Loop Representations
نویسنده
چکیده
The loop representation plays an important role in canonical quantum gravity because loop variables allow a natural treatment of the constraints. In these lectures we give an elementary introduction to (i) the relevant history of loops in knot theory and gauge theory, (ii) the loop representation of Maxwell theory, and (iii) the loop representation of canonical quantum gravity.
منابع مشابه
Fusion Rings of Loop Group Representations
We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.
متن کاملv 1 [ m at h . FA ] 1 5 A ug 2 00 0 Representations of Cuntz algebras , loop groups and wavelets Palle
A theorem of Glimm states that representation theory of an NGCR C∗-algebra is always intractable, and the Cuntz algebra ON is a case in point. The equivalence classes of irreducible representations under unitary equivalence cannot be captured with a Borel cross section. Nonetheless, we prove here that wavelet representations correspond to equivalence classes of irreducible representations of ON...
متن کاملNew Loop Representations for 2+1 Gravity
Since the gauge group underlying 2+1-dimensional general relativity is non-compact, certain difficulties arise in the passage from the connection to the loop representations. It is shown that these problems can be handled by appropriately choosing the measure that features in the definition of the loop transform. Thus, “old-fashioned” loop representations – based on ordinary loops – do exist. I...
متن کاملTowards new background independent representations for Loop Quantum Gravity
Recently, uniqueness theorems were constructed for the representation used in Loop Quantum Gravity. We explore the existence of alternate representations by weakening the assumptions of the so called LOST uniqueness theorem. The weakened assumptions seem physically reasonable and retain the key requirement of explicit background independence. For simplicity, we restrict attention to the case of...
متن کاملOperator algebras and conformal ®eld theory III. Fusion of positive energy representations of LSU(N) using bounded operators
I. Positive energy representations of LSU(N) ................................................................. 477 II. Local loop groups and their von Neumann algebras ................................................. 491 III. The basic ordinary dierential equation ..................................................................... 505 IV. Vector and dual vector primary ®elds ..................
متن کاملGerbal Representations of Double Loop Groups
A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac–Moody groups and Lie algebras). Loop groups embed into the group GL∞ of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL∞. In a sim...
متن کامل