17.5 More general notions of quantization

17.5 More general notions of quantization

Concept links · terms present in this machine draft; source roles are unverified: dual space · Lie algebra · adjoint representation · irreducible representation

The definition given here of quantization using the Schr¨odinger representation of ${\mathfrak{h}}_{2d+1}$ only allows the construction of a quantum system based on a classical phase space for the linear case of $M={\mathbf{R}}^{2d}$ . For other sorts of classical systems one needs other methods to get a corresponding quantum system. One possible approach is the path integral method, which starts with a choice of configuration space and Lagrangian, and will be discussed in chapter 35.

Digression. The name “geometric quantization” refers to attempt to generalize quantization to the case of any symplectic manifold $M$, starting with the idea of prequantization (see equation 15.8). This gives a representation of the Lie algebra of functions on $M$ on a space $of$ sections of a line bundle with connection ∇, with ∇ a connection with curvature $\omega ,$ where $\omega$ is the symplectic form on $M$. One then has to deal with two problems:

• The space of all functions on $M$ is far too big, allowing states localized in both position and coordinate variables in the case $M$ = $R^{2d}$. One needs some way to cut down this space to something like a space of functions depending on only half the variables (e.g., just the positions, or just the momenta). This requires finding an appropriate choice of a so-called “polarization” that will accomplish this.

• To get an inner product on the space of states, one needs to introduce a twist by a “square root” of a certain line bundle, something called the “metaplectic correction”.

For more details, see for instance [41] or [104].

Geometric quantization focuses on finding an appropriate state space. Another general method, the method of “deformation quantization” focuses instead on the algebra of operators, with a quantization given by finding an appropriate non-commutative algebra that is in some sense a deformation of a commutative algebra of functions. To first order the deformation in the product law is determined by the Poisson bracket.

Starting with any Lie algebra $g$, in principle 15.14 can be used to get a Poisson bracket on functions on the dual space $g^{\ast }$, and then one can take the quantization of this to be the algebra of operators known as the universal enveloping algebra $U(g)$. This will in general have many diferent irreducible representations and corresponding possible quantum state spaces. The co-adjoint orbit philosophy posits an approximate matching between orbits in $g^{\ast }$ under the dual of the adjoint representation (which are symplectic manifolds) and irreducible representations. Geometric quantization provides one possible method for trying to associate representations to orbits. For more details, see [51].

None of the general methods of quantization is fully satisfactory, with each running into problems in certain cases, or not providing a construction with all the properties that one would want.

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