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September 30, 2008

Quantum Theory and Analysis

Posted by John Baez

Besides my seminar and an undergrad calculus couse, I’m also teaching a graduate math class on quantum mechanics. I won’t blog about this class, but you can see my lecture notes:

I wrote these notes the first time I taught the course, back in… 1989! I’d just been hired by UCR. I considered myself a mathematical physicist, but I was officially classified as an ‘analyst’ — since my specialty was nonlinear wave equations. Seems like a long time ago. But, I still like quantum theory and analysis.

The main topics are:

  • The spectral theorem for unbounded self-adjoint operators.
  • Stone’s theorem relating one-parameter unitary groups to self-adjoint operators.
  • The Kato–Rellich theorem for proving that perturbations of self-adjoint operators are self-adjoint, with applications to Schrödinger operators.
Posted at September 30, 2008 3:28 AM UTC

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Re: Quantum Theory and Analysis

I’m no scientist, not by a long shot, but i have an idea about creation. My theory is short, full of holes, and i have nothing to back it up with but i think it is worth some thought.
There are an unlimited number of questions. For every answer there is always another question. If white holes made the universe and are made by black holes which intern are made by dieing stars etc. What is the force that started these never ending phenomena? But after we find out what caused the universe to come into being then the question “what caused the force or phenomena that caused everything?” and so on and so forth.
There will never be a definite answer to everything. We will always have unanswered questions. No mater how many answers we may come up with. But then again i could be and probably am wrong. I just thought that maybe somone would find it intriguing.

Posted by: justin kelly on October 8, 2008 10:00 AM | Permalink | Reply to this

Re: Quantum Theory and Analysis

I would really enjoy to read comments if you have any on what i have said. I am just a person interested in learning more.

Posted by: jsutin kelly on October 8, 2008 10:44 AM | Permalink | Reply to this

Re: Quantum Theory and Analysis

Justin wrote:

There will never be a definite answer to everything. We will always have unanswered questions. No matter how many answers we may come up with.

I suspect that’s true. I’m sure it’s true in the realm of mathematics, where one can pretty much prove that there are infinitely many questions. In physics one can imagine a ‘theory of everything’ that in principle allows us to calculate answers to all physics questions about our universe, but 1) it seems pretty clear that we can’t use such a theory to do this in practice, thanks to computational complexity issues, and 2) we don’t know if such a theory actually exists, much less whether we’ll find it.

Anyway, regardless of whether there’ll always be more questions, there are certainly plenty of unanswered questions now — and there will continue to be more for the foreseeable future. So, we’ve got plenty of fun in store.

Posted by: John Baez on October 8, 2008 6:10 PM | Permalink | Reply to this

Re: Quantum Theory and Analysis

Or, as that great American philosopher Yogi Berra said, “In theory there is no difference between theory and practice. In practice there is.”

Posted by: Charlie C on October 9, 2008 2:53 AM | Permalink | Reply to this

Re: Quantum Theory and Analysis

I have a difficult time putting what my thoughts down. I know mathematically that there are certain equations that have exact answers with no holes, but when attempting to answer questions on creation there are an unending number of questions in my mind, because once you answer where did the universe come from you have to answer where did what made the universe come from and then what made whatever made the universe and so on. Unless somewhere in some way there is a law of constant being. Hear is the way i look at it. Cause and effect. Universe=effect, “theoretically” big bang=cause. For every cause there is an effect and vice versa. Answers are the effects of questions, and questions are the causes of answers. If every cause has to have an effect and every effect is itself a cause then to me it seems like that would mean there are and unlimited number of questions and answers.

Posted by: justin on October 11, 2008 11:46 AM | Permalink | Reply to this

Isabella and Cantor; Re: Quantum Theory and Analysis

justin gives a conventional Medieval argument. I’ve read a book on medieval Jewish, Christian, and Islamic philosophy, and was struck by these things among others.

(1) Many of the qualitative arguments of modern cosmology were clearly anticipated a millennium ago – as to whether there was an infinite past, and might be an infinite future.

(2) The singularity was Aristotle. Each of the Abrahamic religions had to reinterpret their old arguments as either in favor of or opposed to Aristotle, but adopted his framework in either case. Then ideas sloshed around between the 3 branches in fascinating ways, which discourse was dramatically curtailed in 1492 when Jews were expelled from Spain, and the tripartite civilization was ethnically cleansed.

By justin’s approach, I cannot see why there are a countable number of questions or an uncountable number. That’s an interesting meta-question, implicit in some prior n-Category formulations.

Cantor trumps Aristotle.

Posted by: Jonathan Vos Post on October 11, 2008 9:49 PM | Permalink | Reply to this

Re: Quantum Theory and Analysis

Well, sure you can always ask more questions; the real problem is whether they are meaningful questions.

As Feynman points out:

‘We can’t define anything precisely. If we attempt to, we get into that paralysis of thought that comes to philosophers…one saying to the other: “you don’t know what you are talking about!”. The second one says: “what do you mean by talking? What do you mean by you? What do you mean by know?”’

Remember also that causality just allows relations between events (events x,y,z “caused” event a means that when such conditions occur, the consequence will occur).

But the universe != effect…the universe is a collection of events. That is, what goes on in space and time defines the universe. So it’s a bit hairy to say “Ah yes well, universe = effect, big bang = cause”…cause of what? Effect of what? What about the important physical processes going on? What caused the big bang? Etc. etc. etc.

Posted by: AngryPhysicist on October 18, 2008 11:18 PM | Permalink | Reply to this

Re: Quantum Theory and Analysis

Here is a very late question.
I am a mathematician, and I wanted to learn the fundamentals of quantum physics for a long time. At first, I was really excited about these notes, because they seem to provide a very conceptual approach. However, I was was lost already with the first equation on the top of page 3. I know that mathematicians and physicians use quite different notations, and this seems to be a typical case.

Here is what I understand: $v_H$ is a vector-field and should therefore be of type $R^{2n}\to R^{2n}$. The formula has free variables in the denominator, therefore we should be able to build a vector from that. However, this would only give dimension $n$. And what I do not understand at all is what to do with the differential operators that are multiplied on the _right_ of the terms with the $H$.

I would really really appreciate if somebody could give me a hint how to read this, or a pointer to a place where it is explained.

Posted by: anon on October 17, 2008 3:13 PM | Permalink | Reply to this

Re: Quantum Theory and Analysis

If you want, you can skip all the classical mechanics stuff and read the rest of the notes.

But if you want a lightning review of the Hamiltonian approach to classical mechanics, I’ll be glad to help you out. I’m using some notation that’s common in differential geometry. In differential geometry, we think of a vector field on a manifold as a 1st-order differential operator — that is, a recipe for differentiating real-valued functions on that manifold.

For example, on n\mathbb{R}^n the standard coordinates x ix_i give a basis of vector fields called

x i \frac{\partial}{\partial x_i}

We can use such a vector field to differentiate a function f: nf : \mathbb{R}^n \to \mathbb{R}, getting this:

fx i \frac{\partial f}{\partial x_i}

We can also multiply this vector field by a function g: ng : \mathbb{R}^n \to \mathbb{R} to get a new vector field:

gx i g \frac{\partial }{\partial x_i}

We can then use this to differentiate the function ff:

gfx i g \frac{\partial f}{\partial x_i}

On page 3 of my notes, I’m working on 2n\mathbb{R}^{2n} with coordinates q i,p iq_i, p_i where 1in1 \le i \le n. Given a smooth function H: 2nH : \mathbb{R}^{2n} \to \mathbb{R}, the Hamiltonian vector field is

v H= iHp iq iHq ip i v_H = \sum_i \frac{\partial H}{\partial p_i} \frac{\partial }{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial }{\partial p_i}

Following the ‘Einstein summation convention’ (widely used in differential geometry), we can omit the summation sign:

v H=Hp iq iHq ip i v_H = \frac{\partial H}{\partial p_i} \frac{\partial }{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial }{\partial p_i}

The idea is that we always sum over repeated indices.

I should improve my notes so that they either explain this stuff or use notation that more people will understand! Luckily, I don’t think you need to understand this classical mechanics business to follow the rest of what I’ll discuss.

Posted by: John Baez on October 17, 2008 10:26 PM | Permalink | Reply to this

Re: Quantum Theory and Analysis

Thank you very much for your quick reply. I can see a bit clearer now :-).

So if I understand it right, we are dealing with an autonomous ordinary differential differential equation

$$y’ = f(y)$$

or rather

$$(q’,p’) = f(q,p)$$

where $f$ is the vector field and the action $U$ of $R$ is the totality of solutions.

But why this differential operator notation? We do not really use the vector field as a differential operator, do we? Or is it because in Leibniz notation, we can pretend to differentiate something where we are really just substituting?

Would that mean that in this notation, we could rewrite an ODE of the form $y’=f(y)$ where y is 1-dimensional as

$$d/dt = f d/dy$$ ?

Is it relevant whether the equation is autonomous (i.e. if $f$ depends on on $t$)?

Posted by: anon on October 19, 2008 3:08 PM | Permalink | Reply to this

Re: Quantum Theory and Analysis

If you click on the ‘text filter’ called ‘itex to mathml with parbreaks’ before submitting your post, the tex in your equations will actually come out looking good.

So if I understand it right, we are dealing with an autonomous ordinary differential differential equation

y=f(y)y’ = f(y)

or rather

(q,p)=f(q,p)(q’,p’) = f(q,p)

where ff is the vector field…

Right. And when the vector field is what I’m calling v Hv_H, the ‘Hamiltonian vector field’, then this autonomous ordinary differential equation is called ‘Hamilton’s equations’.

But why this differential operator notation?

Because modern differential geometers think of vector fields that way. Don’t worry about it — if I were writing a book on differential geometry and physics (which I have), I’d explain this stuff, but it doesn’t matter much here.

Posted by: John Baez on October 19, 2008 6:35 PM | Permalink | Reply to this

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