Dirac came up with a general theory of constraints, including second-class constraints. To quantize such systems, he first computed the Dirac bracket classically, and only then "promoted" the classical Dirac bracket into a commutator. However, this leads to operator ordering ambiguities over and above what already exists for the Poisson bracket.

Is there any more direct way of coming up with a quantum Dirac bracket and Hamiltonian operator which resolves such ambiguities? If the system is symmetrical, symmetry considerations can often single out the "correct" quantization, but what about asymmetrical systems?

Ideally, the correct quantum prescription ought to match what we get from using path integrals.

Similarly, if the second-class constraint is linear, the quantum commutator and Hamiltonian is also straightforward. However, even relatively simple models can have subtleties. Consider the toy model $$L=\frac{\dot{q}^2}{2} -f[q]\frac{F^2}{2}$$ where F is an auxiliary variable and $f$ is a function of $q$. The classical second-class constraints are given by $p_F \approx 0$ and $fF\approx 0$. Using path integrals as a guide, $$\int \mathcal{D}q\, \mathcal{D}F\, \exp\left( \frac{i\dot{q}^2}{2} - \frac{ifF^2}{2} \right) \propto \int f^{-1/2}\mathcal{D}q\, \exp\left( \frac{i\dot{q}^2}{2} \right),$$ which corresponds to the following operator ordering for the Hamiltonian: $$\widehat{H} = \frac{1}{2} \widehat{f}^{1/4}\widehat{p}\widehat{f}^{-1/2}\widehat{p}\widehat{f}^{1/4}.$$

But what about the generic nonlinear case?

On another note, is there a formalism where we can impose the Dirac bracket after quantization, rather than before?

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    $\begingroup$ I think that there's no "God-given" ordering you should apply. A general quantum theory is not specified "purely" by its classical limit. In the context of quantum field theory, all the renormalization issues may be interpreted as "generalized ordering effects", so all the uncertainty about the total coupling and the right subtraction of the infinite parts may be interpreted as an ordering effect that doesn't have any unique canonical solution. $\endgroup$ Commented Jan 25, 2011 at 18:12
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    $\begingroup$ A professor once said that going from the classical Poisson bracket to the quantum mechanical commutator, or that promoting classical observables to qm operators, always includes something akin to a leap of faith, because there is no rigorous way to get from classical mechanics to quantum mechanics. $\endgroup$
    – Lagerbaer
    Commented Jan 27, 2011 at 17:55

2 Answers 2


If you think that there are simple and unambiguous ways to construct theories, then you were misled by someone.

Every achievement was reached by trying different possibilities. Only more-or-less successful constructions were left in the end (different in different particular cases).


There's a nice little book that deals with this question of yours in a mathematically rigorous way: Mathematical Topics between Classical and Quantum Mechanics (Springer Monographs in Mathematics).

In the end of the day, it boils down to studying Symplectic geometry and defining more appropriately what is meant by 'quantization'.

You can look at Deformation (Weyl) quantization and Geometric Quantization, they attack your questions heads-on.


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