How do I deal with a quantum field in the denominator?

I am wondering how to deal with an expression like $$\int d^4\theta \frac{1}{T + T^\dagger} \big( \dots \big)$$ If the denominator was of the form $1 + T + T^\dagger$, I could assume that $T \ll 1$ and expand the denominator in a Taylor series.

If more context helps, this expression pops up in $5D$ SuGra abelian gauge theory (http://arxiv.org/abs/hep-th/0106256, eq. (5) on page 2).

The authors of the above mentioned paper assume the modulus to be stable ($<T> \equiv R$) before performing the superspace integration. I do not want to do this and keep the dependence on the modulus.

In general, if you have a class of operators $\{A(s)\}_{s\in S}$, with $A(s) : D \to H$ for some common domain $D \subset H$ ($H$ being the Hilbert space of the theory), and with $S$ given by some set equipped with a positive measure $\mu$, it is possible to define an integrated operator: $$\int_S A(s) d\mu(s)$$ with the following steps.
Assuming that, for every $\phi\in D$, the map $$S \ni s \mapsto ||A(s) \phi||$$ is $\mu$ integrable and that for every $\phi\in D$ and $\psi \in H$, the map $$S \ni s \mapsto \langle \psi | A(s) \phi \rangle$$ is measurable, then $$H \times D \ni (\psi,\phi) \mapsto Q(\psi, \phi):= \int_S \langle \psi | A(s) \phi \rangle d\mu(s)\:.$$ is well defined, since $$|\langle \psi | A(s) \phi \rangle| \leq ||\psi|| ||A(s) \phi||\:$$ It is simply proved that $Q(\cdot, \cdot)$ is linear in the right-hand slot and it is anti-linear in the left-hand one, moreover: $$|Q(\psi, \phi)|\leq C||\psi|| \:.$$ Riesz' theorem easily implies that, for every $\phi\in D$ there is a unique vector, indicated by $$\int_S A(s) \phi d\mu(s)$$ such that, if $\psi\in H$: $$Q(\psi, \phi) = \left\langle \psi |\int_S A(s) \phi d\mu(s) \right\rangle \:.$$ By construction, since $Q$ is right-linear, one finds that $$D \ni \phi \mapsto \int_S A(s) \phi d\mu(s)$$ is linear, too.
Summing up, under quite mild hypotheses, there exists a unique linear operator, indicated by $\int_S A(s) d\mu(s)$ and defined on $D$, such that: $$\left\langle \psi |\int_S A(s) d\mu(s) \phi \right\rangle = \left\langle \psi |\int_S A(s)\phi d\mu(s) \right\rangle = \int_S \langle \psi | A(s) \phi \rangle d\mu(s)\tag{1}$$ for every $\phi \in D$ and $\psi \in H$. From these identities you can induce some properties from $A(s)$ to $\int_S A(s) d\mu(s)$. For instance, if the $A(s)$s are Hermitian, $\int_S A(s) d\mu(s)$ is. If $||A(s)|| <K <+\infty$ for some $K$ and all $s\in S$, then $\int_S A(s) d\mu(s)$ is bounded, and so on.
In your case $s=\theta$, I expect that all involved operators depend on $\theta$ in some way (the paper is quite obscure to me on these details), and you should use (1) to define the wanted operator: There is only one operator satisfying it. Obviously $1/(T+T^\dagger)$ has to be understood as $(T+T^\dagger)^{-1}$ (inverse operator).
• Thank you for this very interesting and well-written post. The integral is taken over Grassmann-valued superspace and the superfields like $T$ are defined as an exansion in these Grassmannian variables. What I am interested in is not directly the action of the integrated operator on some Hilbert space, but rather its exansion in superspace coordinates such that I can perform the Berezin integration. (This is also why I removed the tags you suggested, since they do not describe what I am looking for). Commented May 31, 2014 at 13:35