Natural units

In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed. A purely natural system of units has all of its units defined in this way, and usually such that the numerical values of the selected physical constants in terms of these units are exactly 1. These constants are then typically omitted from mathematical expressions of physical laws, and while this has the apparent advantage of simplicity, it may entail a loss of clarity due to the loss of information for dimensional analysis. It precludes the interpretation of an expression in terms of fundamental physical constants, such as e and c, unless it is known which units (in dimensionful units) the expression is supposed to have. In this case, the reinsertion of the correct powers of e, c, etc., can be uniquely determined.[1][2]

Introduction

Natural units are intended to elegantly simplify particular algebraic expressions appearing in the laws of physics or to normalize some chosen physical quantities that are properties of universal elementary particles and are reasonably believed to be constant. However, there is a choice of which quantities to set to unity in a natural system of units, and quantities which are set to unity in one system may take a different value or even be assumed to vary in another natural unit system.

Natural units are "natural" because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are often, without qualification, called "natural units", although they constitute only one of several systems of natural units, albeit the best known such system. Planck units (up to a simple multiplier for each unit) might be considered one of the most "natural" systems in that the set of units is not based on properties of any prototype, object, or particle but are solely derived from the properties of free space.

As with other systems of units, the base units of a set of natural units will include definitions and values for length, mass, time, temperature, and electric charge (in lieu of electric current). It is possible to disregard temperature as a fundamental physical quantity, since it states the energy per degree of freedom of a particle, which can be expressed in terms of energy (or mass, length, and time). Virtually every system of natural units normalizes Boltzmann's constant kB to 1, which can be thought of as simply a way of defining the unit of temperature.

In SI, electric charge is a separate fundamental dimension of physical quantity, but in natural unit systems charge is expressed in terms of the mechanical units of mass, length, and time, similarly to cgs. There are two common ways to relate charge to mass, length, and time: In Lorentz–Heaviside units (also called "rationalized"), Coulomb's law is F = q1q2/r2, and in Gaussian units (also called "non-rationalized"), Coulomb's law is F = q1q2/r2.[3] Both possibilities are incorporated into different natural unit systems.

Summary table

Quantity / Symbol Planck
(with L-H)
Planck
(with Gauss)
Stoney Hartree Rydberg "Natural"
(with L-H)
"Natural"
(with Gauss)
Quantum chromodynamics
(original)
Quantum chromodynamics
(with L-H)
Quantum chromodynamics
(with Gauss)
Speed of light
Reduced Planck constant
Elementary charge
Vacuum permittivity
Vacuum permeability
Impedance of free space
Josephson constant
von Klitzing constant
Coulomb constant
Gravitational constant
Boltzmann constant
Proton rest mass
Electron rest mass

where:

Notation and use

Natural units are most commonly used by setting the units to one. For example, many natural unit systems include the equation c = 1 in the unit-system definition, where c is the speed of light. If a velocity v is half the speed of light, then as v = c/2 and c = 1, hence v = 1/2. The equation v = 1/2 means "the velocity v has the value one-half when measured in Planck units", or "the velocity v is one-half the Planck unit of velocity".

The equation c = 1 can be plugged in anywhere else. For example, Einstein's equation E = mc2 can be rewritten in Planck units as E = m. This equation means "The energy of a particle, measured in Planck units of energy, equals the mass of the particle, measured in Planck units of mass."

Advantages and disadvantages

Compared to SI or other unit systems, natural units have both advantages and disadvantages:

  • Simplified equations: By setting constants to 1, equations containing those constants appear more compact and in some cases may be simpler to understand. For example, the special relativity equation E2 = p2c2 + m2c4 appears somewhat complicated, but the natural units version, E2 = p2 + m2, appears simpler.
  • Physical interpretation: Space and time are put on equal footing and are both measured in the same units. Natural unit systems automatically subsume dimensional analysis. For example, in Planck units, the units are defined by properties of quantum mechanics and gravity. Not coincidentally, the Planck unit of length is approximately the distance at which quantum gravity effects become important. Likewise, atomic units are based on the mass and charge of an electron, and not coincidentally the atomic unit of length is the Bohr radius describing the "orbit" of the electron in a hydrogen atom.
  • No prototypes: A prototype is a physical object that defines a unit, such as the International Prototype Kilogram, a physical cylinder of metal whose mass is by definition exactly one kilogram. A prototype definition always has imperfect reproducibility between different places and between different times, and it is an advantage of natural unit systems that they use no prototypes. (They share this advantage with other non-natural unit systems, such as conventional electrical units.)
  • Less precise measurements: SI units are designed to be used in precision measurements. For example, the second is defined by an atomic transition frequency in cesium atoms, because this transition frequency can be precisely reproduced with atomic clock technology. Natural unit systems are generally not based on quantities that can be precisely reproduced in a lab. Therefore, in order to retain the same degree of precision, the fundamental constants used still have to be measured in a laboratory in terms of physical objects that can be directly observed. If this is not possible, then a quantity expressed in natural units can be less precise than the same quantity expressed in SI units. For example, Planck units use the gravitational constant G, which is measurable in a laboratory only to four significant digits.

Choosing constants to normalize

Out of the many physical constants, the designer of a system of natural unit systems must choose a few of these constants to normalize (set equal to 1). It is not possible to normalize just any set of constants. For example, the mass of a proton and the mass of an electron cannot both be normalized: if the mass of an electron is defined to be 1, then the mass of a proton has to be approximately 1836. In a less trivial example, the fine-structure constant, α1/137, cannot be set to 1 because it is a dimensionless number defined in terms of other quantities. The fine-structure constant is related to other physical constants through α = kee2/ħc, where ke is the Coulomb constant, e is the elementary charge, ħ is the reduced Planck constant, and c is the speed of light. Thus, we cannot set all of ke, e, ħ, and c to 1, we can normalize at most three of this set to 1.

Electromagnetism units

In SI units, electric charge is expressed in coulombs, a separate unit which is additional to the "mechanical" units (mass, length, time), even though the traditional definition of the ampere refers to some of these other units. In natural unit systems, however, electric charge has units of [mass]12 [length]32 [time]−1.

In order to build natural units in electromagnetism one can use:

Of these, Lorentz–Heaviside is somewhat more common,[4] mainly because Maxwell's equations are simpler in Lorentz–Heaviside units than they are in Gaussian units.

In the two unit systems, the elementary charge e satisfies:

  • e = αħc (Lorentz–Heaviside),
  • e = αħc (Gaussian)

where ħ is the reduced Planck constant, c is the speed of light, and α1/137 is the fine-structure constant.

In a natural unit system where c = 1, Lorentz–Heaviside units can be derived from SI units by setting ε0 = μ0 = 1. Gaussian units can be derived from SI units by a more complicated set of transformations, such as multiplying all electric fields by (4πε0)−​12, multiplying all magnetic susceptibilities by , and so on.[5]

Systems of natural units

Planck units

Quantity Expression Metric value Name
Length (L) (L–H) 5.729×10−35 m Planck length
(G) 1.616×10−35 m
Mass (M) (L–H) 6.140×10−9 kg Planck mass
(G) 2.176×10−8 kg
Time (T) (L–H) 1.911×10−43 s Planck time
(G) 5.391×10−44 s
Temperature (Θ) (L–H) 3.997×1031 K Planck temperature
(G) 1.417×1032 K
Electric charge (Q) (L–H) 5.291×10−19 C Planck charge
(G) 1.876×10−18 C

Planck units are defined by

c = ħ = G = ke = kB = 1,

where c is the speed of light, ħ is the reduced Planck constant, G is the gravitational constant, ke is the Coulomb constant, and kB is the Boltzmann constant.

Planck units are a system of natural units that is not defined in terms of properties of any prototype, physical object, or even elementary particle. They only refer to the basic structure of the laws of physics: c and G are part of the structure of spacetime in general relativity, and ħ captures the relationship between energy and frequency which is at the foundation of quantum mechanics. This makes Planck units particularly useful and common in theories of quantum gravity, including string theory.

Planck units may be considered "more natural" even than other natural unit systems discussed below, as Planck units are not based on any arbitrarily chosen prototype object or particle. For example, some other systems use the mass of an electron as a parameter to be normalized. But the electron is just one of 16 known massive elementary particles, all with different masses, and there is no compelling reason, within fundamental physics, to emphasize the electron mass over some other elementary particle's mass.

The original Planck units are based on Gaussian units, thus and thus and . However, the Planck units can also be based on Lorentz–Heaviside units, thus and (this is often called rationalized Planck units, e.g. rationalized Planck energy). Both conventions of Planck units set .

Stoney units

Quantity Expression Metric value
Length (L) 1.381×10−36 m
Mass (M) 1.859×10−9 kg
Time (T) 4.605×10−45 s
Temperature (Θ) 1.210×1031 K
Electric charge (Q) 1.602×10−19 C

Stoney units are defined by:

c = G = ke = e = kB = 1,

where c is the speed of light, G is the gravitational constant, ke is the Coulomb constant, e is the elementary charge, and kB is the Boltzmann constant.

George Johnstone Stoney was the first physicist to introduce the concept of natural units. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.[6] Stoney units differ from Planck units by fixing the elementary charge at 1, instead of the Planck constant (only discovered after Stoney's proposal).

Stoney units are rarely used in modern physics for calculations, but they are of historical interest.

Atomic units

Quantity Expression Metric value
Length (L) (both Hartree and Rydberg) 5.292×10−11 m
Mass (M) (Hartree) 9.109×10−31 kg
(Rydberg) 1.822×10−30 kg
Time (T) (Hartree) 2.419×10−17 s
(Rydberg) 4.838×10−17 s
Temperature (Θ) (Hartree) 3.158×105 K
(Rydberg) 1.579×105 K
Electric charge (Q) (Hartree) 1.602×10−19 C
(Rydberg) 1.133×10−19 C

There are two types of atomic units, closely related.

Hartree atomic units:

e = me = ħ = ke = kB = 1
c = 1/α

Rydberg atomic units:[7]

e/2 = 2me = ħ = ke = kB = 1
c = 2/α

Coulomb's constant is generally expressed as

ke = 1/ε0.

These units are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom, and are widely used in these fields. The Hartree units were first proposed by Douglas Hartree, and are more common than the Rydberg units.

The units are designed especially to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, using the Hartree convention, in the Bohr model of the hydrogen atom, an electron in the ground state has orbital velocity = 1, orbital radius = 1, angular momentum = 1, ionization energy = 1/2, etc.

The unit of energy is called the Hartree energy in the Hartree system and the Rydberg energy in the Rydberg system. They differ by a factor of 2. The speed of light is relatively large in atomic units (137 in Hartree or 274 in Rydberg), which comes from the fact that an electron in hydrogen tends to move much slower than the speed of light. The gravitational constant is extremely small in atomic units (around 10−45), which comes from the fact that the gravitational force between two electrons is far weaker than the Coulomb force. The unit length, lA, is the Bohr radius, a0.

The values of c and e shown above imply that e = αħc, as in Gaussian units, not Lorentz–Heaviside units.[8] However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units.[9]

Quantum chromodynamics (QCD) units

Quantity Expression Metric value
Length (L) 2.103×10−16 m
Mass (M) 1.673×10−27 kg
Time (T) 7.015×10−25 s
Temperature (Θ) 1.089×1013 K
Electric charge (Q) (original) 1.602×10−19 C
(L–H) 5.291×10−19 C
(G) 1.876×10−18 C
c = mp = ħ = kB = 1 (in the original QCD units, e is also 1, if the QCD units are based on Lorentz–Heaviside units, then is 1, and if the QCD units are based on Gaussian units, then is 1)

The Electron rest mass is replaced with that of the proton. Strong units are "convenient for work in QCD and nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".[10]

"Natural units" (particle physics and cosmology)

Unit Metric value Derivation
1 eV−1 of length 1.97×10−7 m
1 eV of mass 1.78×10−36 kg
1 eV−1 of time 6.58×10−16 s
1 eV of temperature 1.16×104 K with
1 unit of electric charge
(L–H)
5.29×10−19 C
1 unit of electric charge
(G)
1.88×10−18 C

In particle physics and cosmology, the phrase "natural units" generally means:[11][12]

ħ = c = kB = 1.

where ħ is the reduced Planck constant, c is the speed of light, and kB is the Boltzmann constant.

Both Planck units and QCD units are this type of Natural units. Like the other systems, the electromagnetism units can be based on either Lorentz–Heaviside units or Gaussian units. The unit of charge is different in each.

Finally, one more unit is needed to construct a usable system of units that includes energy and mass. Most commonly, electronvolt (eV) is used, despite the fact that this is not a "natural" unit in the sense discussed above – it is defined by a natural property, the elementary charge, and the anthropogenic unit of electric potential, the volt. (The SI prefixed multiples of eV are used as well: keV, MeV, GeV, etc.)

With the addition of eV (or any other auxiliary unit with the proper dimension), any quantity can be expressed. For example, a distance of 1.0 cm can be expressed in terms of eV, in natural units, as:[12]

1.0 cm = 1.0 cm/ħc ≈ 51000 eV−1

Geometrized units

c = G = 1

The geometrized unit system, used in general relativity, is not a completely defined system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are set equal to unity. Other units may be treated however desired. Planck units and Stoney units are examples of geometrized unit systems.

See also

Notes and references

  1. ^ What are natural units?, Sabine Hossenfelder, 2011-11-07.
  2. ^ Planck Units - Part 1 of 3, DrPhysicistA, 2012-02-14.
  3. ^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, Archived 2009-04-29 at the Wayback Machine" The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)
  4. ^ Walter Greiner; Ludwig Neise; Horst Stöcker (1995). Thermodynamics and Statistical Mechanics. Springer-Verlag. p. 385. ISBN 978-0-387-94299-5.
  5. ^ See Gaussian units#General rules to translate a formula and references therein.
  6. ^ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.
  7. ^ Turek, Ilja (1997). Electronic structure of disordered alloys, surfaces and interfaces (illustrated ed.). Springer. p. 3. ISBN 978-0-7923-9798-4.
  8. ^ Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, by Markus Reiher, Alexander Wolf, p7 [books.google.com/books?id=YwSpxCfsNsEC&pg=PA7 link]
  9. ^ A note on units lecture notes. See the atomic units article for further discussion.
  10. ^ Wilczek, Frank, 2007, "Fundamental Constants," Frank Wilczek web site.
  11. ^ Gauge field theories: an introduction with applications, by Guidry, Appendix A
  12. ^ a b An introduction to cosmology and particle physics, by Domínguez-Tenreiro and Quirós, p422

External links

Atomic units

Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units, where the numerical values of the following four fundamental physical constants are all unity by definition:

In Hartree units, the speed of light is approximately . Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical units, arbitrary units, and absorbance units in different contexts.

Base unit (measurement)

A base unit (also referred to as a fundamental unit) is a unit adopted for measurement of a base quantity. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the subset can be expressed in terms of the others. The SI units, or Systeme International d'unites which consists of the metre, kilogram, second, ampere, Kelvin, mole and candela are base units.

'

A base unit is one that has been explicitly so designated; a secondary unit for the same quantity is a derived unit. For example, when used with the International System of Units, the gram is a derived unit, not a base unit.

In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, temperature, energy, and weight, and units are used to describe their magnitude or quantity. Many of these quantities are related to each other by various physical laws, and as a result the units of a quantities can be generally be expressed as a product of powers of other units; for example, momentum is mass multiplied by velocity, while velocity is measured in distance divided by time. These relationships are discussed in dimensional analysis. Those that can be expressed in this fashion in terms of the base units are called derived units.

Conventional electrical unit

A conventional electrical unit (or conventional unit where there is no risk of ambiguity) is a unit of measurement in the field of electricity which is based on the so-called "conventional values" of the Josephson constant and the von Klitzing constant agreed by the International Committee for Weights and Measures (CIPM) in 1988. These units are very similar in scale to their corresponding SI units, but are not identical because of their different definition. They are distinguished from the corresponding SI units by setting the symbol in italic typeface and adding a subscript "90" – e.g., the conventional volt has the symbol V90 – as they came into international use on 1 January 1990.

This system was developed to increase the precision of measurements: The Josephson and von Klitzing constants can be realized with great precision, repeatability and ease. The conventional electrical units have achieved acceptance as an international standard and are commonly used outside of the physics community in both engineering and industry.

The conventional electrical units are "quasi-natural" in the sense that they are completely and exactly defined in terms of universal constants. They represent a significant step towards using "natural" fundamental physics for practical measurement purposes. However, the conventional electrical units are unlike other systems of natural units in that some physical constants are not set to unity but rather set to fixed numerical values that are very close to (but not precisely the same as) those in the SI system of units.

Several significant steps have been taken in the last half century to increase the precision and utility of measurement units:

In 1967, the thirteenth General Conference on Weights and Measures (CGPM) defined the second of atomic time in the International System of Units as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.

In 1983, the seventeenth CGPM redefined the metre in terms of the second and the speed of light, thus fixing the speed of light at exactly 299792458 m/s.

In 1988, the CIPM recommended adoption of conventional values for the Josephson constant as exactly KJ-90 = 483597.9×109 Hz/V and for the von Klitzing constant as exactly RK-90 = 25812.807 Ω as of 1 January 1990.

In 1991, the eighteenth CGPM noted the conventional values for the Josephson constant and the von Klitzing constant.

In 2000, the CIPM approved the use of the quantum Hall effect, with the value of RK-90 to be used to establish a reference standard of resistance.

In 2018, the twenty-sixth CGPM resolved to abrogate the conventional values of the Josephson and von Klitzing constants with the 2019 redefinition of SI base units.

Electronvolt

In physics, the electronvolt (symbol eV, also written electron-volt and electron volt) is a unit of energy equal to approximately 1.6×10−19 joules (symbol J) in SI units.

Historically, the electronvolt was devised as a standard unit of measure through its usefulness in electrostatic particle accelerator sciences, because a particle with electric charge q has an energy E = qV after passing through the potential V; if q is quoted in integer units of the elementary charge and the potential in volts, one gets an energy in eV.

Like the elementary charge on which it is based, it is not an independent quantity but is equal to 1 J/C √2hα / μ0c0. It is a common unit of energy within physics, widely used in solid state, atomic, nuclear, and particle physics. It is commonly used with the metric prefixes milli-, kilo-, mega-, giga-, tera-, peta- or exa- (meV, keV, MeV, GeV, TeV, PeV and EeV respectively). In some older documents, and in the name Bevatron, the symbol BeV is used, which stands for billion (109) electronvolts; it is equivalent to the GeV.

Geometrized unit system

A geometrized unit system or geometric unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.

The geometrized unit system is not a completely defined or unique system: latitude is left to also set other constants to unity. We may, for example, also set vacuum permittivity, ε0, and the electric charge, , to unity.

The reduced Planck constant, ħ, is not equal to 1 in this system (Stoney units), in contrast to Planck units.

This system is useful in physics, especially in the special and general theories of relativity. All physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures.

Many equations in relativistic physics appear far simpler when expressed in geometric units, because all appearances of G or c drop out. For example, the Schwarzschild radius of a nonrotating uncharged black hole with mass m becomes simply r = 2m. Therefore, many books and papers on relativistic physics use geometric units exclusively. An alternative system of geometrized units is often used in particle physics and cosmology, in which G = 1 instead. This introduces an additional factor of 8π into Newton's law of universal gravitation but simplifies Einstein's equations, the Einstein–Hilbert action, the Friedmann equations and the Newtonian Poisson equation by removing the corresponding factor.

Practical measurements and computations are usually done in SI units, but conversions are generally quite straightforward[citation needed].

Gravitational constant

The gravitational constant (also known as the "universal gravitational constant", the "Newtonian constant of gravitation", or the "Cavendish gravitational constant"), denoted by the letter G, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity.

In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies

with the product of their masses and the inverse square of their distance.

In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the energy–momentum tensor.

The measured value of the constant is known with some certainty to four significant digits. In SI units its value is approximately

6.674×10−11 m3⋅kg−1⋅s−2.The modern notation of Newton's law involving G was introduced in the 1890s by C. V. Boys.

The first implicit measurement with an accuracy within about 1% is attributed to Henry Cavendish in a 1798 experiment.

KT (energy)

kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. This product is used in physics as a scale factor for energy values in molecular-scale systems (sometimes it is used as a unit of energy), as the rates and frequencies of many processes and phenomena depend not on their energy alone, but on the ratio of that energy and kT, that is, on E / kT (see Arrhenius equation, Boltzmann factor). For a system in equilibrium in canonical ensemble, the probability of the system being in state with energy E is proportional to e−ΔE / kT.

More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system, in natural units, by one nat. E / kT therefore represents an amount of entropy per molecule, measured in natural units.

In macroscopic scale systems, with large numbers of molecules, RT value is commonly used; its SI units are joules per mole (J/mol): (RT = kT ⋅ NA).

Length scale

In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other[citation needed][clarification needed] and are said to decouple. The decoupling of different length scales makes it possible to have a self-consistent theory that only describes the relevant length scales for a given problem. Scientific reductionism says that the physical laws on the shortest length scales can be used to derive the effective description at larger length scales. The idea that one can derive descriptions of physics at different length scales from one another can be quantified with the renormalization group.

In quantum mechanics the length scale of a given phenomenon is related to its de Broglie wavelength where is the reduced Planck's constant and is the momentum that is being probed. In relativistic mechanics time and length scales are related by the speed of light. In relativistic quantum mechanics or relativistic quantum field theory, length scales are related to momentum, time and energy scales through Planck's constant and the speed of light. Often in high energy physics natural units are used where length, time, energy and momentum scales are described in the same units (usually with units of energy such as GeV).

Length scales are usually the operative scale (or at least one of the scales) in dimensional analysis. For instance, in scattering theory, the most common quantity to calculate is a cross section which has units of length squared and is measured in barns. The cross section of a given process is usually the square of the length scale.

Lexical item

In lexicography, a lexical item (or lexical unit/ LU, lexical entry) is a single word, a part of a word, or a chain of words (catena) that forms the basic elements of a language's lexicon (≈ vocabulary). Examples are cat, traffic light, take care of, by the way, and it's raining cats and dogs. Lexical items can be generally understood to convey a single meaning, much as a lexeme, but are not limited to single words. Lexical items are like semes in that they are "natural units" translating between languages, or in learning a new language. In this last sense, it is sometimes said that language consists of grammaticalized lexis, and not lexicalized grammar. The entire store of lexical items in a language is called its lexis.

Lexical items composed of more than one word are also sometimes called lexical chunks, gambits, lexical phrases, lexical units, lexicalized stems, or speech formulae. The term polyword listemes is also sometimes used.

Material balance planning

Material balances are a method of economic planning where material supplies are accounted for in natural units (as opposed to using monetary accounting) and used to balance the supply of available inputs with targeted outputs. Material balancing involves taking a survey of the available inputs and raw materials in an economy and then using a balance sheet to balance the inputs with output targets specified by industry to achieve a balance between supply and demand. This balance is used to formulate a plan for resource allocation and investment in a national economy.The method of material balances is contrasted with the method of input-output planning developed by Wassily Leontief.

Nat (unit)

The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the bit. This unit is also known by its unit symbol, the nat. The nat is the coherent unit for information entropy. The International System of Units, by assigning the same units (joule per kelvin) both to heat capacity and to thermodynamic entropy implicitly treats information entropy as a quantity of dimension one, with 1 nat = 1. Physical systems of natural units that normalize the Boltzmann constant to 1 are effectively measuring thermodynamic entropy in nats.

When the Shannon entropy is written using a natural logarithm,

it is implicitly giving a number measured in nats.

One nat is equal to 1/ln 2 shannons (or bits) ≈ 1.44 Sh or, equivalently, 1/ln 10 hartleys ≈ 0.434 Hart. The factors 1.44 and 0.434 arise from the relationships

, and
.

One nat is the information content of an event if the probability of that event occurring is 1/e.

Planck temperature

Planck temperature, denoted by TP, is the unit of temperature in the system of natural units known as Planck units.

It serves as the defining unit of the Planck temperature scale. In this scale the magnitude of the Planck temperature is equal to 1, while that of absolute zero is 0.

Other temperatures can be converted to Planck temperature units. For example, 0 °C = 273.15 K = 1.9279×10−30 TP.

In SI units, the Planck temperature is about 1.417×1032 kelvin (equivalently, degrees Celsius, since the difference is trivially small at this scale), or 2.55×1032 degrees Fahrenheit or Rankine.

Planck time

In quantum mechanics, the Planck time (tP) is the unit of time in the system of natural units known as Planck units. A Planck time unit is the time required for light to travel a distance of 1 Planck length in a vacuum, which is a time interval of approximately 5.39 × 10 −44 s. The unit is named after Max Planck, who was the first to propose it.

The Planck time is defined as:

where:

ħ = ​h2π is the reduced Planck constant (sometimes h is used instead of ħ in the definition)
G = gravitational constant
c = speed of light in vacuum

Using the known values of the constants, the approximate equivalent value in terms of the SI unit, the second, is

where the two digits between parentheses denote the standard error of the approximated value.

Planck units

In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five 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 also known as natural units because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are only one system of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (that would be arbitrarily chosen), but rather on only the properties of free space. Planck units have significance for theoretical physics since they simplify several recurring algebraic expressions of physical law by nondimensionalization. They are relevant in research on unified theories such as quantum gravity.

The term "Planck scale" refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 1.22×1019 GeV (the Planck energy), time intervals around 5.39×10−44 s (the Planck time) and lengths around 1.62×10−35 m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists no longer have any scientific model whatsoever to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10−43 seconds of our universe after the Big Bang, approximately 13.8 billion years ago.

The five universal constants that Planck units, by definition, normalize to 1 are:

the speed of light in a vacuum, c,

the gravitational constant, G,

the reduced Planck constant, ħ,

the Coulomb constant, ke = 1/4πε0

the Boltzmann constant, kBEach of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ħ with quantum mechanics, ε0 with electromagnetism, and kB with the notion of temperature/energy (statistical mechanics and thermodynamics).

Radian

The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at OEIS: A072097). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.Separately, the SI unit of solid angle measurement is the steradian.

The radian is most commonly represented by the symbol rad. An alternative symbol is c, the superscript letter c (for "circular measure"), the letter r, or a superscript R, but these symbols are infrequently used as it can be easily mistaken for a degree symbol (°) or a radius (r). So, for example, a value of 1.2 radians could be written as 1.2 rad, 1.2 r, 1.2rad, 1.2c, or 1.2R.

Steradian

The steradian (symbol: sr) or square radian is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a length on the circumference, a solid angle in steradians, projected onto a sphere, gives an area on the surface. The name is derived from the Greek στερεός stereos 'solid' + radian.

The steradian, like the radian, is a dimensionless unit, the area subtended and the square of its distance from the center: both the numerator and denominator of this ratio have dimension length squared (i.e. L2/L2 = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol "sr" is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian (W⋅sr−1). The steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit.

Stoney units

In physics the Stoney units form a system of units named after the Irish physicist George Johnstone Stoney, who first proposed them in 1881. They are the first historical example of natural units, i.e. units of measurement designed so that certain fundamental physical constants serve as base units.

System of measurement

A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement in use include the International System of Units (SI), the modern form of the metric system, the imperial system, and United States customary units.

Vacuum expectation value

In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by One of the most widely used, but controversial, examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect.

This concept is important for working with correlation functions in quantum field theory. It is also important in spontaneous symmetry breaking. Examples are:

The observed Lorentz invariance of space-time allows only the formation of condensates which are Lorentz scalars and have vanishing charge.[citation needed] Thus fermion condensates must be of the form , where ψ is the fermion field. Similarly a tensor field, Gμν, can only have a scalar expectation value such as .

In some vacua of string theory, however, non-scalar condensates are found.[which?] If these describe our universe, then Lorentz symmetry violation may be observable.

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Planck's natural units
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