From what I understand, canonical quantization of a classical theory consists of replacing the observables by abstract operators, of which only the commutation rules, which have to correspond to the Poisson brackets, are given.

I assume this ensures that in the macroscopic limit we recover classical mechanics (through the Ehrenfest theorem). From these abstract operators we can also recover the dynamics, the time evolution operator is $e^{iHt/\hbar}$, the uncertainty relation is $$\sigma_A\sigma_B ~\ge~ {1\over2}\left|\langle [A,B]\rangle\right|$$ and (in the Heisenberg picture) the time evolution of an observable is $$i\hbar\frac{dA}{dt} = [A,H]+i\hbar\frac{\partial A}{\partial t}.$$

Alternatively one can start by explicitly constructing the state space as a function space, the observables as operators on that space (by making some standard substitutions, taking care of Hermiticity, etc) and one observes that the same commutation relations hold.

My question is if in the first approach, does one completely let go of the explicit description of Hilbert space as a function space and the operators as explicit operators on this space, and instead one works with "the" (abstract) Hilbert space and operators on it of which we only need to specify the commutation relations? Or is it really the same thing, and ultimately we always will need the explicit description to derive some of the properties of the system.

I've been struggling to make my question clear, if it isn't, please let me know.


1 Answer 1


Let's take the canonical commutation relations (CCR), in their exponentiated form (Weyl's relations):

$$V(\eta)T(q)=e^{-i\eta\cdot q}T(q)V(\eta)\; ,$$

where $\{V(\eta)\}_{\eta\in \mathbb{R}^d}$ and $\{T(q)\}_{q\in \mathbb{R}^d}$ are objects of a given normed algebra with involution. This is a very general notion, that is nowadays taken as the definition of the CCR. If we take the exponentials of the position and momentum operators $V(\eta)=e^{i\eta\cdot x}$ and $T(q)=e^{i q\cdot (-i\nabla_x)}$ in $L^2(\mathbb{R}^d)$ we see that they satisfy the Weyl's relations, and they are objects of the $C^*$ algebra of bounded operators on that space.

Now let's start with $W=\{V(\eta), T(q)\}_{\eta, q\in \mathbb{R}^d}$, and construct the $C^*$ algebra $V$ that contains $W$, i.e. $V=\overline{W}$ (the closure of $W$ in the given norm $\lVert \cdot\rVert$ of our obects). This is called the CCR $C^*$ algebra. So, as you can see, the starting point is very abstract, and is given by this CCR $C^*$ algebra.

Now it is possible to show that each $C^*$ algebra has at least one faithful representation as a subalgebra of the bounded operators on some Hilbert space (the so-called GNS construction).

Another remarkable result is the Stone-von Neumann theorem, that says that all the irreducible (i.e. such that the only subspace invariant under the action of the operators is the zero vector) representations of the CCR algebra are unitarily equivalent (i.e. related by a unitary transformation) and in turn equivalent to the representation given by the usual position and momentum operators on $L^2(\mathbb{R}^d)$ I gave above.

Putting together the results, it is then evident that it is sufficient to give the CCR algebra, for it is always represented irreductibly (up to unitary isomorphisms) by the canonical position and momentum operators on $L^2$. Also, the concept of quantum states is directly related to the $C^*$ algebra of observables (it is a subset of its topological dual); and the normal states (a subset of the predual of the von Neumann algebra $V''$, and of the quantum states) are in one-to-one correspondence with the density matrices in the corresponding representation.

Concerning evolution and classical limit (in relation to classical dynamics), these concepts are more easily understood using the point of view of semiclassical analysis, i.e. studying the (Weyl, Wick, anti-Wick) quantization of classical symbols into pseudodifferential operators, and their semiclassical expansions. Nevertheless the quantum evolution may be seen as an automorphism of the $C^*$ algebra of observables (or of the quantum states) that satisfies some regularity assumptions.

Remark: the Stone-von Neumann theorem is only valid for "finite dimensional" Weyl relations, i.e. if $\eta,q \in \mathbb{R}^d$ (the result can be extended by Mackey theory to any locally compact group). If e.g. we consider the analogous "infinite dimensional" CCR algebra generated by $$W(\psi)W(\phi)=e^{-i\mathrm{Im}\langle\psi,\phi\rangle}W(\psi+\phi)\; ,$$ where $\psi,\phi\in \mathscr{H}$, $\mathscr{H}$ infinite dimensional Hilbert space, we have infinitely many unitarily inequivalent irreducible representations. In that situation (this is the case of bosonic quantum field CCR), we have other representations that are inequivalent to the Schrödinger (or the Fock) one; and thus becomes really crucial to see the quantum theory as the theory generated by the algebra of (noncommutative) observables.

  • $\begingroup$ Hi. This is a very interesting answer. While I' m currently beginning to study QFT, that is the quantization of fields, I also came up with a question on the idea behind canonical quantization. The difficulty arising for me is concerned with the fact that your post here is to mathematical- something by no means bad or wrong. Could I ask from you a comment giving some reference for an introductory study of such concepts as you present them and some more advanced? By reference I mean online but also book titles. I would be thankful for such a help. Thanks. $\endgroup$ May 8, 2016 at 10:39
  • $\begingroup$ @ConstantineBlack I would suggest this book. It is also rather mathematical, but it presents some of the above in details (focused on quantum fields). I don't know many other presentations that contain the above ideas "bundled", they're rather scattered across the literature. On the subject of quantization in quantum mechanics, you may find semiclassical analysis interesting, e.g. this book $\endgroup$
    – yuggib
    May 9, 2016 at 9:12

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