Undergraduate Lecture Notes in Topological Quantum Field Theory

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Basics of non-relativistic quantum mechanics
Quantum states and operators
Non-relativistic quantum-mechanical systems have two modes of evolution in time [13, 14]. The first, governed by standard, time–dependent Schr¨ odinger equation : ˆ |ψ , i ∂t |ψ = H (1)
arXiv:0810.0344v1 [math-ph] 2 Oct 2008
Undergraduate Lecture Notes in Topological Quantum Field Theory
Vladimir G. Ivancevic∗ Tijana T. Ivancevic†
Abstract These third–year lecture notes are designed for a 1–semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second–year subjects: multivariable calculus, introduction to quantum mechanics and basic electromagnetism. Keywords: quantum mechanics/field theory, path integral, Hodge decomposition, Chern–Simons and Yang–Mills gauge theories
describes the time evolution of quantum systems when they are undisturbed by measurements. ‘Measurements’ are defined as interactions of the quantum system with its classical environment. As long as the system is sufficiently isolated from the environment, it follows Schr¨ odinger equation. If an interaction with the environment takes place, i.e., a measurement is performed, the system abruptly decoheres i.e., collapses or reduces to one of its classically allowed states. A time–dependent state of a quantum system is determined by a normalized, complex, wave psi–function ψ = ψ (t). In Dirac’s words [13], this is a unit ket vector |ψ , which is an element of the Hilbert space L2 (ψ ) ≡ H, with a coordinate basis (q i ).1 The state ket–vector
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Introduction
There is a number of good textbooks in quantum field theory (QFT, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. However, they are all designed for the graduate-level study and we can only hope that undergraduate students can read some easy parts of them. Moreover, there are certainly no undergraduate-level textbooks for TQFT, so pure students are forced to try to read the original papers from its inventors, Ed Witten [11] and Michael Atiyah [12]. The goal of the present lecture notes is to try to fill in this gap, to give the talented undergraduates the very first glimpse of the mathematical physics of the XXI Century. Throughout these lecture notes we will use the following conventions : (i) natural √ units, in which (some or all of) the following definitions are used: c = = m = 1; ˙ = dz/dt, ∂z = ∂/∂z ; (ii) i = −1, z (iii) Einstein’s summation convention over repeated indices, while nD means n−dimensional. 2
∗ Land Operations Division, Defence Science & Technology Organisation, P.O. Box 1500, Edinburgh SA 5111, Australia (Vladimir.Ivancevic@.au) † School of Electrical and Information Engineering, University of South Australia, Mawson Lakes, S.A. 5095, Australia (Tijana.Ivanceath Integral 2.1 The action–amplitude formalism . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Correlation functions and generating functional . . . . . . . . . . . . . . . . . 2.3 Quantization of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . 3 Path–Integral TQFT 3.1 Schwarz–type and Witten–type theories . . . . . . . . . . . 3.2 Hodge decomposition theorem . . . . . . . . . . . . . . . . . 3.3 Hodge decomposition and gauge path integral . . . . . . . . 3.3.1 Functional measure on the space of differential forms 3.3.2 Abelian Chern–Simons theory . . . . . . . . . . . . . 4 Non-Abelian Gauge Theories 4.1 Intro to non-Abelian theories . . . . 4.2 Yang–Mills theory . . . . . . . . . . 4.2.1 Yang–Mills action . . . . . . 4.2.2 Gauge transformations . . . . 4.3 Quantization of Yang–Mills theory . 4.3.1 Faddeev–Popov determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Appendix 5.1 Manifolds and bundles . . . . . . . . . 5.2 Lie groups . . . . . . . . . . . . . . . . 5.3 Differential forms and Stokes theorem 5.4 De Rham cohomology . . . . . . . . .