[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki\/planck-units-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki\/planck-units-wikipedia\/","headline":"Planck units – Wikipedia","name":"Planck units – Wikipedia","description":"before-content-x4 Units defined only by physical constants after-content-x4 In particle physics and physical cosmology, Planck units are a set of","datePublished":"2022-08-22","dateModified":"2022-08-22","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/07f2f9c2687d8c7a616c3409542a39cd6972e09d","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/07f2f9c2687d8c7a616c3409542a39cd6972e09d","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki\/planck-units-wikipedia\/","wordCount":20598,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Units defined only by physical constants       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of four universal physical constants, in such a manner that these physical constants take on the numerical value of 1 when expressed in terms of these units. Originally proposed in 1899 by German physicist Max Planck, these units are a system of natural units because their definition is based on properties of nature, more specifically the properties of free space, rather than a choice of prototype object. They are relevant in research on unified theories such as quantum gravity.The term Planck scale refers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units.  This region may be characterized by particle energies of around 1019\u00a0GeV or 109\u00a0J, time intervals of around 10\u221243\u00a0s and lengths of around 10\u221235\u00a0m (approximately the energy-equivalent of the Planck mass, the Planck time and the Planck length, respectively). At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate. The best-known example is represented by the conditions in the first 10\u221243 seconds of our universe after the Big Bang, approximately 13.8 billion years ago.The four universal constants that, by definition, have a numeric value 1 when expressed in these units are:       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Planck units do not incorporate an electromagnetic dimension. Some authors choose to extend the system to electromagnetism by, for example, adding either the Coulomb constant (ke = 1\/4\u03c0\u03b50) or the electric constant (\u03b50) to this list. Similarly, authors choose to use variants of the system that give other numeric values to one or more of the four constants above.Table of ContentsIntroductionHistory and definitionDerived unitsSignificancePlanck scaleRelationship to gravityIn cosmologyAnalysis of the unitsPlanck lengthPlanck timePlanck energyPlanck unit of forcePlanck temperatureNondimensionalized equationsAlternative choices of normalizationGravitational constantSee alsoExplanatory notesReferencesExternal linksIntroductionAny system of measurement may be assigned a mutually independent set of base quantities and associated base units, from which all other quantities and units may be derived. In the International System of Units, for example, the SI base quantities include length with the associated unit of the metre. In the system of Planck units, a similar set of base quantities and associated units may be selected, in terms of which other quantities and coherent units may be expressed.[1][2]:\u200a1215\u200a The Planck unit of length has become known as the Planck length, and the Planck unit of time is known as the Planck time, but this nomenclature has not been established as extending to all quantities.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4All Planck units are derived from the dimensional universal physical constants that define the system, and in a convention in which these units are omitted (i.e. treated as having the dimensionless value 1), these constants are then eliminated from equations of physics in which they appear. For example, Newton’s law of universal gravitation,F=Gm1m2r2=(FPlP2mP2)m1m2r2,{displaystyle F=G{frac {m_{1}m_{2}}{r^{2}}}=left({frac {F_{text{P}}l_{text{P}}^{2}}{m_{text{P}}^{2}}}right){frac {m_{1}m_{2}}{r^{2}}},}can be expressed as:FFP=(m1mP)(m2mP)(rlP)2.{displaystyle {frac {F}{F_{text{P}}}}={frac {left({dfrac {m_{1}}{m_{text{P}}}}right)left({dfrac {m_{2}}{m_{text{P}}}}right)}{left({dfrac {r}{l_{text{P}}}}right)^{2}}}.}Both equations are dimensionally consistent and equally valid in any system of quantities, but the second equation, with G absent, is relating only dimensionless quantities since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that each physical quantity is the corresponding ratio with a coherent Planck unit (or “expressed in Planck units”), the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:F\u2032=m1\u2032m2\u2032r\u20322.{displaystyle F’={frac {m_{1}’m_{2}’}{r’^{2}}}.}This last equation (without G) is valid with F\u2032, m1\u2032, m2\u2032, and r\u2032 being the dimensionless ratio quantities corresponding\u00a0to the standard quantities, written e.g. F\u2032 \u2258 F or F\u2032 = F\/FP, but not as a direct equality of quantities. This may seem to be “setting the constants c, G, etc., to 1″ if the correspondence of the quantities is thought of as equality.  For this reason, Planck or other natural units should be employed with care. Referring to “G = c = 1“, Paul S. Wesson wrote that, “Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion.”[3]History and definitionThe concept of natural units was introduced in 1874, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor. Stoney chose his units so that G, c, and the electron charge e would be numerically equal to 1.[4] In 1899, one year before the advent of quantum theory, Max Planck introduced what became later known as the Planck constant.[5][6] At the end of the paper, he proposed the base units that were later named in his honor. The Planck units are based on the quantum of action, now usually known as the Planck constant, which appeared in the Wien approximation for black-body radiation. Planck underlined the universality of the new unit system, writing:… die M\u00f6glichkeit gegeben ist, Einheiten f\u00fcr L\u00e4nge, Masse, Zeit und Temperatur aufzustellen, welche, unabh\u00e4ngig von speciellen K\u00f6rpern oder Substanzen, ihre Bedeutung f\u00fcr alle Zeiten und f\u00fcr alle, auch au\u00dferirdische und au\u00dfermenschliche Culturen notwendig behalten und welche daher als \u00bbnat\u00fcrliche Ma\u00dfeinheiten\u00ab\u00a0bezeichnet werden k\u00f6nnen.… it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called “natural units of measure”.Planck considered only the units based on the universal constants G{displaystyle G}, h{displaystyle h}, c{displaystyle c}, and kB{displaystyle k_{rm {B}}} to arrive at natural units for length, time, mass, and temperature.[6] His definitions differ from the modern ones by a factor of 2\u03c0{displaystyle {sqrt {2pi }}}, because the modern definitions use \u210f{displaystyle hbar } rather than h{displaystyle h}.[5][6]Table 1: Modern values for Planck’s original choice of quantitiesNameDimensionExpressionValue (SI units)Planck lengthlength (L)lP=\u210fGc3{displaystyle l_{text{P}}={sqrt {frac {hbar G}{c^{3}}}}}1.616255(18)\u00d710\u221235\u00a0m[7]Planck massmass (M)mP=\u210fcG{displaystyle m_{text{P}}={sqrt {frac {hbar c}{G}}}}2.176434(24)\u00d710\u22128\u00a0kg[8]Planck timetime (T)tP=\u210fGc5{displaystyle t_{text{P}}={sqrt {frac {hbar G}{c^{5}}}}}5.391247(60)\u00d710\u221244\u00a0s[9]Planck temperaturetemperature (\u0398)TP=\u210fc5GkB2{displaystyle T_{text{P}}={sqrt {frac {hbar c^{5}}{Gk_{text{B}}^{2}}}}}1.416784(16)\u00d71032\u00a0K[10]Unlike the case with the International System of Units, there is no official entity that establishes a definition of a Planck unit system. Some authors define the base Planck units to be those of mass, length and time, regarding an additional unit for temperature to be redundant.[note 1] Other tabulations add, in addition to a unit for temperature, a unit for electric charge, so that either the Coulomb constant ke{displaystyle k_{e}}[12][13] or the vacuum permittivity \u03f50{displaystyle epsilon _{0}}[14] is normalized to 1. Thus, depending on the author’s choice, this charge unit is given byqP=4\u03c0\u03f50\u210fc\u22481.875546\u00d710\u221218\u00a0C\u224811.7\u00a0e{displaystyle q_{text{P}}={sqrt {4pi epsilon _{0}hbar c}}approx 1.875546times 10^{-18}{text{ C}}approx 11.7 e}for ke=1{displaystyle k_{text{e}}=1}, orqP=\u03f50\u210fc\u22485.290818\u00d710\u221219\u00a0C\u22483.3\u00a0e.{displaystyle q_{text{P}}={sqrt {epsilon _{0}hbar c}}approx 5.290818times 10^{-19}{text{ C}}approx 3.3 e.}for \u03b50=1{displaystyle varepsilon _{0}=1}.[note 2] Some of these tabulations also replace mass with energy when doing so.[15]The Planck charge, as well as other electromagnetic units that can be defined like resistance and magnetic flux, are more difficult to interpret than Planck’s original units and are used less frequently.[13]In SI units, the values of c, h, e and kB are exact and the values of \u03b50 and G in SI units respectively have relative uncertainties of 1.5\u00d710\u221210[16] and 2.2\u00d710\u22125.[17] Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of G.Compared to Stoney units, Planck base units are all 1\u03b1\u224811.7{displaystyle {frac {1}{sqrt {alpha }}}approx 11.7} times larger.Derived unitsIn any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Planck units, some of which are seldom used. As with the base units, their use is mostly confined to theoretical physics because most of them are too large or too small for empirical or practical use and there are large uncertainties in their values.Some Planck units, such as of time and length, are many orders of magnitude too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where present-day theories of physics apply.[18] For example, our understanding of the Big Bang does not extend to the Planck epoch, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist.Several quantities are not “extreme” in magnitude, such as the Planck mass, which is about 22 micrograms: very large in comparison with subatomic particles, and within the mass range of living organisms.[19]:\u200a872\u200a Similarly, the related units of energy and of momentum are in the range of some everyday phenomena.SignificancePlanck units have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the metre and second, which exist as base units in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions. Frank Wilczek puts it succinctly:We see that the question [posed] is not, “Why is gravity so feeble?” but rather, “Why is the proton’s mass so small?” For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton’s mass is the tiny number [1\/(13\u00a0quintillion)].[20]While it is true that the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons, this is not about the relative strengths of the two fundamental forces. From the point of view of Planck units, this is comparing apples with oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.Planck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22\u00d71019\u00a0GeV (the Planck energy, corresponding to the energy equivalent of the Planck mass, is 2.17645\u00d710\u22128\u00a0kg) at which quantum effects of gravity become significant. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of quantum field theory break down and become inadequate, due to the impact of the apparent non-renormalizability of gravity within current theories.Relationship to gravityAt the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it has been theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown.[21] The Planck scale is therefore the point at which the effects of quantum gravity can no longer be ignored in other fundamental interactions, where current calculations and approaches begin to break down, and a means to take account of its impact is necessary.[22] On these grounds, it has been speculated that it may be an approximate lower limit at which a black hole could be formed by collapse.[23]While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level, gravity is problematic, and cannot be integrated with quantum mechanics at very high energies using the usual framework of quantum field theory. At lesser energy levels it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of quantum gravity is necessary. Approaches to this problem include string theory and M-theory, loop quantum gravity, noncommutative geometry, and causal set theory.In cosmologyIn Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, tP, or approximately 10\u221243 seconds.[24] There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the inflationary epoch, which ended after about 10\u221232 seconds (or about 1011\u00a0tP).[25]Table 3 lists properties of the observable universe today expressed in Planck units.[26][27]After the measurement of the cosmological constant (\u039b) in 1998, estimated at 10\u2212122 in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe (T) squared. Barrow and Shaw proposed a modified theory in which \u039b is a field evolving in such a way that its value remains \u039b ~ T\u22122 throughout the history of the universe.[28]Analysis of the unitsPlanck lengthThe Planck length, denoted \u2113P, is a unit of length defined as:\u2113P=\u210fGc3{displaystyle ell _{mathrm {P} }={sqrt {frac {hbar G}{c^{3}}}}}It is equal to 1.616255(18)\u00d710\u221235\u00a0m,[7] where the two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value, or about 10\u221220 times the diameter of a proton.[29] It can be motivated in various ways, such as considering a particle whose reduced Compton wavelength is comparable to its Schwarzschild radius,[29][30][31] though whether those concepts are in fact simultaneously applicable is open to debate.[32] (The same heuristic argument simultaneously motivates the Planck mass.[30])The Planck length is a distance scale of interest in speculations about quantum gravity. The Bekenstein\u2013Hawking entropy of a black hole is one-fourth the area of its event horizon in units of Planck length squared.[11]:\u200a370\u200a Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length.[33][34][35] This is sometimes expressed by saying that “spacetime becomes a foam at the Planck scale”.[36] It is possible that the Planck length is the shortest physically measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes.[37]The strings of string theory are modeled to be on the order of the Planck length.[38][39] In theories with large extra dimensions, the Planck length calculated from the observed value of G{displaystyle G} can be smaller than the true, fundamental Planck length.[11]:\u200a61\u200a[40]Planck timeThe Planck time tP is the time required for light to travel a distance of 1\u00a0Planck length in vacuum, which is a time interval of approximately 5.39\u00d710\u221244\u00a0s. No current physical theory can describe timescales shorter than the Planck time, such as the earliest events after the Big Bang.[24] Some conjecture that the structure of time need not remain smooth on intervals comparable to the Planck time.[41]Planck energyThe Planck energy EP is approximately equal to the energy released in the combustion of the fuel in an automobile fuel tank (57.2\u00a0L at 34.2\u00a0MJ\/L of chemical energy). The ultra-high-energy cosmic ray observed in 1991 had a measured energy of about 50\u00a0J, equivalent to about 2.5\u00d710\u22128\u00a0EP.[42][43]Proposals for theories of doubly special relativity posit that, in addition to the speed of light, an energy scale is also invariant for all inertial observers. Typically, this energy scale is chosen to be the Planck energy.[44][45]Planck unit of forceThe Planck unit of force may be thought of as the derived unit of force in the Planck system if the Planck units of time, length, and mass are considered to be base units.FP=mPctP=c4G\u22481.2103\u00d71044\u00a0N{displaystyle F_{text{P}}={frac {m_{text{P}}c}{t_{text{P}}}}={frac {c^{4}}{G}}approx mathrm {1.2103times 10^{44}~N} }It is the gravitational attractive force of two bodies of 1 Planck mass each that are held 1 Planck length apart. One convention for the Planck charge is to choose it so that the electrostatic repulsion of two objects with Planck charge and mass that are held 1 Planck length apart balances the Newtonian attraction between them.[46]Some authors have argued that the Planck force is on the order of the maximum force that can occur between two bodies.[47][48] However, the validity of these conjectures has been disputed.[49][50]Planck temperatureThe Planck temperature TP is 1.416784(16)\u00d71032\u00a0K.[10] At this temperature, the wavelength of light emitted by thermal radiation reaches the Planck length. There are no known physical models able to describe temperatures greater than TP; a quantum theory of gravity would be required to model the extreme energies attained.[51] Hypothetically, a system in thermal equilibrium at the Planck temperature might contain Planck-scale black holes, constantly being formed from thermal radiation and decaying via Hawking evaporation. Adding energy to such a system might decrease its temperature by creating larger black holes, whose Hawking temperature is lower.[52]Nondimensionalized equationsPhysical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (e.g., 1 second is not the same as 1 metre). In theoretical physics, however, this scruple may be set aside, by a process called nondimensionalization. The effective result is that many fundamental equations of physics, which often include some of the constants used to define Planck units, become equations where these constants are replaced by a 1.Examples include the energy\u2013momentum relation \u00a0E2=(mc2)2+(pc)2{displaystyle  E^{2}=(mc^{2})^{2}+(pc)^{2}}, which becomes E2=m2+p2{displaystyle E^{2}=m^{2}+p^{2}}, and the Dirac equation \u00a0(i\u210f\u03b3\u03bc\u2202\u03bc\u2212mc)\u03c8=0{displaystyle  (ihbar gamma ^{mu }partial _{mu }-mc)psi =0}, which becomes \u00a0(i\u03b3\u03bc\u2202\u03bc\u2212m)\u03c8=0{displaystyle  (igamma ^{mu }partial _{mu }-m)psi =0}.Alternative choices of normalizationAs already stated above, Planck units are derived by “normalizing” the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible nor necessarily the best. Moreover, the choice of what factors to normalize, among the factors appearing in the fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.The factor 4\u03c0 is ubiquitous in theoretical physics because in three-dimensional space, the surface area of a sphere of radius r is 4\u03c0r2. This, along with the concept of flux, are the basis for the inverse-square law, Gauss’s law, and the divergence operator applied to flux density. For example, gravitational and electrostatic fields produced by point objects have spherical symmetry, and so the electric flux through a sphere of radius r around a point charge will be distributed uniformly over that sphere. From this, it follows that a factor of 4\u03c0r2 will appear in the denominator of Coulomb’s law in rationalized form.[26]:\u200a214\u201315\u200a (Both the numerical factor and the power of the dependence on r would change if space were higher-dimensional; the correct expressions can be deduced from the geometry of higher-dimensional spheres.[11]:\u200a51\u200a) Likewise for Newton’s law of universal gravitation: a factor of 4\u03c0 naturally appears in Poisson’s equation when relating the gravitational potential to the distribution of matter.[11]:\u200a56\u200aHence a substantial body of physical theory developed since Planck’s 1899 paper suggests normalizing not G but 4\u03c0G (or 8\u03c0G) to 1. Doing so would introduce a factor of 1\/4\u03c0 (or 1\/8\u03c0) into the nondimensionalized form of the law of universal gravitation, consistent with the modern rationalized formulation of Coulomb’s law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of 1\/4\u03c0 in the nondimensionalized form of Coulomb’s law as well, so that the nondimensionalized Maxwell’s equations for electromagnetism and gravitoelectromagnetism both take the same form as those for electromagnetism in SI, which do not have any factors of 4\u03c0.  When this is applied to electromagnetic constants, \u03b50, this unit system is called “rationalized“.  When applied additionally to gravitation and Planck units, these are called rationalized Planck units[53] and are seen in high-energy physics.[54]The rationalized Planck units are defined so that c = 4\u03c0G = \u0127 = \u03b50 = kB = 1.There are several possible alternative normalizations.Gravitational constantIn 1899, Newton’s law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for “small” velocities and masses (the approximate nature of Newton’s law was shown following the development of general relativity in 1915). Hence Planck normalized to 1 the gravitational constant G in Newton’s law. In theories emerging after 1899, G nearly always appears in formulae multiplied by 4\u03c0 or a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4\u03c0 appearing in the equations of physics are to be eliminated via the normalization.Normalizing 4\u03c0G to 1 (and therefore setting G = 1\/4\u03c0):Setting 8\u03c0G = 1 (and therefore setting G = 1\/8\u03c0). This would eliminate 8\u03c0G from the Einstein field equations, Einstein\u2013Hilbert action, and the Friedmann equations, for gravitation. Planck units modified so that 8\u03c0G = 1 are known as reduced Planck units, because the Planck mass is divided by \u221a8\u03c0. Also, the Bekenstein\u2013Hawking formula for the entropy of a black hole simplifies to SBH = (mBH)2\/2 = 2\u03c0ABH.See alsoExplanatory notesReferences^ a b Wilczek, Frank (2005). “On Absolute Units, I: Choices”. Physics Today. American Institute of Physics. 58 (10): 12\u201313. Bibcode:2005PhT….58j..12W. doi:10.1063\/1.2138392.^ a b Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973). Gravitation. New York. ISBN\u00a00-7167-0334-3. OCLC\u00a0585119.^ Wesson, P. S. (1980). “The application of dimensional analysis to cosmology”. Space Science Reviews. 27 (2): 117. Bibcode:1980SSRv…27..109W. doi:10.1007\/bf00212237. S2CID\u00a0120784299.^ Barrow, J. D. (1 March 1983). “Natural Units Before Planck”. Quarterly Journal of the Royal Astronomical Society. 24: 24. Bibcode:1983QJRAS..24…24B. ISSN\u00a00035-8738. Archived from the original on 20 January 2022. Retrieved 16 April 2022.^ a b Planck, Max (1899). “\u00dcber irreversible Strahlungsvorg\u00e4nge”. Sitzungsberichte der K\u00f6niglich Preu\u00dfischen Akademie der Wissenschaften zu Berlin (in German). 5: 440\u2013480. Archived from the original on 17 November 2020. Retrieved 23 May 2020. pp.\u00a0478\u201380 contain the first appearance of the Planck base units, and of the Planck constant, which Planck denoted by b. a and f in this paper correspond to the k and G in this article.^ a b c Tomilin, K. A. (1999). Natural Systems of Units. To the Centenary Anniversary of the Planck System (PDF). Proceedings Of The XXII Workshop On High Energy Physics And Field Theory. pp.\u00a0287\u2013296. Archived from the original (PDF) on 12 December 2020. Retrieved 31 December 2019.^ a b “2018 CODATA Value: Planck length”. The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.^ “2018 CODATA Value: Planck mass”. The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.^ “2018 CODATA Value: Planck time”. The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.^ a b “2018 CODATA Value: Planck temperature”. The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.^ a b c d e Zwiebach, Barton (2004). A First Course in String Theory. Cambridge University Press. ISBN\u00a0978-0-521-83143-7. 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The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.^ “2018 CODATA Value: Newtonian constant of gravitation”. The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.^ Zee, Anthony (2010). Quantum Field Theory in a Nutshell (second\u00a0ed.). Princeton University Press. pp.\u00a0172, 434\u2013435. ISBN\u00a0978-0-691-14034-6. OCLC\u00a0659549695. Just as in our discussion of the Fermi theory, the nonrenormalizability of quantum gravity tells us that at the Planck energy scale … new physics must appear. Fermi’s theory cried out, and the new physics turned out to be the electroweak theory. Einstein’s theory is now crying out.^ Penrose, Roger (2005). The Road to Reality. New York: Alfred A. Knopf. ISBN\u00a0978-0-679-45443-4.^ Wilczek, Frank (2001). “Scaling Mount Planck I: A View from the Bottom”. Physics Today. 54 (6): 12\u201313. 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Providence, Rhode Island: American Mathematical Society. ISBN\u00a0978-1-4704-6313-7. OCLC\u00a01202475208.External links     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki\/planck-units-wikipedia\/#breadcrumbitem","name":"Planck units – Wikipedia"}}]}]