Schreiber scratch paper

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Think of the space below as unlimited electronic scratch paper. Hit “edit” below, to make notes.

Preliminary plan for:

Goals

• learn basic differential geometry (de Rham cohomology, covariant differentiation)

• through examples introduce classical mechanics, electromagnetism, quantum mechanics

• learn about the fibre bundle formalism in which one can study QFT

Lecture 1

• preliminaries from linear algebra, analysis, topology and homological algebra

(Chapter 0. in CCC).

• manifolds, submanifolds, tangent bundle, vector fields (Frankel, Ch. 1, 1.2-1.4)

Lecture 2

• introduction to classical mechanics and examples (Frankel 1.1.d)

• Lagrangian vs. Hamiltonian formalism (Takhtajan)

Lecture 3

• calculus of differential forms (ch 3. and 4. in CCC)

• recalling and expanding from bachelors course: integration of differential forms and Stokes theorem (compare with the chapters in CCC also the chapters 2 and 3 of Frankel)

Lecture 4

Maxwell equations

Question: Lecture 5

Question: do we need more time and should we put Lecture 5:

• vector bundles and fibre bundles (16.1. and 17.1. in Frankel).

Lecture 6:

• Poincare lemma, de Rham cohomology (Ch. 5. in CCC, Ch.5 in Frankel)

Lecture 7:

• Riemannian geometry, covariant differentiation (ch 9., 9.1.-9.4 in Frankel)

Lecture 8:

• curvature (9.5, 9.6. in Frankel), maybe 9.7.

Lecture 9:

• geodesics, Hamiltonian mechanics (Ch.10.)

Question: Do we need Lie derivatives, Ch 4.? What do we need from chapters 7 and 8?

Lecture 10:

• introduction to quantum mechanics: two-slit experiment?, first quantization?, Schrödinger equation

Question: Lecture 11

examples: harmonic oscillator, hydrogen atom, Heisenberg formalism ?

Question: Lecture 12

special relativity/general relativity (Ch 11.) ?

Question: Lecture 13

on Lie groups and Lie algebras

(I guess this cannot and should not be omitted)

Lecture 14

• vector bundles (ch 16. in Frankel), electormagnetic connection (=circle bundle with connection)

Lecture 15:

• topological quantization (Ch. 17. in Frankel)

By this time we understand how to get from Maxwell's equations to gauge theories

Lecture 16:

• introduction to QFT, second quantization, path integral formalism (maybe the first chapter of Zee?)

Lecture 17:

• Noether's theorem, introduction to Yang-Mills theory

Lecture 18:

• Chern forms and homotopy groups

Lecture 19:

• cohomology (of vector bundles??? Ch IX in CCC)

Lecture 20:

• check what has not been covered so far from notes “Preparation for Gauge Theories”

• Question: Clifford algebras, Dirac operator (???)

Remarks

Seminar will be held in Croatian, lecture notes hopefully in English. Once a week, 90 minutes.

References

• Theodore Frankel, The Geometry of Physics - An Introduction (Ch. 1,2,3,5,8?,9,10,11?,15?16,17,20,22)

• Werner Greub, Stephen Halperin, Ray Vanstone Connections, Curvature, and Cohomology (VOl 1., Chapters I-V)

• L. Takhtajan, Quantum Mechanics for Mathematicians (lecture notes, shorter version)

• G. Svetlichny, Preparation for Gauge Theory

• A. Zee, Quantum Field Theory in a Nutshell (Ch. I, maybe some parts of III and IV)

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