Teleparallelism

From Wikipedia, the free encyclopedia

Teleparallelism (also called teleparallel gravity), was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field.

Teleparallel spacetimes

The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set {X₁, X₂, X₃, X₄} of four vector fields defined on all of M such that for every p ∈ M the set {X₁(p), X₂(p), X₃(p), X₄(p)} is a basis of T_pM, where T_pM denotes the fiber over p of the tangent vector bundle TM. Hence, the four-dimensional spacetime manifold M must be a parallelizable manifold. The tetrad field was introduced to allow the distant comparison of the direction of tangent vectors at different points of the manifold, hence the name distant parallelism. His attempt failed because there was no Schwarzschild solution in his simplified field equation.

In fact, one can define the connection of the parallelization (also called the Weitzenböck connection) {X_i} to be the linear connection ∇ on M such that

$${\displaystyle {\mathrm{\nabla }}_v\left(f^i{\mathrm{X}}_i\right)=\left(v{f}^i\right){\mathrm{X}}_i(p),}$$

where v ∈ T_pM and fⁱ are (global) functions on M; thus fⁱX_i is a global vector field on M. In other words, the coefficients of Weitzenböck connection ∇ with respect to {X_i} are all identically zero, implicitly defined by:

$${\displaystyle {\mathrm{\nabla }}_{{\mathrm{X}}_i}{\mathrm{X}}_j=0,}$$

hence

$${\displaystyle {W^k}_{ij}={\omega }^k\left({\mathrm{\nabla }}_{{\mathrm{X}}_i}{\mathrm{X}}_j\right)\equiv 0,}$$

for the connection coefficients (also called Weitzenböck coefficients) in this global basis. Here ω^k is the dual global basis (or coframe) defined by ωⁱ(X_j) = δⁱ
_j.

This is what usually happens in Rⁿ, in any affine space or Lie group (for example the 'curved' sphere S³ but 'Weitzenböck flat' manifold).

Using the transformation law of a connection, or equivalently the ∇ properties, we have the following result.

Proposition. In a natural basis, associated with local coordinates (U, x^μ), i.e., in the holonomic frame ∂_μ, the (local) connection coefficients of the Weitzenböck connection are given by:

$${\displaystyle {{\mathrm{\Gamma }}^{\beta }}_{\mu \nu }=h_i^{\beta }{\mathrm{\partial }}_{\nu }{h}_{\mu }^i,}$$

where X_i = h^μ
_i∂_μ for i, μ = 1, 2,… n are the local expressions of a global object, that is, the given tetrad.

The Weitzenböck connection has vanishing curvature, but – in general – non-vanishing torsion.

Given the frame field {X_i}, one can also define a metric by conceiving of the frame field as an orthonormal vector field. One would then obtain a pseudo-Riemannian metric tensor field g of signature (3,1) by

$${\displaystyle g\left({\mathrm{X}}_i,{\mathrm{X}}_j\right)={\eta }_{ij},}$$

where

$${\displaystyle {\eta }_{ij}=\mathrm{diag}(-1,-1,-1,1).}$$

The corresponding underlying spacetime is called, in this case, a Weitzenböck spacetime.

It is worth noting to see that these 'parallel vector fields' give rise to the metric tensor as a byproduct.

New teleparallel gravity theory

New teleparallel gravity theory (or new general relativity) is a theory of gravitation on Weitzenböck spacetime, and attributes gravitation to the torsion tensor formed of the parallel vector fields.

In the new teleparallel gravity theory the fundamental assumptions are as follows:

1. Underlying spacetime is the Weitzenböck spacetime, which has a quadruplet of parallel vector fields as the fundamental structure. These parallel vector fields give rise to the metric tensor as a by-product. All physical laws are expressed by equations that are covariant or form invariant under the group of general coordinate transformations.
2. The equivalence principle is valid only in classical physics.
3. Gravitational field equations are derivable from the action principle.
4. The field equations are partial differential equations in the field variables of not higher than the second order.

In 1961 Christian Møller revived Einstein's idea, and Pellegrini and Plebanski found a Lagrangian formulation for absolute parallelism.

Møller tetrad theory of gravitation

In 1961, Møller showed that a tetrad description of gravitational fields allows a more rational treatment of the energy-momentum complex than in a theory based on the metric tensor alone. The advantage of using tetrads as gravitational variables was connected with the fact that this allowed to construct expressions for the energy-momentum complex which had more satisfactory transformation properties than in a purely metric formulation. In 2015, it was shown that the total energy of matter and gravitation is proportional to the Ricci scalar of three-space up to the linear order of perturbation.

New translation teleparallel gauge theory of gravity

Independently in 1967, Hayashi and Nakano revived Einstein's idea, and Pellegrini and Plebanski started to formulate the gauge theory of the spacetime translation group. Hayashi pointed out the connection between the gauge theory of the spacetime translation group and absolute parallelism. The first fiber bundle formulation was provided by Cho. This model was later studied by Schweizer et al., Nitsch and Hehl, Meyer; more recent advances can be found in Aldrovandi and Pereira, Gronwald, Itin, Maluf and da Rocha Neto, Münch, Obukhov and Pereira, and Schucking and Surowitz.

Nowadays, teleparallelism is studied purely as a theory of gravity without trying to unify it with electromagnetism. In this theory, the gravitational field turns out to be fully represented by the translational gauge potential B^a_μ, as it should be for a gauge theory for the translation group.

If this choice is made, then there is no longer any Lorentz gauge symmetry because the internal Minkowski space fiber—over each point of the spacetime manifold—belongs to a fiber bundle with the Abelian R⁴ as structure group. However, a translational gauge symmetry may be introduced thus: Instead of seeing tetrads as fundamental, we introduce a fundamental R⁴ translational gauge symmetry instead (which acts upon the internal Minkowski space fibers affinely so that this fiber is once again made local) with a connection B and a "coordinate field" x taking on values in the Minkowski space fiber.

More precisely, let π : M → M be the Minkowski fiber bundle over the spacetime manifold M. For each point p ∈ M, the fiber M_p is an affine space. In a fiber chart (V, ψ), coordinates are usually denoted by ψ = (x^μ, x^a), where x^μ are coordinates on spacetime manifold M, and x^a are coordinates in the fiber M_p.

Using the abstract index notation, let a, b, c,… refer to M_p and μ, ν,… refer to the tangent bundle TM. In any particular gauge, the value of x^a at the point p is given by the section

$${\displaystyle x^{\mu }\to \left(x^{\mu },x^a={\xi }^a(p)\right).}$$

The covariant derivative

$${\displaystyle D_{\mu }{\xi }^a\equiv {\left(d{\xi }^a\right)}_{\mu }+{B^a}_{\mu }={\mathrm{\partial }}_{\mu }{\xi }^a+{B^a}_{\mu }}$$

is defined with respect to the connection form B, a 1-form assuming values in the Lie algebra of the translational abelian group R⁴. Here, d is the exterior derivative of the ath component of x, which is a scalar field (so this isn't a pure abstract index notation). Under a gauge transformation by the translation field α^a,

$${\displaystyle x^a\to x^a+{\alpha }^a}$$

and

$${\displaystyle {B^a}_{\mu }\to {B^a}_{\mu }-{\mathrm{\partial }}_{\mu }{\alpha }^a}$$

and so, the covariant derivative of x^a = ξ^a(p) is gauge invariant. This is identified with the translational (co-)tetrad

$${\displaystyle {h^a}_{\mu }={\mathrm{\partial }}_{\mu }{\xi }^a+{B^a}_{\mu }}$$

which is a one-form which takes on values in the Lie algebra of the translational Abelian group R⁴, whence it is gauge invariant. But what does this mean? x^a = ξ^a(p) is a local section of the (pure translational) affine internal bundle M → M, another important structure in addition to the translational gauge field B^a_μ. Geometrically, this field determines the origin of the affine spaces; it is known as Cartan’s radius vector. In the gauge-theoretic framework, the one-form

$${\displaystyle h^a={h^a}_{\mu }d{x}^{\mu }=\left({\mathrm{\partial }}_{\mu }{\xi }^a+{B^a}_{\mu }\right)d{x}^{\mu }}$$

arises as the nonlinear translational gauge field with ξ^a interpreted as the Goldstone field describing the spontaneous breaking of the translational symmetry.

A crude analogy: Think of M_p as the computer screen and the internal displacement as the position of the mouse pointer. Think of a curved mousepad as spacetime and the position of the mouse as the position. Keeping the orientation of the mouse fixed, if we move the mouse about the curved mousepad, the position of the mouse pointer (internal displacement) also changes and this change is path dependent; i.e., it does not depend only upon the initial and final position of the mouse. The change in the internal displacement as we move the mouse about a closed path on the mousepad is the torsion.

Another crude analogy: Think of a crystal with line defects (edge dislocations and screw dislocations but not disclinations). The parallel transport of a point of M along a path is given by counting the number of (up/down, forward/backwards and left/right) crystal bonds transversed. The Burgers vector corresponds to the torsion. Disinclinations correspond to curvature, which is why they are neglected.

The torsion—that is, the translational field strength of Teleparallel Gravity (or the translational "curvature")—

$${\displaystyle {T^a}_{\mu \nu }\equiv {\left(D{B}^a\right)}_{\mu \nu }=D_{\mu }{B^a}_{\nu }-D_{\nu }{B^a}_{\mu },}$$

is gauge invariant.

We can always choose the gauge where x^a is zero everywhere, although M_p is an affine space and also a fiber; thus the origin must be defined on a point-by-point basis, which can be done arbitrarily. This leads us back to the theory where the tetrad is fundamental.

Teleparallelism refers to any theory of gravitation based upon this framework. There is a particular choice of the action that makes it exactly equivalent to general relativity, but there are also other choices of the action which are not equivalent to general relativity. In some of these theories, there is no equivalence between inertial and gravitational masses.

Unlike in general relativity, gravity is due not to the curvature of spacetime but to the torsion thereof.

Non-gravitational contexts

There exists a close analogy of geometry of spacetime with the structure of defects in crystal. Dislocations are represented by torsion, disclinations by curvature. These defects are not independent of each other. A dislocation is equivalent to a disclination-antidisclination pair, a disclination is equivalent to a string of dislocations. This is the basic reason why Einstein's theory based purely on curvature can be rewritten as a teleparallel theory based only on torsion. There exists, moreover, infinitely many ways of rewriting Einstein's theory, depending on how much of the curvature one wants to reexpress in terms of torsion, the teleparallel theory being merely one specific version of these.

A further application of teleparallelism occurs in quantum field theory, namely, two-dimensional non-linear sigma models with target space on simple geometric manifolds, whose renormalization behavior is controlled by a Ricci flow, which includes torsion. This torsion modifies the Ricci tensor and hence leads to an infrared fixed point for the coupling, on account of teleparallelism ("geometrostasis").

Relativistic quantum chemistry

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Relativistic_quantum_chemistry

Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of gold: due to relativistic effects, it is not silvery like most other metals.

The term relativistic effects was developed in light of the history of quantum mechanics. Initially, quantum mechanics was developed without considering the theory of relativity. Relativistic effects are those discrepancies between values calculated by models that consider relativity and those that do not. Relativistic effects are important for heavier elements with high atomic numbers, such as lanthanides and actinides.

Relativistic effects in chemistry can be considered to be perturbations, or small corrections, to the non-relativistic theory of chemistry, which is developed from the solutions of the Schrödinger equation. These corrections affect the electrons differently depending on the electron speed compared with the speed of light. Relativistic effects are more prominent in heavy elements because only in these elements do electrons attain sufficient speeds for the elements to have properties that differ from what non-relativistic chemistry predicts.

History

Beginning in 1935, Bertha Swirles described a relativistic treatment of a many-electron system, despite Paul Dirac's 1929 assertion that the only imperfections remaining in quantum mechanics "give rise to difficulties only when high-speed particles are involved and are therefore of no importance in the consideration of the atomic and molecular structure and ordinary chemical reactions in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass and velocity and assumes only Coulomb forces between the various electrons and atomic nuclei".

Theoretical chemists by and large agreed with Dirac's sentiment until the 1970s, when relativistic effects were observed in heavy elements. The Schrödinger equation had been developed without considering relativity in Schrödinger's 1926 article. Relativistic corrections were made to the Schrödinger equation (see Klein–Gordon equation) to describe the fine structure of atomic spectra, but this development and others did not immediately trickle into the chemical community. Since atomic spectral lines were largely in the realm of physics and not in that of chemistry, most chemists were unfamiliar with relativistic quantum mechanics, and their attention was on lighter elements typical for the organic chemistry focus of the time.

Dirac's opinion on the role relativistic quantum mechanics would play for chemical systems is wrong for two reasons. First, electrons in s and p atomic orbitals travel at a significant fraction of the speed of light. Second, relativistic effects give rise to indirect consequences that are especially evident for d and f atomic orbitals.

Qualitative treatment

Relativistic γ as a function of velocity. For a small velocity, the ${\displaystyle E_{\text{rel}}}$ (ordinate) is equal to ${\displaystyle E_0=m{c}^2,}$ but as ${\displaystyle v_{\text{e}}\to c}$, the ${\displaystyle E_{\text{rel}}}$ goes to infinity.

One of the most important and familiar results of relativity is that the relativistic mass of the electron increases as

${\displaystyle m_{\text{rel}}=\frac{m_{\text{e}}}{\sqrt{1-(v_{\text{e}}/c{)}^2}},}$

where ${\displaystyle m_e,v_e,c}$ are the electron rest mass, velocity of the electron, and speed of light respectively. The figure at the right illustrates this relativistic effect as a function of velocity.

This has an immediate implication on the Bohr radius (${\displaystyle a_0}$), which is given by

${\displaystyle a_0=\frac{\hslash }{m_{\text{e}}c\alpha },}$

where ${\displaystyle \hslash }$ is the reduced Planck's constant, and α is the fine-structure constant (a relativistic correction for the Bohr model).

Arnold Sommerfeld calculated that, for a 1s orbital electron of a hydrogen atom with an orbiting radius of 0.0529 nm, α ≈ 1/137. That is to say, the fine-structure constant shows the electron traveling at nearly 1/137 the speed of light. One can extend this to a larger element with an atomic number Z by using the expression ${\displaystyle v\approx \frac{Zc}{137}}$ for a 1s electron, where v is its radial velocity, i.e., its instantaneous speed tangent to the radius of the atom. For gold with Z = 79, v ≈ 0.58c, so the 1s electron will be moving at 58% of the speed of light. Plugging this in for v/c in the equation for the relativistic mass, one finds that m_rel = 1.22m_e, and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%.

If one substitutes the "relativistic mass" into the equation for the Bohr radius it can be written

${\displaystyle a_{\text{rel}}=\frac{\hslash \sqrt{1-(v_{\text{e}}/c{)}^2}}{m_{\text{e}}c\alpha }.}$

Ratio of relativistic and nonrelativistic Bohr radii, as a function of electron velocity

It follows that

${\displaystyle \frac{a_{\text{rel}}}{a_0}=\sqrt{1-(v_{\text{e}}/c{)}^2}.}$

At right, the above ratio of the relativistic and nonrelativistic Bohr radii has been plotted as a function of the electron velocity. Notice how the relativistic model shows the radius decreases with increasing velocity.

When the Bohr treatment is extended to hydrogenic atoms, the Bohr radius becomes

${\displaystyle r=\frac{n^2}{Z}{a}_0=\frac{n^2{\hslash }^{2}4\pi {\epsilon }_0}{m_{\text{e}}Z{e}^2},}$

where ${\displaystyle n}$ is the principal quantum number, and Z is an integer for the atomic number. In the Bohr model, the angular momentum is given as ${\displaystyle m{v}_{\text{e}}r=n\hslash }$. Substituting into the equation above and solving for ${\displaystyle v_{\text{e}}}$ gives

${\displaystyle r=\frac{n^2{a}_0}{Z}=\frac{n\hslash }{m{v}_{\text{e}}},}$

${\displaystyle v_{\text{e}}=\frac{Z}{n^2{a}_0}\frac{n\hslash }{m},}$

${\displaystyle \frac{v_{\text{e}}}{c}=\frac{Z\alpha }{n}=\frac{Z{e}^2}{4\pi {\epsilon }_0\hslash cn}.}$

From this point, atomic units can be used to simplify the expression into

${\displaystyle v_{\text{e}}=\frac{Z}{n}.}$

Substituting this into the expression for the Bohr ratio mentioned above gives

${\displaystyle \frac{a_{\text{rel}}}{a_0}=\sqrt{1-{\left(\frac{Z}{nc}\right)}^2}.}$

At this point one can see that a low value of ${\displaystyle n}$ and a high value of ${\displaystyle Z}$ results in ${\displaystyle \frac{a_{\text{rel}}}{a_0}<1}$. This fits with intuition: electrons with lower principal quantum numbers will have a higher probability density of being nearer to the nucleus. A nucleus with a large charge will cause an electron to have a high velocity. A higher electron velocity means an increased electron relativistic mass, and as a result the electrons will be near the nucleus more of the time and thereby contract the radius for small principal quantum numbers.

Periodic-table deviations

The periodic table was constructed by scientists who noticed periodic trends in known elements of the time. Indeed, the patterns found in it are what give the periodic table its power. Many of the chemical and physical differences between the 5th period (Rb–Xe) and the 6th period (Cs–Rn) arise from the larger relativistic effects of the latter. These relativistic effects are particularly large for gold and its neighbors – platinum and mercury. An important quantum relativistic effect is the van der Waals force.

Mercury

Mercury (Hg) is a liquid down to −39 °C (see the Melting point). Bonding forces are weaker for Hg–Hg bonds than for their immediate neighbors such as cadmium (m.p. 321 °C) and gold (m.p. 1064 °C). The lanthanide contraction only partially accounts for this anomaly. Mercury in the gas phase is alone among metals in that it is quite typically found in a monomeric form as Hg(g). Hg₂²⁺(g) also forms, and it is a stable species due to the relativistic shortening of the bond.

Hg₂(g) does not form because the 6s² orbital is contracted by relativistic effects and may therefore only weakly contribute to any bonding; in fact, Hg–Hg bonding must be mostly the result of van der Waals forces, which explains why the bonding for Hg–Hg is weak enough to allow for Hg to be a liquid at room temperature.

Au₂(g) and Hg(g) are analogous with H₂(g) and He(g) with regard to having the same nature of difference. The relativistic contraction of the 6s² orbital leads to gaseous mercury sometimes being referred to as a pseudo noble gas.

Color of gold and caesium

Spectral reflectance curves for aluminum (Al), silver (Ag), and gold (Au) metal mirrors

Alkali-metal coloration: rubidium (silvery) versus caesium (golden)

The reflectivity of aluminium (Al), silver (Ag), and gold (Au) is shown in the graph to the right. The human eye sees electromagnetic radiation with a wavelength near 600 nm as yellow. Gold absorbs blue light more than it absorbs other visible wavelengths of light; the reflected light reaching the eye is therefore lacking in blue compared with the incident light. Since yellow is complementary to blue, this makes a piece of gold under white light appear yellow to human eyes.

The electronic transition from the 5d orbital to the 6s orbital is responsible for this absorption. An analogous transition occurs in silver, but the relativistic effects are smaller than in gold. While silver's 4d orbital experiences some relativistic expansion and the 5s orbital contraction, the 4d–5s distance in silver is much greater than the 5d–6s distance in gold. The relativistic effects increase the 5d orbital's distance from the atom's nucleus and decrease the 6s orbital's distance.

Caesium, the heaviest of the alkali metals that can be collected in quantities sufficient for viewing, has a golden hue, whereas the other alkali metals are silver-white. However, relativistic effects are not very significant at Z = 55 for caesium (not far from Z = 47 for silver). The golden color of caesium comes from the decreasing frequency of light required to excite electrons of the alkali metals as the group is descended. For lithium through rubidium, this frequency is in the ultraviolet, but for caesium it reaches the blue-violet end of the visible spectrum; in other words, the plasmonic frequency of the alkali metals becomes lower from lithium to caesium. Thus caesium transmits and partially absorbs violet light preferentially, while other colors (having lower frequency) are reflected; hence it appears yellowish.

Lead–acid battery

Without relativity, lead (Z = 82) would be expected to behave much like tin (Z = 50), so tin–acid batteries should work just as well as the lead–acid batteries commonly used in cars. However, calculations show that about 10 V of the 12 V produced by a 6-cell lead–acid battery arises purely from relativistic effects, explaining why tin–acid batteries do not work.

Inert-pair effect

Main article: Inert-pair effect

In Tl(I) (thallium), Pb(II) (lead), and Bi(III) (bismuth) complexes a 6s² electron pair exists. The inert pair effect is the tendency of this pair of electrons to resist oxidation due to a relativistic contraction of the 6s orbital.

Other effects

Additional phenomena commonly caused by relativistic effects are the following:

• The effect of relativistic effects on metallophilic interactions is uncertain. Although Runeberg et al. (1999) calculated an attractive effect, Wan et al. (2021) instead calculated a repulsive effect.
• The stability of gold and platinum anions in compounds such as caesium auride.
• The slightly reduced reactivity of francium compared with caesium.
• About 10% of the lanthanide contraction is attributed to the relativistic mass of high-velocity electrons and the smaller Bohr radius that results.

Thesaurus

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Thesaurus

A thesaurus (PL: thesauri or thesauruses), sometimes called a synonym dictionary or dictionary of synonyms, is a reference work which arranges words by their meanings (or in simpler terms, a book where you can find different words with same meanings to other words), sometimes as a hierarchy of broader and narrower terms, sometimes simply as lists of synonyms and antonyms. They are often used by writers to help find the best word to express an idea:

...to find the word, or words, by which [an] idea may be most fitly and aptly expressed

— Peter Mark Roget, 1852

Synonym dictionaries have a long history. The word 'thesaurus' was used in 1852 by Peter Mark Roget for his Roget's Thesaurus.

While some works called "thesauri", such as Roget's Thesaurus, group words in a hierarchical hypernymic taxonomy of concepts, others are organised alphabetically or in some other way.

Most thesauri do not include definitions, but many dictionaries include listings of synonyms.

Some thesauri and dictionary synonym notes characterise the distinctions between similar words, with notes on their "connotations and varying shades of meaning". Some synonym dictionaries are primarily concerned with differentiating synonyms by meaning and usage. Usage manuals such as Fowler's Dictionary of Modern English Usage or Garner's Modern English Usage often prescribe appropriate usage of synonyms.

Writers sometimes use thesauri to avoid repetition of words – elegant variation – which is often criticised by usage manuals: "wrWritersometimes use them not just to vary their vocabularies but to dress them up too much".

Etymology

The word "thesaurus" comes from Latin thēsaurus, which in turn comes from Greek θησαυρός (thēsauros) 'treasure, treasury, storehouse'. The word thēsauros is of uncertain etymology.

Until the 19th century, a thesaurus was any dictionary or encyclopedia, as in the Thesaurus Linguae Latinae (Dictionary of the Latin Language, 1532), and the Thesaurus Linguae Graecae (Dictionary of the Greek Language, 1572). It was Roget who introduced the meaning "collection of words arranged according to sense", in 1852.

History

Peter Mark Roget, author of Roget's thesaurus.

In antiquity, Philo of Byblos authored the first text that could now be called a thesaurus. In Sanskrit, the Amarakosha is a thesaurus in verse form, written in the 4th century.

The study of synonyms became an important theme in 18th-century philosophy, and Condillac wrote, but never published, a dictionary of synonyms.

Some early synonym dictionaries include:

• John Wilkins, An Essay Towards a Real Character, and a Philosophical Language and Alphabetical Dictionary (1668) is a "regular enumeration and description of all those things and notions to which names are to be assigned". They are not explicitly synonym dictionaries — in fact, they do not even use the word "synonym" — but they do group synonyms together.
• Gabriel Girard, La Justesse de la langue françoise, ou les différentes significations des mots qui passent pour synonymes (1718)
• John Trusler, The Difference between Words esteemed Synonyms, in the English Language; and the proper choice of them determined (1766)
• Hester Lynch Piozzi, British Synonymy (1794)
• James Leslie, Dictionary of the Synonymous Words and Technical Terms in the English Language (1806)
• George Crabb, English Synonyms Explained (1818)

Roget's Thesaurus, first compiled in 1805 by Peter Mark Roget, and published in 1852, follows John Wilkins' semantic arrangement of 1668. Unlike earlier synonym dictionaries, it does not include definitions or aim to help the user choose among synonyms. It has been continuously in print since 1852 and remains widely used across the English-speaking world. Roget described his thesaurus in the foreword to the first edition:

It is now nearly fifty years since I first projected a system of verbal classification similar to that on which the present work is founded. Conceiving that such a compilation might help to supply my deficiencies, I had, in the year 1805, completed a classed catalogue of words on a small scale, but on the same principle, and nearly in the same form, as the Thesaurus now published.

Organization

Conceptual

Roget's original thesaurus was organized into 1000 conceptual Heads (e.g., 806 Debt) organized into a four-level taxonomy. For example, debt is classed under V.ii.iv:

Class five, Volition: the exercise of the will

Division Two: Social volition

Section 4: Possessive Relations

Subsection 4: Monetary relations.

Each head includes direct synonyms: Debt, obligation, liability, ...; related concepts: interest, usance, usury; related persons: debtor, debitor, ... defaulter (808); verbs: to be in debt, to owe, ... see Borrow (788); phrases: to run up a bill or score, ...; and adjectives: in debt, indebted, owing, .... Numbers in parentheses are cross-references to other Heads.

The book starts with a Tabular Synopsis of Categories laying out the hierarchy, then the main body of the thesaurus listed by the Head, and then an alphabetical index listing the different Heads under which a word may be found: Liable, subject to, 177; debt, 806; duty, 926.

Some recent versions have kept the same organization, though often with more detail under each Head. Others have made modest changes such as eliminating the four-level taxonomy and adding new heads: one has 1075 Heads in fifteen Classes.

Some non-English thesauri have also adopted this model.

In addition to its taxonomic organization, the Historical Thesaurus of English (2009) includes the date when each word came to have a given meaning. It has the novel and unique goal of "charting the semantic development of the huge and varied vocabulary of English".

Different senses of a word are listed separately. For example, three different senses of "debt" are listed in three different places in the taxonomy:

A sum of money that is owed or due; a liability or obligation to pay

Society

Trade and Finance

Management of Money

Insolvency

Indebtedness [noun]

An immaterial debt; is an obligation to do something

Society

Morality

Duty or obligation

[noun]

An offence requiring expiation (figurative, Biblical)

Society

Faith

Aspects of faith

Spirituality

Sin

[noun]

instance of

Alphabetical

Other thesauri and synonym dictionaries are organized alphabetically.

Most repeat the list of synonyms under each word.

Some designate a principal entry for each concept and cross-reference it.^[

A third system interfiles words and conceptual headings. Francis March's Thesaurus Dictionary gives for liability: CONTINGENCY, CREDIT–DEBT, DUTY–DERELICTION, LIBERTY–SUBJECTION, MONEY, each of which is a conceptual heading. The CREDIT—DEBT article has multiple subheadings, including Nouns of Agent, Verbs, Verbal Expressions, etc. Under each are listed synonyms with brief definitions, e.g. "Credit. Transference of property on promise of future payment." The conceptual headings are not organized into a taxonomy.

Benjamin Lafaye's Synonymes français (1841) is organized around morphologically related families of synonyms (e.g. logis, logement), and his Dictionnaire des synonymes de la langue française (1858) is mostly alphabetical, but also includes a section on morphologically related synonyms, which is organized by prefix, suffix, or construction.

Contrasting senses

Before Roget, most thesauri and dictionary synonym notes included discussions of the differences among near-synonyms, as do some modern ones.

Merriam-Webster's Dictionary of Synonyms is a stand-alone modern English synonym dictionary that does discuss differences. In addition, many general English dictionaries include synonym notes.

Several modern synonym dictionaries in French are primarily devoted to discussing the precise demarcations among synonyms.

Additional elements

Some include short definitions.

Some give illustrative phrases.

Some include lists of objects within the category (hyponyms), e.g. breeds of dogs.

Bilingual

Bilingual synonym dictionaries are designed for language learners. One such dictionary gives various French words listed alphabetically, with an English translation and an example of use. Another one is organized taxonomically with examples, translations, and some usage notes.

Information science and natural language processing

Main article: Thesaurus (information retrieval)

In library and information science, a thesaurus is a kind of controlled vocabulary.

A thesaurus can form part of an ontology and be represented in the Simple Knowledge Organization System (SKOS).

Thesauri are used in natural language processing for word-sense disambiguation and text simplification for machine translation systems.