September, 1994

Reading Guide to the New Variables

Our intention in creating this list is to compile a guide to the new variables. We hope this will help you to find the ``pre-requisites'' as well as some of the more contemporary work on this approach to quantum gravity. We've divided the list into ten fields. In each field we've listed references roughly in order of difficulty/sophistication.

You're encouraged not to read linearly (this is, after-all, a non-linear field!) For instance, you might start with the more recent book by Abhay known as ``Lectures...'' (the blue book), the older ``New Perspectives...'' or the shorter, more inspirational, ``Primer'' by Jorge and then backtrack to pick up the background you find you need. For information not covered here, ask around! Let us know about your favorite references.

-- Seth Major and Troy Schilling

Since we completed this list, two books have been brought to our attention:

John Baez has a new introductory book! Gauge Fields, Knots and Gravity (World Scientific 1994 ISBN 981-02-2034-0 (paperback))

Volume 434 of Springer-Verlag's Lecture Notes in Physics , Canonical Gravity From Classical to Quantum is the proceedings of a conference held in September 1993.

Classical Mechanics - Symplectic structures:

• Joe Romano's notes or Geroch's notes, Geometrical Quantum Mechanics [unpublished Chicago lecture notes - there is a copy in the preprint lounge].
• V.I. Arnold, Mathematical Methods of Classical Mechanics 2nd ed., pg. 201 (Springer-Verlag, New York, 1989) [MATH QA805.A6813 1989].
• A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, Appendix B, (World Scientific, Singapore, 1991) [QC178.A48 1991].

Constrained Systems:

• P.A.M. Dirac, Lectures on Quantum Mechanics, (Yeshiva, New York, 1964) [Annex QC174.1.D55].
• M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Chs. 1, 2 and 4, (Princeton, 1992). [QC793.3.F5H46 1992].
• Joe Romano, ``Geometrodynamics vs. Connection Dynamics'' in General Relativity and Gravitation, 25 (1993) 759 (This is Joe's Ph.D. thesis, Syracuse 1991).
• A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, Appendix B, (World Scientific, Singapore, 1991) [QC178.A48 1991].

Hamiltonian Formulation of GR - 3+1:

• R. Wald, General Relativity, Apendix E, (Chicago, 1984) [It's all in the book but you have to hop back and forth from chaper to appendix].
• A. Ashtekar, New Perspectves in Canonical Gravity , pg. 37, (Bibliopolis, Napoli, 1988) [QC178.A73 1988].

New Variables:

• Joe Romano, ``Geometrodynamics vs. Connection Dynamics'' in General Relativity and Gravitation, 25 (1993) 759 (This is Joe's Ph.D. thesis, Syracuse 1991).
• A. Ashtekar, Mathematical problems of non-perturbative quantum general relativity in Les Houches 1992, [B. Julia ed., Les Houches 1992 (1994)].
• A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, (World Scientific, Singapore, 1991) [QC178.A48 1991].

Loop Representation:

• J. Pullin Knot Theory and Quantum Gravity in Loop Space: A Primer , hep-th/9301028 (breezy but inspiring!).
• For a broad review try C. Rovelli. Class. and Quantum Grav. 8 (1991) 1613 or L. Smolin in Proceedings of the XXII Gift International Seminar (World Scientific 1992). Or, read the orginal paper: C. Rovelli and L. Smolin. Nuc.Phys. B331 (1990) 80.
• B. Brügmann, On the constraints of quantum general relativity in the loop representation, Ph.D. thesis, Syracuse University (May 1993).
• B. Brügmann, Loop Reprentations , gr-qc/9312001.
• A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, Ch. 15 and 16, (World Scientific, Singapore, 1991) [QC178.A48 1991].

Quantization:

• R. Geroch, Geometrical Quantum Mechanics, unpublished lecture notes, Univ. of Chicago.
• A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, Ch. 10, (World Scientific, Singapore, 1991) [QC178.A48 1991].

Fibre Bundles:

• C. Nash and S. Sen, Topology and Geometry for Physicists , pg. 140, (Academic, London, 1983) [QA611.N35].
• C. Isham, Modern Differential Geometry for Physicists, pg. 111 (World Scientific, 1989).
• M. Nakahara, Geometry, Topology and Physics, Ch. 9, (Adam Hilger, 1990) [QA641.N35 1990].
• A. Ashtekar, G. Horowitz, and A. Magnon-Ashtekar, General Relativity and Gravitation, 14 (1982) 411.
• Y. Choquet-Bruhat and C DeWitt-Morette, Analysis, Manifolds and Physics, Amsterdam; N.Y., North-Holland Pub. (1981) [QC20.7.A5C48 1981].

Connections in Field Theory/on Principal Bundles:

• Your favorite field theory book or L.D. Faddeev and A.A. Slavnov, Gauge Fields: Introduction to Quantum Theory, Ch 1 and 3 (an old style for field theory but closer to the language used around here) (Benjamin/Cummings, Reading, MA, 1980), [QC793.3.F5S5213].
• L. Kauffman, Knots and Physics, pg. 293 (just after the little bit on SU(2)) (World Scientific, Singapore, 1991) [QC20.7.K56K38 1991] .
• Jackiw in Les Houches, Relativity, B. DeWitt and R. Stora, eds. [QC174.13.R45 1984].
• M. Nakahara, Geometry, Topology and Physics, Ch. 10, (Adam Hilger, 1990) [QA641.N35 1990].

Tetrads:

• S. Weinberg, Gravitation and Cosmology, pg. 385 (Wiley, New York, 1972).
• R. Wald, General Relativity, pg. 49, (Chicago, 1984).

Spinors:

• W. Bede and H. Jehle, Rev. Mod. Phys. 25 (1953) 714 (old and from the point of view of the Dirac equation).
• L. Kauffman, Knots and Physics, pg. 392 (esp. 398).
• R. Penose and W. Rindler, Spinors and Spacetime Volume 1, (Cambridge, 1984) [Math QC20.7.S65P46 1984].
• A. Trautmann, F. Pirani, and H. Bondi, Lectures on General Relativity, Ch. 3, (Prentice-Hall, 1965) [Annex QC6.B675 1964].
• G. Sterman, An Introduction to Quantum Field Theory, pg. 119, (Cambridge, 1993).
• A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, Appendix A, (World Scientific, Singapore, 1991) [QC178.A48 1991].

• About this document ...