I have a problem with understanding how the resolution of the identity of an operator is presented in some literature for physicists.


I'm a student of mathematics, and I understand the notion of a spectral measure (which is somethimes called the resolution of identity) and also have some knowledge in spectral theory (for normal operators).


Here is my brief explanation what do I understand:
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Let $H$ be Hilbert space (with an inner product linear w.r.t 2nd coordinate) with an orthonormal basis $(e\_{n})$
and define a linear operator (diagonal operator)
$$ A = \sum\_{i \geq 1} \lambda\_i \left|e\_i\right>\left<e\_i\right|,$$
where $(\lambda\_i)$ is a sequence of complex numbers (the properties of this sequences determine the properties of $A$ such as boundedness, selfadjointness, compactness etc.), to make sense of $A$ we assume that the above series converges in SOT, I also used Dirac bra-ket notation. The associated spectral measure of $A$ is defined via
$$ E(\Delta) = \sum\_i \mathbf{1}\_{\Delta}(\lambda\_i) \left|e\_i\right>\left<e\_i\right|,$$
where $\Delta$ is an element of Borel sigma field over the spectrum of $A$, and we have that
$$\left<x| Ay\right> = \int\_{\sigma(A)} \lambda \left<x| E(d\lambda)y\right> \ \ (x,y \in H).$$


Very often physicists would use the following notation for $A$ which acts on an element $\psi \in H$
$$ A\left|\psi\right> = \sum\_{i} \lambda\_i \left|i\right>\left<i\right|\left|\psi\right>.$$


My problems with notation
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I started reading some notes, books about quantum field theory, and often it is written that, the identity operator $I$ on some (separable) Hilbert space $H$, has the expansion, called the resolution of the identity
$$ I= \int dq^{\prime} \left|q^{\prime}\right>\left<q^{\prime}\right|,$$
I don't know whether it matters here but $\{\left|q\right>\}$ is supposed to be a complete set of states.


Reference: <http://eduardo.physics.illinois.edu/phys582/582-chapter5.pdf> bottom of p. 129.


My question
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Is the notion of the above resolution of the identity the same as an integral w.r.t a spectral measure ($I$ is a diagonal operator)? If yes, how should I understand the above notation. If no, what do they actually mean by this resolution of the identity and how do they define this integral. I noticed that in a lot of book concerning quantum mechanics there are many calculations, but not very many definitions and assumptions, which makes stuff hard to understand for a mathematician.



Quantum Field theory - integral notation
The answer is Yes. The interpretation of the notation is quite straight forward: $dq'|q'\rangle\langle q'| = E(dq')$. We need to presume that $E$ is the spectral measure of an operator $Q' = \int q' E(dq') = \int dq'\, q' |q'\rangle\langle q'|$. The only aspect that doesn't necessarily mesh well with your question, as written, is the fact that you've defined a spectral measure $E$ only for operators whose spectrum consists of eigenvalues, since $A|e\_i\rangle = \lambda\_i |e\_i\rangle$. Spectral measures can be defined for operators with any kind of spectrum, including continuous.


