3 research outputs found
Infinite dimensional Lie algebras in 4D conformal quantum field theory
The concept of global conformal invariance (GCI) opens the way of applying
algebraic techniques, developed in the context of 2-dimensional chiral
conformal field theory, to a higher (even) dimensional space-time. In
particular, a system of GCI scalar fields of conformal dimension two gives rise
to a Lie algebra of harmonic bilocal fields, V_m(x,y), where the m span a
finite dimensional real matrix algebra M closed under transposition. The
associative algebra M is irreducible iff its commutant M' coincides with one of
the three real division rings. The Lie algebra of (the modes of) the bilocal
fields is in each case an infinite dimensional Lie algebra: a central extension
of sp(infty,R) corresponding to the field R of reals, of u(infty,infty)
associated to the field C of complex numbers, and of so*(4 infty) related to
the algebra H of quaternions. They give rise to quantum field theory models
with superselection sectors governed by the (global) gauge groups O(N), U(N),
and U(N,H)=Sp(2N), respectively.Comment: 16 pages, with minor improvements as to appear in J. Phys.
Conformal invariance: from Weyl to SO(2,d)
The present work deals with two different but subtilely related kinds of
conformal mappings: Weyl rescaling in dimensional spaces and SO(2,d)
transformations. We express how the difference between the two can be
compensated by diffeomorphic transformations. This is well known in the
framework of String Theory but in the particular case of spaces. Indeed,
the Polyakov formalism describes world-sheets in terms of two-dimensional
conformal field theory. On the other hand, B. Zumino had shown that a classical
four-dimensional Weyl-invariant field theory restricted to live in Minkowski
space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to
relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS).
This allows us to assert that a classical -invariant field does not
distinguish, at least locally, between two different -dimensional CFSs.Comment: 5 pages, no figures. There are slight modifications to match with the
published versio