### Combinatorics and quantum theory

#### (supervisor: John Gough)

Nature of project: **theory**, software

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

#### Project description and methodology

This project concerns Wick ordering. This is a standard trick in quantum theory and is widely used in quantum mechanics and quantum field theory. It is also at the heart of Feynman diagrams.

The first occurrence of this principle will be for normal ordering of creation and annihilation operators associated with the harmonic oscillator. The computation of moments of observables in the ground state, or more generally, in coherent states.

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

Wick's Theorem then deals with dynamical models and involves developing a perturbation series of the unitary evolution operator, followed by Wick ordering. The project will show how this leads to Feynman diagrams.

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:* Wick's Theorem may be extended to thermal states and determining the form of this result in this case would be an additional step for a 4th year project.

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

*Initial literature for students:*

- J Gough, J Kupsch, Quantum Fields and Processes: A Combinatorial Approach, Cambridge 2018
- J. Glimme and A. Jaffe, Quantum Physics (Springer-Verlag, New York, 1981), 2nd ed
- S. Weinberg, The Quantum Theory of Fields (Volume I) Cambridge University Press (1995)
- T.S. Evans, D.A. Steer, Wick's theorem at finite temperature, Nucl. Phys B 474, 481-496 (1996) arXiv:hep-ph/9601268

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

The coherent states are currently 4th year material, but accessible to students who have done Yr2 QM. Wick's Theorem is however extra-curricular. The topic will be of interest to anyone looking to develop a background in quantum field theory.

An understanding of basic QM and the underlying mathematical foundations is essential.

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

milestone | to be completed by |

Locating source material | end of October |

Working out Wick's Theorem for the quantum HO | Christmas |

Extend to quantum fields, and Feynman diagrams | mid-March |

Further directions and Analysis | Easter |