Adv. Appl. Cliﬀord Algebras 27 (2017), 2457–2471
2017 Springer International Publishing
published online April 17, 2017
Applied Cliﬀord Algebras
Four-Dimensional Conformal Field Theory
Communicated by Jayme Vaz
Abstract. We have built a constrained four-dimensional quaternion-par-
ametrized conformal ﬁeld theory using quaternion holomorphic func-
tions as the generators of quaternionic conformal transformations. With
the two-dimensional complex-parametrized conformal ﬁeld theory as our
model, we study the stress tensor, the conserved charge, the symmetry
generators, the quantization conditions and several operator product
expansions. Future applications are also addressed.
Conformal symmetry generalizes the symmetry operation of dilation, and
consequently preserves angles of plane ﬁgures. Translations and rotations also
preserve angles, but dilations do not necessarily preserve areas. These simple
symmetry properties are employed in the theory of Lie groups, where the
metric preserving groups and the volume preserving groups are the most im-
portant [16,19]. Conversely, a conformal group is possible, and its generators
are translations, rotations, dilations and a special conformal transformation.
The angle preserving property makes conformal symmetry fundamental in
complex analysis and in hyperbolic geometry , and physical applications
include condensed matter physics and string theory.
However, the applications in physics depend on a conformal quantum
ﬁeld theory (CFT). The two-dimensional conformal ﬁeld theory [3,17]isthe
most widely used, and it involves many features that explain this widespread
use. The inﬁnite-dimensional conformal algebra and its central extension are
certainly important applicability factors. Another feature is the parametriza-
tion of two-dimensional CFT using complex numbers, and thus the employ-
ment of the full machinery of complex analysis in order to build a complex-
parametrized conformal ﬁeld theory (CCFT).