Abstract
We use the extension problem proposed by Caffarelli and Silvestre to study the quantization of a scalar nonlocal quantum field theory built out of the fractional Laplacian. We show that the quantum behavior of such a nonlocal field theory in $d$-dimensions can be described in terms of a local action in $d+1$ dimensions which can be quantized using the canonical operator formalism though giving up local commutativity. In particular, we discuss how to obtain the two-point correlation functions and the vacuum energy density of the nonlocal fractional theory as a brane limit of the bulk correlators. We show explicitly how the quantized extension problem reproduces exactly the same particle content of other approaches based on the spectral representation of the fractional propagator. We also briefly discuss the inverse fractional Laplacian and possible applications of this approach in general relativity and cosmology.
Highlights
During the past years, there has been an increasing interest in the study of nonlinear and quasilinear equations that involve fractional powers of the Laplacian
We use the extension problem proposed by Caffarelli and Silvestre to study the quantization of a scalar nonlocal quantum field theory built out of the fractional Laplacian
We show that the quantum behavior of such a nonlocal field theory in d dimensions can be described in terms of a local action in d þ 1 dimensions which can be quantized using the canonical operator formalism though giving up local commutativity
Summary
There has been an increasing interest in the study of nonlinear and quasilinear equations that involve fractional powers of the Laplacian (see e.g., [1,2,3,4]). The extension problem described above is the mathematical tool that we are going to use in order to develop a description of a nonlocal d-dimensional scalar field theory, based on the fractional Laplacian, in terms of a local (d þ 1)-dimensional field theory. By using the extension problem, it will be shown that quantizing a local field theory in a bounded spacetime (bulk) with an extra spacelike transverse dimension (y ∈ 1⁄20; þ∞) via the operator formalism, but waiving local commutativity, reproduces, through a limiting procedure (y → 0), the correlation functions of the nonlocal theory showing a complete equivalence with the fractional propagator obtained independently via analyticity considerations in terms of the spectral representation.
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