Properties of the operator product expansion in quantum field theory

Holland, Jan W. 2013. Properties of the operator product expansion in quantum field theory. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

We prove that the operator product expansion (OPE), which is usually thought of as an asymptotic short distance expansion, actually converges at arbitrary finite distances within perturbative quantum field theory. The result is derived for the massive scalar field with $\varphi^{4}$-interaction on Euclidean spacetime. This constitutes a generalisation of an earlier result by Hollands and Kopper, which states that the OPE of exactly two quantum fields converges. We also show that the OPE coefficients satisfy factorisation conditions for certain configurations of the spacetime arguments. Such conditions are known to encode information on the algebraic structure of the underlying quantum field theory. Both results rely on modified versions of the renormalisation group flow equations, which allow us to derive explicit bounds on the remainder of these expansions. Within this framework, we also derive a new formula for the perturbation of OPE coefficients, i.e. an equation relating coefficients at a given perturbation order to those of lower order. By iteration of this formula, a new constructive method for the computation of OPE coefficients in perturbation theory is obtained, which only requires the coefficients of the free theory as initial data. Finally, we investigate a strategy to restrict renormalisation ambiguities in quantum field theory via the condition that the OPE coefficients depend analytically on the coupling constant(s) of the respective model. We apply this strategy to the computation of the vacuum expectation value of the stress energy operator in the two dimensional Gross-Neveu model and we obtain a unique prediction for the non-perturbative contribution to this expectation value, which is of the order $\exp(-2\pi/g^{2})$ (here $g$ is the coupling constant). We discuss the possibility that a similar effect, if present in the Standard Model of particle physics, could account for the ''unnatural'' smallness of the cosmological constant.

Item Type:	Thesis (PhD)
Status:	Unpublished
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Quantum field theory; Operator product expansion; renormalisation group; dark energy
Funders:	Leverhulme Trust
Date of First Compliant Deposit:	30 March 2016
Last Modified:	19 Mar 2016 23:28
URI:	https://orca.cardiff.ac.uk/id/eprint/53230

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