• Getting to the bottom of Noether’s theorem.

though it may not make sense without reading Section 4.

]]>Thanks! I see the KMS condition

at the top of page 4, but this is different than what I’m talking about.

]]>Entropy and the spectral action ]]>

Okay, now I get it. I like this “modular flow” idea a lot.

]]>Ah crap, my bad: apparently I’ve been hard-wired to use for the density matrix. To conform to your notation, should be replaced by everywhere in my previous post. Anyways, given a quantum system described by the density matrix , the modular hamiltonian is defined to be , while “modular time” is the parameter I called in my previous post. By this def’n, different density matrices on the same hilbert space have different associated modular hamiltonians and correspondingly generate different modular flows.

Spacetime doesn’t play any role here except in the rindler discussion, which I only mentioned because I find it helpful in building some geometric intuition for the action of modular flow. For example, in QM you could take an EPR pair and consider the reduced density matrix obtained by tracing over the first partner. Modular flow will then mix the pauli matrices acting on the second partner.

]]>I’ll have to think about what you’re saying. What’s ? I need to take what you’re saying and remove everything about spacetime, causal diamonds and see what’s left. All I’ve got is a density matrix and a bounded-below self-adjoint operator on the same Hilbert space. What’s “modular time” or the “modular Hamiltonian” in this context?

Glad to see that zero temperature corresponds to infinite coolness!

Yes: now the law of thermodynamics that used to say “it would require an infinite number of steps to reach absolute zero” becomes “it would require an infinite number of steps to reach infinite coolness”, which seems quite plausible.

]]>By modular time, I mean unitarily evolving operators using the modular hamiltonian rather than the ordinary hamiltonian (which is often referred to as “modular flow”):

,

where O is any operator on the hilbert space on which acts and I have suppressed all spacetime labels, which just go along for the ride. One nice property of modular flow is that it preserves the algebra of operators associated to any causal diamond. My impression is that this was first discussed in the 70s in the axiomatic QFT literature but has recently found use in holography and in proving the quantum null energy condition.

Anyways, to connect to rindler: when is the density matrix obtained by tracing the minkowski vacuum over a half-space, the modular hamiltonian is proportionate to the generator of boosts around the splitting surface. In this special case modular flow acts geometrically on the algebra, just like ordinary time evolution, except with the evolution generated by boosts instead. I’m not aware of a geometric interpretation in the general case.

Re: “quench in modular time”, I may have been too glib: in your procedure, there is no sense in which the sudden change of the modular hamiltonian occurs at a particular value of the modular flow parameter. So it’s not literally a quench in modular time.

Let me try to give a more accurate description instead. Consider some system that is controlled by an idealized experimentalist. At some particular lab time , suppose the experimentalist changes the system’s density matrix from to , without changing the hamiltonian describing the system. At this lab time the modular hamiltonian suddenly changes, so one might be tempted to call what happened a quench in the modular hamiltonian. I’m not sure how useful this language is though, since it is not really a quench in the usual sense of suddenly changing the generator of dynamical evolution at some point along its flow (by pouring a bucket of icewater on the system, for example)

Thanks for the nickname for , by the way. Glad to see that zero temperature corresponds to infinite coolness!

]]>I forget what “modular time” is, though I know about Rindler spacetime. In particular I don’t know how a “quench in modular time” is, or how it would achieve what we want here.

]]>I want to understand this operation as a natural operation on positive-semidefinite self-adjoint operators. Given two such operators A and B, we'd like to use their product AB. Disaster strikes immediately, as this may be neither semidefinite nor self-adjoint. To make something self-adjoint, a common way is to take its average with its adjoint, so we would be using . Unfortunately, this is still not semidefinite, so we need something different.

However, the arithmetic mean is not the only way to average things. We could instead use the geometric mean, . Of course, defining the principal square root of a matrix is a little tricky if it’s not semidefinite, so let's use instead. Now this is well-defined, positive-semidefinite, and self-adjoint (as you can verify from these properties of A and B, remembering that a principal square root is also positive-semidefinite). Of course, this formula simplifies to , which is the formula above for , or B ⋆ A in plain text.

This is not a kind of conjugation but rather a kind of multiplication. You can think of it as multiplying square roots. Of course, is not the principal square root of B ⋆ A (nor of A ⋆ B, which is different). But besides its *principal* square root, every positive-semidefinite self-adjoint operator A has various *hermitian* square roots: operators a such that . And is a hermitian square root of A ⋆ B; that is basically how A ⋆ B is defined. (More generally, if a is a hermitian square root of A, then is a hermitian square root of A ⋆ B. But this isn't quite general enough to be interesting.)

Now, A ⋆ B is not quite what you (John) want; to get a density operator, you still have to normalize this by dividing by its trace. Unfortunately, even if A and B are density operators, the trace of A ⋆ B may well be zero. (Essentially, this happens when A and B, thought of as information about a quantum system, are mutually contradictory.) But when one of these is a thermal density matrix, then this cannot happen, and so then you can normalize it. (Leifer & Spekkens apply this in a different context, where A is not a density operator at all but instead satisfies a trace condition related to B that guarantees that A ⋆ B is a density operator; thus, they have no need to normalize.)

]]>(replying to John below)

I agree that it is more like a kind of chilling or cooling, though not necessarily with respect to the original notion of temperature defined by D.

Actually, I think your procedure is a special case of something closely analogous to what I would ordinarily call a quench (a sudden change in the hamiltonian). The only difference here is that the sudden change is in the modular hamiltonian, from to , i.e. it is a quench in modular rather than physical time. There is probably a way to think about this more physically in terms of the experience of an accelerating observer, whose proper time is modular time.

]]>