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/content/aip/journal/jmp/57/8/10.1063/1.4960823
1.
A. Stottmeister and T. Thiemann, “Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations,” J. Math. Phys. 57, 063509 (2016).
http://dx.doi.org/10.1063/1.4954228
2.
A. Stottmeister and T. Thiemann, “Coherent states, quantum gravity, and the Born- Oppenheimer approximation. II. Compact Lie groups,” J. Math. Phys. 57, 073501 (2016).
http://dx.doi.org/10.1063/1.4954803
3.
G. Panati, H. Spohn, and S. Teufel, Adv. Theor. Math. Phys. 7, 145 (2003).
http://dx.doi.org/10.4310/ATMP.2003.v7.n1.a6
4.
K. Giesel, J. Tambornino, and T. Thiemann, e-print arXiv:0911.5331 (2009).
5.
T. Thiemann, Classical Quantum Gravity 18, 3293 (2001).
http://dx.doi.org/10.1088/0264-9381/18/17/301
6.
S. Lanéry and T. Thiemann, preprint arXiv:1411.3592 (2014).
7.
A. Stottmeister and T. Thiemann, preprint arXiv:1312.3657 (2013).
8.
An elementary introduction to the semi-analytic category can be found in Ref. 12.
9.
N. Bodendorfer, T. Thiemann, and A. Thurn, Classical Quantum Gravity 30, 045001 (2013).
http://dx.doi.org/10.1088/0264-9381/30/4/045001
10.
N. Bodendorfer, T. Thiemann, and A. Thurn, Classical Quantum Gravity 30, 045002 (2013).
http://dx.doi.org/10.1088/0264-9381/30/4/045002
11.
C. Rovelli, in Quantum Gravity, Cambridge Monographs on Mathematical Physics, edited byP. V. Landshoff, D. R. Nelson, and S. Weinberg (Cambridge University Press, 2007).
12.
T. Thiemann, in Modern Canonical Quantum General Relativity, edited byP. V. Landshoff, D. R. Nelson, and S. Weinberg, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2008).
13.
T. Thiemann, Classical Quantum Gravity 15, 839 (1998).
http://dx.doi.org/10.1088/0264-9381/15/4/011
14.
T. Thiemann, Classical Quantum Gravity 15, 875 (1998).
http://dx.doi.org/10.1088/0264-9381/15/4/012
15.
T. Thiemann, Classical Quantum Gravity 15, 1207 (1998).
http://dx.doi.org/10.1088/0264-9381/15/5/010
16.
T. Thiemann, Classical Quantum Gravity 15, 1249 (1998).
http://dx.doi.org/10.1088/0264-9381/15/5/011
17.
T. Thiemann, Classical Quantum Gravity 15, 1281 (1998).
http://dx.doi.org/10.1088/0264-9381/15/5/012
18.
T. Thiemann, Classical Quantum Gravity 15, 1463 (1998).
http://dx.doi.org/10.1088/0264-9381/15/6/005
19.
H. Amann, “Vector-Valued Distributions and Fourier Multipliers” (unpublished), http://user.math.uzh.ch/amann/files/distributions.ps.
20.
Unfortunately, this excludes fractal graphs, which are sometimes assumed to be of relevance for loop quantum gravity.
21.
G. Morchio and F. Strocchi, Ann. Phys. 324, 2236 (2009).
http://dx.doi.org/10.1016/j.aop.2009.07.005
22.
N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics (Springer, 1998).
23.
D. P. Williams, Crossed Products of C*-algebras (American Mathematical Society, 2007), Vol. 134.
24.
I. Raeburn and D. P. Williams, Morita Equivalence and Continuous-trace C*-Algebras (American Mathematical Society, 1998), Vol. 60.
25.
The partial order ≤ is essentially the one, ≺, defined in Ref. 5.
26.
It seems that this non-trivial condition has been overlooked in the main part of the literature with the exception of Ref. 6, where the partial orders L and R are defined making the use of edge inversions obsolete at the phase space level.
27.
T. Thiemann, Classical Quantum Gravity 18, 2025 (2001).
http://dx.doi.org/10.1088/0264-9381/18/11/304
28.
T. Thiemann and O. Winkler, Classical Quantum Gravity 18, 2561 (2001).
http://dx.doi.org/10.1088/0264-9381/18/14/301
29.
T. Thiemann and O. Winkler, Classical Quantum Gravity 18, 4997 (2001).
http://dx.doi.org/10.1088/0264-9381/18/23/302
30.
L. Freidel, M. Geiller, and J. Ziprick, Classical Quantum Gravity 30, 085013 (2013).
http://dx.doi.org/10.1088/0264-9381/30/8/085013
31.
We consider only *-morphisms corresponding to and , because generic are not Poisson maps. Furthermore, we have compatibility with transitivity of ≤, and the commutative diagram (2.24) with respect to edge inversion for and .
32.
O. Bratteli and D. W. Robinson, in Operator Algebras and Quantum Statistical Mechanics 1: C-and W-Algebras, Symmetry Groups, Decomposition of States, 2nd ed. edited by R. Balian, W. Beiglböck, H. Grosse, E. H. Lieb, N. Reshetikhin, H. Spohn, and W. Thirring, Texts and Monographs in Physics (Springer, 1987).
33.
D. C. Taylor, Trans. Am. Math. Soc. 150, 633 (1970).
http://dx.doi.org/10.1090/S0002-9947-1970-0290117-2
34.
R. V. Kadison and J. R. Ringrose, in Fundamentals of the Theory of Operator Algebras, Vol. II, Advanced Theory, edited byS. Eilenberg and H. Bass, Pure and Applied Mathematics (Academic Press, Inc., 1986).
35.
Z. Takeda, Tohoku Math. Soc. 7, 67 (1955).
http://dx.doi.org/10.2748/tmj/1178245105
36.
T. Thiemann and O. Winkler, Classical Quantum Gravity 18, 4629 (2001).
http://dx.doi.org/10.1088/0264-9381/18/21/315
37.
H. Sahlmann, T. Thiemann, and O. Winkler, Nucl. Phys. B 606, 401 (2001); e-print arXiv:gr-qc/0102038.
http://dx.doi.org/10.1016/S0550-3213(01)00226-7
38.
H. Sahlmann and T. Thiemann, Classical Quantum Gravity 23, 867 (2006).
http://dx.doi.org/10.1088/0264-9381/23/3/019
39.
H. Sahlmann and T. Thiemann, Classical Quantum Gravity 23, 909 (2006).
http://dx.doi.org/10.1088/0264-9381/23/3/020
40.
K. Giesel and T. Thiemann, Classical Quantum Gravity 24, 2465 (2007).
http://dx.doi.org/10.1088/0264-9381/24/10/003
41.
K. Giesel and T. Thiemann, Classical Quantum Gravity 24, 2499 (2007).
http://dx.doi.org/10.1088/0264-9381/24/10/004
42.
K. Giesel and T. Thiemann, Classical Quantum Gravity 24, 2565 (2007).
http://dx.doi.org/10.1088/0264-9381/24/10/005
43.
K. Giesel and T. Thiemann, Class. Quantum Grav. 32, 135015 (2015).
http://dx.doi.org/10.1088/0264-9381/32/13/135015
44.
M. Domaga ła, K. Giesel, W. Kamiński, and J. Lewandowski, Phys. Rev. D 82, 104038 (2010).
http://dx.doi.org/10.1103/PhysRevD.82.104038
45.
This means, that l should contain as least one pair of subgraphs, which are changed into one another.
46.
T. Thiemann, Classical Quantum Gravity 15, 1487 (1998).
http://dx.doi.org/10.1088/0264-9381/15/6/006
47.
A. Stottmeister and T. Thiemann, J. Math. Phys. 57, 022303 (2016).
http://dx.doi.org/10.1063/1.4940052
48.
M. Assanioussi, A. Dapor, and J. Lewandowski, Phys. Lett. B 751, 302305 (2015).
http://dx.doi.org/10.1016/j.physletb.2015.10.043
49.
I. Agullo, A. Ashtekar, and W. Nelson, Phys. Rev. Lett. 109, 251301 (2012).
http://dx.doi.org/10.1103/PhysRevLett.109.251301
50.
I. Agullo, A. Ashtekar, and W. Nelson, Phys. Rev. D 87, 043507 (2013).
http://dx.doi.org/10.1103/PhysRevD.87.043507
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/content/aip/journal/jmp/57/8/10.1063/1.4960823
2016-08-16
2016-09-26

Abstract

In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the proposed Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity.

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