Path Integrals on Curved Spaces: Geometry, SU(2) and SO(3)

This course contains the use of artificial intelligence. I will carefully and openly disclose this in the introductory section, but let me state here as well that the course structure, mathematical derivations, explanations, and final verification were created and reviewed by me, the instructor.

COURSE DESCRIPTION

Path integrals become more subtle when the configuration space of a quantum system is curved. The metric affects the kinetic energy, the invariant measure, the quantum Hamiltonian, the short-time propagator, and the contribution of classical paths to quantum evolution.

Many advanced books state the resulting formulas without showing all the steps that connect them. I created this course for learners who want to see what happens in between. The calculations are developed in detail, while animated visualizations help make their geometrical meaning clearer, and PDF documents are attached to follow the lectures at one’s own individual and comfortable pace.

The course begins with the classical mechanics of a particle on a Riemannian manifold. Starting from the metric and the Lagrangian, we derive the canonical momentum and Hamiltonian. We then introduce the geometrical tools needed for quantization: covariant derivatives, the Levi-Civita connection, geodesics, curvature tensors, and normal coordinates. This leads to the invariant volume element, the Laplace-Beltrami operator, and the relationship between operator ordering and path-integral discretization.

Particular attention is given to the short-time propagator, the Van Vleck-Morette determinant, and the role of curvature. These ideas are then applied to SU(2) and SO(3), where summations on paths and spectral representations are compared to show how geometry and topology affect quantum evolution.

The final lectures broaden the discussion to canonical gravity and quantum geometry, introducing the ADM formulation, Ashtekar–Barbero variables, holonomies, fluxes, and intertwiners. An explicit heat-kernel calculation is followed by a discussion of functional determinants and of how one-loop quantum fluctuations contribute curvature-dependent terms to an effective action.

Manim animations accompany the derivations, but they do not replace the mathematics. Their purpose is to make the geometry visible and the transitions between equations easier to follow.

If you have ever felt that a textbook skipped the very step you needed, this course was designed for you.

LEARNING OBJECTIVES

By the end of the course, students will be able to:

• derive the classical Hamiltonian on a curved configuration space;

• work with covariant derivatives, geodesics, curvature tensors, the invariant measure, and the Laplace-Beltrami operator;

• explain the connection between operator ordering and path-integral discretization;

• derive and interpret the main "ingredients" of the curved-space short-time propagator;

• compare the summation on paths and spectral representations studied for SU(2) and SO(3);

• follow the main steps leading to the ADM formulation and the Ashtekar-Barbero variables;

• connect heat kernels, functional determinants, and curvature terms in one-loop effective actions.

REQUIREMENTS

Students should be familiar with multivariable calculus, linear algebra, Lagrangian or Hamiltonian mechanics, and quantum mechanics. Familiarity with tensor notation is definitely helpful. Prior knowledge of general relativity, heat kernels, representation theory, or loop quantum gravity is not required, although the later lectures are more demanding.

INTENDED LEARNERS

This course is intended for advanced students and independent learners interested in mathematical physics, curved-space quantum mechanics, differential geometry, path integrals, gravitation, and quantum geometry. It is especially suitable for those who want to see the intermediate calculations that advanced textbooks often omit.