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If $| \alpha >$ represents a coherent state (the normalized right eigenstate of the destruction operator $a$ in Quantum Mechanics; $\alpha$ is a complex number), then it is known that:

\begin{equation} \int | \alpha >< \alpha| \frac{d^2\alpha}{\pi} = I \end{equation} where $I$ refers to the identity operator.

Can any operator acting on the appropriate Hilbert space be represented in the Glauber-Sudarshan P-representation, and if it can, how to prove that this is the case? (I am especially interested about the representation of density operators)

By the Glauber-Sudarshan representation, I mean the following: \begin{equation} \int P(\alpha,\alpha^*)|\alpha><\alpha|\frac{d^2 \alpha}{\pi} \end{equation}

Both the integrals are over the entire complex plane.

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  • $\begingroup$ I have tried doing so, but I am unable to get anything out of it. Please guide me with this; thanks @AccidentalFourierTransform. $\endgroup$ – Harsha Mar 29 '16 at 18:52
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    $\begingroup$ I think my answer here has the formulae you want. $\endgroup$ – ACuriousMind Mar 29 '16 at 19:09
  • $\begingroup$ Optical equivalence theorem . $\endgroup$ – Cosmas Zachos May 1 '16 at 22:55
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The answer is no, and the details are clearly spelled out in Glauber's Les Houches lectures (circa 1964). Glauber introduces a "T-representation" which can represent any operator in the Fock space of harmonic oscillator states, a less general "R-representation" which can represent any density operator, and the still less general "P-representation" which can "represent virtually all [states] studied in optics".

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    $\begingroup$ Thanks for answering. Could you please include a link for the same, as I am unable to locate those lectures? $\endgroup$ – Harsha May 3 '16 at 14:17

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