Challenges in Merging GR and QM from a Lagrangian Perspective

[bolbteppa]

My first of many GR questions :

If electromagnetism can be incorporated into the standard model, where I think of electromagnetism as derivable from an action principle ala Landau-Lifshitz, at what point does this process stop working when you use gravitation? I was thinking the difficulty lies in the fact that the theory of gravitation in Landau & Lifshitz derives from the equivalence principle implying that one can incorporate curved space into your lagrangian by varying your metric (i.e. this difference in exposition of theory causes problems) yet this link develops the basic theory of gravitation completely analogously to the path followed by Landau & Lifshitz in their electromagnetism section, so I guess there is a more substantial reason problems arise & I'm just wondering what these are. Another way to ask this question would be: why don't GR & QM merge when you look at things from the perspective of Lagrangians? Thanks!

(If I give the impression of knowing what I'm talking about I barely do, still just learning!)

 

[WannabeNewton]

GR has a well-defined classical action from which the EFEs follow as a result of the variational principle (e.g. the Hilbert action plus the action for matter fields). The existence of a classical action doesn't imply that it has a natural foray into QFT. See for example part VIII of Zee's QFT book wherein one of the most basic issues is the non-terminating nature of the expansion of the Hilbert action in powers of $h_{\mu\nu}$.

 

[bolbteppa]

Wow! Thanks, I get the gist of it now. That's an answer to a three year problem in my head right there, whoosh... Zee/WBN rocks! :thumbs:

 

[rubi]

When you (canonically) quantize a classical theory, you usually start from it's Hamiltonian formulation. You have position and momentum variables, a Hamiltonian and some contraints. All of these satisfy some Poisson bracket relations. For example if you quantize the harmonic oscillator, you have $x$ and $p$ and $H=\frac{p^2}{2m}+\frac{m\omega^2 x^2}{2}$ and $\{x,p\}=1$. Quantizing this theory means (very roughly speaking) choosing a Hilbert space and representing these variables on this Hilbert space, keeping their Poisson bracket relations intact (as far as possible) and then solving the constraints if there are any. In the case of the harmonic oscillator for example, you simply choose $\mathcal H = L^2(\mathbb R)$, $\hat x$ as the multiplication operator with $x$, $\hat p=-\mathrm i\hbar\frac{\mathrm d}{\mathrm dx}$ (which gives you $[\hat x,\hat p]=\mathrm i\hbar$) and $\hat H=\frac{\hat p^2}{2m}+\frac{m\omega^2 \hat x^2}{2}$. In the case of GR, the first thing one would try would be to use the so called ADM variables. It turns out that the Hamiltonian is $0$ and you get additional constraints (the "Hamiltonian constraint" and the "Diffeomorphism constraint") that satisfy complicated Poisson brackets. Up to the present day, nobody has succeeded in finding a representation of this algebra and thus there is no well-defined quantum theory.

Approaches like Loop Quantum Gravity try to solve this problem by using different variables (Ashtekar variables).

 

[bolbteppa]

I don't understand that process, in classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & symmetric"]), then extremizing it gives the equations of motion. Alternatively one can find a first order PDE for the action as a function of it's endpoints to obtain the Hamilton-Jacobi equation, & the Poisson bracket formulation is merely a means of changing variables in your PDE so as to ensure your new variables are still characteristics of the H-J PDE (i.e. solutions of the EOM - see No. 37). All that makes sense to me, we're extremizing a functional to get the EOM or solving a PDE which implicitly assumes we've already got the solution (path of the particle) inside of the action that leads to the PDE. However in quantum mechanics you apparently just take the Hamiltonian (the Lagrangian in canonical coordinates) & mish-mash this with ideas from changing variables in the Hamilton-Jacobi equation representation of your problem so that you ensure the coordinates are characteristics of your Hamilton-Jacobi equation (i.e. the solutions of the EOM), then you put these ideas in some new space for some reason (Hilbert space) & have a theory of QM. Based on what I've written you are literally doing the exact same thing you do in classical mechanics in the beginning, you're sneaking in classical ideas & for some reason you make things into an algebra - I don't see why this is necessary, or why you can't do exactly what you do in classical mechanics? Furthermore I think my questions have some merit when you note that Schrödinger's original derivation involved an action functional using the Hamilton-Jacobi equation. Again we see Schrödinger doing a similar thing to the modern idea's, here he's mish-mashing the Hamilton-Jacobi equation with extremizing an action functional instead of just extremizing the original Lagrangian or Hamiltonian, analogous to modern QM mish-mashing the Hamiltonian with changes of variables in the H-J PDE (via Poisson brackets).

What's going on in this big Jigsaw? Why do we need to start mixing up all our pieces, why can't we just copy classical mechanics exactly - we are on some level anyway, as far as I can see... I can understand doing these things if they are just convenient tricks, the way you could say that invoking the H-J PDE is just a trick for dealing with Lagrangians & Hamiltonians, but I'm pretty sure the claim is that the process of quantization simply must be done, one step is just absolutely necessary, you simply cannot follow the classical ideas, even though from what I've said we basically are just doing the classical thing - in a roundabout way. It probably has something to do with complex numbers, at least partially, as mentioned in the note on page 276 here, but I have no idea as to how to see that & Schrödinger's original derivation didn't assume them so I'm confused about this.

Would all this jumbling about not be at the root of some of the problems? If things like the Yang-Mills action are constructed in terms of a theory which is built on a jumbling of ideas, a jumbling that at one step adds something new which leads to QM & QFT etc... differing from classical ideas, I'd love to know what it is, why it's absolutely necessary or if it isn't & that you can in fact construct QM completely analogously to classical mechanics then what causes problems with GR down that route. For instance, in Zee he says the Yang-Mills action terminates, but this termination may be due to the sneaking in of ideas, maybe they all don't terminate if you construct those theories from a theory of QM that copies field theory as closely as it can (by this I mean a field theory where the E-M & Gravitational field theory are developed along the lines in L&L or here).

 

[rubi]

The goal is to construct a quantum theory that has general relativity as it's classical limit. There are usually many such theories and there is no way to decide which one of these is right from just theoretical considerations (as far as we know), so there's a lot of arbitraryness in the process of quantization. We have to explore as many theories as we can in order to have a chance to find one that actually describes real experiments correctly.

What is a quantum theory? A quantum theory is a theory that has a Hilbert space of states that describe your physical system and a number of observables that are represented as self-adjoint operators on that Hilbert space. These operators form an algebra (you can add them and multiply them). The point is that once you know the algebraic relations between your operators, you have already fixed a large part of your quantum theory, so it is a good idea to start from guessing the right algebra. There are some hints on how to choose the right algebra: If you postulate $[\widehat A,\widehat B] = \mathrm i\hbar\widehat{\{A,B\}} + O(\hbar^2)$, the theory has a chance to have the right classical limit and it also ensures a Heisenberg uncertainty principle for your canonical variables.

There are of course other approaches to quantization that may also work. I'm just describing the canonical approach here, because it is the traditional approach that people have tried first (see Wheeler-DeWitt gravity).

 

[bolbteppa]

Thanks a lot for the input, if there are multiple methods of quantization then my questions may have some something to them, I guess I'll go to the QM forum with them & return if anything relevant occurs

 

[atyy]

GR and QM do merge from the perspective of Lagrangians, and GR is fine as a quantum field theory at low energies (well below the Planck scale).

http://arxiv.org/abs/1209.3511

From the point of view of GR as a quantum field theory of massless spin 2, there is even an argument that the equivalence principle can be derived (whereas in classical GR it has to be postulated).

http://arxiv.org/abs/1007.0435 (section 2.2.2)