17.3 Canonical quantization in d dimensions

17.3 Canonical quantization in d dimensions

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · unitary representation

The above can easily be generalized to the case of $d$ dimensions, with the Schr¨odinger representation ${\Gamma }_S^{\prime }$ now giving a unitary representation of the Heisenberg Lie algebra ${\mathfrak{h}}_{2d+1}$ determined by

$${\Gamma }_S^{\prime }(q_j)=-i{Q}_j,{\Gamma }_S^{\prime }(p_j)=-i{P}_j$$

which satisfy the Heisenberg relations

$$[Q_j,P_k]=i{\delta }_{jk}$$

Generalizing to quadratic polynomials in the phase space coordinate functions, we have

$${\Gamma }_S^{\prime }(q_j{q}_k)=-i{Q}_j{Q}_k,{\Gamma }_S^{\prime }(p_j{p}_k)=-i{P}_j{P}_k,{\Gamma }_S^{\prime }(q_j{p}_k)=-\frac{i}{2}(Q_j{P}_k+P_k{Q}_j)\text{\hspace{1em}(17.1)}$$

These operators can be exponentiated to get a representation on the same H of $\mathcal{M}p(2d,\mathbf{R})$ , a double cover of the symplectic group $Sp(2d,\mathbf{R})$ This phenomenon will be examined carefully in later chapters, starting with chapter 20 and the calculation in section 20.3.2, followed by a discussion in chapters 24 and 25 using a diferent (but unitarily equivalent) representation that appears in the quantization of the harmonic oscillator. The Groenewold-van Hove theorem implies that we cannot find a unitary representation of a larger group of canonical transformations extending this one of the Heisenberg and metaplectic groups.

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