# LOG#076. Pedro, in memoriam.

Today, I am going to remember a cool Spanish physicist that is not longer with us since the past summer ( he passed away relatively young, suddenly when he was only 65). His name was Pedro F. González-Díaz. And no, he was not Pedro Almodóvar but his papers had also a fresh and original spirit from time to time, like the movies by the Spanish film maker, something that sometimes physicists obsessed with publishing loose. I was trying to contact him around September of the past year, and his wife Carmen told me the bad and sad news by e-mail. It is a pity that bright people and scientists in all over the world are not remembered more than some shallow people you find in the big media (TV and newspapers). And I wish it will change in the future, since it is nonsense that people who work for the advance of the Human Kind are not more remembered than a nasty soccer match. What a world where people are yet distracted from true knowledge with a new kind of circus! I wish people will wake up some day, and maybe they will realized that Science is what will allow the Human Kind to survive and reach the stars, not precisely a sport competition, even if that competition move money, power and sex. Of course, it is my personal opinion, and unfortunately, it is not shared yet (it seems) by the most of my compatriots.

My friend Francis, in his spanish blog devoted to the spreading of Science, remembered him too. If you understand the Spanish language, you can also read this own dedicatory here:

http://francisthemulenews.wordpress.com/2012/12/10/breve-homenaje-a-pedro-f-gonzalez-diaz-y-su-energia-fantasma

Pedro was a great man. He made his first contributions in Optics, but he was also a potential theoretical physicist and he became one with global interests and he published on many different topics: optics and lasers, decoherence and foundations of Quantum Physics, entanglement, the holographic principle, bohmian Quantum Mechanics, phantom energy, dark energy, the generalized Chaplygin gas, theoretical cosmology, wormholes, and many others. I was thinking what kind of humble tribute I could do for him, and I believe that his works, a complete list from the arxiv ( he has more published papers and works but arxiv rules in these times) will do the job much better than any other further comment ( I did not meet him in life so his scientific works -linked here for you- are the best contribution I can do for him right now):

1. arXiv:1111.4128 [pdf, ps, other]
Quantum entanglement in the multiverse
Subjects: General Relativity and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)
2. arXiv:1111.3178 [pdf, ps, other]
Decoherence in an accelerated universe
S. Robles-Perez, A. Alonso-Serrano, P. F. Gonzalez-Diaz
Subjects: General Relativity and Quantum Cosmology (gr-qc)
3. arXiv:1107.4627 [pdf, ps, other]
Isotropic extensions of the vacuum solutions in general relativity
C. Molina, Prado Martín-Moruno, Pedro F. González-Díaz
Comments: 12 pages, 6 figures. Version to be published in Physical Review D
Journal-ref: Phys. Rev. D 84, 104013 (2011)
Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Extragalactic Astrophysics (astro-ph.CO); High Energy Physics – Theory (hep-th)
4. arXiv:1102.3784 [pdf, ps, other]
Observing other universe through ringholes and Klein-bottle holes
Pedro F. González-Díaz, Ana Alonso-Serrano
Comments: 9 pages, 3 figures, Latex
Journal-ref: Phys.Rev.D84:023008,2011
Subjects: Cosmology and Extragalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc)
5. arXiv:1102.2771 [pdf, ps, other]
Is there a Bigger Fix in the Multiverse?
Pedro F. González-Díaz
Comments: 4 pages, 1 figure, LaTex
Subjects: High Energy Physics – Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
6. arXiv:1005.2147 [pdf, ps, other]
Quantum state of the multiverse
Journal-ref: Phys.Rev.D81:083529,2010
Subjects: General Relativity and Quantum Cosmology (gr-qc)
7. arXiv:1001.3799 [pdf, ps, other]
Multidimensional quantum cosmic models: New solutions and gravitational waves
Pedro F. Gonzalez-Diaz, Alberto Rozas-Fernandez (IFF, CSIC, Madrid, Spain)
Comments: 7 pages, 1 Figure, LaTex
Subjects: General Relativity and Quantum Cosmology (gr-qc)
8. arXiv:1001.3778 [pdf, ps, other]
Is the 2008 NASA/ESA double Einstein ring actually a ringhole signature?
Pedro F. Gonzalez-Diaz (IFF, CSIC, Madrid, Spain)
Comments: 5 pages, 1 figure, LaTex
Subjects: Cosmology and Extragalactic Astrophysics (astro-ph.CO)
9. arXiv:0909.3063 [pdf, ps, other]
Coherent states in the quantum multiverse
S. Robles-Perez, Y. Hassouni, P. F. Gonzalez-Diaz
Journal-ref: Phys.Lett.B683:1-6,2010
Subjects: General Relativity and Quantum Cosmology (gr-qc)
10. arXiv:0908.3244 [pdf, ps, other]
Life originated during accelerating expansion in the multiverse
Pedro F. Gonzalez-Diaz
Comments: 4 pages, no figures, LaTex
Subjects: General Physics (physics.gen-ph); Cosmology and Extragalactic Astrophysics (astro-ph.CO)
11. arXiv:0908.1165 [pdf, ps, other]
The entangled accelerating universe
Journal-ref: Phys.Lett.B679:298-301,2009
Subjects: General Relativity and Quantum Cosmology (gr-qc)
12. arXiv:0907.4055 [pdf, ps, other]
Thermal radiation from Lorentzian traversable wormholes
Journal-ref: Phys. Rev. D {\bf 80}, 024007 (2009)
Subjects: General Relativity and Quantum Cosmology (gr-qc)
13. arXiv:0904.0099 [pdf, ps, other]
Lorentzian wormholes generalizes thermodynamics still further
Comments: Fast Track Communication Class. Quant. Grav., in press, 2009
Journal-ref: Class. Quant. Grav. 26 (2009) 215010
Subjects: General Relativity and Quantum Cosmology (gr-qc)
14. arXiv:0901.1213 [pdf, ps, other]
Unified dark energy thermodynamics: varying w and the -1-crossing
Emmanuel N. Saridakis, Pedro F. Gonzalez-Diaz, Carmen L. Siguenza
Comments: 5 pages, version published in Class. Quant. Grav
Journal-ref: Class.Quant.Grav.26:165003,2009
Subjects: Cosmology and Extragalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics – Theory (hep-th)
15. arXiv:0812.4856 [pdf, ps, other]
Holographic kinetic k-essence model
Norman Cruz, Pedro F. Gonzalez-Diaz, Alberto Rozas-Fernandez, Guillermo Sanchez
Comments: 6 pages, 4 figures, revised version, accepted for publication in Phys.Lett.B
Journal-ref: Phys.Lett.B679:293-297,2009
Subjects: General Relativity and Quantum Cosmology (gr-qc)
16. arXiv:0811.2948 [pdf, ps, other]
On the onset of the dark energy era
Pedro F. Gonzalez-Diaz, Alberto Rozas-Fernandez
Comments: 4 pages, no figures. Contribution to “The Problems of Modern Cosmology”, special volume on the occasion of Prof. S.D. Odintsov’s 50th birthday. Editor: Prof. P. M. Lavrov, Tomsk State Pedagogical University; ISBN: 978-5- 89428-313-5
Subjects: General Relativity and Quantum Cosmology (gr-qc)
17. arXiv:0807.2055 [pdf, ps, other]
Quantum cosmic models and thermodynamics
Pedro F. González-Díaz, Alberto Rozas-Fernández
Comments: 15 pages, 1 figure, accepted for publication in Class. Quantum Grav
Journal-ref: Class.Quant.Grav.25:175023,2008
Subjects: General Relativity and Quantum Cosmology (gr-qc)
18. arXiv:0803.2005 [pdf, ps, other]
Dark Energy Accretion onto black holes in a cosmic scenario
Journal-ref: Gen.Rel.Grav.41:2797-2811,2009
Subjects: General Relativity and Quantum Cosmology (gr-qc)
19. arXiv:0709.4038 [pdf, ps, other]
Quantum theory of an accelerating universe
P. F. Gonzalez-Diaz, S. Robles-Perez
Journal-ref: Int.J.Mod.Phys.D17:1213-1228,2008
Subjects: General Relativity and Quantum Cosmology (gr-qc)
20. arXiv:0709.3302 [pdf, ps, other]
Coherent states in quantum cosmology
S. Robles-Perez, Y. Hassouni, P. F. Gonzalez-Diaz
Subjects: General Relativity and Quantum Cosmology (gr-qc)
21. arXiv:0707.2390 [pdf, ps, other]
On the generalised Chaplygin gas: worse than a big rip or quieter than a sudden singularity?
Comments: 19 pages, 6 figures. Discussion expanded and references added. Version to appear in the International Journal of Modern Physics D
Journal-ref: Int.J.Mod.Phys.D17:2269-2290,2008
Subjects: General Relativity and Quantum Cosmology (gr-qc)
22. arXiv:0705.4347 [pdf, ps, other]
A graceful multiversal link of particle physics to cosmology
Pedro F. Gonzalez-Diaz, Prado Martin-Moruno, Artyom V. Yurov
Subjects: Astrophysics (astro-ph)
23. arXiv:0705.4108 [pdf, ps, other]
Astronomical bounds on future big freeze singularity
Artyom V. Yurov, Artyom V. Astashenok, Pedro F. Gonzalez-Diaz
Journal-ref: Grav.Cosmol.14:205-212,2008
Subjects: Astrophysics (astro-ph)
24. arXiv:0704.1731 [pdf, ps, other]
Wormholes in the accelerating universe
Pedro F Gonzalez-Diaz, Prado Martin-Moruno (IMAFF, CSIC)
Comments: 2 pages, LaTex, to appear in the Proceedings of the 11th Marcel Grossmann Conference, 2006
Subjects: Astrophysics (astro-ph)
25. arXiv:gr-qc/0701127 [pdf, ps, other]
A dark energy multiverse
Journal-ref: Class.Quant.Grav.24:F41-F45,2007
Subjects: General Relativity and Quantum Cosmology (gr-qc)
26. arXiv:gr-qc/0612135 [pdf, ps, other]
Worse than a big rip?
Comments: 6 pages, 4 figures, RevTeX 4. Discussion expanded and references added. Version to appear in PLB
Journal-ref: Phys.Lett.B659:1-5,2008
Subjects: General Relativity and Quantum Cosmology (gr-qc)
27. arXiv:astro-ph/0609263 [pdf, ps, other]
Accelerating Hilbert-Einstein universe without dynamic dark energy
Pedro F. Gonzalez-Diaz, Alberto Rozas-Fernandez (IMAFF, CSIC, Madrid)
Comments: 7 pages, LaTex, to appear in Phys. Lett. B
Journal-ref: Phys.Lett. B641 (2006) 134-138
Subjects: Astrophysics (astro-ph)
28. arXiv:hep-th/0608204 [pdf, ps, other]
Dark energy without dark energy
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 4 pages, 1 figure, to appear in AIP proceedings of “Dark side of the Universe”
Journal-ref: AIP Conf.Proc.878:227-231,2006
Subjects: High Energy Physics – Theory (hep-th)
29. arXiv:hep-th/0607137 [pdf, ps, other]
Some notes on the Big Trip
Pedro F. Gonzalez-Diaz (IMAFF, Csic, Madrid)
Comments: 8 pages, no figures, LaTex
Journal-ref: Phys.Lett. B635 (2006) 1-6
Subjects: High Energy Physics – Theory (hep-th)
30. arXiv:astro-ph/0606529 [pdf, ps, other]
New “Bigs” in cosmology
Artyom V. Yurov, Prado Martin Moruno, Pedro F. Gonzalez-Diaz
Comments: 7 figures, 13 pages, RevTeX
Journal-ref: Nucl.Phys.B759:320-341,2006
Subjects: Astrophysics (astro-ph)
31. arXiv:astro-ph/0603761 [pdf, ps, other]
Will black holes eventually engulf the universe?
Journal-ref: Phys.Lett.B640:117-120,2006
Subjects: Astrophysics (astro-ph)
32. arXiv:astro-ph/0510771 [pdf, ps, other]
On the accretion of phantom energy onto wormholes
Pedro F. Gonzalez-Diaz (IMAFF, CSIC)
Comments: 4 pages, LaTex, to appear in Phys. Lett. B
Journal-ref: Phys.Lett.B632:159-161,2006
Subjects: Astrophysics (astro-ph)
33. arXiv:astro-ph/0510051 [pdf, ps, other]
Evolution of a Kerr-Newman black hole in a dark energy universe
Comments: 11 figures added. Some explanations extended. E-mails updated. References updated. Conclusions unchanged. Accepted in Gravitation &amp; Cosmology
Journal-ref: Grav.Cosmol.14:213-225,2008
Subjects: Astrophysics (astro-ph)
34. arXiv:astro-ph/0507714 [pdf, ps, other]
Holographic cosmic energy, fundamental physics and the future of the universe
Journal-ref: Grav.Cosmol. 12 (2006) 29-36
Subjects: Astrophysics (astro-ph)
35. arXiv:astro-ph/0506717 [pdf, ps, other]
Wiggly cosmic strings accrete dark energy
Pedro F. Gonzalez-Diaz, Jose A. Jimenez Madrid
Journal-ref: Int.J.Mod.Phys.D15:603-614,2006
Subjects: Astrophysics (astro-ph)
36. arXiv:hep-th/0411070 [pdf, ps, other]
Fundamental theories in a phantom universe
Pedro F. Gonzalez-Diaz
Subjects: High Energy Physics – Theory (hep-th)
37. arXiv:astro-ph/0408450 [pdf, ps, other]
Dark energy and supermassive black holes
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 13 pages, RevTex, accepted for publication in Phys. Rev. D
Journal-ref: Phys.Rev. D70 (2004) 063530
Subjects: Astrophysics (astro-ph)
38. arXiv:hep-th/0408225 [pdf, ps, other]
On tachyon and sub-quantum phantom cosmologies
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 8 pages, LaTex, to appear in TSPU Vestnik
Journal-ref: TSPU Vestnik 44N7 (2004) 36-40
Subjects: High Energy Physics – Theory (hep-th)
39. arXiv:astro-ph/0407421 [pdf, ps, other]
Phantom thermodynamics
Pedro F. Gonzalez-Diaz (IMAFF, CSIC), Carmen L. Siguenza
Comments: 19 pages, LaTex, 2 figures, accepted for publication in Nuclear Physics B
Journal-ref: Nucl.Phys.B697:363-386,2004
Subjects: Astrophysics (astro-ph)
40. arXiv:hep-th/0406261 [pdf, ps, other]
Phantom inflation and the “Big Trip”
Pedro F. Gonzalez-Diaz (IMAFF, CSIC), Jose A. Jimenez-Madrid (IAA, CSIC)
Comments: 23 pages, 5 figures, LaTex, Phys. Lett. B (in press)
Journal-ref: Phys.Lett. B596 (2004) 16-25
Subjects: High Energy Physics – Theory (hep-th)
41. arXiv:astro-ph/0404045 [pdf, ps, other]
Achronal cosmic future
Pedro F. Gonzalez-Diaz (IMAFF, CSIC)
Comments: 5 pages, 3 figures, RevTex, a comment and one reference added to match the version accepted for publication in Physical Review Letters
Journal-ref: Phys.Rev.Lett. 93 (2004) 071301
Subjects: Astrophysics (astro-ph)
42. arXiv:hep-th/0401082 [pdf, ps, other]
Axion Phantom Energy
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 6 pages, RevTex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D69 (2004) 063522
Subjects: High Energy Physics – Theory (hep-th)
43. arXiv:astro-ph/0312579 [pdf, ps, other]
K-Essential Phantom Energy: Doomsday around the corner?
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 10 pages, Latex, 1 Figure, to appear in Physics Letters B
Journal-ref: Phys.Lett. B586 (2004) 1-4
Subjects: Astrophysics (astro-ph)
44. arXiv:astro-ph/0311244 [pdf, ps, other]
Sub-Quantum Dark Energy
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 7 pages, RevTex, some minor changes and references added, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D69 (2004) 103512
Subjects: Astrophysics (astro-ph)
45. arXiv:astro-ph/0308382 [pdf, ps, other]
Wormholes and Ringholes in a Dark-Energy Universe
Pedro F. Gonzalez-Diaz
Comments: 11 pages, RevTex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev.D68:084016,2003
Subjects: Astrophysics (astro-ph)
46. arXiv:hep-th/0308148 [pdf, ps, other]
Tensorial perturbations in the bulk of inflating brane worlds
Pedro F. Gonzalez-Diaz
Comments: 4 pages, RevTex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D68 (2003) 084009
Subjects: High Energy Physics – Theory (hep-th)
47. arXiv:astro-ph/0305559 [pdf, ps, other]
You need not be afraid of phantom energy
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 4 pages, RevTex, to appear as a Rapid Communication in Phys. Rev. D
Journal-ref: Phys.Rev. D68 (2003) 021303
Subjects: Astrophysics (astro-ph)
48. arXiv:astro-ph/0212414 [pdf, ps, other]
Unified Model for Dark Energy
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 5 pages, some misprints corrected, a comment on the initial equation of state inserted, one reference added
Journal-ref: Phys.Lett.B562:1-8,2003
Subjects: Astrophysics (astro-ph)
49. arXiv:astro-ph/0210177 [pdf, ps, other]
Accelerating universe without event horizon
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 4 pages, contribution to the XVIIIth IAP Meeting “On the Nature of Dark Energy”
Subjects: Astrophysics (astro-ph)
50. arXiv:hep-th/0208226 [pdf, ps, other]
On new gravitational instantons describing creation of brane-worlds
M. Bouhmadi-Lopez, P.F. Gonzalez-Diaz, A. Zhuk
Comments: 11 pages, 3 figures, LaTeX2e, accepted for publication in Classical and Quantum Gravity
Journal-ref: Class.Quant.Grav. 19 (2002) 4863-4876
Subjects: High Energy Physics – Theory (hep-th); Astrophysics (astro-ph); General Relativity and Quantum Cosmology (gr-qc)
51. arXiv:hep-th/0207170 [pdf, ps, other]
Topological defect brane-world models
M. Bouhmadi-Lopez, P. F. Gonzalez-Diaz, A. Zhuk
Comments: 10 pages, LateX2e, a few comments added and misprints corrected. To be published in Gravitation and Cosmology
Journal-ref: Grav.Cosmol. 8 (2002) 285-293
Subjects: High Energy Physics – Theory (hep-th); Astrophysics (astro-ph); General Relativity and Quantum Cosmology (gr-qc)
52. arXiv:gr-qc/0205057 [pdf, ps, other]
Can black holes exist in an accelerating universe?
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Subjects: General Relativity and Quantum Cosmology (gr-qc)
53. arXiv:gr-qc/0204072 [pdf, ps, other]
Quantum behavior of FRW radiation-filled universes
M. Bouhmadi-Lopez, L. J. Garay, P. F. Gonzalez-Diaz
Comments: RevTeX 4, 16 pages, 2 figures
Journal-ref: Phys.Rev. D66 (2002) 083504
Subjects: General Relativity and Quantum Cosmology (gr-qc)
54. arXiv:hep-th/0203210 [pdf, ps, other]
Stable accelerating universe with no hair
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 9 pages, RevTex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D65 (2002) 104035
Subjects: High Energy Physics – Theory (hep-th)
55. arXiv:hep-th/0111291 [pdf, ps, other]
Perfect fluid brane-world model
M. Bouhmadi-Lopez, P. F. Gonzalez-Diaz, A. Zhuk
Comments: 14 pages, LateX2e, 4 figures
Subjects: High Energy Physics – Theory (hep-th)
56. arXiv:astro-ph/0110335 [pdf, ps, other]
Eternally accelerating universe without event horizon
Pedro F. González-Díaz (IMAFF, CSIC, Madrid)
Journal-ref: Phys.Lett. B522 (2001) 211-214
Subjects: Astrophysics (astro-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics – Theory (hep-th)
57. arXiv:astro-ph/0106421 [pdf, ps, other]
Classical and quantum quintessence cosmology
Pedro F. González-Diáz (IMAFF, CSIC, Madrid)
Comments: 12 pages, RevTex, 1 figure incorporated
Journal-ref: Nucl.Phys. B619 (2001) 646-666
Subjects: Astrophysics (astro-ph)
58. arXiv:hep-th/0105088 [pdf, ps, other]
Brane russian doll
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Journal-ref: Phys.Lett. B512 (2001) 127-130
Subjects: High Energy Physics – Theory (hep-th)
59. arXiv:astro-ph/0103194 [pdf, ps, other]
Quintessence and the first Doppler peak
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Journal-ref: Mod.Phys.Lett. A16 (2001) 2003-2012
Subjects: Astrophysics (astro-ph)
60. arXiv:hep-th/0008193 [pdf, ps, other]
Cosmological predictions from the Misner brane
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Journal-ref: Int.J.Mod.Phys. D12 (2003) 985-1014
Subjects: High Energy Physics – Theory (hep-th)
61. arXiv:gr-qc/0007065 [pdf, ps, other]
Kinks, energy conditions and closed timelike curves
Pedro F. Gonzalez-Diaz (IMAFF, Csic)
Comments: 11 pages, LaTex, 1 figure, to appear in IJMPD
Journal-ref: Int.J.Mod.Phys. D9 (2000) 531-541
Subjects: General Relativity and Quantum Cosmology (gr-qc)
62. arXiv:astro-ph/0004125 [pdf, ps, other]
Cosmological models from quintessence
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 9 pages, RevTex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D62 (2000) 023513
Subjects: Astrophysics (astro-ph)
63. arXiv:gr-qc/0004078 [pdf, ps, other]
Misner-brane cosmology
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Journal-ref: Phys.Lett. B486 (2000) 158-164
Subjects: General Relativity and Quantum Cosmology (gr-qc)
64. arXiv:hep-th/0002033 [pdf, ps, other]
Quintessence in brane cosmology
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 10 pages, additional physical motivation and connections to high energy physics and observations, to appear in Phys. Lett. B
Journal-ref: Phys.Lett. B481 (2000) 353-359
Subjects: High Energy Physics – Theory (hep-th)
65. arXiv:gr-qc/9909094 [pdf, ps, other]
Generalized De Sitter Space
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 10 pages, RevTex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D61 (2000) 024019
Subjects: General Relativity and Quantum Cosmology (gr-qc)
66. arXiv:gr-qc/9907026 [pdf, ps, other]
On the warp drive space-time
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 7 pages, minor comment on chronology protection added, RevTex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D62 (2000) 044005
Subjects: General Relativity and Quantum Cosmology (gr-qc)
67. arXiv:gr-qc/9901016 [pdf, ps, other]
Nonorientable spacetime tunneling
Pedro F. Gonzalez-Diaz, Luis J. Garay
Comments: 11 pages, RevTex, Accepted in Phys. Rev. D
Journal-ref: Phys.Rev. D59 (1999) 064026
Subjects: General Relativity and Quantum Cosmology (gr-qc)
68. arXiv:gr-qc/9901012 [pdf, ps, other]
Perdurance of multiply connected de Sitter space
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Journal-ref: Phys.Rev. D59 (1999) 123513
Subjects: General Relativity and Quantum Cosmology (gr-qc)
69. arXiv:gr-qc/9810084 [pdf, ps, other]
Inflationary cosmology of the extreme cosmic string
Pedro F Gonzalez-Diaz (IMAFF, CSIC, Spain)
Comments: 15 pages, RevTex, to appear in Int. J. Mod. Phys. D
Journal-ref: Int.J.Mod.Phys. D7 (1998) 793-813
Subjects: General Relativity and Quantum Cosmology (gr-qc)
70. arXiv:cond-mat/9809140 [pdf, ps, other]
Closed timelike curves in superfluid $^{3}$He
Pedro F. Gonzalez-Diaz (IMAFF,CSIC,Madrid), Carmen L. Siguenza (Univ. CEU, San Pablo)
Comments: 7 pages, LaTex, to appear in Phys. Lett. A
Subjects: Condensed Matter (cond-mat)
71. arXiv:hep-th/9805012 [pdf, ps, other]
Nonsingular instantons for the creation of open universes
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Spain)
Comments: 8 pages, RevTex. A new section on the instantonic global structure and a figure have been added. To appear in Phys. Rev. D
Journal-ref: Phys.Rev. D59 (1999) 043509
Subjects: High Energy Physics – Theory (hep-th); Astrophysics (astro-ph); General Relativity and Quantum Cosmology (gr-qc)
72. arXiv:gr-qc/9804013 [pdf, ps, other]
Anti-de Sitter wormhole kink
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 10 pages, to appear in IJMPD
Journal-ref: Int.J.Mod.Phys.D7:587-602,1998
Subjects: General Relativity and Quantum Cosmology (gr-qc)
73. arXiv:hep-th/9803178 [pdf, ps, other]
On the geometry of the Hawking Turok instanton
Pedro F. Gonzalez-Diaz (IMAFF, CSIC)
Subjects: High Energy Physics – Theory (hep-th)
74. arXiv:quant-ph/9803043 [pdf, ps, other]
On multiphoton absorption by molecules
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Spain)
Comments: 4 pages, RevTex, to appear in Theochem
Subjects: Quantum Physics (quant-ph)
75. arXiv:gr-qc/9712099 [pdf, ps, other]
Representations of relativity, quantum gravity and cosmology
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 16 pages, RevTex, To appear in Grav. and Cosmol
Journal-ref: Grav.Cosmol. 3 (1997) 217-232
Subjects: General Relativity and Quantum Cosmology (gr-qc)
76. arXiv:gr-qc/9712033 [pdf, ps, other]
Quantum time machine
Pedro F. Gonzalez-Diaz
Journal-ref: Phys.Rev. D58 (1998) 124011
Subjects: General Relativity and Quantum Cosmology (gr-qc)
77. arXiv:quant-ph/9709018 [pdf, ps, other]
Quantum gravity and the problem of measurement
Pedro F. Gonzalez-Diaz (IMAFF, CSIC, Madrid)
Comments: 9 pages, LaTex, to appear in Int. J. Theor. Phys
Journal-ref: Int.J.Theor.Phys.37:249-256,1998
Subjects: Quantum Physics (quant-ph)
78. arXiv:gr-qc/9708044 [pdf, ps, other]
Observable effects from spacetime tunneling
Pedro F. Gonzalez-Diaz
Comments: 5 pages, RevTex, 2 figures (available upon request), to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D56 (1997) 6293-6297
Subjects: General Relativity and Quantum Cosmology (gr-qc); Astrophysics (astro-ph)
79. arXiv:astro-ph/9706040 [pdf, ps, other]
Protein folding and cosmology
P.F. Gonzalez-Diaz (IMAFF, CSIC), C.L. Siguenza (CEU)
Subjects: Astrophysics (astro-ph)
80. arXiv:gr-qc/9610003 [pdf, ps, other]
On the black-hole kink
Pedro F. Gonzalez-Diaz (IMAFF, Madrid, Spain)
Comments: 20 pages, LaTex, to appear in Int. J. Mod. Phys. D
Journal-ref: Int.J.Mod.Phys. D6 (1997) 57-68
Subjects: General Relativity and Quantum Cosmology (gr-qc)
81. arXiv:gr-qc/9608059 [pdf, ps, other]
Ringholes and closed timelike curves
Pedro F. Gonzalez-Diaz
Comments: 11 pages, RevTex, 4 figures available upon request
Journal-ref: Phys.Rev. D54 (1996) 6122-6131
Subjects: General Relativity and Quantum Cosmology (gr-qc)
82. arXiv:hep-th/9608004 [pdf, ps, other]
Circular strings, wormholes and minimum size
Luis J. Garay, Pedro F. Gonzalez-Diaz, Guillermo A. Mena Marugan, Jose M. Raya
Comments: 11 pages, LaTeX2e, minor changes
Journal-ref: Phys.Rev. D55 (1997) 7872-7877
Subjects: High Energy Physics – Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
83. arXiv:gr-qc/9606036 [pdf, ps, other]
Quantum black-hole kinks
Pedro F. Gonzalez-Diaz
Comments: 22 pages, RevTex, figures available under request
Journal-ref: Grav.Cosmol. 2 (1996) 122-142
Subjects: General Relativity and Quantum Cosmology (gr-qc)
84. arXiv:gr-qc/9510047 [pdf, ps, other]
Asymptotically anti-de Sitter wormholes
Carlos Barcelo, Luis J. Garay, Pedro F. Gonzalez-Diaz, Guillermo A. Mena Marugan
Comments: 10 pages, RevTeX 3.0, LaTeX 2.09
Journal-ref: Phys.Rev. D53 (1996) 3162-3171
Subjects: General Relativity and Quantum Cosmology (gr-qc)
85. arXiv:gr-qc/9508055 [pdf, ps, other]
Extreme Cosmic String
P.F. Gonzalez-Diaz
Comments: 18 pages, latex, to appear in Phys. Rev. D
Journal-ref: Phys.Rev. D52 (1995) 5698-5706
Subjects: General Relativity and Quantum Cosmology (gr-qc)
86. arXiv:gr-qc/9503011 [pdf, ps, other]
The Schwarzschild Black-Hole Pair
Pedro F. Gonzalez-Diaz
Comments: 17 pages, plain latex, figure available upon request
Subjects: General Relativity and Quantum Cosmology (gr-qc)
87. arXiv:gr-qc/9502027 [pdf, ps, other]
Pedro F. Gonzalez-Diaz
Comments: 9 pages, standard latex, no figures
Subjects: General Relativity and Quantum Cosmology (gr-qc)
88. arXiv:gr-qc/9412034 [pdf, ps, other]
Time Walk Through the Quantum Cosmic String
Pedro F. Gonzalez-Diaz
Comments: 26 pages, 6 figures (available upon request)
Subjects: General Relativity and Quantum Cosmology (gr-qc)
89. arXiv:gr-qc/9408022 [pdf, ps, other]
Wormholes and Cosmic Strings
Pedro F. Gonzalez-Diaz
Subjects: General Relativity and Quantum Cosmology (gr-qc)
90. arXiv:gr-qc/9408021 [pdf, ps, other]
Quantum State of Wormholes and Topological Arrow of Time
Pedro F. Gonzalez-Diaz
Journal-ref: Int.J.Mod.Phys. D3 (1994) 549-568
Subjects: General Relativity and Quantum Cosmology (gr-qc)
91. arXiv:gr-qc/9404012 [pdf, ps, other]
Complementary Relativity
P.F. Gonzalez-Diaz
Subjects: General Relativity and Quantum Cosmology (gr-qc)
92. arXiv:gr-qc/9403037 [pdf, ps, other]
P.F. Gonzalez-Diaz
Subjects: General Relativity and Quantum Cosmology (gr-qc)
93. arXiv:gr-qc/9309009 [pdf, ps, other]
Blackbody Distribution for Wormholes
P.F. González-Díaz
Journal-ref: Class.Quant.Grav.10:2505-2510,1993
Subjects: General Relativity and Quantum Cosmology (gr-qc)
94. arXiv:gr-qc/9309008 [pdf, ps, other]
Quantum State and Spontaneous Symmetry Breaking in Gravity
Pedro F. Gonzalez-Diaz
Journal-ref: Int.J.Mod.Phys.D3:191-194,1994
Subjects: General Relativity and Quantum Cosmology (gr-qc)
95. arXiv:gr-qc/9307007 [src]
2D dilaton-gravity from 5D Einstein equations
P.F. González-Díaz
Subjects: General Relativity and Quantum Cosmology (gr-qc)
96. arXiv:gr-qc/9306031 [pdf, ps, other]
Elliptic and circular wormholes
P.F. González-Díaz
Subjects: General Relativity and Quantum Cosmology (gr-qc)
97. arXiv:gr-qc/9305006 [pdf, ps, other]
The Quantum Sphaleron
P.F. Gonzalez-Diaz
Journal-ref: Phys.Lett. B307 (1993) 362-366
Subjects: General Relativity and Quantum Cosmology (gr-qc)
R.I.P. Pedro F. González Díaz. In memoriam.

# LOG#058. LHC: last 2012 data/bounds.

Today, 12/12/12, the following paper  arised in the arxiv http://arxiv.org/abs/1212.2339

This interesting paper reviews the last bounds about Beyond Stantard Model particles (both, fermions and bosons) for a large class of models until the end of this year, 2012. Particle hunters, some theoretical physicists are! The fundamental conclusions of this paper are encoded in a really beautiful table:

There, we have:

1. Extra gauge bosons $W', Z'$. They are excluded below 1-2 TeV, depending on the channel/decay mode.

2. Heavy neutrinos $N$. They are excluded with softer lower bounds.

3. Fourth generation quarks $t', b'$ and $B, T$ vector-like quarks are also excluded with $\sim 0.5 TeV$ bounds.

4. Exotic quarks with charge $Q= 5/3$ are also excluded below 0.6 TeV.

We continue desperately searching for deviations to the Standard Model (SM). SUSY, 4th family, heavy likely right-handed neutrinos, tecnifermions, tecniquarks, new gauge bosons, Kalula-Klein resonances (KK particles), and much more are not appearing yet, but we are seeking deeply insight the core of the matter and the deepest structure of the quantum vacuum. We know we have to find “something” beyond the Higgs boson/particle, but what and where is not clear for any of us.

Probably, we wish, some further study of the total data in the next months will clarify the whole landscape, but these data are “bad news” and “good news” for many reasons. They are bad, since they point out to no new physics beyond the Higgs up to 1 TeV (more or less). They are good since we are collecting many data and hopefully, we will complement the collider data with cosmological searches next year, and then, some path relative to the Standard Model extension and the upcoming quantum theory of gravity should be enlightened, or at least, some critical models and theories will be ruled out! Of course, I am being globally pesimist but some experimental hint beyond the Higgs (beyond collider physics) is necessary in order to approach the true theory of this Universe.

And if it is not low energy SUSY (it could if one superparticle is found, but we have not found any superparticle yet), what stabilizes the Higgs potential and provides a $M_H\sim 127GeV$ Higgs mass, i.e, what does that “job”/role? What is forbidding the Higgs mass to receive Planck mass quantum corrections? For me, as a theoretical physicist, this question is mandatory! If SUSY fails to be the answer, we really need some good theoretical explanation for the “light” mass the Higgs boson seems to have!

Stay tuned!

# LOG#023. Math survey.

What is a triangle? It is a question of definition in Mathematics. Of course you could disagree, but it is true. Look the above three “triangles”. Euclidean geometry is based in the first one. The second “triangle” is commonly found in special relativity. Specially, hyperbolic functions. The third one is related to spherical/elliptical geometry.

Today’s summary: some basic concepts in arithmetics, complex numbers and functions. We are going to study and review the properties of some elementary and well known functions. We are doing this in order to prepare a better background for the upcoming posts, in which some special functions will appear. Maybe, this post can be useful for understanding some previous posts too.

First of all, let me remember you that elementary arithmetics is based on seven basic “operations”: addition, substraction, multiplication, division, powers, roots, exponentials and logarithms. You are familiar with the 4 first operations, likely you will also know about powers and roots, but exponentials and logarithms are the last kind of elementary operations taught in the school ( high school, in the case they are ever explained!).

Let me begin with addition/substraction of real numbers (it would be also valid for complex numbers $z=a+bi$ or even more general “numbers”, “algebras”, “rings” or “fields”, with suitable extensions).

$a+b=b+a$

$(a+b)+c=a+(b+c)$

$a+(-a)=0$

$a+0=a$

Multiplication is a harder operation. We have to be careful with the axioms since there are many places in physics where multiplication is generaliz loosing some of the following properties:

$kA=\underbrace{A+..+A}_\text{k-times}$

$AB=BA$

$(AB)C=A(BC)=ABC$

$1A=A1=A$

$(A+B)C=AC+BC$

$A(B+C)=AB+AC$

$A^{-1}A=AA^{-1}=1$

Indeed the last rule can be undestood as the “division” rule, provided $A\neq 0$ since in mathematics or physics there is no sense to “divide by zero”, as follows.

$A^{-1}=\dfrac{1}{A}$

Now, we are going to review powers and roots.

$x^a=\underbrace{x\cdots x}_\text{a-times}$

$(x^a)^b=x^{ab}$

$x^{-a}=\dfrac{1}{x^a}$

$x^ax^b=x^{a+b}$

$\sqrt[n]{x}=x^{1/n}$

Note that the identity $x^ay^a=(x+y)^a$ is not true in general. Moreover, if $x\neq 0$ then $x^0=1$ as well, as it can be easily deduced from the previous axioms. Now, the sixth operation is called exponentiation. It reads:

$\exp (a+b)=\exp (a)\exp (b)$

Sometimes you can read $e^{a+b}=e^{a}e^{b}$, where $\displaystyle{e=\lim_{x\to\infty}\left(a+\dfrac{1}{n}\right)^n}$ is the so-called “e” number. The definitions is even more general, since the previous property is the key feature for any exponential. I mean that,

$a^x=\underbrace{a\cdots a}_\text{x-times}$

$a^xb^x=(ab)^x$

$\left(\dfrac{a}{b}\right)^x=\dfrac{a^x}{b^x}$

We also get that for any $x\neq 0$, then $0^x=0$. Finally, the 7th operation. Likely, the most mysterious for the layman. However, it is very useful in many different places. Recall the definition of the logarithm in certain base “a”:

$\log_{(a)} x=y \leftrightarrow x=a^y$

Please, note that this definition has nothing to do with the “deformed” logarithm of my previous log-entry. Notations are subtle, but you must always be careful about what are you talking about!

Furthermore, there are more remarks:

1st. Sometimes you write $\log_e=\ln x$. Be careful, some books use other notations for the Napier’s logarithm/natural logarithm. Then, you can find out there $\log_e=L$ or even $\log_e=\log$.

2nd. Whenever you are using a calculator, you can generally find $\log_e=\ln$ and $\log_{(10)}=\log$. Please, note that in this case $\log$ is not the natural logarithm, it is the decimal logarithm.

Logarithms (caution: logarithms of real numbers, since the logarithms of  complex numbers are a bit more subtle) have some other cool properties:

$\log_{(a)}(xy)=\log_{(a)}x+\log_{(a)}y$

$\log_{(a)}\dfrac{x}{y}=\log_{(a)}x-\log_{(a)}y$

$\log_{(a)}x^y=y\log_{(a)}x$

$\log_{(a)}=\dfrac{\log_{(b)}}{\log_{(b)}a}$

Common values of the logarithm are:

$\ln 0^+=-\infty;\; \ln 1=0;\; \ln e=1;\; \ln e^x=e^{\ln x}=x$

Indeed, logarithms are also famous due to a remarkable formula by Dirac to express any number in terms of 2’s as follows:

$\displaystyle{N=-\log_2\log_2 \sqrt{\sqrt{\underbrace{\cdots}_\text{(N-1)-times}2}}=-\log_2\log_2\sqrt{\underbrace{\cdots}_\text{(N)-times}2}}$

However, it is quite a joke, since it is even easier to write $N=\log_a a^N$, or even $N=\log_{(1/a)}a^{-N}$

Are we finished? NO! There are more interesting functions to review. In particular, the trigonometric functions are the most important functions you can find in the practical applications.

EUCLIDEAN TRIGONOMETRY

Triangles are cool! Let me draw the basic triangle in euclidean trigonometry.

The trigonometric ratios/functions you can define from this figure are:

i)The function (sin), defined as the ratio of the side opposite the angle to the hypotenuse:
$\sin A=\dfrac{\textrm{opposite}}{\textrm{hypotenuse}}=\dfrac{a}{\,c\,}$
ii) The function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
$\cos A=\dfrac{\textrm{adjacent}}{\textrm{hypotenuse}}=\dfrac{b}{\,c\,}$
iii) The function (tan), defined as the ratio of the opposite leg to the adjacent leg.
$\tan A=\dfrac{\textrm{opposite}}{\textrm{adjacent}}=\dfrac{a}{\,b\,}=\dfrac{\sin A}{\cos A}$

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle ”A”. The ”’adjacent leg”’ is the other side that is adjacent to angle ”A”. The ”’opposite side”’ is the side that is opposite to angle ”A”. The terms ”’perpendicular”’ and ”’base”’ are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA ( a mnemonics rule whose derivation and meaning is left to the reader).

The multiplicative inverse or reciprocals of these functions are named the cosecant (csc or cosec), secant(sec), and cotangent (cot), respectively:
$\csc A=\dfrac{1}{\sin A}=\dfrac{c}{a}$
$\sec A=\dfrac{1}{\cos A}=\dfrac{c}{b}$
$\cot A=\dfrac{1}{\tan A}=\dfrac{\cos A}{\sin A}=\dfrac{b}{a}$

The inverse trigonometric functions/inverse functions are called the arcsine, arccosine, and arctangent, respectively. These functions are what in common calculators are given by $\sin^{-1},\cos^{-1},\tan^{-1}$. Don’t confuse them with the multiplicative inverse trigonometric functions.

There are arithmetic relations between these functions, which are known as trigonometric identities.  The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to “co-“. From the goniometric circle (a circle of radius equal to 1) you can read the Fundamental Theorem of (euclidean) Trigonometry:

$\cos^2\theta+\sin^2\theta=1$

Indeed, from the triangle above, you can find out that the pythagorean theorem implies

$a^2+b^2=c^2$

or equivalently

$\dfrac{a^2}{c^2}+\dfrac{b^2}{c^2}=\dfrac{c^2}{c^2}$

So, the Fundamental Theorem of Trigonometry is just a dressed form of the pythagorean theorem!
The fundamental theorem of trigonometry can be rewritten too as follows:

$\tan^2\theta+1=\sec^2\theta$

$\cot^2+1=\csc^2\theta$

These equations can be easily derived geometrically from the goniometric circle:

The trigonometric ratios are also related geometrically to this circle, and it can seen from the next picture:

Other trigonometric identities are:

$\sin (x\pm y)=\sin x \cos y\pm \sin y\cos x$

$\cos (x\pm y)=\cos x \cos y -\sin x \sin y$

$\tan (x\pm y)=\dfrac{\tan x\pm \tan y}{1\mp \tan x \tan y}$

$\cot (x\pm y)=\dfrac{\cot x \cot y\mp 1}{\cot x\pm \cot y}$

$\sin 2x=2\sin x \cos x$

$\cos 2x=\cos^2 x-\sin^2 x$

$\tan 2x=\dfrac{2\tan x}{1-\tan^2 x}$

$\sin \dfrac{x}{2}=\sqrt{\dfrac{1-\cos x}{2}}$

$\cos \dfrac{x}{2}=\sqrt{\dfrac{1+\cos x}{2}}$

$\tan \dfrac{x}{2}=\sqrt{\dfrac{1-\cos x}{1+\cos x}}$

$\sin x\sin y=\dfrac{\cos(x-y)-\cos(x+y)}{2}$

$\sin x\cos y=\dfrac{\sin(x+y)+\sin(x-y)}{2}$

$\cos x\cos y=\dfrac{\cos (x+y)+\cos(x-y)}{2}$

The above trigonometric functions are also valid for complex numbers with care enough. Let us write a complex number as either a binomial expression $z=a+bi$ or like a trigonometric expression $z=re^{i\theta}$. The famous Euler identity:

$e^{i\theta}=\cos \theta + i\sin \theta$

allows us to relate both two expressions for a complex number since

$z=r(\cos \theta + i\sin \theta)$

implies that $a=r\cos\theta$ and $b=r\sin\theta$. The Euler formula is also useful to recover the identities for the sin and cos of a sum/difference, since $e^{iA}e^{iB}=e^{i(A+B)}$

The complex conjugate of a complex number is $\bar{z}=a-bi$, and the modulus is

$z\bar{z}=\vert z\vert^2=r^2$

with $\theta =\arctan \dfrac{b}{a}, \;\; \vert z \vert=\sqrt{a^2+b^2}$

Moreover, $\overline{\left(z_1\pm z_2\right)}=\bar{z_1}\pm\bar{z_2}$, $\vert \bar{z}\vert=\vert z\vert$, and if $z_2\neq 0$, then

$\overline{\left(\dfrac{z_1}{z_2}\right)}=\dfrac{\bar{z_1}}{\bar{z_2}}$

We also have the so-called Moivre’s formula

$z^n=r^n(\cos n\theta+i\sin n\theta)$

and for the complex roots of complex numbers with $w^n=z$ the identity:

$w=z^{1/n}=r^{1/n}\left(cos\left(\dfrac{\theta +2\pi k}{n}\right)+i\sin\left(\dfrac{\theta +2\pi k}{n}\right)\right)\forall k=0,1,\ldots,n-1$

The complex logarithm (or the complex power) is a multivalued functions (be aware!):

$\ln( re^{i\theta})=\ln r +i\theta +2\pi k,\forall k\in \mathbb{Z}$

The introduction of complex numbers and complex values of trigonometric functions are fun. You can check that

$\cos z=\dfrac{\exp (iz)+\exp (-iz)}{2}$ and $\sin z=\dfrac{\exp (iz)-\exp (-iz)}{2i}$ and $\tan z=\dfrac{\exp (iz)-\exp (-iz)}{i(\exp (iz)+\exp(-iz))}$

thanks to the Euler identity.

In special relativity, the geometry is “hyperbolic”, i.e., it is non-euclidean. Let me review the so-called hyperbolic trigonometry. More precisely, we are going to review the hyperbolic functions related to special relativity now.

HYPERBOLIC TRIGONOMETRY

We define the functions sinh, cosh and tanh ( sometimes written as sh, ch, th):

$\sinh x=\dfrac{\exp (x) -\exp (-x)}{2}$

$\cosh x=\dfrac{\exp (x) +\exp (-x)}{2}$

$\tanh x=\dfrac{\exp (x) -\exp (-x)}{\exp (x)+\exp (-x)}$

The fundamental theorem of hyperbolic trigonometry is

$\cosh^2 x-\sinh^2 x=1$

The hyperbolic triangles are objects like this:

The hyperbolic inverse functions are

$\sinh^{-1} x=\ln (x+\sqrt{x^2+1})$

$\cosh^{-1} x=\ln (x+\sqrt{x^2-1})$

$\tanh^{-1} x=\dfrac{1}{2}\ln\dfrac{1+x}{1-x}$

Two specially useful formulae in Special Relativity (related to the gamma factor, the velocity and a parameter called rapidity) are:

$\boxed{\sinh \tanh^{-1} x=\dfrac{x}{\sqrt{1-x^2}}}$

$\boxed{\cosh \tanh^{-1} x=\dfrac{1}{\sqrt{1-x^2}}}$

In fact, we also have:

$\exp(x)=\sinh x+\cosh x$

$\exp(-x)=-\sinh x+\cosh x$

$\sec\mbox{h}^2x+\tanh^2 x=1$

$\coth ^2 x-\csc\mbox{h}^2 x=1$

There are even more identities to be known. The most remarkable and important are likely to be:

$\sinh (x\pm y)=\sinh (x)\cosh (y)\pm\sinh (y)\cosh (x)$

$\cosh (x\pm y)=\cosh (x)\cosh (y)\pm\sinh (x)\sinh (y)$

$\tanh (x\pm y)=\dfrac{\tanh x\pm \tanh y}{1\pm \tanh x\tanh y}$

$\coth (x\pm y)=\dfrac{\coth x\coth y\pm 1}{\coth y\pm \coth x}$

You can also relate euclidean trigonometric functions with hyperbolic trigonometric functions with the aid of complex numbers. For instance, we get

$\sinh x=-i\sin ix$

$\cosh x=\cos ix$

$\tanh x= -i\tan ix$

and so on. The hyperbolic models of geometry/trigonometry are also very known in arts. Escher’s drawings are very beautiful and famous:

or the colorful variation of this theme

I love Escher’s drawings. And I also love Mathematics, Physics, Physmatics, and Science. Equations are cool. And hyperbolic functions, and other functions we have reviewed here today, will arise naturally in the next posts.

# LOG#004. Feynmanity.

The dream of every theoretical physicist, perhaps the most ancient dream of every scientist, is to reduce the Universe ( or the Polyverse if you believe we live in a Polyverse, also called Multiverse by some quantum theorists) to a single set of principles and/or equations. Principles should be intuitive and meaningful, while equations should be as simple as possible but no simpler to describe every possible phenomenon in the Universe/Polyverse.

What is the most fundamental equation?What is the equation of everything? Does it exist? Indeed, this question was already formulated by Feynman himself  in his wonderful Lectures on Physics! Long ago, Feynman gave us other example of his physical and intuitive mind facing the First Question in Physics (and no, the First Question is NOT “(…)Dr.Who?(…)” despite many Doctors have faced it in different moments of the Human History).

Today, we will travel through this old issue and the modest but yet intriguing and fascinating answer (perhaps too simple and general) that R.P. Feynman found.

Well, how is it?What is the equation of the Universe? Feynman idea is indeed very simple. A nullity condition! I call this action a Feynmann nullity, or feynmanity ( a portmanteau), for brief. The Feynman equation for the Universe is a feynmanity:

$\boxed{U=0}$

Impressed?Indeed, it is very simple. What is the problem then?As Feynman himself said, the problem is really a question of “order” and a “relational” one. A question of what theoretical physicists call “unification”. No matter you can put equations together, when they are related, they “mix” somehow through the suitable mathematical structures.  Gluing “different” pieces and objects is not easy.  I mean, if you pick every equation together and recast them as feynmanities, you will realize that there is no relation a priori between them. However, it can not be so in a truly unified theory. Think about electromagnetism. In 3 dimensions, we have 4 laws written in vectorial form, plus the gauge condition and electric charge conservation through a current. However, in 4D you realize that they are indeed more simple. The 4D viewpoint helps to understand electric and magnetic fields as the the two sides of a same “coin” (the coin is a tensor). And thus, you can see the origin of the electric and magnetic fields through the Faraday-Maxwell tensor $F_{\mu \nu }$. Therefore, a higher dimensional picture simplifies equations (something that it has been remarked by physicists like Michio Kaku or Edward Witten) and helps you to understand the electric and magnetic field origin from a second rank tensor on equal footing.

You can take every equation describing the Universe set it equal to zero. But of course, it does not explain the origin of the Universe (if any), the quantum gravity (yet to be discovered) or whatever. However, the remarkable fact is that every important equation can be recasted as a Feynmanity! Let me put some simple examples:

Example 1. The Euler equation in Mathematics. The most famous formula in complex analysis is a Feynmanity $e^{i\pi}+1=0$ or $e^{2\pi i}=1+0$ if you prefer the constant $\tau=2\pi$.

Example 2. The Riemann’s hypothesis. The most important unsolved problem in Mathematics(and number theory, Physics?) is the solution to the equation $\zeta (s)=0$, where $\zeta(s)$ is the celebrated riemann zeta function in complex variable $s=\kappa + i \lambda$, $\kappa, \lambda \in \mathbb{R}$. Trivial zeroes are placed in the real axis $s=-2n$ $\forall n=1,2,3,...,\infty$. Riemann hypothesis is the statement that every non-trivial zero of the Riemann zeta function is placed parallel to the imaginary axis and they have all real part equal to 1/2. That is, Riemann hypothesis says that the feynmanity $\zeta(s)=0$ has non-trivial solutions iff $s=1/2\pm i\lambda _n$, $\forall n=1,2,3,...,\infty$, so that

$\displaystyle{\lambda_{1}=14.134725, \lambda_{2}= 21.022040, \lambda_{3}=25.010858, \lambda _{4}=30.424876, \lambda_{5}=32.935062, ...}$

I generally prefer to write the Riemann hypothesis in a more symmetrical and “projective” form. Non-trivial zeroes have the form $s_n=\dfrac{1\pm i \gamma _n}{2}$ so that for me, non-trivial true zeroes are derived from projective-like operators $\hat{P}_n=\dfrac{1\pm i\hat{\gamma} _n}{2}$, $\forall n=1,2,3,...,\infty$. Thus

$\gamma_1 =28.269450, \gamma_2= 42.044080, \gamma_3=50.021216, \gamma _4=60.849752, \gamma_5=65.870124,...$

Example 3. Maxwell equations in special relativity. Maxwell equations have been formulated in many different ways along the history of Physics. Here a picture of that. Using tensor calculus, they can be written as 2 equations:

$\partial _\mu F^{\mu \nu}-j^\nu=0$

and

$\epsilon ^{\sigma \tau \mu \nu} \partial _\tau F_{\mu\nu}=\partial _\tau F_{\mu \nu}+ \partial _\nu F_{\tau \mu}+\partial_\mu F_{\nu \tau}=0$

Using differential forms:

$dF=0$

and

$d\star F-J=0$

Using Clifford algebra (Clifford calculus/geometric algebra, although some people prefer to talk about the “Kähler form” of Maxwell equations) Maxwell equations are a single equation: $\nabla F-J=0$ where the geometric product is defined as $\nabla F=\nabla \cdot F+ \nabla \wedge F$.

Indeed, in the Lorentz gauge  $\partial_\mu A^\mu=0$, the Maxwell equations reduce to the spin one field equations:

$\square ^2 A^\nu=0$

where we defined

$\square ^2=\square \cdot \square = \partial_\mu \partial ^\mu =\dfrac{\partial^2}{\partial x^i \partial x_i}-\dfrac{\partial ^2}{c^2\partial t^2}$

Example 4. Yang-Mills equations. The non-abelian generalization of electromagnetism can be also described by 2 feynmanities:

The current equation for YM fields is $(D^{\mu}F_{\mu \nu})^a-J_\nu^a=0$

The Bianchi identities are $(D _\tau F_{\mu \nu})^a+( D _\nu F_{\tau \mu})^a+(D_\mu F_{\nu \tau})^a=0$

Example 5. Noether’s theorems for rigid and local symmetries. Emmy Noether proved that when a r-paramatric Lie group leaves the lagrangian quasiinvariant and the action invariant, a global conservation law (or first integral of motion) follows. It can be summarized as:

$D_iJ^i=0$ for suitable (generally differential) operators $D^i,J^i$ depending on the particular lagrangian (or lagrangian density) and $\forall i=1,...,r$.

Moreover, she proved another theorem. The second Noether’s theorem applies to infinite-dimensional Lie groups. When the lagrangian is invariant (quasiinvariant is more precise) and the action is invariant under the infinite-dimensional Lie group parametrized by some set of arbitrary (gauge) functions ( gauge transformations), then some identities between the equations of motion follow. They are called Noether identities and take the form:

$\dfrac{\delta S}{\delta \phi ^i}N^i_\alpha=0$

where the gauge transformations are defined locally as

$\delta \phi ^i= N^i_\alpha \epsilon ^\alpha$

with $N^i_\alpha$ certain differential operators depending on the fields and their derivatives up to certain order. Noether theorem’s are so general that can be easily generalized for groups more general than those of Lie type. For instance, Noether’s theorem for superymmetric theories (involving lie “supergroups”) and many other more general transformations can be easily built. That is one of the reasons theoretical physicists love Noether’s theorems. They are fully general.

Example 6. Euler-Lagrange equations for a variational principle in Dynamics take the form $\hat{E}(L)=0$, where L is the lagrangian (for a particle or system of particles and $\hat{E}(L)$ is the so-called Euler operator for the considered physical system, i.e., if we have finite degrees of freedom, L is a lagrangian) and a lagrangian “density” in the more general “field” theory framework( where we have infinite degrees of freedom and then L is a lagrangian density $\mathcal{L}$. Even the classical (and quantum) theory of (super)string theory follows from a lagrangian (or more precisely, a lagrangian density). Classical actions for extended objects do exist, so it does their “lagrangians”. Quantum theory for p-branes $p=2,3,...$ is not yet built but it surely exists, like M-theory, whatever it is.

Example 7.  The variational approach to Dynamics or Physics implies  a minimum ( or more generally a “stationary”) condition for the action. Then the feynmanity for the variational approach to Dynamics is simply $\delta S=0$. Every known fundamental force can be described through a variational principle.

Example 8. The Schrödinger’s equation in Quantum Mechanics $H\Psi-E\Psi=0$, for certain hamiltonian operator H. Note that the feynmanity is itself $H=0$ when we studied special relativity from the hamiltonian formalism. Even more, in Loop Quantum Gravity, one important unsolved problem is the solution to the general hamiltonian constraint for the gauge “Wilson-like” loop variables, $\hat{H}=0$.

Example 9. The Dirac’s equation $(i\gamma ^\mu \partial_\mu - m) \Psi =0$ describing free spin 1/2 fields. It can be also easily generalized to interacting fields and even curved space-time backgrounds. Dirac equation admits a natural extension when the spinor is a neutral particle and it is its own antiparticle through the Majorana equation

$i\gamma^\mu\partial_\mu \Psi -m\Psi_c=0$

Example 10. Klein-Gordon’s equation for spin 0 particles: $(\square ^2 +m^2 )\phi=0$.

Example 11. Rarita-Schwinger spin 3/2 field equation: $\gamma ^{\mu \nu \sigma}\partial_{\nu}\Psi_\sigma+m\gamma^{\mu\nu}\Psi_\nu=0$. If $m=0$ and the general conventions for gamma matrices, it can be also alternatively writen as

$\gamma ^\mu (\partial _\mu \Psi_\nu -\partial_\nu\Psi_\mu)=0$

Note that antisymmetric gamma matrices verify:

$\gamma ^{\mu \nu}\partial_{\mu}\Psi_\nu=0$

More generally, every local (and non-local) field theory equation for spin s can be written as a feynmanity or even a theory which contains interacting fields of different spin( s=0,1/2,1,3/2,2,…).  Thus, field equations have a general structure of feynmanity(even with interactions and a potential energy U) and they are given by $\Lambda (\Psi)=0$, where I don’t write the indices explicitely). I will not discuss here about the quantum and classical consistency of higher spin field theories (like those existing in Vasiliev’s theory) but field equations for arbitrary spin fields can be built!

Example 12. SUSY charges. Supersymmetry charges can be considered as operators that satisfy the condicion $\hat{Q}^2=0$ and $\hat{Q}^{\dagger 2}=0$. Note that Grassman numbers, also called grassmanian variables (or anticommuting c-numbers) are “numbers” satisfying $\theta ^2=0$ and $\bar{\theta}^2=0$.

The Feynman’s conjecture that everything in a fundamental theory can be recasted as a feynmanity seems very general, perhaps even silly, but  it is quite accurate for the current state of Physics, and in spite of the fact that the list of equations can be seen unordered of unrelated, the simplicity of the general feynmanity (other of the relatively unknown neverending Feynman contributions to Physics)

$something =0$

is so great that it likely will remain forever in the future of Physics. Mathematics is so elegant and general that the Feynmanity will survive further advances unless  a  Feynman “inequality” (that we could call perhaps, unfeynmanity?) shows to be more important and fundamental than an identity. Of course, there are many important results in Physics, like the uncertainty principle or the second law of thermodynamics that are not feynmanities (since they are inequalities).

Do you know more examples of important feynmanities?

Do you know any other fundamental physical laws or principles that can not be expressed as feynmanities, and then, they are important unfeynmanities?

# LOG#001. A brand new blog.

Hello, world! Hello, blogosphere!

You are surely invited  to share my digital Odyssey through the Neverending Story of Science…