Consider the real scalar field $\phi(x,t)$ on 1+1 dimensional space-time with some action, for instance
$$ S[\phi] = \frac{1}{4\pi\nu} \int dx\,dt\, (v(\partial_x \phi)^2 – \partial_x\phi\partial_t \phi), $$
where $v$ is some constant and $1/\nu\in \mathbb Z$. (This example describes massless edge excitations in the fractional quantum Hall effect.)
To obtain the quantum mechanics of this field, there are two possibilities:
- Perform canonical quantization, i.e. promote the field $\phi$ to an operator and set $[\phi,\Pi] = i\hbar$ where $\Pi$ is the canonically conjugate momentum from the Lagrangian.
- Use the Feynman path integral to calculate all expectation values of interest, like $\langle \phi(x,t)\phi(0,0) \rangle$, and forget about operators altogether.
My question is
Are the approaches 1. and 2. always equivalent?
It appears to me that the Feynman path integral is a sure way to formulate a quantum field theory from an action, while canonical quantization can sometimes fail.
For instance, the commutation relations for the field $\phi$ in the example above look really weird; it is conjugate to its own derivative $\Pi(x,t) = -\frac{1}{2\pi\nu}\partial_x\phi(x,t)$. The prefactor is already a little off. For this to make sense, we have to switch to Fourier transformation and regard the negative field modes as conjugate momenta, $\Pi_k=\frac{1}{2\pi\nu}(-ik)\phi_{-k}$.
A more serious example: it seems to me that the Feynman integral easily provides a quantum theory of the electromagnetic gauge field $A_\mu$ whereas in canonical quantization, we must first choose an appropriate gauge and hope that the quantization does not depend on our choice.
Could you give a short argument why 2. gives the right quantum theory of the electromagnetic field? (standard action $-\frac1{16\pi} F^{\mu\nu}F_{\mu\nu})$
Best Answer
This type of problems is often referred to as constrained mechanical system. It was studied by Dirac, who developed the theory of constrained quantization. This theory was formalized and further developed by Marseden and Weinstein to what is called "Symplectic reduction". A particularly illiminating chapter for finite dimensional systems may be found in Marsden and Ratiu's book: "Introduction to mechanics and symmetry".
When the phase space of a dynamical system is a cotangent bundle, one can use the usual methods of canonical quantization, and the corresponding path integral. However, this formalism does not work in general for nonlinear phase spaces. One important example is when the phase space is defined by a nonlinear surface in a larger linear phase space.
Basically, Given a symmetry of a phase space, one can reduce the problem to a smaller phase space in two stages
This procedure reduces by 2 the dimensions of the phase space, and the reduced dimension remains even. One can prove that if the original phase space is symplectic, so will be the reduced phase space.
May be the most simple example is the elimination of the center of mass motion in a two particle system and working in the reduced dynamics.
There is a theorem by Guillemin and Sternberg for certain types of finite dimensional phase spaces which states that quantization commutes with reduction. That is, one can either quantize the original theory and impose the constraints on the quantum Hilbert space to obtain the "physical" states. Or, on can reduce the classical theory, then quantize. In this case the reduced Hilbert space is automatically obtained. The second case is not trivuial because the reduced phase space becomes a non-linear symplectic manifold and in many cases it is not even a manifold (because the group action is not free).
Most of the physics applications treat however field theories which correspond to infinite dimensional phase spaces and there is no counterpart of the Guillemin-Sternberg theorem. (There are works trying to generalize the theorem to some infinite dimensional spaces by N. P. Landsman). But in general, the commutativity of the reduction and quantization is used in the physics literature, even though a formal proof is still lacking. The most known example is the quantization of the moduli space of flat connections in relation to the Chern-Simons theory.
The most known example of constrained dynamics in infinite dimensional spaces is the Yang-Mills theory where the momentum conjugate to $A_0$ vanishes. It should be mentioned that there is an alternative (and equivalent) approach to treat the constraints and perform the Marsden-Weinstein reduction through BRST, and this is the usual way in which the Yang-Mills theory is treated. In this approach, the phase space is extended to a supermanifold instead of being reduced. The advantage of this approach is that the resulting supermanifold is flat and methods of canonical quantization can be used.
In the mentioned case of the scalar field, the phase space may be considered as an infinite numer of copies of $T^{*}\mathbb{R}$. The relation $\Pi = \partial_x \phi$ is the constraint surface. In a naive dimension count of the reduced phase space dimensions one finds that in every space point the 1+1 dimensions the phase space (The field and its conjugate momentum) are fully reduced by the constraint and its symmetry generator. Thus we are left with a "zero-dimensional" theory. I haven't worked out this example, but I am quite sure that if this case is done carefully we would have been left with a finite number of residual parameters. This is a sign that this theory is topological - which can be seen through the quantization of the global coefficient".
Update:
In response to Greg's comments here are further references and details.
The following review article (Aspects of BRST Quantization) by J.W. van Holten explains the BRST quantization of electrdynamics and Yang-Mills theory (Faddeev-Popov theory) as constrained mechanical systems.The article contains other example from (finite dimensioal phase space) quantum mechanics as well.
The following article by Phillial Oh. (Classical and Quantum Mechanics of Non-Abelian Chern-Simons Particles) describes the quantization of a (finite dimensional) mechanical system performing the symplectic reduction directly without using BRST. Here, the reduced spaces are coadjoint orbits (such as flag manifolds or projective spaces). The beautiful geometry of these spaces is very well known and this is the reason why the reduction can be performed here directly. For most of the reduced phase spaces, such an explicit knowledge of the geometry is lacking. In field theory, problems such as quantization of the two dimensional Yang-Mills theory possess such an explicit description, but for higher dimensional I don't know of an explicit treatment (besides BRST).
The following article by Kostant and Sternberg, describes the equivalence between the BRST theory and the direct symplectic reduction.
Now, concerning the path integral. I think that most of the recent physics achievements were obtained by means of the path integral, even if it has some loose points. I can refer you to the following book by Cartier and Cecile DeWitt-Morette, where they treated path integrals on non-flat symplectic manifolds and in addition, they formulated the oscillatory path integral in terms of Poisson processes.
There is a very readable reference by Orlando Alvarez describing the quantization of the global coefficients of topological terms in, Commun. Math. Phys. 100, 279-309 (1985)(Topological Quantization and Cohomology). I think that the Lagrangian given in the question can be treated by the same methods. basically, the quantization of these terms is due to the same physical reason that the product of electric and magnetic charges of magnetic monopoles should be quantized. This is known as the Dirac quantization condition. In the path integral formulation, it follows from the requirement that a gauge transformation should produce a phase shift of multiples of $2\pi$. In geometric quantization, this condition follows from the requirement that the prequantization line bundle should correspond to an integral symplectic form.