Discreteness of area and volume in quantum gravity. (English) Zbl 0925.83013

Nucl. Phys., B 442, No. 3, 593-619 (1995); erratum ibid. 456 No. 3, 753-754 (1995).
Summary: We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.


83C45 Quantization of the gravitational field
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[1] Isham, C. J., Prima facia questions in quantum gravity, gr-qc/9310031, (Ehlers, J.; Friedrich, H., Canonical Gravity: From Classical to Quantum. Canonical Gravity: From Classical to Quantum, Lecture Notes in Physics, 434 (1994), Springer-Verlag: Springer-Verlag Berlin)
[2] Rovelli, C.; Smolin, L., Nucl. Phys. B, 133, 80 (1990)
[3] Ashtekar, A.; Rovelli, C.; Smolin, L., Phys. Rev. Lett., 69, 237 (1992) · Zbl 0968.83510
[4] Rovelli, C.; Smolin, L., Phys. Rev. Lett., 72, 446 (1994) · Zbl 0973.83523
[5] Morales-Tecotl, H.; Rovelli, C., Phys. Rev. Lett., 72, 3642 (1994) · Zbl 0973.83525
[6] Ashtekar, A.; Isham, C., Class. and Quant. Grav., 9, 1069 (1992) · Zbl 0749.53042
[7] Ashtekar, A.; Lewandowski, J.; Marlof, D.; Mourãu, J.; Thiemann, T., Quantum geometrodynamics and Coherent state transform on the space of connections, Penn. State University preprints (1994)
[8] Rayner, D., Class. and Quant. Grav., 7, 651 (1990) · Zbl 0825.58064
[9] Bruegmann, B.; Gambini, R.; Pullin, J., Phys. Rev. Lett., 68, 431 (1992) · Zbl 0969.83507
[10] Rovelli, C., Class. Quant. Grav., 8, 1613 (1991) · Zbl 0733.53050
[11] Ashtekar, A., Non perturbative canonical gravity (1991), World Scientific: World Scientific Singapore · Zbl 0948.83500
[12] Smolin, L., (Pérez-Mercader, J.; etal., Quantum Gravity and Cosmology (1992), World Scientific: World Scientific Singapore)
[13] Rovelli, C., Class and Quantum Grav., 8, 317 (1991)
[14] Rovelli, C., Nucl. Phys. B, 405, 797 (1993) · Zbl 1006.83501
[15] Rovelli, C., A physical prediction from Quantum Gravity: the quantization of the area, (Akerlof, C. W.; Srednicki, M. A., Texas/Pascos 92: Relativistic Astrophysics and Particle Cosmology, Volume 688 (1993), Annals of the New York Academy of Science), New York · Zbl 0956.83526
[16] Smolin, L., Phys. Rev. D, 49, 4028 (1994)
[17] Smolin, L., (Hu, B. L.; Jacobson, T., Directions in General Relativity, Vol. 2, papers in honour of Dieter Brill (1994), Cambridge University Press: Cambridge University Press Cambridge)
[18] Kuchař, K.; Brown, J. D., Dust as a Standard of Space and Time in canonical Quantum Gravity, Utah University preprint (1994)
[19] Penrose, R., (Bastin, E. A., Quantum Theory and beyond (1971), Cambridge University Press) · Zbl 1205.00067
[20] Atiyah, M., The Geometry and Physics of Knots (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0729.57002
[21] (Baez, J., Knots and Quantum Gravity (1994), Oxford University Press: Oxford University Press Oxford) · Zbl 0796.00020
[22] Rovelli, C.; Smolin, L., Spin-network basis in quantum gravity (1994), (unpublished)
[23] Carlip, S., Geometric structures and loop variables in (2+1) dimensional gravity, (Baez, J., Knots and Quantum Gravity (1994), Oxford University Press: Oxford University Press Oxford) · Zbl 0820.58010
[24] Ashtekar, A., Phys. Rev. D, 36, 1587 (1987)
[25] Rovelli, C., Phys. Rev. D, 47, 1703 (1993)
[26] Junichi Iwasaki, personal communication.; Junichi Iwasaki, personal communication.
[27] Greensite, J., Phys. Lett. B, 255, 375 (1991)
[28] Amati, D.; Ciafaloni, M.; Veneziano, G.; Guida, R.; Konishi, K.; Provero, P., Mod. Phys. Lett. A, 6, 1487 (1991)
[29] Gross, D.; Mende, P. F., Nucl. Phys. B, 303, 407 (1988)
[30] Atick, J. J.; Witten, E., Nucl. Phys. B, 310, 291 (1988)
[31] Klebanov, I.; Susskind, L., Nucl. Phys. B, 309, 175 (1988)
[32] Horowitz, G., (talk delivered at the conference on Quantum Gravity. talk delivered at the conference on Quantum Gravity, Durham (1994))
[33] Baez, J., Strings, Loops, Knots and Gauge Fields, (Baez, J., Knots and Quantum Gravity (1994), Oxford University Press: Oxford University Press Oxford) · Zbl 0815.58007
[34] Bekenstein, J. D., Lett. Nuovo Cim., 11, 467 (1974)
[35] Jacobson, T., Phys. Rev. D, 44, 1731 (1991)
[36] Hawking, S. W., Nucl. Phys. B, 144, 349 (1978)
[37] Garay, L. J., Quantum gravity and minimum length, Imperial College preprint/TP/93-94/20, grqc/9403008 (1994)
[38] Iwasaki, J.; Rovelli, C., Class. and Quantum Grav., 11, 1653 (1994)
[39] Di Bartolo, C.; Gambini, R.; Griego, J., The extended loop representation of quantum gravity, Phys. Rev. D (1994), to appear · Zbl 0973.83524
[40] Rovelli, C., Class. and Quantum Grav., 10, 1567 (1993) · Zbl 0821.46086
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