# Mathematical Aspects of Quantum Mechanics

## General Information

### Dozent: Prof. Dr. Roland Speicher

Assistent: Felix Leid

Time, place and format: Tuesday, 12-14 online meeting via Teams to discuss the recorded lectures and in particular the assignments

## Videos

Here is the link to the youtube channel with the videos.

## Scans of the Handwritten Manuscript

- Introduction
- 1. Hilbert Spaces
- 2. Bounded Operators on Hilbert Spaces
- 3. Unbounded Operators
- 4. Symmetric and Selfadjoint Operators
- 5. Spectrum and Spectral Theorem: part 1, part 2
- 6. Theorem of Stone
- 7. CCR and Weyl Relations
- 8. Stone-von Neumann Uniqueness Theorem
- 9. Symmetric Fock Space and Second Quantization
- 10. Infinite Tensor Product of Hilbert Spaces and Inequivalence of Representations of CCR(infinity)
- 11. CCR Algebra as C*-Algebra of the Weyl Relations
- Appendix: Lebesgue-Stieltjes Integral and Absolutely Continuous Functions

## Lecture Notes

I am putting here some lecture notes which are taken by a student and will evolve parallel to the lectures.

## Problem Sheets

### News

- I am now also putting the problem sheets here.
- I plan to put the videos of the lectures also on a public youtube channel, here is the link to the playlist
- There is now an introduction video which gives a survey of what I want to cover in the course.
- Enrolment for the class is now open in Moodle; you have to login in Moodle, then you can search for the course, and selfenrol yourself; for this you need the password "vonNeumann".
- I have added some literature for the course at the bottom.
- The administration of the course will be done in the moodle system of the UdS.

### General Information

This is a 2h topics lecture course which aims at a mathematical rigorous description of some of the mathematical foundations of quantum mechanics. In particular, unbounded operators on Hilbert spaces and canonical commutation relations will be treated. For more concrete info see below.

It is to be expected that the lectures and the exercise sessions will happen online. I am thinking about recording lectures at the blackboard, possibly even streaming them live - depending on the corona situation and the number of participants one might also think about onsite attendance. But it's too early to make definite statements ....

### Content

**Unbounded operators on Hilbert spaces**

- In the usual mathematical description of quantum mechanics observables are given by selfadjoint operators on Hilbert spaces. Unfortunately, it is a fact of life that typical observables correspond to unbounded operators. We will thus first address the basic theory of unbounded operators, and, in particular, elaborate on the difference between symmetric and selfadjoint operators, and the relevance of this in physical terms.
- The relevant part of the theory of bounded operators on Hilbert spaces will be recalled, where we can adapt to the prerequisites of the audience. In any case, participants should have seen Hilbert spaces and bounded operators before, either in an functional analysis course or in some lectures on quantum mechanics.

**Canonical commutation relations**

- The basic dictum for position and momentum observables is that they should satisfy the canonical commutation relations
*p*δ_{i}q_{j}-q_{j}p_{i}=ihThis raises the question whether operators satisfying those relations are essentially unique (i.e., unitarily equivalent). The answer to this is very different for finitely many degrees of freedom (only finitely many_{ij}.*i*s) and for the infinite situation. - Since the operators are unbounded, one has to rewrite the commutation relations in the form of Weyl relations to avoid mathematical pathologies. Then there is the famous uniqueness result of von Neumann that in the finite degree situation all irreducible representations of the Weyl relations are equivalent.
- In the infinite degree case the situation is very different; there is an abundance of inequivalent representations. This is the reason for the occurence of the notorious infinities in quantum field theories. Different theories (like those without and with interaction) are non-equivalent and can thus not be connected by a unitary transformation, as perturabation theory is trying to do.

**From concrete operators to abstract operator algebras**

- This gives then also a change in perspective for a mathematical frame of quantum mechanics and quantum field theory: instead of working with concrete operators in a fixed Hilbert space, the more relevant object is the operator algebra generated by the abstract operators, and concrete physical situations correspond then to different representations of those operator algebras.

### Literature

I have been asked for some literature for the course. There is not one main book that I will follow, but I will collect material from different sources and put together some lecture notes, which will develop during the term. But I can point out some of the materials which I find useful and interesting for the various topics.

For the part on (un)bounded operators on Hilbert spaces:

- A very good and classic introduction is:
*M. Reed and B. Simon: Methods of Modern Mathematical Physics, I: Functional Analysis*. - Chapter X.1 of part II of the same series will also be relevant.
- If you like it more compact and concise (and probably incomprehensible if you are not already an expert) then you might have a look at:
*W. Thirring: Lehrbuch der Mathematischen Physik, III: Quantenmechanik von Atomen und Molekuelen*. There is also an Englisch version of this and we will try to get one of those before the lectures start. In any case you should not be scared away by this, one goal of the class is that you should be able to read and understand at least some parts of his Section 2 after we are done with the course

For the canonical commutation relations:

- If you can read German then the original paper of von Neumann (from 1931) on his uniqueness theorem is still worth reading; as often with von Neumann, his original proof is still the best and has not been surpassed; here is the reference:
*J. von Neumann: Die Eindeutigkeit der Schroedingerschen Operatoren, Math. Ann. 104 (1931), 570-578.* - The books
*D. Petz: An invitation to the algebra of CCR*and*I. Putnam: Commutation properties of Hilbert space operators and related topics*might also be useful.

to be continued ...

There is now also a Semesterapparat with a collection of relevant books.

### Postal address

Saarland University

Department of Mathematics

Postfach 15 11 50

66041 Saarbrücken

Germany

### Visitors

Saarland University

Campus building E 2 4

66123 Saarbrücken

Germany