A spin network primer

American Journal of Physics

Volumn 67, Issue 11, 1999, Pages 972-980

  Major, Seth A ^a  

^a UNIVERSITY OF VIENNA   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033443155     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19175     Document Type: Article

References (40)

1. Roger Penrose, "Angular momentum: An approach to combinatorial space-time," in Quantum Theory and Beyond, edited by T. Bastin (Cambridge U.P., Cambridge, 1971); "Combinatorial quantum theory and quantized directions," in Advances in Twistor Theory, edited by L. P. Hughston and R. J. Ward, Research Notes in Mathematics, Vol. 37 (Pitman, San Francisco, 1979), pp. 301-307; and "Theory of Quantized Directions" (unpublished notes).
2. Roger Penrose, "Angular momentum: An approach to combinatorial space-time," in Quantum Theory and Beyond, edited by T. Bastin (Cambridge U.P., Cambridge, 1971); "Combinatorial quantum theory and quantized directions," in Advances in Twistor Theory, edited by L. P. Hughston and R. J. Ward, Research Notes in Mathematics, Vol. 37 (Pitman, San Francisco, 1979), pp. 301-307; and "Theory of Quantized Directions" (unpublished notes).
3. Roger Penrose, "Angular momentum: An approach to combinatorial space-time," in Quantum Theory and Beyond, edited by T. Bastin (Cambridge U.P., Cambridge, 1971); "Combinatorial quantum theory and quantized directions," in Advances in Twistor Theory, edited by L. P. Hughston and R. J. Ward, Research Notes in Mathematics, Vol. 37 (Pitman, San Francisco, 1979), pp. 301-307; and "Theory of Quantized Directions" (unpublished notes).
4. John C. Baez, "Spin networks in gauge theory," Adv. Math. 117, 253-272 (1996), Online Preprint Archive: http://xxx.lanl.gov/abs/gr-qc/ 9411007; "Spin networks in nonperturbative quantum gravity," in The Interface of Knots and Physics, edited by Louis Kauffman (American Mathematical Society, Providence, RI, 1996), pp. 167-203, Online Preprint Archive: http://xxx.lanl.gov/abs/gr-qc/9504036.
5. John C. Baez, "Spin networks in gauge theory," Adv. Math. 117, 253-272 (1996), Online Preprint Archive: http://xxx.lanl.gov/abs/gr-qc/ 9411007; "Spin networks in nonperturbative quantum gravity," in The Interface of Knots and Physics, edited by Louis Kauffman (American Mathematical Society, Providence, RI, 1996), pp. 167-203, Online Preprint Archive: http://xxx.lanl.gov/abs/gr-qc/9504036.
6. There are more philosophic motivations for this model as well. Mach advocated an interdependence of phenomena: "The physical space I have in mind (which already includes time) is therefore nothing but the dependence of the phenomena on one another. A completed physics that knew of this dependence would have no need of separate concepts of space and time because these would already have be encompassed" [Ernst Mach, Fichtes Zeitschrift für Philosophie 49, 227 (1866) cited by Lee Smolin in Conceptual Problems of Quantum Gravity, edited by A. Ashtekar and J. Stachel (Birkhäuser, Boston, 1991)] -echoing Leibniz's much earlier critique of Newton's concept of absolute space and time. Penrose invokes such a Machian principle: A background space on which physical events unfold should not play a role; only the relationships of objects to each other can have significance.
7. There are more philosophic motivations for this model as well. Mach advocated an interdependence of phenomena: "The physical space I have in mind (which already includes time) is therefore nothing but the dependence of the phenomena on one another. A completed physics that knew of this dependence would have no need of separate concepts of space and time because these would already have be encompassed" [Ernst Mach, Fichtes Zeitschrift für Philosophie 49, 227 (1866) cited by Lee Smolin in Conceptual Problems of Quantum Gravity, edited by A. Ashtekar and J. Stachel (Birkhäuser, Boston, 1991)] -echoing Leibniz's much earlier critique of Newton's concept of absolute space and time. Penrose invokes such a Machian principle: A background space on which physical events unfold should not play a role; only the relationships of objects to each other can have significance.
8. John P. Moussouris, "Quantum models as spacetime based on recoupling theory," Oxford Ph.D. dissertation, unpublished, 1983.
9. Carlo Rovelli and Lee Smolin, "Discreteness of area and volume in quantum gravity," Nucl. Phys. B 442, 593-622 (1995).
10. Roberto De Pietri and Carlo Rovelli, "Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity," Phys. Rev. D 54 (4), 2664-2690 (1996). See also, Ref. 5.
11. μν,hence the name.
12. There is some redundancy in notation. Numerical (or more general) labels associated with edges are frequently called weights or labels. The term "tag" encompasses the meaning of these labels as well as operators on lines.
13. A mathematical knot is a knotted, closed loop. One often encounters collections of, possibly knotted, knots. These are called links.
14. K. Reidemeister, Knotentheorie (Chelsea, New York, 1948), original printing (Springer, Berlin, 1932). See also Louis Kauffman, Knots and Physics, Series on Knots and Everything Vol. 1 (World Scientific, Singapore, 1991), p. 16.
15. K. Reidemeister, Knotentheorie (Chelsea, New York, 1948), original printing (Springer, Berlin, 1932). See also Louis Kauffman, Knots and Physics, Series on Knots and Everything Vol. 1 (World Scientific, Singapore, 1991), p. 16.
16. Louis H. Kauffman, in Ref. 10, pp. 125-130, 443-447. Also, Louis H. Kauffman and Sóstenes L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Annals of Mathematics Studies No. 134 (Princeton U.P., Princeton, 1994), pp. 1-100.
17. Louis H. Kauffman, in Ref. 10, pp. 125-130, 443-447. Also, Louis H. Kauffman and Sóstenes L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Annals of Mathematics Studies No. 134 (Princeton U.P., Princeton, 1994), pp. 1-100.
18. Note that, because of the additional sign associated with crossings, the "antisymmetrizer" symmetrizes the indices in the δ∈ world.
19. n.
20. See Ref. 4, p. 17.
21. Roberto De Pietri, "On the relation between the connection and the loop representation of quantum gravity," Class. Quantum Grav. 14, 53-70 (1990).
22. See Ref. 6, p. 2671.
23. This may be shown by noticing that (equation presented) as may be derived using Eqs. (4) and (7).
24. i⊗⋯⊗1 where the sum is over the possible positions of the Pauli matrix.
25. Since the Clebsch-Gordan symbols are complete, any map from a ⊗ b to c must be a multiple of the Clebsch-Gordan or 3j symbol.
26. See, for instance, Albert Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 2, p. 573 or H. F. Jones, Groups, Representations and Physics (Hilger, Bristol, 1990), p. 116.
27. See, for instance, Albert Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 2, p. 573 or H. F. Jones, Groups, Representations and Physics (Hilger, Bristol, 1990), p. 116.
28. See Carlo Rovelli, "Loop Quantum Gravity," in Living Reviews in Relativity, http://www.livingreviews.org/Articles/Volume1/1998-Irovelli for a recent review.
29. Abhay Ashtekar, "New variables for classical and quantum gravity," Phys. Rev. Lett. 57 (18), 2244-2247 (1986); New Perspectives in Canonical Gravity (Bibliopolis, Naples, 1988); Lectures on Non-perturbative Canonical Gravity, Advanced Series in Astrophysics and Cosmology Vol. 6 (World Scientific, Singapore, 1991).
30. Abhay Ashtekar, "New variables for classical and quantum gravity," Phys. Rev. Lett. 57 (18), 2244-2247 (1986); New Perspectives in Canonical Gravity (Bibliopolis, Naples, 1988); Lectures on Non-perturbative Canonical Gravity, Advanced Series in Astrophysics and Cosmology Vol. 6 (World Scientific, Singapore, 1991).
31. Abhay Ashtekar, "New variables for classical and quantum gravity," Phys. Rev. Lett. 57 (18), 2244-2247 (1986); New Perspectives in Canonical Gravity (Bibliopolis, Naples, 1988); Lectures on Non-perturbative Canonical Gravity, Advanced Series in Astrophysics and Cosmology Vol. 6 (World Scientific, Singapore, 1991).
32. For those readers familiar with general relativity the potential determines the parallel transport of spin-1/2 particles.
33. edt ė(t) · A(e(t))) associated with it.; See, for example, Abhay Ashtekar, "Quantum mechanics of Riemannian geometry," http:// vishnu.nirvana.phys.psu.edu/riem_qm/riem_qm.html.
34. 2 sin(θ)dΦdθ.
35. See Ref. 5; Abhay Ashtekar and Jerzy Lewandowski, "Quantum Theory of Geometry. I. Area operators," Class. Quantum Grav. 14, A55-A81 (1997); S. Fittelli, L. Lehner, and C. Rovelli, "The complete spectrum of the area from recoupling theory in loop quantum gravity," ibid. 13, 2921-2932 (1996).
36. See Ref. 5; Abhay Ashtekar and Jerzy Lewandowski, "Quantum Theory of Geometry. I. Area operators," Class. Quantum Grav. 14, A55-A81 (1997); S. Fittelli, L. Lehner, and C. Rovelli, "The complete spectrum of the area from recoupling theory in loop quantum gravity," ibid. 13, 2921-2932 (1996).
37. See Ref. 5.
38. See A. Ashtekar and J. Lewandowski in Ref. 26.
39. See L. Kauffman and S. Lins in Ref. 11.
40. See L. Kauffman and S. Lins in Ref. 11. p. 98.