Mathematical Quantum Field Theory - Field Variations - Comments

[Urs Schreiber]

Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Field Variations

Continue reading the Original PF Insights Post.

 

[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"

 

[Urs Schreiber]

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?

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.

Or should we say "jet sub-variety"?

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.

In leadup to Prop 4.5: maybe say "physical jet bundle" for a jet bundle restricted by equations of motion?
...
the distinction needs clearer terminology, imho.

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.

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

Thanks! Fixed now.

 

[strangerep]

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 to Prop.4.5".

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. (?)

 

[Urs Schreiber]

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. (?)

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 $p + 1 = 0 + 1$.)

 

[strangerep]

Near the end of example 4.11:

Urs Schreiber[/URL]]$$d(\delta\phi^a) ~=~ - \delta(d\phi^a) ~=~ \dots $$In words this says that “the spacetime derivative of the variation of the field is the variation of its spacetime derivative”.

Is this just sloppy wording, or does the minus sign somehow not mean anything here?

 

[Urs Schreiber]

does the minus sign somehow not mean anything here?

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

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