The functional calculus allows you to apply any function to any self-adjoint matrix (and thus any self-adjoint operator on a finite-dimensional Hilbert space), or any holomorphic function to any matrix (and thus to any linear operator on a finite-dimensional space).

Also, when we have f(O), does f being smooth mean that, it can be expanded in the form of a power series in O?

We can do that when f is **holomorphic**. This means that

and the power series converges for all You can then prove that

converges for all matrices , so we can define to be this.

When is self-adjoint we can go further, and define for any function , simply by saying that has the same eigenvectors as , and

We need to be self-adjoint here because that guarantees that we can choose a *basis* of eigenvectors of .

For some reason I chose to focus on the case where is smooth, but there was no need to do this. Since is self-adjoint, I could have assumed is any function whatsoever.

]]>Also, When we have f(O), does f being smooth mean that, it can be expanded in the form of a power series in O? ]]>

Very cool. I look forward to seeing that.

]]>Dear all,

For this week’s OASIS seminar we have the pleasure of a talk by Harvey Brown, the professor in philosophy of physics at Oxford who is well-known for his work on the foundations of quantum mechanics, relativity theory, and the role of symmetry principles in physics, including several books. Moreover, he is a very clear and entertaining speaker! This Friday he will convince us that we need to take symmetries and their subtleties more seriously.

Time and place: This Friday, 2pm, Lecture Theatre B, Department of Computer Science.

Title: Noether’s famous 1918 symmetry theorem — what does it prove?

Abstract: Recently, Brendan Fong and John Baez have provided an analogue in stochastic mechanics to what they call Noether’s theorem in quantum mechanics. Noether’s original theorem, relating symmetries and conservation principles, was the first in a series of theorems she proved in 1918 within a program in the calculus of variations, inspired by interpretational problems related to conservations laws in general relativity. I will sketch the background to Noether’s work and give special emphasis to the form and meaning of her “first” theorem. An unusual application of the theorem to quantum mechanics will be exploited.

Philosophers of physics being as they are, the phrase “what they call” makes me afraid he’s planning to chide me for using the term “Noether’s theorem” in a very extended sense, not very close to that of her original 1918 paper. Physicists being as they are, such chiding wouldn’t stop me. But I’m curious to hear what he actually says. The talk will be videotaped and put on the OASIS website. Furthermore, Brendan is now at Oxford and can hear the talk in person!

]]>**Puzzle.** Suppose is a stochastic operator and is an observable. Show that commutes with iff the expected values of *and its square* don’t change when we evolve our state one time step using In other words, show that

if and only if

and

for all stochastic states

**Answer.** One direction is easy: if then for all so

where in the last step we use the fact that is stochastic.

For the converse direction we can use the same tricks that worked for Markov processes. Assume that

and

for all stochastic states . These imply that

and

We wish to show that . Note that

To show this is always zero, we’ll show that when , then . This says that when our system can hop from one state to another, the observable must take the same value on these two states.

For this, in turn, it’s enough to show that the following sum vanishes for any :

Why? The matrix elements are nonnegative since is stochastic. Thus the sum can only vanish if each term vanishes, meaning that whenever .

To show the sum vanishes, let’s expand it:

Now, since (1) and (2) hold for all stochastic states , this equals

But this is zero because is stochastic, which implies

So, we’re done!

]]>Tomate wrote:

I’ve always been here…

Oh, good! Sometimes I think everyone is leaving, or falling asleep.

This is precisely what I had in mind. I can send you via email a couple of pages from my master thesis if you want: they are in Italian, but the formulas are quite clear.

If it mainly says what we’ve already discussed, I guess I won’t make you bother. I guess this is some sort of ‘folk wisdom’.

“as if it’s dreaming of the jump before it goes”: this is always the effect it has when we project statistical arguments onto the individuals, like when people play long-overdue numbers at the lotteries…

Okay, good point!

]]>Blake wrote:

Does it still count as a “nascent idea” if it’s been around since 1976?

I’d say the mathematical trick has been around since 1976. The nascent *idea* lurking in this trick is that we can think of a probability distribution as a quantum state if we normalize it in a nonstandard way and promise to only ask about its transition amplitudes to a certain ‘default’ state . Mathematical tricks often conceal ideas that are too strange for people to say in words.

More seriously, I think the main issue is that most of the people involved just weren’t that concerned with quantum-to-classical transitions.

Yes, that’s one part of it. But even if we don’t try to describe the *same system* both classically and quantumly, there’s also the question of the *logical relation* between the classical and quantum descriptions: that’s what I’m especially interested in. But this is not the sort of question that ‘practical’ people tend to enjoy—perhaps because they can’t imagine what one might do with the answer.

Second, I think that the people who study diffusion-limited reactions, active-to-absorbing phase transitions, directed percolation and the like are generally eager to skip past the first steps of defining the formalism and get to a Lagrangian they can play with.

Right. For me the murky beginning steps are the most interesting part, because they hint at a relation between quantum mechanics and probability theory that seems a bit different than the ‘obvious’ one, where rather than the wavefunction acts like a probability distribution. I’ve got a bunch of ideas about this that I’ll reveal as soon as I can.

]]>This is precisely what I had in mind. I can send you via email a couple of pages from my master thesis if you want: they are in Italian, but the formulas are quite clear. Funnily, I don’t have references… I wrote that chapter out of some personal notes of my professor, which didn’t have references neither. In the field, it’s like everybody knows about it but nobody knows exactly where it comes from…

“as if it’s dreaming of the jump before it goes”: this is always the effect it has when we project statistical arguments onto the individuals, like when people play long-overdue numbers at the lotteries…

]]>