1003.1366 (Oliver J. Rosten)

Fundamentals of the Exact Renormalization Group    [PDF]

Oliver J. Rosten

Various aspects of the Exact Renormalization Group (ERG) are explored,
starting with a review of the concepts underpinning the framework and the
circumstances under which it is expected to be useful. A particular emphasis is
placed on the intuitive picture provided for both renormalization in quantum
field theory and universality associated with second order phase transitions. A
qualitative discussion of triviality, asymptotic freedom and asymptotic safety
is presented.
Focusing on scalar field theory, the construction of assorted flow equations
is considered using a general approach, whereby different ERGs follow from
field redefinitions. It is recalled that Polchinski's equation can be cast as a
heat equation, which provides intuition and computational techniques for what
follows. The analysis of properties of exact solutions to flow equations
includes a proof that the spectrum of the anomalous dimension at critical
fixed-points is quantized.
Two alternative methods for computing the beta-function in lambda phi^4
theory are considered. For one of these it is found that all explicit
dependence on the non-universal differences between a family of ERGs cancels
out, exactly. The Wilson-Fisher fixed-point is rediscovered in a rather novel
way.
The discussion of nonperturbative approximation schemes focuses on the
derivative expansion, and includes a refinement of the arguments that, at the
lowest order in this approximation, a function can be constructed which
decreases monotonically along the flow.
A new perspective is provided on the relationship between the
renormalizability of the Wilsonian effective action and of correlation
functions, following which the construction of manifestly gauge invariant ERGs
is sketched, and some new insights are given. Drawing these strands together
suggests a new approach to quantum field theory.

View original: http://arxiv.org/abs/1003.1366