# Representing operators in the Glauber-Sudarshan P-representation

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:

$$\int | \alpha >< \alpha| \frac{d^2\alpha}{\pi} = I$$ 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: $$\int P(\alpha,\alpha^*)|\alpha><\alpha|\frac{d^2 \alpha}{\pi}$$

Both the integrals are over the entire complex plane.

• I have tried doing so, but I am unable to get anything out of it. Please guide me with this; thanks @AccidentalFourierTransform. – Harsha Mar 29 '16 at 18:52
• I think my answer here has the formulae you want. – ACuriousMind Mar 29 '16 at 19:09
• – Cosmas Zachos May 1 '16 at 22:55