Abstract
A large body of work over several decades indicates that, in the presence of gravitational interactions, there is loss of localization resolution within a fundamental (∼ Planck) length scale ℓ. We develop a general formalism based on wavelet decomposition of fields that takes this UV ‘opaqueness’ into account in a natural and mathematically well-defined manner. This is done by requiring fields in a local Lagrangian to be expandable in only the scaling parts of a (complete or, in a more general version, partial) wavelet Multi-Resolution Analysis. This delocalizes the interactions, now mediated through the opaque regions, inside which they are rapidly decaying. The opaque regions themselves are capable of discrete excitations of ∼ 1/ℓ spacing. The resulting effective Feynman rules, which give UV regulated and (perturbatively) unitary physical amplitudes, resemble those of string field theory.
Highlights
This implies that working with particular subsets of components φn, φA can be done by using suitable projection operators acting on the full field φ
A general scheme was developed for incorporating the physical requirement of loss of resolution inside some fundamental length scale in a mathematically well-defined and natural way
The formalism is based on the mathematics of wavelets, which allows separation of length scales in orthogonal decompositions
Summary
We fix some UV length scale , which here may be naturally taken to be of the order of Planck length or some unification scale. Here we take the scaling space, denoted Vl, to refer to the scale This means that, given a mother scaling function σ(x) and corresponding 2d − 1 mother wavelet functions υq(x) on Rd, the basis set is given by with x ∈ Rd , σln(x) = 2dl/2σ(2ˆlx − bn) υmq n(x) = 2dm/2υq(2mx − bn) n ∈ Zd , ˆl ≤ m ∈ Z , 1 ≤ q ≤ 2d − 1. The wavelet parts probe inside such regions down to arbitrarily small scales with successively finer resolution: the υmq n wavelets terms probe scales of order m, m ≥ 1. The set of coefficients {φn, φqmn} gives a discrete representation of φ(x) with the coefficients as the dynamical degrees of freedom This representation, with integration over φ replaced by integration over the infinite set {φn, φqmn} in the functional integral, is the natural one in applications such as real space renormalization group [7, 13]. Where D(Λ) denotes the transformation matrix in the appropriate tensor representation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
