An ERGE primer
An
“Exact Renormalization Group Equation” or “Functional
Renormalization Group Equation” is an equation describing the
dependence of some effective action as a cutoff is varied.
There are various forms of ERGE, depending on what functional
one is considering and on the way the cutoff is imposed.
Polchinski's equation refers to the Wilson action, Wetterich's
equation to the generating functional of 1PI Green functions,
also known as the "effective average action". This picture shows
Wetterich and Polchinski in Corfu, september 2010.
The
following files contain a very quick introduction to the
Wetterich equation and its use to find fixed points. The context
is scalar theory and the fixed point in question is the
Wilson-Fisher fixed point. The applications of the ERGE to
gravity are technically more complicated in several ways, but
the basic principles are not very different.
Original papers
Kenneth G. Wilson and Michael G Fischer
Critical exponents in 3.99 dimensions
Phys. Rev. Lett. 28, 240 (1972)
Joseph Polchinski
Renormalization and Effective
Lagrangians.
Nucl. Phys. B231,
269-295
Christoph Wetterich
Exact evolution equation for the effective
potential.
Phys. Lett. B 301, 90 (1993)
Derivative expansion of the
exact renormalization group.
Phys.Lett.B329:241-248 (1994)
Tim R. Morris
On truncations of the exact
renormalization group.
Phys. Lett. B334:355-362 (1994)
Tim R. Morris, Michael D. Turner
Derivative expansion of the renormalization group
in O(N) scalar field theory.
Nucl. Phys. B509, 637-661 (1998)
Daniel F. Litim (2001)
Optimised renormalisation group flows.
Phys. Rev. D 64, 105007.
(This paper introduces a very useful type
of cutoff that gives
the beta functions in closed form, aside from other good
properties.)
Reviews
Tim R. Morris (1998)
Elements of the continuous renormalization group
Prog. Theor. Phys. Suppl.131 395-414
C. Bagnuls and C. Bervillier (2001)
Exact renormalization group equations: An
introductory review.
Phys. Rept. 348, 91.
J. Berges, N. Tetradis, and C. Wetterich
Nonperturbative renormalization
flow in quantum field theory and statistical physics.
Phys. Rept. 363, 223 92002)
Jan M. Pawlowski
Aspects of the functional
renormalization group
Annals Phys. 322, 2831-2915 (2007)
Fundamentals of the functional
renormalization group.