Mathematical Quantum Field Theory - Observables - Comments

[Urs Schreiber]

Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Observables

Continue reading the Original PF Insights Post.

 

[strangerep]

Typo near "linear off-shell observables": $$A(\Phi_1+\Phi_2) ~ A(\Phi_1)+A(\phi_2)$$ I.e., "=" sign is missing, and 2nd phi is lower case.

And further down... "field field histories"

Also: "...in that that subset of field histories..."

 

[Urs Schreiber]

Typo

Thanks! All fixed now.

 

[strangerep]

Another typo: hiorizontal

 

[Urs Schreiber]

Another typo

Thanks again! Also fixed now.

 

[Duong]

I have troubles with the comment about observables on fermionic fields (under the definition of observables)
1. Why $Obs = [\Gamma_\Sigma(E_{odd}) , \mathbb{C}]$ has only 1 point?

I assume that when we talk about "points", we're using $Obs(\mathbb{R}^0)$. However,
$$
\begin{align*}
Obs(\mathbb{R}^0) &= \{\mathbb{R}^0 \rightarrow Obs \} \\
&= \{\mathbb{R}^0 \rightarrow [\Gamma_\Sigma(E) , \mathbb{C}] \}\\
&= \{\mathbb{R}^0 \times \Gamma_\Sigma(E) \rightarrow \mathbb{C} \} \\
&= \{\Gamma_\Sigma(E) \rightarrow \mathbb{C} \}
\end{align*}
$$
(this is true for both $E = E_{even}$ and $E=E_{odd}$.) We know that $\Gamma_\Sigma(E_{odd})$ has only 1 "point" (the zero-section), and so hand-wavvily I would assume that there are several morphisms going from that unique "point" to $\mathbb{C}$ which correspond to several "points" in $Obs = [\Gamma_\Sigma(E_{odd}) , \mathbb{C}]$.

2. What is the map $(\theta \mapsto \theta A): \mathbb{R}^{0|1} \rightarrow Obs$? (The notation seems suggesting but I could not yield any conclusion.)

I am inclined to think this is an extension/consequence of the correspondence between a morphism $\psi_{(-)}:\mathbb{R}^{0|1} \rightarrow \Gamma_\Sigma(E_{odd})$ and a morphism $\phi_{(-)}:\mathbb{R}^{0} \rightarrow \Gamma_\Sigma(E_{even})$, but I haven't been successful to see how.

 

[Urs Schreiber]

I have troubles with the comment about observables on fermionic fields (under the definition of observables)

Thanks for catching this. Indeed this comment was wrong as stated. I was trying to say that these global points of the space of field histories of the Dirac field always yield the zero linear observable for the fermions (i.e cannot observe $\mathbf{\Psi}^\alpha(x)$), but of course they do see the even powers, such as $\mathbf{\Psi}^\alpha(x) \cdot \mathbf{\Psi}^\beta(x)$. I have corrected the statement in that comment and added a new example that says this in some detail, now here

 

[Duong]

Thanks! I think I got what you meant. Then perhaps still in that comment there're a few typos:

1. The plot $\mathbb{R}^{0|1} \rightarrow Obs$ should be specified by $c\mapsto \theta \mathbf{\Psi}^\alpha$.
2. (in the line below) The bosonic-observable component should be $\mathbf{\Psi}^\alpha$ instead of $\Pi$.

I think it is useful to add that the bosonic observable which originally is of even degree now is regarded as of odd-degree (so that $\theta \mathbf{\Psi}^\alpha$ is of even degree), and this fact is made possible by Prop. 3.51. Also, I think it is very nice to keep the mentioning of "the zero linear observable" if we insist on thinking of the observables as "points".

 

[Urs Schreiber]

Then perhaps still in that comment there're a few typos:

Right, thanks again. Should be fixed now.