2,139 research outputs found

### Elements of the Continuous Renormalization Group

These two lectures cover some of the advances that underpin recent progress
in deriving continuum solutions from the exact renormalization group. We
concentrate on concepts and on exact non-perturbative statements, but in the
process will describe how real non-perturbative calculations can be done,
particularly within derivative expansion approximations. An effort has been
made to keep the lectures pedagogical and self-contained. Topics covered are
the derivation of the flow equations, their equivalence, continuum limits,
perturbation theory, truncations, derivative expansions, identification of
fixed points and eigenoperators, and the role of reparametrization invariance.
Some new material is included, in particular a demonstration of
non-perturbative renormalizability, and a discussion of ultraviolet
renormalons.Comment: Invited lectures at the Yukawa International Seminar '97. 20 pages
including 6 eps figs. LaTeX. PTPTeX style files include

### The Renormalization Group and Two Dimensional Multicritical Effective Scalar Field Theory

Direct verification of the existence of an infinite set of multicritical
non-perturbative FPs (Fixed Points) for a single scalar field in two
dimensions, is in practice well outside the capabilities of the present
standard approximate non-perturbative methods. We apply a derivative expansion
of the exact RG (Renormalization Group) equations in a form which allows the
corresponding FP equations to appear as non-linear eigenvalue equations for the
anomalous scaling dimension $\eta$. At zeroth order, only continuum limits
based on critical sine-Gordon models, are accessible. At second order in
derivatives, we perform a general search over all $\eta\ge.02$, finding the
expected first ten FPs, and {\sl only} these. For each of these we verify the
correct relevant qualitative behaviour, and compute critical exponents, and the
dimensions of up to the first ten lowest dimension operators. Depending on the
quantity, our lowest order approximate description agrees with CFT (Conformal
Field Theory) with an accuracy between 0.2\% and 33\%; this requires however
that certain irrelevant operators that are total derivatives in the CFT are
associated with ones that are not total derivatives in the scalar field theory.Comment: Note added on "shadow operators". Version to be published in Phys.
Lett.

### Renormalization group properties of the conformal sector: towards perturbatively renormalizable quantum gravity

The Wilsonian renormalization group (RG) requires Euclidean signature. The
conformal factor of the metric then has a wrong-sign kinetic term, which has a
profound effect on its RG properties. Generically for the conformal sector,
complete flows exist only in the reverse direction (i.e. from the infrared to
the ultraviolet). The Gaussian fixed point supports infinite sequences of
composite eigenoperators of increasing infrared relevancy (increasingly
negative mass dimension), which are orthonormal and complete for bare
interactions that are square integrable under the appropriate measure. These
eigenoperators are non-perturbative in $\hbar$ and evanescent. For
$\mathbb{R}^4$ spacetime, each renormalised physical operator exists but only
has support at vanishing field amplitude. In the generic case of infinitely
many non-vanishing couplings, if a complete RG flow exists, it is characterised
in the infrared by a scale $\Lambda_\mathrm{p}>0$, beyond which the field
amplitude is exponentially suppressed. On other spacetimes, of length scale
$L$, the flow ceases to exist once a certain universal measure of inhomogeneity
exceeds $O(1)+2\pi L^2\Lambda^2_\mathrm{p}$. Importantly for cosmology, the
minimum size of the universe is thus tied to the degree of inhomogeneity, with
spacetimes of vanishing size being required to be almost homogeneous. We
initiate a study of this exotic quantum field theory at the interacting level,
and discuss what the full theory of quantum gravity should look like, one which
must thus be perturbatively renormalizable in Newton's constant but
non-perturbative in $\hbar$.Comment: 52 pages, 4 figures; fixed typos; improved explanation of the sign of
V, and the use of Sturm-Liouville theory. To be publ in JHE

### Background independent exact renormalization group for conformally reduced gravity

Within the conformally reduced gravity model, where the metric is
parametrised by a function $f(\phi)$ of the conformal factor $\phi$, we keep
dependence on both the background and fluctuation fields, to local potential
approximation and $\mathcal{O}(\partial^2)$ respectively, making no other
approximation. Explicit appearances of the background metric are then dictated
by realising a remnant diffeomorphism invariance. The standard non-perturbative
Renormalization Group (RG) scale $k$ is inherently background dependent, which
we show in general forbids the existence of RG fixed points with respect to
$k$. By utilising transformations that follow from combining the flow equations
with the modified split Ward identity, we uncover a unique background
independent notion of RG scale, $\hat k$. The corresponding RG flow equations
are then not only explicitly background independent along the entire RG flow
but also explicitly independent of the form of $f$. In general $f(\phi)$ is
forced to be scale dependent and needs to be renormalised, but if this is
avoided then $k$-fixed points are allowed and furthermore they coincide with
$\hat k$-fixed points.Comment: 53 pages, broken reference correcte

### Convergence of derivative expansions of the renormalization group

We investigate the convergence of the derivative expansion of the exact
renormalization group, by using it to compute the beta function of scalar field
theory. We show that the derivative expansion of the Polchinski flow equation
converges at one loop for certain fast falling smooth cutoffs. The derivative
expansion of the Legendre flow equation trivially converges at one loop, but
also at two loops: slowly with sharp cutoff (as a momentum-scale expansion),
and rapidly in the case of a smooth exponential cutoff. Finally, we show that
the two loop contributions to certain higher derivative operators (not involved
in beta) have divergent momentum-scale expansions for sharp cutoff, but the
smooth exponential cutoff gives convergent derivative expansions for all such
operators with any number of derivatives.Comment: Latex inc axodraw. 20 page

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