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{{About|the beta functions of theoretical physics|other beta functions|Beta function (disambiguation)}}
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{{quantum field theory}}
 
In [[theoretical physics]], specifically [[quantum field theory]], a '''beta function''', ''β(g)'', encodes the dependence of a [[Coupling constant|coupling parameter]], ''g'', on the [[energy scale]],  ''μ'',  of a given physical process described by [[quantum field theory]].
It is defined as
:: <math>\beta(g) = \frac{\partial g}{\partial \log(\mu)} ~,</math>
and, by dint of the underlying  [[renormalization group]], it has no explicit dependence on ''μ'', so it only depends on ''μ'' implicitly through  ''g''.
This dependence on the energy scale thus specified is known as the [[Coupling constant#Running coupling|running]] of the coupling parameter, a fundamental
feature of  scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques.
 
==Scale invariance==
If the beta functions of a quantum field theory vanish, usually at particular values of the coupling parameters, then the theory is said to be [[Scale invariance|scale-invariant]]. Almost all scale-invariant QFTs are also [[Conformal symmetry|conformally invariant]]. The study of such theories is [[conformal field theory]].
 
The coupling parameters of a quantum field theory can run even if the corresponding [[classical field theory]] is scale-invariant. In this case, the non-zero beta function tells us that the classical scale invariance is [[Conformal anomaly|anomalous]].
 
==Examples==
Beta functions are usually computed in some kind of approximation scheme. An example is [[Perturbation theory (quantum mechanics)|perturbation theory]], where one assumes that the coupling parameters are small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher [[Feynman graph|loop]] contributions, due to the number of loops in the corresponding [[Feynman graph]]s).
 
Here are some examples of beta functions computed in perturbation theory:
 
===Quantum electrodynamics===
 
The one-loop beta function in [[quantum electrodynamics]] (QED) is
 
*<math>\beta(e)=\frac{e^3}{12\pi^2}~,</math>
 
or
 
*<math>\beta(\alpha)=\frac{2\alpha^2}{3\pi}~,</math>
 
written in terms of the [[fine structure constant]],  ''α'' =  ''e''<sup>2</sup>/4π .
 
This beta function tells us that the coupling increases with increasing energy scale, and QED becomes strongly coupled at high energy. In fact, the coupling apparently becomes infinite at some finite energy, resulting in a [[Landau pole]]. However, one cannot expect the perturbative beta function to give accurate results at strong coupling, and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid.
 
===Quantum chromodynamics===
 
The one-loop beta function in [[quantum chromodynamics]] with <math>n_f</math> {{pad|.1em}}[[Flavour (particle physics)#Quantum chromodynamics|flavours]] is
 
*<math>\beta(g)=-\left(11-\frac{2n_f}{3}\right)\frac{g^3}{16\pi^2}~,</math>
 
or
 
*<math>\beta(\alpha_s)=-\left(11-\frac{2n_f}{3}\right)\frac{\alpha_s^2}{2\pi}~,</math>
 
written in terms of ''α<sub>s</sub>'' = <math>\frac{g^2}{4\pi}</math> .
 
If ''n''<sub>''f''</sub> ≤ 16,  the ensuing beta function dictates that the coupling decreases with increasing energy scale, a phenomenon known as [[asymptotic freedom]]. Conversely, the coupling increases with decreasing energy scale. This means that the coupling becomes large at low energies, and one can no longer rely on perturbation theory.
 
===SU(N) Non-Abelian gauge theory===
While the gauge group of QCD is <math>SU(3)</math>, we can generalize to any number of colors, <math>N_c</math>, with gauge group <math>G=SU(N_c)</math>. Then for this gauge group, with Dirac fermions in a [[Representations of Lie groups|representation]] <math>R</math> of <math>G</math>, the one-loop beta function is
 
*<math>\beta(g)=-\left(\tfrac{11}{3}C_2(G)-\frac{4}{3}n_fC(R)\right)\frac{g^3}{16\pi^2}~,</math>
 
where <math>C_2(G)</math> is the [[Casimir invariant|quadratic Casimir]] of <math>G</math> and <math>C(R)</math> is another Casimir invariant defined by <math>Tr (T^a_RT^b_R) = C(R)\delta^{ab}</math> for generators <math>T^{a,b}_R</math> of the Lie algebra in the representation R. For gauge fields (''i.e.'' gluons) in the [[Adjoint representation of a Lie group|adjoint]] of <math>G</math>, <math>C_2(G) = N_c</math>; for fermions in the [[Fundamental representation|fundamental]] (or anti-fundamental) representation of <math>G</math>, <math>C(R) = 1/2</math>. Then for QCD, with <math>N_c = 3</math>, the above equation reduces to that listed for the quantum chromodynamics beta function.
 
This famous result was derived nearly simultaneously in 1973 by [[H. David Politzer|Politzer]],<ref>
{{cite journal
| author=H.David Politzer
| year=1973
| title=Reliable Perturbative Results for Strong Interactions?
| journal=Phys. Rev. Lett.
| volume=30 | pages=1346–1349
| doi=10.1103/PhysRevLett.30.1346
| url=http://inspirehep.net/record/81351?ln=en|bibcode = 1973PhRvL..30.1346P }}</ref> [[David Gross|Gross]] and [[Frank Wilczek|Wilczek]],<ref>
{{cite journal
| author=D.J. Gross and F. Wilczek
| year=1973
| title=Asymptotically Free Gauge Theories. 1
| journal=Phys. Rev. D
| volume=8 | pages=3633–3652
| doi=10.1103/PhysRevD.8.3633
| url=http://inspirehep.net/record/81404 |bibcode = 1973PhRvD...8.3633G }}</ref> and [[Gerard 't Hooft|'t Hooft]],<ref>
{{cite journal
| author=G. 't Hooft
| year=1973
| title=
| postscript=. Unpublished.}}</ref> for which the first three were awarded the [[List of Nobel laureates in Physics|Nobel Prize in Physics]] in 2004.
 
=== Minimal Supersymmetric Standard Model ===
{{Main|Minimal_Supersymmetric_Standard_Model#Gauge-Coupling Unification}}
 
==See also==
*[[Callan-Symanzik equation]]
*[[Quantum triviality]]
*[[Banks–Zaks fixed point|Banks-Zaks fixed point]]
 
==References==
{{Reflist}}
 
==Further reading==
* Peskin, M and Schroeder, D.; ''An Introduction to Quantum Field Theory,'' Westview Press (1995). A standard introductory text, covering many topics in QFT including calculation of beta functions; see especially chapter 16.
* Weinberg, Steven; ''The Quantum Theory of Fields,'' (3 volumes) Cambridge University Press (1995). A monumental treatise on QFT.
* Zinn-Justin, Jean; ''Quantum Field Theory and Critical Phenomena,'' Oxford University Press (2002). Emphasis on the renormalization group and related topics.
 
==External links==
 
[[Category:Quantum field theory]]
[[Category:Renormalization group]]
[[Category:Scaling symmetries]]

Latest revision as of 08:24, 18 December 2014

SINGAPORE — Angered by what they allege to be "defects" on their property — which embrace termite-infested pool decking, flooded staircases in widespread areas and incorrectly-installed electrical fittings — 108 house owners of an upscale condominium on Sentosa Cove have taken property developer Ho Bee Investment and three contractors to court.

Since Ground to Floor height is always > Ground to Ceiling heights, some builders prefer to quote the floor to flooring peak. Once more, we must always compare between developments utilizing floor to ceiling heights. Core Central Area (CCR) > Central (or RCR, Rest of central area) > Remainder of Singapore (or OCR, Outdoors Central Region) Since students that live within 1km or 2km to a college gets entry priority to that college, proximity to good faculties provides to the worth of the event. Information to the shopping for process Information me by means of the process of shopping for accomplished properties (condominium/property already built liao). I need to have no less than an concept of it! The lawyer's fees do not include stamp charges or disbursements properly incurred bythe lawyer. Selling Your Home or Flat

Frasers Centrepoint Properties ranks amongst one of many top-promoting developers in Singapore residential property market. With greater than 11,000 local properties built in over 40 developments since the launch of The Anchorage, its inaugural undertaking in 1992, its success is due largely to a mixture of robust attributes - good places, revolutionary product options and engaging pricing. The continued success of Frasers Centrepoint Homes is basically due to the inherent capacity of the Group to satisfy and exceed the expectations of the market.

We count on Singapore builders to commerce at a wider P/RNAV discount and cut our worth targets by 7-12%. We downgrade CAPL and KPLD to In-Line and CIT to Underperform (from Outperform). We want workplace REITs. Listed on the Primary Board of the Singapore Exchange since 5 July 2006, FCT is managed by Frasers Centrepoint Asset Administration, the actual property and fund management division of FCL. The Institute of Property Brokers (IEA), Singapore, is organising the first Asia Pacific Real Property Conference & Expo (APRECE) 2013, to be held at Marina Bay Sands, Singapore. Posted on Apr 15, 2012 in Articles , Funding Strategy , Singapore Property Investment by Gerald Tay 0 Feedback "Fools rush in when Angels concern to tread" TEHO BANKS ON PROPERTY AS A PART OF CORE ENTERPRISE

SILVERSCAPE NON-PUBLIC CONDOMINIUM APARTMENT GEYLANG LORONG 32, SINGAPORE (DISTRICT 14) SILVERSEA PRIVATE CONDOMINIUM CONDO MARINE PARADE ROAD, SINGAPORE (DISTRICT 15) SKYLINE @ ORCHARD BOULEVARD NON-PUBLIC CONDOMINIUM HOUSE ANGULLIA PARK, SINGAPORE (DISTRICT 09) SKYLINE RESIDENCES PRIVATE CONDOMINIUM CONDO TELOK BLANGAH ROAD, SINGAPORE (DISTRICT 04) SKYSUITES 17 PERSONAL CONDOMINIUM CONDO JALAN RAJAH, SINGAPORE (DISTRICT 12) GOOD SUITES PERSONAL CONDOMINIUM RESIDENCE SIMS AVENUE, SINGAPORE (DISTRICT 14) SORRENTO, THE NON-PUBLIC CONDOMINIUM CONDOMINIUM WEST COAST HIGHWAY, SINGAPORE (DISTRICT 05) SPOTTISWOODE RESIDENCES PRIVATE CONDOMINIUM CONDOMINIUM SPOTTISWOODE PARK HIGHWAY, SINGAPORE (DISTRICT 02) EASTTREE EAST COAST TERRACE, SINGAPORE (DISTRICT 15)

MY MANHATTAN NON-PUBLIC CONDOMINIUM CONDOMINIUM SIMEI AVENUE three, SINGAPORE (DISTRICT 18) NATURA @ HILLVIEW PRIVATE CONDOMINIUM CONDOMINIUM HILLVIEW TERRACE, SINGAPORE (DISTRICT 23) NATURALIS NON-PUBLIC CONDOMINIUM CONDO TELOK KURAU LORONG M, SINGAPORE (DISTRICT 15) NEWTON IMPERIAL (READY PROPERTIES) PRIVATE CONDOMINIUM HOUSE NEWTON HIGHWAY, SINGAPORE (DISTRICT eleven) NOTTINGHILL SUITES PERSONAL CONDOMINIUM RESIDENCE TOH TUCK STREET, SINGAPORE (DISTRICT 21) ONE ROSYTH PRIVATE CONDOMINIUM HOUSE ROSYTH STREET, SINGAPORE (DISTRICT 19) ONE SHENTON (PREPARED HOMES) PERSONAL CONDOMINIUM CONDO SHENTON METHOD, SINGAPORE (DISTRICT 01) OXLEY BIZHUB INDUSTRIAL ENTERPRISE AREA (B1, MANUFACTURING FACILITY, WAREHOUSE) UBI HIGHWAY 1, PAYA LEBAR HIGHWAY, SINGAPORE (DISTRICT 14) Killiney Road, Singapore 239519 Map

One more reason why the property market could come down is over-supply of properties, or to be more particular, more supply of properties than demand. Economics 101 tells us that value is decided by the interaction of demand and provide. When there may be extra demand than supply, prices go up. Conversely, if provide exceeds demand, prices come down. With a view to decide the risk of over-provide, we need to split it up into the totally different categories of housing.

Keppel Land is the property arm of the Keppel Group. Keppel Land is one of the largest property teams listed on the Singapore inventory exchange. Its real property activities involve property growth, funding, mission and property management, in Singapore and abroad. Riversails Condominium @ Higher Serangoon Identification of the property supplied for sale SingaporePropertyCycle.com.sg is a Singapore Property website devoted to providing traders with the graphs and charts to trace the Singapore Property Market Cycle and make higher real estate investment choices. www.the-hillview-peak.sg (latest new launch in Oct/Nov 2012) Tips on how to Discover A Singapore Property Agent? Ferra Rental @ Leonie Hill Singapore Land Limited Property Tax Property tax Sorts of Properties in Singapore Condo

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