# How do I quantize a classical field theory

I have not been able to find any information about this on the Internet. I am a middle-schooler, 14, who self-studies physics, and I know up to and including ODEs, and some of the calculus of variations, enough so that I can take a variation of the action. However, I have absolutely zero budget, and no ability to get textbooks. So, I was wondering how a physicist would quantize a classical field theory, such as Gauss's law for gravity, which I'm currently trying to create a quantum field theory for, as part of my plan to create a possible theory of quantum gravity, by adding in the relativistic corrections later.

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Some of my favorite online lecture notes on QFT damtp.cam.ac.uk/user/tong/qft.html, pa.ucla.edu/content/eric-dhoker-lecture-notes (there is a link to the QFT notes in the second linked page). –  joshphysics Apr 17 '13 at 18:24
Great to see such enthusiasm for the subject! I would just say that you should make sure that you thoroughly understand how a mechanical system is quantized (if you haven't already done so) before you move onto fields. For QFT the links Josh gave are as good as you can get. Good luck! –  twistor59 Apr 17 '13 at 18:50
Love the enthusiasm as well. Don't forget Srednicki! If there is a university library near by check out the physics section. I recommend you master the operator formalism of quantum mechanics before moving to QFT. You'll need linear algebra, some PDEs, and you'll pick up some group theory on the way. If you don't know special relativity learn it now - using 4-vectors and tensors. If you get stuck because of gaps in your knowledge check out 't Hooft's subject list to see what needs to be filled in. –  Michael Brown Apr 30 '13 at 15:31
+1 for baller status. –  DilithiumMatrix Apr 30 '13 at 17:37
DUDE THATS AWESOME.... I'm 15 (just recently) and I am in almost the exact same place as you are... I know calculus and should be in calculus 2. (I have to do the boring stuff first, though, sadly.... –  Damon Blevins Mar 5 at 1:34

First off, props for even knowing the proper use of the terminology. I dare say most middle-schoolers probably think classical field theory is about listening to Mozart in a meadow.

Now, if I may start at the beginning...

You seem to know this already, but I'll restate it for pedagogical reasons. When someone says they have a classical field theory they want to quantize, that means they usually have a Lagrangian density in terms of a scalar, gauge, vector, spinor, or other type of field. Additionally, the Lagrangian usually obeys a known (that is known to them) type of symmetry (eg rotational invariance) and they usually have the equations of motion for it as well. Specific to your example, Gauss' Law for gravity is the equation of motion for the Newtonian gravity Lagrangian density (Wikipedia can provide this equation). In this Lagrangian, the scalar field $\phi$ is the important component, not a gravity field.

Now, assuming a physicist has a Classical field theory, one of the standard methods of quantizing it (promoting it to a Quantum Field Theory) is this:
1. Solve for a general case solution to $\phi$ if possible, this greatly eases the process.
2. Write the operator $\phi$ in terms of annihilation and creation operators (let's call them $a_k$ and $a_k^\dagger$ respectively).
As an example, if I managed to find a general solution for my $\phi$ was:
$$\phi(x)=e^{ikx}+e^{-ikx}$$ Then I could rewrite it as: $$\phi(x)=A(a_ke^{-ikx}+a_k^\dagger e^{ikx})$$ 3. Impose the cannonical (anti-)commutation relations on the theory. If the theory is written using bosons (and I believe yours is), then use commutators. If it uses fermions, then use anti-commutators. For your case, the cannonical commutation relations should be:

$$[\phi_a(\mathbf x),\Pi_b^0(\mathbf y)]~=~i\delta_{ab}\delta^3(\vec x-\vec y)$$ $$[a_{k},a_{k'}^\dagger]~=~\delta^3(\vec k-\vec k')$$

When I say "impose these relations" what I mean is that in step 2, you need to put normalization coefficients (coefficients like "A" simply used to get the right outcome in step 3) in front of the creation and annihilation operator terms. Then, when imposing the relations, you need to set LHS=RHS and solve for these coefficients such that the relations hold. It is important to note that these coefficients need not be constant, just that they be c-numbers (not functions of x nor operators). In fact, one of the most basic coefficients ends up with a $\omega_k$ in it, which is a function of the momentum, k.

Then you're done! If you can do steps 2 and 3, then your resulting theory is a quantum field theory. I'm sure others will let me know I've missed a method or two. If you think of something I've forgotten, let me know and I'll add it in.

Edit
It has been pointed out that I (somehow) forgot to mention the path integral formulation method of quantizing a theory. This is a conceptually harder method that requires a more advanced understanding (i.e. formal training) of theoretical physics. I am not including a full description of it here because I don't want to increase the level of complexity of this answer too much. In general, path integral is found to work better than the one I listed, but it is certainly not as straight forward (debatable). It also is inherently relativistic, which is an advantage over the more basic method (unless you intend to add in relativity later).

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You might also want to discuss how Gauss' Law in EM is a constraint in the Hamiltonian formulation, and how similarly Gauss' Law in -- say -- the Ashtekar formalism for GR is also a constraint... (Heh, sorry, I realized I might as well have asked you to explain how to canonically quantize gauge systems instead!) –  Alex Nelson Apr 30 '13 at 15:27
I honestly thought about doing that, but I felt the answer should be simple enough to be understood with relatively minimal formal schooling. I didn't know how to put that into simple wording. If you can and want to add it to the answer, go right ahead. –  Jim Apr 30 '13 at 15:31
I just like to watch you work ;) –  Alex Nelson May 1 '13 at 2:23
That's great and all, but what do you mean by imposing these relations? I've looked everywhere, and I just can't find an explanation. –  Darius Goad May 1 '13 at 15:55
Good point, I'll add that in. –  Jim May 1 '13 at 16:06