### Path Integrals

#### (supervisor: John Gough)

Nature of project: **theory**, software

Available to
students on full-time physics degree schemes only.

#### Project description and methodology

Quantum Mechanics was originally formulated in terms of Hamiltonians, however, Dirac suggested a tentative Lagrangian formulation. This was put in place by Richard Feynman in his "sum-over-all-histories" version of quantum theory. The basic concept he introduced was the path integral.

This project will look at Feynman's approach. It will present the Hamiltonian and Lagrangian formulations of classical mechanics and show how Feynman's theory replicates standard QM.

*A successful project will develop beyond the above in one/some of the following directions:*

The path integral theory can be extended to statistical mechanics and quantum field theory. Describing the advantages of the theory for these branches of physics will be the required for a top grade.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*Additional scope or challenge if taken as a Year-4 project:* Determining the mathematical side of path integrals and the role of generating functions.

Please speak to **John Gough** if you consider doing this project.

*Initial literature for students:*

- Feynman & Hibbs, Quantum Mechanics & Path Integrals
- R.P Feynman, Statistical Mechanics: A Set Of Lectures, WEsley
- C. Grosche, An Introduction into the Feynman Path Integral, https://arxiv.org/pdf/hep-th/9302097.pdf

#### Novelty, degree of difficulty and amount of assistance required

The concept of integration over paths is novel and challenging.

The project assumes extensive knowledge of calculus and requires a familiarity with PDEs, Fourier transforms and a QM.

#### Project milestones and deliverables (including timescale)

milestone | to be completed by |

Locating source material/additional material/gaining understanding of the background concepts | end of November |

overview of theory/ able to formulate QM/heat equation in path integral language | Christmas |

analysis of more sophisticated models - harmonic oscillator | end of February |

general applications to modern physics | Easter |