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[[Image:Dirac 4.jpg|thumb|Paul Dirac]]
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The '''Dirac large numbers hypothesis''' ('''LNH''') is an observation made by [[Paul Dirac]] in 1937 relating ratios of size scales in the [[Universe]] to that of force scales. The ratios constitute very large, dimensionless numbers: some [[Orders of_magnitude (numbers)#1039|40 orders of magnitude]] in the present cosmological epoch. According to Dirac's hypothesis, the apparent equivalence of these ratios might not be a mere coincidence but instead could imply a [[physical cosmology|cosmology]] with these unusual features:
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*The strength of gravity, as represented by the [[gravitational constant]], is inversely proportional to the [[age of the universe]]: <math>G \propto 1/t\,</math>
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*The mass of the universe is proportional to the square of the universe's age: <math>M \propto t^2</math>.
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Neither of these two features has gained wide acceptance in mainstream physics and, though some proponents{{who|date=September 2012}} of [[non-standard cosmology|non-standard cosmologies]] refer to Dirac's cosmology as a foundational basis for their own ideas and studies, some physicists{{who|date=September 2012}} dismiss the large numbers in LNH as mere coincidences. A coincidence, however, may be defined optimally as 'an event that provides support for an alternative to a currently favoured causal theory, but not necessarily enough support to accept that alternative in light of its low prior probability.'<ref>
{{cite journal
|author1=T. L. Griffiths
|author2=J. B. Tenenbaum
|year=2007
|title=From mere coincidences to meaningful discoveries
|journal=[[Cognition (journal)|Cognition]]
|volume=103 |issue=2 |pages=180–226
|doi=10.1016/j.cognition.2006.03.004
|pmid=16678145
}}</ref> Research into LNH, or the large number of coincidences that underpin it, appears to have gained new impetus from failures in standard cosmology to account for anomalies such as the recent discovery that the universe might be expanding at an accelerated rate.<ref name=Ray2007>
{{cite arxiv
|author1=S. Ray
|author2=U. Mukhopadhyay
|author3=P. P. Ghosh
|year=2007
|title=Large Number Hypothesis: A Review
|class=gr-qc
|eprint=0705.1836
}}</ref>


== Background ==
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LNH was Dirac's personal response to a set of large number 'coincidences' that had intrigued other theorists at about the same time. The 'coincidences' began with [[Hermann Weyl]] (1919),<ref name=Ray2007/><ref>
'''MathML'''
{{cite journal
:<math forcemathmode="mathml">E=mc^2</math>
|author=H. Weyl
|year=1917
|title=Zur Gravitationstheorie
|journal=[[Annalen der Physik]]
|volume=359 |issue=18 |pages=117
|bibcode=1917AnP...359..117W
|doi=10.1002/andp.19173591804
}}</ref><ref>
{{cite journal
|author=H. Weyl
|year=1919
|title=Eine neue Erweiterung der Relativitätstheorie
|journal=[[Annalen der Physik]]
|volume=364 |issue=10 |pages=101
|bibcode=1919AnP...364..101W
|doi=10.1002/andp.19193641002
}}</ref> who speculated that the observed radius of the universe, ''R''<sub>U</sub>, might also be the hypothetical radius of a particle whose rest energy is equal to the gravitational self-energy of the electron:
:<math>\frac {R_U}{r_e}\approx \frac{r_H}{r_e} \approx 10^{42} ,</math>
:<math>r_e = \frac {e^2}{4 \pi \epsilon_0 m_e c^2},</math>
:<math>r_H = \frac {e^2}{4 \pi \epsilon_0 m_H c^2},</math>
:<math>m_H c^2 = \frac {Gm_e^2}{r_e}</math>
where ''r''<sub>e</sub> is the [[classical electron radius]], ''m''<sub>e</sub> is the mass of the electron, ''m''<sub>H</sub> denotes the mass of the hypothetical particle, and ''r''<sub>H</sub> is its electrostatic radius.


The coincidence was further developed by [[Arthur Eddington]] (1931)<ref>
<!--'''PNG''' (currently default in production)
{{cite journal
:<math forcemathmode="png">E=mc^2</math>
|author=A. Eddington
|year=1931
|title=Preliminary Note on the Masses of the Electron, the Proton, and the Universe
|journal=[[Proceedings of the Cambridge Philosophical Society]]
|volume=27 |issue= |pages=15
|bibcode=1931PCPS...27...15E
|doi=10.1017/S0305004100009269
}}</ref> who related the above ratios to '''N''', the estimated number of charged particles in the Universe:
:<math>\frac {e^2}{4 \pi \epsilon_0 Gm_e^2} \approx \sqrt {N} \approx 10^{42}.</math>


In addition to the examples of Weyl and Eddington, Dirac was influenced also by the primeval-atom hypothesis of [[Georges Lemaître]], who lectured on the topic in Cambridge in 1933.<ref name=Ray2007/> The notion of a varying-G cosmology first appears in the work of [[Edward Arthur Milne]] a few years before Dirac formulated LNH. Milne was inspired not by large number coincidences but by a dislike of Einstein's [[general theory of relativity]].<ref>
'''source'''
{{cite book
:<math forcemathmode="source">E=mc^2</math> -->
|author=E. A. Milne
|year=1935
|title=Relativity, Gravity and World Structure
|publisher=[[Oxford University Press]]
}}</ref><ref>
{{cite book
|author=H. Kragh
|year=1996
|title=Cosmology and Controversy: The historical development of two theories of the universe
|pages=61–62
|publisher=[[Princeton University Press]]
|isbn=978-0-691-02623-7
}}</ref> For Milne, space was not a structured object but simply a system of reference in which Einstein's conclusions could be accommodated by relations such as this:
:<math>G = \left(\frac{c^3}{M_U}\right)t,</math>
where ''M''<sub>U</sub> is the mass of the universe and ''t'' is the age of the universe in seconds. According to this relation, ''G'' increases over time.


== Dirac's interpretation of the large number coincidences==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


The Weyl and Eddington ratios above can be rephrased in a variety of ways, as for instance in the context of time:
==Demos==
:<math>\frac {ct}{r_e} \approx 10^{40},</math>


where ''t'' is the age of the universe, <math>c</math> is the [[speed of light]] and ''r''<sub>e</sub> is the classical electron radius. Hence, in units where ''c''=1 and ''r''<sub>e</sub>&nbsp;=&nbsp;1, the age of the Universe is about 10<sup>40</sup> units of time. This is the same [[order of magnitude]] as the ratio of the [[electromagnetic force|electrical]] to the [[gravitational]] [[force]]s between a [[proton]] and an [[electron]]:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
:<math>\frac{4 \pi \epsilon_0 G m_p m_e}{e^2} \approx 10^{-40}.</math>


Hence, interpreting the [[electric charge|charge]] <math>e</math> of the [[electron]], the [[mass]] <math>m_p</math>/<math>m_e</math> of the proton/electron, and the permittivity factor <math> 4 \pi \epsilon_0</math> in atomic units (equal to 1), the value of the [[gravitational constant]] is approximately 10<sup>−40</sup>. Dirac interpreted this to mean that <math>G</math> varies with time as <math>G \approx 1/t\,</math>. Although [[George Gamov]] noted that such a temporal variation does not necessarily follow from Dirac's assumptions,<ref>
{{cite book
|author=H. Kragh
|year=1990
|title=Dirac: A Scientific Biography
|publisher=[[Cambridge University Press]]
|page=177
|isbn=978-0-521-38089-8
}}</ref> a corresponding change of G has not been found.<ref>
{{cite journal
|author=J. P.Uzan
|year=2003
|title=The fundamental constants and their variation, Observational status and theoretical motivations
|journal=[[Reviews of Modern Physics]]
|volume=75 |issue=2 |page=403
|arxiv=hep-ph/0205340
|bibcode=2003RvMP...75..403U
|doi=10.1103/RevModPhys.75.403
}}</ref>
According to general relativity, however, G is constant, otherwise the law of conserved energy is violated. Dirac met this difficulty by introducing into the [[Einstein field equations]] a gauge function {{lang|grc|β}} that describes the structure of spacetime in terms of a ratio of gravitational and electromagnetic units. He also provided alternative scenarios for the continuous creation of matter, one of the other significant issues in LNH:<ref name=Ray2007/>
*'additive' creation (new matter is created uniformly throughout space) and
*'multiplicative' creation (new matter is created where there are already concentrations of mass).


== Later developments and interpretations ==
* accessibility:
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Dirac's theory has inspired and continues to inspire a significant body of scientific literature in a variety of disciplines. In the context of geophysics, for instance, [[Edward Teller]] seemed to raise a serious objection to LNH in 1948<ref>
==Test pages ==
{{cite journal
|author=E. Teller
|year=1948
|title=On the change of physical constants
|journal=[[Physical Review]]
|volume=73 |issue=7 |pages=801–802
|bibcode=1948PhRv...73..801T
|doi=10.1103/PhysRev.73.801
}}</ref> when he argued that variations in the strength of gravity are not consistent with paleontological data. However, [[George Gamow]] demonstrated in 1962<ref>
{{cite book
|author=G. Gamow
|year=1962
|title=Gravity
|pages=138–141
|publisher=[[Doubleday (publisher)|Doubleday]]
|lccn=62008840
}}</ref> how a simple revision of the parameters (in this case, the age of the solar system) can invalidate Teller's conclusions. The debate is further complicated by the choice of LNH [[Cosmology|cosmologies]]: In 1978, G. Blake<ref>
{{cite journal
|author=G. Blake
|year=1978
|title=The Large Numbers Hypothesis and the rotation of the Earth
|journal=[[Monthly Notices of the Royal Astronomical Society]]
|volume=185 |pages=399
|bibcode=1978MNRAS.185..399B
|doi=10.1093/mnras/185.2.399
}}</ref> argued that paleontological data is consistent with the 'multiplicative' scenario but not the 'additive' scenario. Arguments both for and against LNH are also made from astrophysical considerations. For example, D. Falik<ref>
{{cite journal
|author=D. Falik
|year=1979
|title=Primordial Nucleosynthesis and Dirac's Large Numbers Hypothesis
|journal=[[The Astrophysical Journal]]
|volume=231 |page=L1
|bibcode=1979ApJ...231L...1F
|doi=10.1086/182993
}}</ref> argued that LNH is inconsistent with experimental results for [[microwave background radiation]] whereas Canuto and Hsieh<ref>
{{cite journal
|author=V. Canuto, S. Hsieh
|year=1978
|title=The 3 K blackbody radiation, Dirac's Large Numbers Hypothesis, and scale-covariant cosmology
|journal=[[The Astrophysical Journal]]
|volume=224 |pages=302
|bibcode=1978ApJ...224..302C
|doi=10.1086/156378
}}</ref><ref>
{{cite journal
|author=V. Canuto, S. Hsieh
|year=1980
|title=Primordial nucleosynthesis and Dirac's large numbers hypothesis
|journal=[[The Astrophysical Journal]]
|volume=239 |pages=L91
|bibcode=1980ApJ...239L..91C
|doi=10.1086/183299
}}</ref> argued that it ''is'' consistent. One argument that has created significant controversy was put forward by [[Robert Dicke]] in 1961. Known as the [[anthropic coincidence]] or [[fine-tuned universe]], it simply states that the large numbers in LNH are a necessary coincidence for intelligent beings since they parametrize [[nuclear fusion|fusion]] of [[hydrogen]] in [[star]]s and hence carbon-based [[life]] would not arise otherwise.


Various authors have introduced new sets of numbers into the original 'coincidence' considered by Dirac and his contemporaries, thus broadening or even departing from Dirac's own conclusions. Jordan (1947)<ref>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
{{Cite book
*[[Displaystyle]]
|author=P. Jordan
*[[MathAxisAlignment]]
|year=1947
*[[Styling]]
|title=Die Herkunft der Sterne
*[[Linebreaking]]
|publisher=[[Wissenschaftliche Verlagsgesellschaft]]
*[[Unique Ids]]
|bibcode=1947QB801.J65......
*[[Help:Formula]]
|doi=10.1002/asna.19472751012
}}</ref> noted that the mass ratio for a typical star and an electron approximates to 10<sup>60</sup>, an interesting variation on the 10<sup>40</sup> and 10<sup>80</sup> that are typically associated with Dirac and Eddington respectively. Various numbers of the order of 10<sup>60</sup> were arrived at by V. E. Shemi-Zadah (2002)<ref>
{{cite arxiv
|author=V. E. Shemi-Zadah
|year=2002
|title=Coincidence of Large Numbers, exact value of cosmological parameters and their analytical representation
|class=gr-qc
|eprint=gr-qc/0206084
}}</ref> through measuring cosmological entities in [[Planck units]]. P. Zizzi (1998) argued that there might be a modern mathematical interpretation of LNH in a Planck-scale setting in the context of [[quantum foam]].<ref>
{{cite journal
|author=P. Zizzi
|year=1998
|title=Quantum Foam and de Sitter-like Universes
|volume=38 |issue=9 |pages=2333–2348
|journal=[[International Journal of Theoretical Physics]]
|arxiv=hep-th/9808180
|bibcode=1999IJTP...38.2333Z
|doi=10.1023/A:1026675702309
}}</ref> The relevance of the Planck scale to LNH was further demonstrated by S. Caneiro and G. Marugan (2002)<ref name="arxiv.org">
{{cite journal
|author=S. Carneiro, G. Marugan
|year=2001
|title=Holography and the large number hypothesis
|journal=[[Physical Review D]]
|volume=65 |issue=8 |page=087303
|arxiv=gr-qc/0111034
|bibcode=2002PhRvD..65h7303M
|doi=10.1103/PhysRevD.65.087303
}}</ref> by reference to the [[holographic principle]]. Previously, Carneiro (1997)<ref>
{{cite journal
|author=S. Carneiro
|year=1997
|title=The Large Numbers Hypothesis and Quantum Mechanics
|journal=Foundations of Physics Letters
|volume=11 |issue= |pages=95
|arxiv=gr-qc/9712014
|bibcode=1998FoPhL..11...95C
|doi=10.1023/A:1022411021285
}}</ref> arrived at an intermediate scaling factor 10<sup>20</sup> when considering the possible quantization of cosmic structures and a rescaling of [[Planck's constant]].


Several authors have recently identified and pondered the significance of yet another large number, approximately [[Orders of magnitude (numbers)#Larger than 10100|120 orders of magnitude]]. This is for example the ratio of the theoretical and observational estimates of the energy density of the [[vacuum]], which Nottale (1993)<ref>
*[[Inputtypes|Inputtypes (private Wikis only)]]
{{cite web
*[[Url2Image|Url2Image (private Wikis only)]]
|author=L. Nottale
==Bug reporting==
|title=Mach's Principle, Dirac's Large Numbers and the Cosmological Constant Problem
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
|url=http://luth2.obspm.fr/~luthier/nottale/arlambda.pdf
}}</ref> and Matthews (1997)<ref>
{{cite web
|author=R. Matthews
|url=http://ourworld.compuserve.com/homepages/rajm/agdirac.htm
|title=Robert Matthews: ''Dirac's coincidences sixty years on''
}}</ref> associated in an LNH context with a scaling law for the [[cosmological constant]]. [[Carl Friedrich von Weizsäcker]] identified 10<sup>120</sup> with the ratio of the universe's volume to the volume of a typical nucleon bounded by its Compton wavelength, and he identified this ratio with the sum of elementary events or [[bit]]s of [[information]] in the universe.<ref>
{{cite arxiv
|author=H. Lyre
|year=2003
|title=C. F. Weizsäcker's Reconstruction of Physics: Yesterday, Today and Tomorrow
|class=quant-ph
|eprint=quant-ph/0309183
}}</ref> T. Goernitz (1986), building on Weizsäcker's work, posited an explanation for large number 'coincidences' in the context of Bekenstein–Hawking [[entropy]].<ref>
{{cite journal
|author=T. Gornitz
|year=1986
|title=New Look at the Large Numbers
|journal=[[International Journal of Theoretical Physics]]
|volume=25 |issue=8 |pages=897
|bibcode=1986IJTP...25..897G
|doi=10.1007/BF00669925
}}</ref> Genreith (1999)<ref>
{{cite arxiv
|author=H. Genreith
|year=1999
|title=The Large Numbers Hypothesis: Outline of a self-similar quantum cosmological Model
|class=gr-qc
|eprint=gr-qc/9909009
}}</ref> has sketched out a [[fractal]] cosmology in which the smallest mass, which he identified as a [[neutrino]], is about 120 orders of magnitude smaller than the mass of the universe (note: this 'neutrino' approximates in scale to the hypothetical particle m<sub>H</sub> mentioned above in the context of Weyl's work in 1919). Sidharth (2005)<ref>
{{cite arxiv
|author=B. Sidharth
|year=2005
|title=The Planck Scale Underpinning for Spacetime
|class=physics.gen-ph
|eprint=physics/0509026
}}</ref> interpreted a typical electromagnetic particle such as the [[pion]] as a collection of 10<sup>40</sup> [[Planck oscillator]]s and the universe as a collection of 10<sup>120</sup> Planck oscillators. The fact that a number like 10<sup>120</sup> can be represented in a variety of ways has been interpreted by Funkhouser (2006)<ref>
{{cite journal
|author=S. Funkhouser
|year=2006
|title=A New Large Number Coincidence and a Scaling Law for the Cosmological Constant
|volume=464 |issue=2093 |pages=1345–1353
|journal=[[Proceedings of the Royal Society A]]
|arxiv=physics/0611115
|bibcode=2008RSPSA.464.1345F
|doi=10.1098/rspa.2007.0370
}}</ref> as a new large numbers coincidence. Funkhouser claimed to have 'resolved' the LNH coincidences without departing from the [[standard model]] for cosmology. In a similar vein, Carneiro and Marugan (2002) claimed that the scaling relations in LNH can be explained entirely according to basic principles.<ref name="arxiv.org"/>
 
==See also==
*[[Naturalness (physics)]]
 
== References ==
{{reflist|2}}
 
 
==Further reading==
*{{cite journal
|author=P. A. M. Dirac
|year=1938
|title=A New Basis for Cosmology
|journal=[[Proceedings of the Royal Society of London A]]
|volume=165 |issue=921 |pages=199–208
|bibcode=1938RSPSA.165..199D
|doi=10.1098/rspa.1938.0053
}}
*{{cite journal
|author=P. A. M. Dirac
|year=1937
|title=The Cosmological Constants
|journal=[[Nature (journal)|Nature]]
|volume=139 |issue=3512 |pages=323
|bibcode=1937Natur.139..323D
|doi=10.1038/139323a0
}}
*{{cite journal
|author=P. A. M. Dirac
|year=1974
|title=Cosmological Models and the Large Numbers Hypothesis
|journal=[[Proceedings of the Royal Society of London A]]
|volume=338 |issue=1615 |pages=439–446
|bibcode=1974RSPSA.338..439D
|doi=10.1098/rspa.1974.0095
}}
*{{cite journal
|author1=G. A. Mena Marugan
|author2=S. Carneiro
|year=2002
|title=Holography and the large number hypothesis
|journal=[[Physical Review D]]
|volume=65 |issue=8 |page=087303
|arxiv=gr-qc/0111034
|bibcode=2002PhRvD..65h7303M
|doi=10.1103/PhysRevD.65.087303
}}
*{{cite journal
|author1=C.-G. Shao
|author2=J. Shen
|author3=B. Wang
|author4=R.-K. Su
|title=Dirac Cosmology and the Acceleration of the Contemporary Universe
|year=2006
|journal=[[Classical and Quantum Gravity]]
|volume=23 |issue=11 |pages=3707–3720
|arxiv=gr-qc/0508030
|bibcode=2006CQGra..23.3707S
|doi=10.1088/0264-9381/23/11/003
}}
*{{cite arxiv
|author1=S. Ray
|author2=U. Mukhopadhyay
|author3=P. P. Ghosh
|year=2007
|title=Large Number Hypothesis: A Review
|class=gr-qc
|eprint=0705.1836
}}
*{{cite journal
|author1=A. Unzicker
|year=2009
|title=A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke
|journal=[[Annalen der Physik]]
|volume=18 |issue=1 |pages=57–70
|arxiv=0708.3518
|bibcode=2009AnP...521...57U
|doi=10.1002/andp.200810335
}}
 
== External links ==
*[http://www.paricenter.com/library/download/dirac01.mp3 Audio of Dirac talking about the large numbers hypothesis]
*[http://www.fdavidpeat.com/interviews/dirac.htm Full transcript of Dirac's speech.]
*[http://web.archive.org/web/20080203133606/http://ourworld.compuserve.com/homepages/rajm/agdirac.htm Robert Matthews: Dirac's coincidences sixty years on]
*[http://www.jgiesen.de/astro/stars/diracnumber.htm The Mysterious Eddington–Dirac Number]
 
 
{{DEFAULTSORT:Dirac Large Numbers Hypothesis}}
[[Category:Physical cosmology]]
[[Category:Obsolete scientific theories]]
[[Category:Paul Dirac|Large Numbers Hypothesis]]
[[Category:Astronomical hypotheses]]

Latest revision as of 23:52, 15 September 2019

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