Abstract
In this note, we explicitly compute the vacuum expectation value of the commutator of scalar fields in a d-dimensional conformal field theory on the cylinder. We find from explicit calculations that we need smearing not only in space but also in time to have finite commutators except for those of free scalar operators. Thus the equal time commutators of the scalar fields are not well-defined for a non-free conformal field theory, even if which is defined from the Lagrangian. We also have the commutator for a conformal field theory on Minkowski space, instead of the cylinder, by taking the small distance limit. For the conformal field theory on Minkowski space, the above statements are also applied.
Highlights
Studied intensively partly because there are not infinitely many conserved current.2 On the other hand, the commutator of fields in the cylinder R × Sd−1 can be derived from the operator product expansion (OPE) for the higher dimensional CFT recently [15]
We find from explicit calculations that we need smearing in space and in time to have finite commutators except for those of free scalar operators
The equal time commutators of the scalar fields are not well-defined for a non-free conformal field theory, even if which is defined from the Lagrangian
Summary
First we parametrize a position in Euclidean flat space Rd by xμ = r eμ(Ω), where r = |x| and eμ(Ω) is a unit vector, i.e. eμ(Ω)eμ(Ω) = 1. We parametrize unit vector eμ by angular variables Ω. For the normalization of the spherical harmonics, see the appendix A.1. Cl(Ω12), 4We mean “taking most singular part” by ∼. Where Clα(x) is the Gegenbauer polynomial [18].5. For the normalization of the Gegenbauer polynomials, see appendix A.2. We have considered only the most singular part of the OPE, other parts can be expanded in the same way. We continue dealing only with the most singular part. We can compute the commutator formally [15] we need the explicit OPE data to give an explicit result
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