Weyl consistency conditions in non-relativistic quantum field theory
Received: June
Weyl consistency conditions in non-relativistic quantum eld theory
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c The Authors. 0 1
0 San Diego , 9500 Gilman Drive, La Jolla, CA 92093 , U.S.A
1 Department of Physics, University of California
Weyl consistency conditions have been used in unitary relativistic quantum eld theory to impose constraints on the renormalization group ow of certain quantities. We classify the Weyl anomalies and their renormalization scheme ambiguities for generic non-relativistic theories in 2 + 1 dimensions with anisotropic scaling exponent z = 2; the extension to other values of z are discussed as well. We give the consistency conditions among these anomalies. As an application we nd several candidates for a C-theorem. We comment on possible candidates for a C-theorem in higher dimensions.
quantum; Anomalies in Field and String Theories; Conformal and W Symmetry; Renor-
1 Introduction 2 Generalities 2.1
Dynamical exponent
Listing out terms
Using counter-terms
Consistency conditions and vanishing anomalies
Generalisation to arbitrary z value
A candidate for a C-theorem in d + 1D
Summary and discussion
A Consistency conditions for 2 + 1d NRCFT
Anomaly ambiguities
B.1 @t2 sector
C S-theorem: 0 + 1D conformal quantum mechanics
Introduction
lous dimensions of operators.
However, additional information, such as dynamical (or
other methods, e.g., mean
eld approximation, that can give more detailed information.
between these regions.
diverges, jaj ! 1, exhibit non-relativistic conformal symmetry. Ultracold atom gas
exseveral atomic systems, including 85Rb [9],138Cs [10, 11], 39K [12].
In the context of critical dynamics the response function exhibits dynamical scaling.
the presence of ghosts,1 Horava suggested extending Einstein gravity by terms with higher
using holographic methods.
Wess-Zumino consistency conditions for Weyl transformations have been used in
unitary relativistic quantum
eld theory to impose constraints on the renormalization group
Zamolodchikov's C-function [30], that famously decreases monotonically along
Weyl consistency conditions can be used to show that a quantity ~a satis es
is the renormalization group scale, increasing towards short distances. The
equation shows that at xed points, characterized by
dg =d
= 0, a~ is stationary. It can
be shown in perturbation theory that H
is a positive de nite symmetric matrix [31]. By
1Generically, the S-matrix in models with ghosts is not unitary. However, under certain conditions on the
and unitarity is a long-distance emergent phenomenon [17].
= H
by explicit computations of \metric" H
in perturbation theory [34{36]. However it was
quantity a~ of ref. [33] that obey a C-theorem perturbatively.
eld theories. The constraints imposed at
xed points have been studied in ref. [23] for
the critical points along the
ows. As mentioned above, there are questions that can
conditions on
ows can be used to ask a number of questions. For example, we may ask if
there is a suitable candidate for a C-theorem.
A related issue is the possibility of recursive renormalization group ows. Recursive
ows in the perturbative regime have been found in several examples in 4
and in 4
dimensional relativistic quantum
eld theory [40{45]. Since Weyl consistency conditions
in any physically meaningful sense [46, 47]; in fact, they may be removed by a
not show whether generically the \metric" H
has de nite sign. The question of under
further investigation.
The paper is organized as follows. In section 2 we set-up the computation, using a
appendix C.
Generalities
function of elds
(t; ~x), mass parameters m and coupling constants g that parametrize
remains invariant under the rescaling
S[ (~x; t)] =
( ~x; zt) = S[ (~x; t)] :
is the matrix of canonical dimensions of the elds . In a multi- eld model the
term in L is local, so that it entails powers of derivative operators, z counts the mismatch
is a single time derivative and z spatial derivatives so that z is an integer.
For a simple example, useful to keep in mind for orientation, the action for a single
complex scalar eld with anisotropic scaling z in d dimensions is given by
S =
riz=2
riz=2
gmz=d
scaling property (2.1) holds with
the set of dimensionless couplings that we denote by g below.
The above setup is appropriate for studies of, say, quantum criticality. However the
iy in the example of eq. (2.2) the corresponding
energy integral is
H =
riz=2
riz=2
+ gmz=d
The short distance divergences encountered in these models need to be regularized
symmetries explicitly. Thus we consider NR
eld theories in 1 + n dimensions, where
the spatial dimension n = d
, with d an integer. Dimensional regularization requires
the introduction of a parameter
with dimensions of inverse length, L 1
. Invariance
under (2.1) is then broken, but can be formally recovered by also scaling
1 . For an example, consider the dimensionally regularized version of (2.2):
S[ 0(~x; t); ] =
riz=2 0ri1
We have written this i (...truncated)