# Are Matsubara states pure states?

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''?

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.

• I generally agree that Wick rotation + periodic boundary conditions is a nontrivial change in topology. But I am certain that periodic boundary conditions alone would not prevent one from defining a complete set of (unitary) basis vectors, and this leads me to suspect that it is the Wick rotation that is responsible for the mixing of states because it spoils unitarity. However I am still confused on what, exactly, the Matsubara state $e^{i\omega_n\tau-kx}$ (that solves the imaginary time KG equation) represents. Is there a probability density associated with it? If so, what does it measure? – alphanzo Nov 13 '14 at 8:50
• That confuses me too -- I thought it might still be some linear combination of pure states (though any semblance of normalization is now lost) -- which tempts me to believe that the decohering might depend on the compactification. I'm not sure if one can define unitary basis vectors after compactification (without Wick rotation). There's no unique way to get from one time to another (you could move into the past or the future) :-? – Siva Nov 13 '14 at 10:50
• If you think of the "in-out" S-matrix (density-matrix like object) then once you compactify and impose boundary conditions for going around that circle, you're saying that the "initial" and "final" states must be related in a particular way. It sort of removes the off-diagonal elements of the matrix. I interpret that as decohering. – Siva Nov 13 '14 at 10:51
• Fair point, and I think you are right that imposing b.c. can serve to trace out unwanted'' elements, and so provide a source of decoherence. Interestingly, while the matsubara states are unbounded and cannot define a density, they still provide a complete basis that one can construct a Green's function from. My current position is to abandon all hope in interpreting these states as amplitudes and instead look at what their different combinations say about the Green's function they are defining. I will update on anything interesting I find. – alphanzo Nov 13 '14 at 21:01
• One more thing realized: We use Wick rotation all over the place, to calculate loop integrals in zero temperature QFT. At zero temperature, one better not have any decoherence -- that should be a coherent sum. – Siva Nov 14 '14 at 22:44