First quantization, first pass
November 5, 2008 by Moshe
Suppose you want to solve a linear partial differential equation of the form 
, which determines some quantity 
in terms of its source 
. Here x could stand for possibly many variables, and the differential operator 
can be pretty much anything. This is a very general type of problem, not even specific to physics. An example in physics could be the Klein-Gordon equation, or with some more bells and whistles the Maxwell equation, which determines the electric and magnetic fields.
Let us replace this problem with the following equivalent one. If we find a function 
such that:
with the initial condition 
, and assuming the regularity condition 
, then it is easy to see that the function 
satisfies the original equation we set out to solve.
Now, this new equation for looks kind of familiar, if we are willing to overlook a few details. If we wish, we can think about as a time dependent wave function, with the parameter s playing the role of time. The equation for could then be interpreted as a Schrödinger equation, with the original operator playing the role of the Hamiltonian. We are ignoring a few issues to do with convergence, analytic continuation, and the related fact that the Schrödinger equation is complex, and the one we are discussing is not. Never mind, these are subtleties which need to be considered at a later stage.
The point is that we can now use any technique we learned in quantum mechanics to solve the original equation – path integral, canonical quantization, you name it. We can talk about the states 
and the Hilbert space they form, Fourier transform to get another basis for that Hilbert space, even discuss “time” evolution (that is, the dependence of various states on the auxiliary parameter s). We can get the state by summing over all paths of a “particle” with an appropriate worldline action and boundary conditions. Depending on the problem, we may be interested in various (differential) operators acting on , and they of course do not commute, resulting in uncertainty relations. You get the picture.
This technique is sometimes called first quantization, or Schwinger proper time method, or heat Kernel expansion. Whatever you call it, it has a priori nothing to do with quantum mechanics, there are no probabilities, Planck constant or any wavefunctions in any real sense. At this point we may be discussing the financial markets, population dynamics of bacteria, or simply classical field theory.
In the second pass, we can apply this idea to linear fields, generating solutions to various linear differential equations. Some of those equations are Lorentz invariant (Klein-Gordon, Dirac, Maxwell equations), but they have nothing to do with quantum mechanics, despite the original way they were referred to as “relativistic wave equations”. Once we add spin to the game, we start having the fascinating structures of (worldline) fermions and supersymmetry (not to be confused with spacetime fermions and supersymmetry), and we are also in a good shape to make the leap from classical field theory to classical string theory. Maybe I’ll get to that sometime…
Like this:
Like Loading...
There are a few more properties that make the heat-kernel so attractive: Often, you are interested in the special case where j is the delta function, in that case psi is called the Green function of O or the propagator for O. Of course, delta is not really a function but more singular, something mathematicians call a distribution. Similarly, in that case psi is also a distribution.
However, for any s>0, everything is smooth enough and psi(x,s) is actually an ordinary function of x.
Furthermore, if you have some sort of local symmetry like in gauge theory or in gravity, it turns out you can find gauge invariant expressions for psi and thus you do not have to use gauge dependent stuff like potentials.
Finally, it turns out, the one loop effective action can be expressed as integral ds/s psi(0,s). In this expression, all UV divergences show up only as divergences of the indefinite integral for s->0. The singularity can then very elegantly computed in a gauge invariant way using some recursive relation for psi.
As far as I understand, there are two approaches to QFT:
a) we can take a free relativistic particle, write the action for it and quantize it. That’s first quantization and that is how string is quantized (starting from the Nambu-Goto action).
b) we can start with a field from the very beginnin (second quantization).
Nice thing about (b) is that I can understand easily the effect of interactions, i.e., treat processes involving several particles, such as particle production in a strong external field, etc.
How does one deal with multiparticle processes in the first quantization?
Cheers
Dmitry, when you first-quantize the relativistic particle, you generate solutions to the classical Klein-Gordon equation. With worldline fermions you could do the same for the Dirac equation, and presumably also for the Maxwell and linearized Einstein equations (though I would be interested in seeing a reference to the latter, since I am curious how gauge invariance is dealt with). In all those cases this is “fake quantization”, you solve some auxiliary Schrodinger equation for the purpose of finding classical solutions.
Interactions in this formalism come from including explicitly vertices, then building tree graphs from propagators and those vertices. Quantum corrections come when including loops, exactly as in string perturbation theory.
The main advantage for “second” (i.e. field) quantization (annoying counting scheme for quantizations notwithstanding) is that non-perturbative physics is accessible. Unfortunately this is not yet fully accessible in string theory.
Hi Moshe
BTW what is the current status of the second quantization of the string (i.e. string field theory)?
It still attracts the attention of the string community? It is argued that is the only hope for a consistent non-perturbative formulation of a background independent string theory. I have the impression though that after ADS/CFT, the interest for this field is declining. Am i right?
BR
Hi Giotis, there is a constant interest in open string field theory, which in my eyes is similar in many ways to second quantized gauge theory (on a fixed manifold). As far as I know, not much is going on in the closed string field theory front, but maybe I am wrong.
My personal opinion, for what its worth, is that this is the wrong way to go, the approach is too tied to perturbation theory (and as a result, among other things, it is not background independent). This opinion only gets stronger with each example of fully non-perturbative (and background independent) formulation of string theory, like ads/cft, matrix theory, etc. etc., in all of them closed SFT seems to miss the interesting physics.
I hope I haven’t missed all the interesting discussion here, or maybe I should wait for the “second pass”.
In any case, the degrees of freedom in matrix models (probably “old” matrix models) allow for the description of a variable number of fundamental perturbation objects, already at the classical level (in the form of block-diagonal sectors). Is this an example of a system with particle-like excitations that doesn’t follow the first/second quantization pattern, or does the fact that the system have a classical description in terms of infinite-dimensional matrices hint at a hidden first quantization?