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- Photon statistics

In a given mode with an average number of photons ``##\bar{n}##, the photons are distributed around their average according to the formula

$$p_n = e^{-\bar{n}} \frac{\bar{n}^n}{n!}$$

The justification of this formula in quantum field theory involves considering field operators acting on a space with an indefinite metric, because it involves relations like ##[A_\mu, \dot{A}_\nu] = -ig_{\mu\nu} \delta(x - y)##. The S matrix involved in these calculations must leave the positive metric physical subspace invariant.

I wonder what is the right way to think about Fock spaces with an indefinite metric? These spaces occur in string theory because of the commutation rules like ##[\alpha_\mu, \alpha_\nu] = ig_{\mu\nu}##. The n! in the above formula is a consequence of bose statistics, and the ##\bar{n}^n## is a consequence of the fact that (a

$$p_n = e^{-\bar{n}} \frac{\bar{n}^n}{n!}$$

The justification of this formula in quantum field theory involves considering field operators acting on a space with an indefinite metric, because it involves relations like ##[A_\mu, \dot{A}_\nu] = -ig_{\mu\nu} \delta(x - y)##. The S matrix involved in these calculations must leave the positive metric physical subspace invariant.

I wonder what is the right way to think about Fock spaces with an indefinite metric? These spaces occur in string theory because of the commutation rules like ##[\alpha_\mu, \alpha_\nu] = ig_{\mu\nu}##. The n! in the above formula is a consequence of bose statistics, and the ##\bar{n}^n## is a consequence of the fact that (a

^{+})^{n}~ n^{n/2}, but the factor ## e^{-\bar{n}}## is mysterious. The mass shell condition in string theory involves relations of the form m^{2}~ N, where m^{2}is a kind of hamiltonian on Hilbert spaces with indefinite metric. What is the significance of Hilbert spaces with an indefinite metric in QFT and in string theory?
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