# Insights Mathematical Quantum Field Theory - Observables - Comments

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1. Nov 19, 2017

### Urs Schreiber

2. Nov 19, 2017

### 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..."

Last edited: Nov 19, 2017
3. Nov 20, 2017

### Urs Schreiber

Thanks! All fixed now.

4. Nov 21, 2017

### strangerep

Another typo: hiorizontal

5. Nov 21, 2017

### Urs Schreiber

Thanks again! Also fixed now.

6. Jan 28, 2018

### 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.

Last edited: Jan 28, 2018
7. Jan 28, 2018

### Urs Schreiber

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

8. Jan 28, 2018

### 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".

Last edited: Jan 28, 2018
9. Jan 29, 2018

### Urs Schreiber

Right, thanks again. Should be fixed now.