Clavelli-Shapiro Trick

Computing the expected energy for a harmonic oscillator in a heat bath isn't bad. Other observables aren't too bad either, but here is a trick that will generalize.

First introduce new b operators obeying the same commutation relations.

To compute the thermal expectation value with parameter y ( fugacity goes with chemical potential, I mean the one that corresponds to temperature), you take the regular expectation value of your operator, but replace all the a's and their adjoints by

respectively.

3 Proofs in A.6 at least in a useful special case.

This seems insanely useful depending on how far it is pushed.

Spooky Action from the FUTURE!

I have 2 machines that produce entangled pairs of photons. Call them thing 1 and thing 2.

Thing 1 sends a photon to Dick. Thing 2 sends a photon to Sally. They both send the other photons to the cat.

Dick and Sally measure their photons in whatever basis they choose.

Later, the cat decides whether or not to entangle the two photons it got.

If the cat did nothing, Dick and Sally's measurements have nothing to do with each other. There is no way they could talk to each other.

But if the cat did entangle them, the two measurements are correlated. That's weird, the cat acted in the future and influenced what Dick and Sally saw before.

An actual experiment

http://arxiv.org/ftp/arxiv/papers/1203/1203.4834.pdf

It reminds me of those time travelling neutrinos that came back and loosened that cable so they wouldn't be found out. (Credit Kate)