# Mathematical Quantum Field Theory - Observables - Comments

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Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Observables

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bhobba and dextercioby

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..."

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Urs Schreiber
Gold Member
Typo

Thanks! All fixed now.

strangerep
Another typo: hiorizontal

Gold Member
Another typo

Thanks again! Also fixed now.

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.

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Urs Schreiber
Gold Member
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 and dextercioby
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".

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Urs Schreiber and dextercioby