# Insights Mathematical Quantum Field Theory - Field Variations - Comments

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1. Nov 12, 2017

### Urs Schreiber

Last edited by a moderator: Nov 12, 2017
2. Nov 13, 2017

### strangerep

In leadup to Prop 4.5: maybe say "physical jet bundle" for a jet bundle restricted by equations of motion?
Or should we say "jet sub-variety"?

I know you don't really get into this until ch5 where (I guess?) you impose(?) a specific Lagrangian (action?) on a generic jet bundle, restricting the latter to the ("on-shell"?) subspace satisfying the EoM arising from the Lagrangian? Still, the distinction needs clearer terminology, imho.

Btw, there's a typo in defn 4.7: "Monkowski"

3. Nov 14, 2017

### Urs Schreiber

Yes! All this is now in 5. Lagrangians.

There is considered the "shell" as a subspace $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ of the jet bundle, the vanishing locus of the Euler-Lagrange form, as well as the "prolonged shell" $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ which is the smaller vanishing locus also of all the "differential consequences" of the equations of motion.

In good cases this is a sub-manifold/sub-variety (or rather sub-locally-pro-supermanifold), but this is not necessary for the theory to proceed. The subspace always exists as as super smooth set, and that is all we need.

Okay, could you give me more precise coordinates in which line you'd like to see the wording improved? I am not sure I see which line you have in mind "in leadup toProp.4.5". Thanks.

Thanks! Fixed now.

4. Nov 14, 2017

### strangerep

I'm not sure either. Let me read your "Lagrangians" chapter first, and then (possibly) come back to this.

Btw, my background in this comes from dynamical symmetries in ordinary classical mechanics, where one works within a much simpler version of a generic jet bundle, i.e., particle position(s) and all time derivatives thereof. I'm not sure whether one should still call that a "jet bundle", though. (?)

5. Nov 15, 2017

### Urs Schreiber

Yes, this is the jet bundle of the bundle $\mathbb{R}^1 \times X \overset{pr_1}{\longrightarrow}\mathbb{R}^1$ where $\mathbb{R}^1$ is thought of as the time axis, and where $X$ is the manifold inside which the particles roam. (This is field theory in dimension $p + 1 = 0 + 1$.)

6. Nov 29, 2017

### strangerep

Near the end of example 4.11:
Is this just sloppy wording, or does the minus sign somehow not mean anything here?

7. Nov 29, 2017

### Urs Schreiber

We are looking at a differential 2-form, which is the wedge product of a field variation $\delta \phi^a$ with the change $d x^\mu$ in spacetime position. As always for differential forms, we keep track with a sign of the orientation of the infinitesimal volume spanned by this 2-form.

You may change the order of the two 1-forms in the wedge product and equivalently write

$$d (\delta \phi^a) = d x^\mu \wedge \delta \phi^a_{,\mu}$$