrow 0}\pi(U(u))=U(0),\qquad s-\lim_{v\rightarrow 0}\pi(V(v))=V(0)$$ are unitary equivalent to the Schroedinger one: $$H\equiv L^2(\mathbb{R}^n)$$ $$\pi(U(u))\equiv e^{i\overline{u\cdot X}},\qquad \pi(V(v))\equiv e^{i\overline{v\cdot P}}$$ where $X,P$ are the usual position and momentum operators defined (and essentially self-adjoint) on the Schwartz space $S(\mathbb{R}^n)$. For this reason, this algebra is usually associated to the quantization of a spinless non relativistic particle in $\mathbb{R}^n$. But what happen if one considers a spinless particle confined in a subregion $K\subset\mathbb{R}^n$, for example a circle $S^1\subset\mathbb{R}^2$, a sphere $S^2\subset\mathbb{R}^n$ or a generic open subset? The state space could still be chosen as $L^2(K)$, but what about the associated Weyl Algebra? I think it can still be defined abstractly but, while the commutation relations must be conserved in some way, the parameters domain $(u,v)$ cannot still be taken as $\mathbb{R}^n\times\mathbb{R}^n$ otherwise we fall in the previous context.

Is there some prescription for such a general situation? To be more precise: Given a bosonic particle confined in a sub region K, is it possibile to associate a family of abstract elements $U(u),V(v)$, $u,v$ belonging for example to some additive groups, such that the above Weyl properties are satisfied?

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  • $\begingroup$ To start with, the momentum operator is not defined as a self-adjoint extension of -id/dx unless proper boundary conditions are specified. It must be self-adjoint for the exponential to exist. $\endgroup$ – Urgje Jan 16 '14 at 19:17
  • $\begingroup$ Probably my question is not well-posed. I never said this, I wrote that the momentum is given by the self adjoint extension of -id/dx_j over the Schwartz space only for the R^n case. My question is exactly what happens for the general K situation. $\endgroup$ – moppio89 Jan 16 '14 at 19:32
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When the phase space of a system is not $R^{2n}$, the algebra of observables will not be the Weyl algebra, for example, when the phase space is the two sphere $S^2$ then the observables are the angular momenta which satisfy the $SU(2)$ algebra$

However, the Weyl algebra continues to play an important role even in the case of non-flat phase spaces. The main reason is that due to the Darboux theorem, any phase space looks locally like $R^{2n}$. In the case of a cotangent bundle of a general (non-flat) manifold, the Weyl algebra becomes the algebra of symbols of pseudodifferential operators. These operators are important in semiclassical analysis.

Even more generally, there is a general construction of quantization due to Fedosov in which a Weyl algebra is attached to each point in a symplectic manifold to form a Weyl bundle. The several fibers of this bundle are glued by a connecton known as the Fedosov connection. Using this data, one can construct the Fedosov quantization map order by order in the Planck's constant. Please see the following thesis by Philip Tillman where the case of the quantization of the sphere is given explicitly.

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