# Real Ashtekar variables for Lorentzian signature space-times.

@article{Barbero1995RealAV, title={Real Ashtekar variables for Lorentzian signature space-times.}, author={J. Fernando G. Barbero}, journal={Physical Review D}, year={1995}, volume={51}, pages={5507-5510} }

I suggest in this letter a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual $SO(3)$ Yang-Mills phase space I show that it is possible to describe space-times with Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular…

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## References

SHOWING 1-10 OF 18 REFERENCES

"J."

- Philosophy
- 1890

however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)…

MATH

- Biology
- 1992

It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/z, and some other physical models of knots as flattened ropes or strips which exhibit similar length versus complexity power laws are surveyed.

Int. J. Mod. Phys. D3 J. F. Barbero G. Phys. Rev. D49

- Int. J. Mod. Phys. D3 J. F. Barbero G. Phys. Rev. D49
- 1994

Phys. Rev. Lett

- Phys. Rev. Lett
- 1992

Rev

- Lett. 69, 237
- 1992

Stornaiolo Commun. Math. Phys

- Stornaiolo Commun. Math. Phys
- 1992

Phys

- Rev. Lett. 57, 2244 (1986) 9 A. Ashtekar, Phys. Rev. D36, 1587 (1987) A. Ashtekar Non Perturbative Canonical Gravity (Notes prepared in collaboration with R. S. Tate)
- 1991

Phys

- Rev. Lett. 61, 1155 (1988) C. Rovelli and L. Smolin, Nucl. Phys. B331, 80
- 1990

Int. J. Mod. Phys A4

- Int. J. Mod. Phys A4
- 1989

Phys. Rev. Lett

- Phys. Rev. Lett
- 1989