He has some lectures on quantum field theory online. I decided it may be fun to read some of them since: it's Ed Witten! On lecture one he decides to define what he means by a quantum theory:
We will define a quantum theory to be a pair (A,H) where A is a *-algebra (not necessarily commutative), and H is a selfadjoint element of A, defined up to a real number. The algebra A is called the algebra of quantum observables (operators), The element H, as before, is called the Hamiltonian...Now, it turns out if you have taken enough math, quantum mechanics and read through the above slowly you can see that what he is saying does make sense. (Obviously, as if he would be wrong.)
By a realization (or solution) of a quantum theory (A,H) we will mean an irreducible *-representation of the algebra A in some Hilbert space H, such that the spectrum of the operator H is bounded from below (representations are considered up to an isomorphism that preserves H). We will always normalize H so that the lowest point of a spectrum is zero.
But come on! I don't think any normal human thinks about quantum principles as deeply and rigorously as this. Edward Witten is definitely very special and part of me wishes I could understand physics at this level.
I think I get most of what he was saying but it took me about 3 times reading through those two paragraphs and my brain hurts. In astrophysical fluid dynamics our fancy math is perturbation theory, spectral transforms, and matched asymptotic expansions, which most people think is "real math". However I will admit that people like Joe and Ed Witten have me beat - you guys do "REAL MATH".
ReplyDeleteWell, though I admit I enjoy this type of "advanced" math, there is no way I will ever be comparable with Witten.
ReplyDeleteHe, as far as I know, is the only physicst to win a Fields Medal, the highest honor in mathematics. (At least he was the first.)
By the way, I think I am going to use Witten's definition from now on every time I am asked what quantum mechanics is.
ReplyDeleteSeriously, though a first reading was intimidating, the more I think about it the more he has boiled it down in a very useful way:
1. QM says observables are related to operators.
2. H, the Hamiltonian, is a special operator that in some sense is the most important one.
3. What separates QM from everything else is since observables are related to operators everything we observe in nature seem to be eigenstates of these operator where the eigenvalues are what we measure.
4. Operators may not commute giving uncertainty in measurement. (Ie Heisenberg's uncertainty principle.)
5. The representations we observe are always irreducible.
Anyways, my point is it is actually a helpful definition cutting to the core of what a quantum theory is. (Like Witten wouldn't have realized this.)
Joe,
ReplyDeleteYou definately got more out of that than I did. I took away your points 1, 2, and 4 and it would have taken me twice as long to explain them.
Basically what I'm saying is that if I do real math and Ed Witten does REAL math, then I'd put you at something like REaL math.
REal math. I love it.
ReplyDeleteThat is a really nice presentation of what is meant by a quantum theory. I appreciate that people like Witten have taken the time to think so carefully about deeper issues. It's why I love the work of Julian Barbour so much, even though his work freaks me out a little.
Nick, I too thought your "REaL math" quote was very clever.
ReplyDeletebut why do observables relate to operators? In this direct way, and how does the curved spacetime change the relationship. Operators are now measured in the tangent space of the manifold, and relate non-linearly to the Observables,which is not a qm theory
ReplyDeleteAnonymous,
ReplyDeleteGood questions.
First, I am not an expert enough to *prove* from some base axioms that all observables must be related to operators. However, Dirac set up this formal notation for quantum mechanics where all quantum states are elements of a Hilbert space and once you start working in that formalism it becomes obvious that all observables are related to eigenstates of particles.
So I personally can't prove it other to say when working in Dirac's formalism it becomes obvious. Furthermore, in the QM books I used it is actually a postulate maning it is just taken as a given and not something proved. Some things I just believe because it always works and matches experiment.
Second, the curved space question. First, though it is true operators only exist in the tangent space, through a process known as parellel transport you can relate an operators to different points.