Symplectic geometry and Hamiltonian flow of the renormalisation group equation

TL;DR

This paper interprets the renormalisation group flow as a Hamiltonian vector flow within a symplectic geometric framework, linking it to physical quantities like energy-momentum and vacuum expectation values.

Contribution

It introduces a novel symplectic geometric interpretation of renormalisation group flow as a Hamiltonian system with explicit phase space structure.

Findings

Renormalisation group flow can be viewed as Hamiltonian vector flow.

Ward identities correspond to constants of motion in this framework.

The Hamiltonian is related to the vacuum expectation value of the energy-momentum tensor.

Abstract

It is argued that renormalisation group flow can be interpreted as being a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate \lq\lq momenta", which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace of the energy-momentum operator. For theories with massive couplings the identity operator plays a central role and its associated coupling gives rise to a potential in the flow equations. The evolution of any quantity , such as $N$-point Green functions, under renormalisation group flow can be obtained from its Poisson bracket with the Hamiltonian. Ward identities can be represented as constants of the motion which act as symmetry generators on the phase space via the Poisson bracket structure.