# Are Matsubara states pure states?

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Generally in a non-interacting QFT one can solve the Klein-Gordon equation to get a (complete) set of states $\frac{e^{i\omega_k t-ikx}}{\sqrt{2\omega_k}}$. It is not clear to me how to construct the density matrix $\rho_0$ for zero-temperature QFT, but presumably it can be done, and the resulting matrix corresponds to a pure state (zero entropy).

On the other hand, at finite temperature one has to solve the imaginary'' time K-G equation and it yeilds a set of eigenmodes $e^{i(2\pi/\beta)n\tau-k_nx}$ for integer $n$. This state has dispersion, and as well, the entropy corresponding to the free-particle states is that of a Bose gas at temperature $\beta$.

My question is: what do the eigenmodes correspond to in this case? Are they complete, and if so, do they correspond to a mixed state''?

This post imported from StackExchange Physics at 2014-11-13 08:03 (UTC), posted by SE-user alphanzo
retagged Nov 13, 2014

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I would love to hear an authoritative and concrete answer, but till then, here's what I think:

The finite temperature (Matsubara) state is a mixed state (as one should expect, for thermal states). When one Wick rotates and compactifies the time direction into a circle (while imposing suitable BCs), I think one turns the sum in the partition function into a decoherent sum over energy levels -- as a trace over the diagonal elements in a density matrix.

The zero temperature state was a coherent superposition while the finite temperature state is a decoherent superposition. It's not completely clear to me whether this "decoherence" is happening from the compactification, or the Wick rotation. I'm tempted to imagine that the act of compactifying (hence changing the topology) makes the sum decoherent (the size just sets the energy scale for the thermal distribution). The Wick rotation should plausibly only change the weights in the sum from phases to Boltzmann suppression. But if one has a diagonal density matrix, it better have real eigenvalues, so maybe Wick rotation is a must before we compactify the time direction. (That might also help prevent closed time-like curves)

Like I said, I'm just thinking out aloud. I would highly appreciate if there were more responses answering the question or even comments elaborating on my thinking.

This post imported from StackExchange Physics at 2014-11-13 08:03 (UTC), posted by SE-user Siva
answered Nov 13, 2014 by (720 points)

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