# What is the Weyl algebra of a confined bosonic particle?

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?

• 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. – Urgje Jan 16 '14 at 19:17
• 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. – moppio89 Jan 16 '14 at 19:32

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.