# Does the vacuum have a zero Seebeck coefficient?

I was considering how at very high energies (e.g the Schwinger Limit) the vacuum starts having properties we would normally associate with "materials", such as non-linear polarizibility. The Seebeck coefficient measures the degree to which a temperature gradient creates an electric field in a material, and I was wondering whether a similar phenomenon could occur in the vacuum (e.g. through spontaneous pair production).

There are two sort of roadblocks to this question. First, having a meaningful notion of "temperature" of vacuum. To be clear, I mean vacuum as a region of space at its zero point with regards to radiation, no particles besides the virtual particles. But this leaves nothing to have a varying temperature, and certainly introducing charged particles would be cheating -- so let's say, maybe, that we have a gas of neutrinos/anti-neutrinos whose average kinetic energy varies along some gradient. This is clearly not really a vacuum any more, but maybe we could loosely say that it's a vacuum with regards to charge and electromagnetic radiation.

Second, the Seebeck coefficient is a charge asymmetric quantity. This isn't a problem in normal, metal materials, because the electron/proton asymmetry allows there to be a net voltage. In a vacuum, since there's no charge asymmetry in the setup of the matter, the Seebeck coefficient would necessarily be zero. Thus any process that would lead to it having a nonzero coefficient would need to be exploiting CP violation.

If this question is still ill-defined or trivial, I apologize. I might not just know enough QFT yet to understand. :)

• So you want to create an electric field... but without an electromagnetic field? OK... that sounds a bit like a contradiction, though? Other than that the question is interesting... does a high temperature gradient kick off a spatially varying DC field? – CuriousOne Jun 30 '16 at 1:19
• In some sense, the Seebeck effect is "creating an electric field without an electromagnetic field" -- in the sense that there is no macroscale field, just the thermal fluctuations in field. But then somehow making the magnitude of fluctuations vary across space -- causes the average of the fluctuations to become nonzero, and a macroscale field to emerge! The vacuum also has no macroscale field, but always has zero-point fluctuations. – Alex Meiburg Jun 30 '16 at 9:54
• That's a very weird interpretation which I don't share. An em field is an em field is an em field. I don't think your idea of zero-point fluctuations is correct, either. – CuriousOne Jun 30 '16 at 15:46