Abstract
A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ℝ4. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Ashtekar, Phys. Rev. Lett. 57, 2244(1986).
L. Smolin, "The cubic matrix model and duality between strings and loops," hep-th/ 0006137 (2000).
A. Ashtekar, "Quantum geometry and gravity: recent advances," gr-qc/0112038.
P. Benioff, "Language is physical," quant-ph/0210211 (2002).
C. Isham, "Some reflections on the status of conventional quantum theory when applied to quantum gravity," quant-ph/0206090 (2002).
J. Król, "Set theoretical forcing in quantum mechanics and AdS/CFT correspondence," to appear in Int. J. Theor. Phys. quant-ph/0303089 (2003).
A. Kock and C. J. Mikkelsen, "Topos-theoretic factorization of non-standard extensions," in Proc. Victoria Symposium on Non-standard Analysis, A. Hurd and P. Loeb, (eds.) (Springer, Berlin, Heidelberg, New York, 1974), pp. 122-143.
J. von Neuman and G. Birkhoff, Ann. Math. 37, 823(1936).
L. Smolin, "Quantum gravity with a positive cosmological constant," hep-th/0209079 (2002).
J. Väänänen, "Second order logic and foundations of mathematics," Bull. Symb. Logic 7(4), 504(2001).
W. Lawvere, "Continuously variable sets; algebraic geometry=geometric logic," in Logic Coll. Vol. 73 (North-Holland, Bristol, 1975), p. 135.
G. E. Reyes and I. Moerdijk, Models for Smooth Infinitesimal Analysis (Springer, New York, 1991).
P. Benioff, "Models of Zermelo–Frankel set theory as carriers for the mathematics of physics I, II," J. Math. Phys. 17(5), 618-628, 629–640 (1976).
A. Scedrov, "Forcing and classifying topoi," Memoirs of the American Mathematical Society 45(295), (1984).
S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic. A First Introduction to Topos Theory (Springer, New York, 1992).
K. Kunen, Set Theory. An Introduction to Independence Proofs (North-Holland, Amsterdam, 1980).
T. Bartoszynski and H. Judach, Set Theory. On the Structure of the Real Line (Peters, Wellesley, Massachusetts, 1995).
J. L. Bell, Boolean-Valued Models and Independence Proofs in Set Theory, 2nd edn. (Oxford University Press, Oxford, 1985).
G. Takeuti, "Two applications of logic to mathematics," Math. Soc. Japan 13, Kano Memorial Lecture 3(1978).
S. Koppelberg, Handbook of Boolean Algebras, Vol. 1, J. D. Monk, (ed.) (North-Holland, Amsterdam, New York, 1989).
Private communication with G. Takeuti.
T. Jech, Set Theory, 2nd edn. (Springer, Berlin, Heildelberg, 1997).
B. Balcar, T. Jech, and J. Zapletal, "Semi Cohen Boolean algebras," Ann. Pure Appl. Logic 87, 187-208 (1997).
T. Jech, "Abstract theory of Abelian operator algebras: An application of forcing," Trans. Amer. Math. Soc. 289(1), 133-162 (1984).
M. Ozawa, "Forcing in nonstandard analysis," Ann. Pure Appl. Logic 68, 263-297 (1994).
M. Ozawa, "Scott incomplete Boolean ultrapowers of the real line," J. Symb. Logic 60(1), 160-171 (1995).
G. Takeuti, "Quantum set theory," in Current Issues in Quantum Logic, E. Beltrametti and B. C. van Frassen, (eds.) (Plenum, New York, 1981), p. 303.
H. J. Keisler and C. C. Chang, Model Theory, 3rd edn. (North-Holland, Amsterdam, 1990).
M. Murakami, "Standardization principle of nonstandard universes," J. Symb. Logic 64(4), 1645-1655 (1999).
A. Blass, "End extensions, conservative extensions, and the Rudin–Frolik ordering," Trans. Amer. Math. Soc. 225, 325-340 (1977).
T. Thiemann, "Introduction to modern canonical quantum general relativity," gr-qc/0110054 (2001).
A. Robinson, Non-Standard Analysis, Studies in Logic and the Foundations of Mathematics, (North-Holland, Amsterdam, 1966).
J. Polchinski, String Theory. Superstring Theory and Beyond, Vol. 2 (Cambridge University Press, 1999).
J. Polchinski, "Dirichlet branes and Ramond–Ramond charges," Phys. Rev. Lett. 75, 4724(1995).
J. L. Petersen, "Introduction to the Maldacena conjecture on AdS/CFT," preprint Niels Bohr Institute, NBI-HE-99-05 (1999).
R. E. Gompf and A. I. Stipsicz, An Introduction to 4-Manifolds and Kirby Calculus (American Mathematical Society, Rhode Island, 1999).
R. C. Kirby, The Topology of 4-Manifolds, Lectures Notes in Mathematics, Vol. 1374 (Springer, Berlin, Heidelberg, New York, 1989).
C. H. Brans, J. Math. Phys. 35, 5494(1994).
J. Sładkowski, "Gravity on exotic R4's with few symmetries," Int. J. Mod. Phys. D 10, 311(2001).
M. H. Freedman, "The topology of four-dimensional manifolds," J. Diff. Geom. 17, 357-453 (1982).
Ž Bižaca and R. Gompf, "Elliptic surfaces and some simple exotic R4's," J. Diff. Geom. 43, 458-504 (1996).
S. DeMichelis and M. Freedman, "Uncountably many exotic R4's in standard 4-space," J. Diff. Geom. 35, 219-254 (1992).
T. Jech, Multiple Forcing, Cambridge Tracts in Mathematics, Vol. 88 (Cambridge University Press, 1986).
T. Asselmeyer, "Generation of source terms in general relativity by differential structures," Class. Quant. Grav. 14, 749-758 (1997).
A. Robinson, "Topics in non-archimedean mathematics," in Collected Papers (North-Holland, Amsterdam, 1979), pp. 99-112.
M. Arnsdorf and L. Smolin, "The Maldacena conjecture and Rehren duality," hep-th/0106073 (2001).
J. Polchinski and M. J. Strassler, "The string dual of a confining four-dimensional gauge theory," hep-th/0003136 (2000).
J. L. Bell, "A new approach to quantum logic," Brit. J. Phil. Sci. 37, 83-99 (1986).
E. Witten, "Supersymmetric Yang–Mills theory on a four-manifold," J. Math. Phys. 35(10), 5101(1994).
S. Donaldson, "An application of gauge theory to the topology of 4-manifolds," J. Diff. Geom. 18, 269-316 (1983).
R. Mansfield, "The theory of Boolean ultrapowers," Ann. Math. Logic 2, 297-323 (1971).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Król, J. Background Independence in Quantum Gravity and Forcing Constructions. Foundations of Physics 34, 361–403 (2004). https://doi.org/10.1023/B:FOOP.0000019620.04821.a2
Issue Date:
DOI: https://doi.org/10.1023/B:FOOP.0000019620.04821.a2