Classical Field Theory, a precursor

Thanks to Kevin for letting me post some of my thoughts and questions here; I’m a big fan of Kevin’s passion for theory, and I hope to learn a lot from contributing here. The following post can also be seen here

As Kevin mentioned, I’m a physicist by training, but my experience with theory has been limited to a few classes in graduate school. I’m now trying to catch up, so I’m going back to my old text books and starting from scratch… for me, that’s classical field theory. 

The starting point for studying quantum field theory (QFT) is to investigate classical fields, which we all know and love. Fluids, gravity, and electromagnetism can all be described using classical fields, and this comprises most of our introductory physics education. The important aspects to nail down before moving into QFT is that the theories need to be relativistic and formulated using Lagrangian/Hamiltonian formalisms. Here is a brief summary list of some important bits to remember (there are more), but this will form the starting point for some problems from “Introduction to Quantum Field Theory” by Peskin & Schroeder (hereafter P&S) that I’d like to discuss later:

1: A classical scalar field \phi(x^{\mu})  is a function that takes a space-time vector x^{\mu} = (ct, \vec{x}) and returns a scalar value. 

2: The dynamics of the field are governed by the equation(s) of motion, and if we want a relativistic theory (i.e., like nature), then the equation of motion should be Lorentz invariant. This is automatic if one derives the equations of motion from a Lagrangian that is a Lorentz scalar.

3: We also want a local theory, where the dynamics of the field at a point are governed only by the value of the field and it’s derivatives at that point. There is something deep and philosophical here, but at this point I’ll just say it.

4: Once we have our Lagrangian (density), \mathscr{L}(\phi, \partial_{\mu} \phi ) that is local and a Lorentz scalar, we can find the equations of motion by varying the action,  S = \int d^{4}x \mathscr{L}(\phi,\partial_{\mu}\phi), yielding the Euler-Lagrange equations \partial_{\mu}(\frac{\partial \mathscr{L}}{\partial_{\mu} \phi }) - \frac{\partial \mathscr{L}}{\partial \phi} = 0. The derivative operator here is: \partial_{\mu} = \partial/\partial x^{\mu} = (\partial/\partial ct, \vec{\nabla}).

One can raise or lower any of the indices using the metric tensor g_{\mu\nu}, which, following P&S has a signature diag(+1, -1, -1, -1). If we are considering fields in curved space-time, then the metric can deviate from the minkowski metric, and the derivatives would need to be covariant derivatives \partial_{\mu} \rightarrow \nabla_{\mu}. This last statement is simply my intuition at this point, as this is what I would do when considering a classical field in curved space-time.

Of course, the above (1-4) are also true of non-scaler classical field theories, such as the electromagnetic and gravitational fields (where the field variables are the vector potential A_{\mu} and metric tensor g_{\mu\nu} respectively), and there are a number of great problems that are fun and instructive to work through… I’ll be posting some of my favorites next.

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One Response to Classical Field Theory, a precursor

  1. Shea thanks for contributing I’ll be sure to comment on both sites! I have a number of thoughts/questions with regards this topic so I’ll just start:

    (1) I’m rather excited to have a chance to ‘talk’ (i.e. post/comment) about how the invariance of the theory under the Poincare group yields, via representations, particles. I know representation theory plays a large role in non-relativistic QM as well but it’s been on the back burner.

    (2) It should be noted that QFT is background dependent in that it assumes a flat spacetime ahead of time as opposed to GR which presupposes no initial geometry. I’m aware that work has been done over the years to smuggle in quantum mechanics onto curved spacetime, though I know nothing about it other than the assumed metric is static I’m fairly sure.

    (3) I’m glad you posted something about QFT being local because this is something I’d like to learn more about. Also in reading about TQFT it’s been said that TQFTs are not local and that GR in 2+1 dimensions has no local degrees of freedom? There are apparently consequences for causality.

    (4) Have you seen Baez’s book on gauge theories and knots: “Gauge Fields, Knots, and Gravity?” I’m chipping my way through it but it’s very accessible (and easy to find as a pdf online). It has a nice presentation of EM as a gauge theory. So it’s win-win, classical field, Lorentz invariant and an introduction to a gauge theory.

    I’m really looking forward to these posts!

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