The electromagnetic field tensor "F" constructs the electromagnetic stress–energy tensor "T" by the equation:


where "η" is the Minkowski metric tensor (with signature ). Notice that we use the fact that


In vacuum (or for the microscopic equations, not including macroscopic material descriptions), Maxwell's equations can be written as two tensor equations.


The two inhomogeneous Maxwell's equations, Gauss's Law and Ampère's law (with Maxwell's correction) combine into (with metric):


while the homogeneous equations – Faraday's law of induction and Gauss's law for magnetism combine to form formula_26, which may written using Levi-Civita duality as:


where "F""αβ" is the electromagnetic tensor, "J""α" is the four-current, "ε""αβγδ" is the Levi-Civita symbol, and the indices behave according to the Einstein summation convention.


Each of these tensor equations corresponds to four scalar equations, one for each value of "β".


Using the antisymmetric tensor notation and comma notation for the partial derivative (see Ricci calculus), the second equation can also be written more compactly as:


In the absence of sources, Maxwell's equations reduce to:


which is an electromagnetic wave equation in the field strength tensor.


The Lorenz gauge condition is a Lorentz-invariant gauge condition. (This can be contrasted with other gauge conditions such as the Coulomb gauge, which if it holds in one inertial frame will generally not hold in any other.) It is expressed in terms of the four-potential as follows:


In the Lorenz gauge, the microscopic Maxwell's equations can be written as:


Electromagnetic (EM) fields affect the motion of electrically charged matter: due to the Lorentz force. In this way, EM fields can be detected (with applications in particle physics, and natural occurrences such as in aurorae). In relativistic form, the Lorentz force uses the field strength tensor as follows.


Expressed in terms of coordinate time "t", it is:


where "p""α" is the four-momentum, "q" is the charge, and "x""β" is the position.


Expressed in frame-independent form, we have the four-force


where "u""β" is the four-velocity, and "τ" is the particle's proper time, which is related to coordinate time by .


The density of force due to electromagnetism, whose spatial part is the Lorentz force, is given by


and is related to the electromagnetic stress–energy tensor by


Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector


which expresses the conservation of linear momentum and energy by electromagnetic interactions.


In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, "J""ν" Frequently, it is convenient to separate the current into two parts, the free current and the bound current, which are modeled by different equations;


Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H:


where M is the magnetization and P the electric polarization.


The bound current is derived from the P and M fields which form an antisymmetric contravariant magnetization-polarization tensor


If this is combined with "F"μν we get the antisymmetric contravariant electromagnetic displacement tensor which combines the D and H fields as follows:


The three field tensors are related by:


which is equivalent to the definitions of the D and H fields given above.


The bound current and free current as defined above are automatically and separately conserved


In vacuum, the constitutive relations between the field tensor and displacement tensor are:


Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define "F""μν" by


the constitutive equations may, in "vacuum", be combined with the Gauss–Ampère law to get:


The electromagnetic stress–energy tensor in terms of the displacement is:


where "δαπ" is the Kronecker delta. When the upper index is lowered with "η", it becomes symmetric and is part of the source of the gravitational field.


Thus we have reduced the problem of modeling the current, "J""ν" to two (hopefully) easier problems — modeling the free current, "J""ν"free and modeling the magnetization and polarization, formula_55. For example, in the simplest materials at low frequencies, one has


where one is in the instantaneously comoving inertial frame of the material, "σ" is its electrical conductivity, "χ"e is its electric susceptibility, and "χ"m is its magnetic susceptibility.


The constitutive relations between the formula_59 and "F" tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are:


where "u" is the four-velocity of material, "ε" and "μ" are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), formula_62 and denotes the Hodge dual.


The Lagrangian density for classical electrodynamics is composed by two components: a field component and a source component:


In the interaction term, the four-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables; the four-current is not itself a fundamental field.


The Lagrange equations for the electromagnetic lagrangian density formula_64 can be stated as follows:


the expression inside the square bracket is


Therefore, the electromagnetic field's equations of motion are


Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:


Using Lagrange equation, the equations of motion for formula_73 can be derived.


The equivalent expression in non-relativistic vector notation is


In continuum mechanics, the generalized Lagrangian mean (GLM) is a formalism – developed by – to unambiguously split a motion into a mean part and an oscillatory part. The method gives a mixed Eulerian–Lagrangian description for the flow field, but appointed to fixed Eulerian coordinates.


In general, it is difficult to decompose a combined wave–mean motion into a mean and a wave part, especially for flows bounded by a wavy surface: e.g. in the presence of surface gravity waves or near another undulating bounding surface (like atmospheric flow over mountainous or hilly terrain). However, this splitting of the motion in a wave and mean part is often demanded in mathematical models, when the main interest is in the mean motion – slowly varying at scales much larger than those of the individual undulations. From a series of postulates, arrive at the (GLM) formalism to split the flow: into a generalised Lagrangian mean flow and an oscillatory-flow part.


The GLM method does not suffer from the strong drawback of the Lagrangian specification of the flow field – following individual fluid parcels – that Lagrangian positions which are initially close gradually drift far apart. In the Lagrangian frame of reference, it therefore becomes often difficult to attribute Lagrangian-mean values to some location in space.


The specification of mean properties for the oscillatory part of the flow, like: Stokes drift, wave action, pseudomomentum and pseudoenergy – and the associated conservation laws – arise naturally when using the GLM method.


The GLM concept can also be incorporated into variational principles of fluid flow.


The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of stochastic processes. Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system.


Less formally, it is normally straightforward to write down a mathematical description of a system where processes happen randomly (for example, radioactive atoms randomly decay in a physical system, or genes that are expressed stochastically in a cell). However, these mathematical descriptions are often too difficult to solve for the study of the systems statistics (for example, the mean and variance of the number of atoms or proteins as a function of time). The system size expansion allows one to obtain an approximate statistical description that can be solved much more easily than the master equation.


Systems that admit a treatment with the system size expansion may be described by a probability distribution formula_1, giving the probability of observing the system in state formula_2 at time formula_3. formula_2 may be, for example, a vector with elements corresponding to the number of molecules of different chemical species in a system. In a system of size formula_5 (intuitively interpreted as the volume), we will adopt the following nomenclature: formula_6 is a vector of macroscopic copy numbers, formula_7 is a vector of concentrations, and formula_8 is a vector of deterministic concentrations, as they would appear according to the rate equation in an infinite system. formula_9 and formula_6 are thus quantities subject to stochastic effects.


A master equation describes the time evolution of this probability. Henceforth, a system of chemical reactions will be discussed to provide a concrete example, although the nomenclature of "species" and "reactions" is generalisable. A system involving formula_11 species and formula_12 reactions can be described with the master equation:


Here, formula_5 is the system size, formula_15 is an operator which will be addressed later, formula_16 is the stoichiometric matrix for the system (in which element formula_16 gives the stoichiometric coefficient for species formula_18 in reaction formula_19), and formula_20 is the rate of reaction formula_19 given a state formula_9 and system size formula_5.


formula_24 is a step operator, removing formula_16 from the formula_18th element of its argument. For example, formula_27. This formalism will be useful later.


The above equation can be interpreted as follows. The initial sum on the RHS is over all reactions. For each reaction formula_19, the brackets immediately following the sum give two terms. The term with the simple coefficient −1 gives the probability flux away from a given state formula_6 due to reaction formula_19 changing the state. The term preceded by the product of step operators gives the probability flux due to reaction formula_19 changing a different state formula_32 into state formula_6. The product of step operators constructs this state formula_32.


For example, consider the (linear) chemical system involving two chemical species formula_35 and formula_36 and the reaction formula_37. In this system, formula_38 (species), formula_39 (reactions). A state of the system is a vector formula_40, where formula_41 are the number of molecules of formula_35 and formula_36 respectively. Let formula_44, so that the rate of reaction 1 (the only reaction) depends on the concentration of formula_35. The stoichiometry matrix is formula_46.


where formula_48 is the shift caused by the action of the product of step operators, required to change state formula_6 to a precursor state formula_50.


If the master equation possesses nonlinear transition rates, it may be impossible to solve it analytically. The system size expansion utilises the ansatz that the variance of the steady-state probability distribution of constituent numbers in a population scales like the system size. This ansatz is used to expand the master equation in terms of a small parameter given by the inverse system size.


Specifically, let us write the formula_51, the copy number of component formula_18, as a sum of its "deterministic" value (a scaled-up concentration) and a random variable formula_53, scaled by formula_54:


The probability distribution of formula_6 can then be rewritten in the vector of random variables formula_53:


Consider how to write reaction rates formula_59 and the step operator formula_15 in terms of this new random variable. Taylor expansion of the transition rates gives:


The step operator has the effect formula_62 and hence formula_63:


We are now in a position to recast the master equation.


This rather frightening expression makes a bit more sense when we gather terms in different powers of formula_5. First, terms of order formula_54 give


These terms cancel, due to the macroscopic reaction equation


The terms of order formula_70 are more interesting:


The time evolution of formula_75 is then governed by the linear Fokker–Planck equation with coefficient matrices formula_76 and formula_77 (in the large-formula_5 limit, terms of formula_79 may be neglected, termed the linear noise approximation). With knowledge of the reaction rates formula_80 and stoichiometry formula_81, the moments of formula_75 can then be calculated.


The approximation implies that fluctuations around the mean are Gaussian distributed. Non-Gaussian features of the distributions can be computed by taking into account higher order terms in the expansion.


The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.


Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.


The wave equation is a partial differential equation that may constrain some scalar function of a time variable and one or more spatial variables . The quantity may be, for example, the pressure in a liquid or gas, or the displacement, along some specific direction, of the particles of a vibrating solid away from their resting positions. The equation is


where is a fixed non-negative real coefficient.


Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as


where the double dot denotes double time derivative of , is the nabla operator, and is the (spatial) Laplacian operator:


A solution of this equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed . This analysis is possible because the wave equation is linear; so that any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the superposition principle in physics.


The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.


The wave equation is the simplest example of a hyperbolic differential equation. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines.


The wave equation in one space dimension can be written as follows:


This equation is typically described as having only one space dimension , because the only other independent variable is the time . Nevertheless, the dependent variable may represent a second space dimension, if, for example, the displacement takes place in -direction, as in the case of a string that is located in the .


The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string that is vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension.


Another physical setting for derivation of the wave equation in one space dimension utilizes Hooke's Law. In the theory of elasticity, Hooke's Law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).


The wave equation in the one-dimensional case can be derived from Hooke's Law in the following way: imagine an array of little weights of mass interconnected with massless springs of length . The springs have a spring constant of :


Here the dependent variable measures the distance from the equilibrium of the mass situated at , so that essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass at the location are:


The equation of motion for the weight at the location is given by equating these two forces:


where the time-dependence of has been made explicit.


If the array of weights consists of weights spaced evenly over the length of total mass , and the total spring constant of the array we can write the above equation as:


Taking the limit and assuming smoothness one gets:


which is from the definition of a second derivative. is the square of the propagation speed in this particular case.


In the case of a stress pulse propagating longitudinally through a bar, the bar acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law. A uniform bar, i.e. of constant cross-section, made from a linear elastic material has a stiffness given by


Where is the cross-sectional area and is the Young's modulus of the material. The wave equation becomes


is equal to the volume of the bar and therefore


where is the density of the material. The wave equation reduces to


The speed of a stress wave in a bar is therefore .


The one-dimensional wave equation is unusual for a partial differential equation in that a relatively simple general solution may be found. Defining new variables:


In other words, solutions of the 1D wave equation are sums of a right traveling function and a left traveling function . "Traveling" means that the shape of these individual arbitrary functions with respect to stays constant, however the functions are translated left and right with time at the speed . This was derived by Jean le Rond d'Alembert.


Another way to arrive at this result is to note that the wave equation may be "factored":


As a result, if we define thus,


From this, must have the form , and from this the correct form of the full solution can be deduced.


For an initial value problem, the arbitrary functions and can be determined to satisfy initial conditions:


In the classical sense if and then . However, the waveforms and may also be generalized functions, such as the delta-function. In that case, the solution may be interpreted as an impulse that travels to the right or the left.


The basic wave equation is a linear differential equation and so it will adhere to the superposition principle. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the Fourier transform breaks up a wave into sinusoidal components.


Another way to solve for the solutions to the one-dimensional wave equation is to first analyze its frequency eigenmodes. A so-called eigenmode is a solution that oscillates in time with a well-defined "constant" angular frequency , so that the temporal part of the wave function takes the form , and the amplitude is a function of the spatial variable , giving a separation of variables for the wave function:


This produces an ordinary differential equation for the spatial part :


which is precisely an eigenvalue equation for , hence the name eigenmode. It has the well-known plane wave solutions


The total wave function for this eigenmode is then the linear combination


where complex numbers depend in general on any initial and boundary conditions of the problem.


Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor formula_28. so that a full solution can be decomposed into an eigenmode expansion


or in terms of the plane waves,


Scalar wave equation in three space dimensions.


A solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used to obtain the same solution in two space dimensions.


The wave equation can be solved using the technique of separation of variables. To obtain a solution with constant frequencies, let us first Fourier-transform the wave equation in time as


This is the Helmholtz equation and can be solved using separation of variables. If spherical coordinates are used to describe a problem, then the solution to the angular part of the Helmholtz equation is given by spherical harmonics and the radial equation now becomes


Here and the complete solution is now given by


where and are the spherical Hankel functions.


To gain a better understanding of the nature of these spherical waves, let us go back and look at the case when . In this case, there is no angular dependence and the amplitude depends only on the radial distance i.e. . In this case, the wave equation reduces to


where the quantity satisfies the one-dimensional wave equation. Therefore, there are solutions in the form


where and are general solutions to the one-dimensional wave equation, and can be interpreted as respectively an outgoing or incoming spherical wave. Such waves are generated by a point source, and they make possible sharp signals whose form is altered only by a decrease in amplitude as increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.


For physical examples of non-spherical wave solutions to the 3D wave equation that do possess angular dependence, see dipole radiation.


Although the word "monochromatic" is not exactly accurate since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on Plane wave eigenmodes, if we again restrict our solutions to spherical waves that oscillate in time with well-defined "constant" angular frequency , then the transformed function has simply plane wave solutions,


From this we can observe that the peak intensity of the spherical wave oscillation, characterized as the squared wave amplitude


A flexible string that is stretched between two points and satisfies the wave equation for and . On the boundary points, may satisfy a variety of boundary conditions. A general form that is appropriate for applications is


where and are non-negative. The case where u is required to vanish at an endpoint is the limit of this condition when the respective or approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form


The eigenvalue must be determined so that there is a non-trivial solution of the boundary-value problem


This is a special case of the general problem of Sturm–Liouville theory. If and are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for and can be obtained from expansion of these functions in the appropriate trigonometric series.


Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model:


If each mass point has the mass , the tension of the string is , the separation between the mass points is and are the offset of these points from their equilibrium points (i.e. their position on a straight line between the two attachment points of the string) the vertical component of the force towards point is


and the vertical component of the force towards point is


Taking the sum of these two forces and dividing with the mass one gets for the vertical motion:


The wave equation is obtained by letting in which case takes the form where is continuous function of two variables, takes the form and


with formula_53</ref> with all formula_54. The blue curve is the state at time formula_55 i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity would need for one fourth of the length of the string.


Figure 3 displays the shape of the string at the times formula_56. The wave travels in direction right with the speed without being actively constraint by the boundary conditions at the two extremes of the string. The shape of the wave is constant, i.e. the curve is indeed of the form .


Figure 4 displays the shape of the string at the times formula_57. The constraint on the right extreme starts to interfere with the motion preventing the wave to raise the end of the string.


Figure 5 displays the shape of the string at the times formula_58 when the direction of motion is reversed. The red, green and blue curves are the states at the times formula_59 while the 3 black curves correspond to the states at times formula_60 with the wave starting to move back towards left.


Figure 6 and figure 7 finally display the shape of the string at the times formula_61 and formula_62. The wave now travels towards left and the constraints at the end points are not active any more. When finally the other extreme of the string the direction will again be reversed in a way similar to what is displayed in figure 6.


The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain in -dimensional space, with boundary . Then the wave equation is to be satisfied if is in and . On the boundary of , the solution shall satisfy


where is the unit outward normal to , and is a non-negative function defined on . The case where vanishes on is a limiting case for approaching infinity. The initial conditions are


where and are defined in . This problem may be solved by expanding and in the eigenfunctions of the Laplacian in , which satisfy the boundary conditions. Thus the eigenfunction satisfies


In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary . If is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle , multiplied by a Bessel function (of integer order) of the radial component. Further details are in Helmholtz equation.


If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are spherical harmonics, and the radial components are Bessel functions of half-integer order.


The inhomogeneous wave equation in one dimension is the following:


The function is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism.


One method to solve the initial value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point , the value of depends only on the values of and and the values of the function between and . This can be seen in d'Alembert's formula, stated above, where these quantities are the only ones that show up in it. Physically, if the maximum propagation speed is , then no part of the wave that can't propagate to a given point by a given time can affect the amplitude at the same point and time.


In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that casually affects point as . Suppose we integrate the inhomogeneous wave equation over this region.


To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:


The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute


In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus .


For the other two sides of the region, it is worth noting that is a constant, namely , where the sign is chosen appropriately. Using this, we can get the relation , again choosing the right sign:


And similarly for the final boundary segment:


Adding the three results together and putting them back in the original integral:


In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source.


In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation.


The elastic wave equation (also known as the Navier–Cauchy equation) in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:


By using the elastic wave equation can be rewritten into the more common form of the Navier–Cauchy equation.


Note that in the elastic wave equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation.


As an aid to understanding, the reader will observe that if and are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field , which has only transverse waves.


In dispersive wave phenomena, the speed of wave propagation varies with the wavelength of the wave, which is reflected by a dispersion relation


where is the angular frequency and is the wavevector describing plane wave solutions. For light waves, the dispersion relation is , but in general, the constant speed gets replaced by a variable phase velocity:


This article summarizes equations in the theory of fluid mechanics.


Here formula_1 is a unit vector in the direction of the flow/current/flux.


This article summarizes equations in the theory of gravitation.


A common misconception occurs between centre of mass and centre of gravity. They are defined in similar ways but are not exactly the same quantity. Centre of mass is the mathematical description of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts. They are equal if and only if the external gravitational field is uniform.


& = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\sum_i \mathbf{r}_i m_i \left | \mathbf{g} \left ( \mathbf{r}_i \right ) \right | \end{align}\,\!</math>


Centre of gravity for a continuum of mass:


In the weak-field and slow motion limit of general relativity, the phenomenon of gravitoelectromagnetism (in short "GEM") occurs, creating a parallel between gravitation and electromagnetism. The "gravitational field" is the analogue of the electric field, while the "gravitomagnetic field", which results from circulations of masses due to their angular momentum, is the analogue of the magnetic field.


It can be shown that a uniform spherically symmetric mass distribution generates an equivalent gravitational field to a point mass, so all formulae for point masses apply to bodies which can be modelled in this way.


This article summarizes equations in the theory of waves.


Relation between space, time, angle analogues used to describe the phase:


In what follows "n, m" are any integers (Z = set of integers); formula_2.


Gravitational radiation for two orbiting bodies in the low-speed limit.


A common misconception occurs between phase velocity and group velocity (analogous to centres of mass and gravity). They happen to be equal in non-dispersive media. In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency.


Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function called the "Dispersion Relation". The use of the explicit form "ω"("k") is standard, since the phase velocity "ω"/"k" and the group velocity d"ω"/d"k" usually have convenient representations by this function.


Sinusoidal solutions to the 3d wave equation.


Resultant complex amplitude of all "N" waves


The transverse displacements are simply the real parts of the complex amplitudes.


The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The "angle addition" and "sum-to-product" trigonometric formulae are useful; in more advanced work complex numbers and fourier series and transforms are used.


The Vasiliev equations are generating equations and yield differential equations in the space-time upon solving them order by order with respect to certain auxiliary directions. The equations rely on several ingredients: unfolded equations and higher-spin algebras.


The exposition below is organised in such a way as to split the Vasiliev's equations into the building blocks and then join them together. The example of the four-dimensional bosonic Vasiliev's equations is reviewed at length since all other dimensions and super-symmetric generalisations are simple modifications of this basic example.


Three variations of Vasiliev's equations are known: four-dimensional, three-dimensional and d-dimensional. They differ by mild details that are discussed below.


Higher-spin algebras are global symmetries of the higher-spin theory multiplet. The same time they can be defined as global symmetries of some conformal field theories (CFT), which underlies the kinematic part of the higher-spin AdS/CFT correspondence, which is a particular case of the AdS/CFT. Another definition is that higher-spin algebras are quotients of the universal enveloping algebra of the anti-de Sitter algebra formula_1 by certain two-sided ideals. Some more complicated examples of higher-spin algebras exist, but all of them can be obtained by tensoring the simplest higher-spin algebras with matrix algebras and then imposing further constraints. Higher-spin algebras originate as associative algebras and the Lie algebra can be constructed via the commutator.


In the case of the four-dimensional bosonic higher-spin theory the relevant higher-spin algebra is very simple thanks to formula_2 and can be built upon two-dimensional quantum Harmonic oscillator. In the latter case two pairs of creation/annihilation operators formula_3 are needed. These can be packed into the quartet


formula_4 of operators obeying the canonical commutation relations


where formula_6 is the formula_7 invariant tensor, i.e. it is anti-symmetric. As is well known, the bilinears provide an oscillator realization of formula_7:


The higher-spin algebra is defined as the algebra of all even functions formula_10 in formula_11. That the functions are even is in accordance with the bosonic content of the higher-spin theory as formula_11 will be shown to be related to the Majorana spinors from the space-time point of view and even powers of formula_11 correspond to tensors. It is an associative algebra and the product is conveniently realised by the Moyal star product:


with the meaning that the algebra of operators formula_15 can be replaced with the algebra of function formula_16 in ordinary commuting variables formula_17 (hats off) and the product needs to be replaced with the non-commutative star-product. For example, one finds


and therefore formula_19 as it would be the case for the operators. Another representation of the same star-product is more useful in practice:


The exponential formula can be derived by integrating by parts and dropping the boundary terms. The prefactor is chosen as to ensure formula_21. In the Lorentz-covariant base we can split formula_22 and we also split formula_23. Then the Lorentz generators are formula_24, formula_25 and the translation generators are formula_26. The formula_27-automorphism can be realized in two equivalent ways: either as formula_28 or as formula_29. In both the cases it leaves the Lorentz generators untouched and flips the sign of translations.


The higher-spin algebra constructed above can be shown to be the symmetry algebra of the three-dimensional Klein-Gordon equation formula_30. Considering more general free CFT's, e.g. a number of scalars plus a number of fermions, the Maxwell field and other, one can construct more examples of higher-spin algebras.


The Vasiliev equations are equations in certain bigger space endowed with auxiliary directions to be solved for. The additional directions are given by the doubles of formula_17, called formula_32,


which are furthermore entangled with Y. The star-product on the algebra of functions in formula_33 in formula_34-variables is


The integral formula here-above is a particular star-product that corresponds to the Weyl ordering among Y's and among Z's, with the opposite signs for the commutator:


Moreover, the Y-Z star product is normal ordered with respect to Y-Z and Y+Z as is seen from


The higher-spin algebra is an associative subalgebra in the extended algebra. In accordance with the bosonic projection is given by formula_38.


The essential part of the Vasiliev equations relies on an interesting deformation of the Quantum harmonic oscillator, known as deformed oscillators. First of all, let us pack the usual creation and annihilation operators formula_39 in a doublet formula_40. The canonical commutation relations (the formula_41-factors are introduced to facilitate comparison with Vasiliev's equations)


can be used to prove that the bilinears in formula_43 form formula_44 generators


In particular, formula_46 rotates formula_43 as an formula_48-vector with formula_49 playing the role of the formula_48-invariant metric. The deformed oscillators are defined by appending the set of generators with an additional generating element formula_51 and postulating


Again, one can see that formula_46, as defined above, form formula_48-generators and rotate properly formula_43. At formula_56 we get back to the undeformed oscillators. In fact, formula_43 and formula_46 form the generators of the Lie superalgebra formula_59, where formula_43 should be viewed as odd generators. Then, formula_61 is the part of the defining relations of formula_59.


One (or two) copies of the deformed oscillator relations form a part of the Vasiliev equations where the generators are replaced with fields and the commutation relations are imposed as field equations.


The equations for higher-spin fields originate from the Vasiliev equations in the unfolded form.


Any set of differential equations can be put in the first order form by introducing auxiliary fields to denote derivatives. Unfolded approach is an advanced reformulation of this idea that takes into account gauge symmetries and diffeomorphisms. Instead of just formula_63 the unfolded equations are written in the language of differential forms as


where the variables are differential forms formula_65 of various degrees, enumerated by an abstract index formula_66; formula_67 is the exterior derivative formula_68. The structure function formula_69 is assumed to be expandable in exterior product Taylor series as


where formula_71 has form degree formula_72 and the sum is over all forms whose form degrees add up to formula_73. The simplest example of unfolded equations are the zero curvature equations formula_74 for a one-form connection formula_75 of any Lie algebra formula_76. Here formula_66 runs over the base of the Lie algebra, and the structure function formula_78 encodes the structure constants of the Lie algebra.


Since formula_79 the consistency of the unfolded equations requires


which is the Frobenius integrability condition. In the case of the zero curvature equation this is just the Jacobi identity. Once the system is integrable it can be shown to have certain gauge symmetries. Every field formula_71 that is a form of non-zero degree formula_72 possesses a gauge parameter formula_83 that is a form of degree formula_84 and the gauge transformations are


The Vasiliev equations generate the unfolded equations for a specific field content, which consists of a one-form formula_75 and a zero-form formula_87, both taking values in the higher-spin algebra. Therefore, formula_88 and formula_89, formula_90. The unfolded equations that describe interactions of higher-spin fields are


where formula_92 are the interaction vertices that are of higher and higher order in the formula_87-field. The product in the higher-spin algebra is denoted by formula_94. The explicit form of the vertices can be extracted from the Vasiliev equations. The vertices that are bilinear in the fields are determined by the higher-spin algebra. Automorphism formula_27 is induced by the automorphism of the anti-de Sitter algebra that flips the sign of translations, see below.


If we truncate away higher orders in the formula_87-expansion, the equations are just the zero-curvature condition for a connection formula_75 of the higher-spin algebra and the covariant constancy equation for a zero-form formula_87 that takes values in the twisted-adjoint representation (twist is by the automorphism formula_27).


The field content of the Vasiliev equations is given by three fields all taking values in the extended algebra of functions in Y and Z:


As to avoid any confusion caused by the differential forms in the auxiliary Z-space and to reveal the relation to the deformed oscillators the Vasiliev equations are written below in the component form.


The Vasiliev equations can be split into two parts. The first part contains only zero-curvature or covariant constancy equations:


where the higher-spin algebra automorphism formula_112 is extended to the full algebra as


the latter two forms being equivalent because of the bosonic projection imposed on formula_114.


Therefore, the first part of the equations implies that there is no nontrivial curvature in the x-space since formula_115 is flat. The second part makes the system nontrivial and determines the curvature of the auxiliary connection formula_108:


The existence of the Klein operators is of utter importance for the system. They realise the formula_112 automorphism as an inner one


In other words, the Klein operator formula_121 behave as formula_122, i.e. it anti-commutes to odd functions and commute to even functions in y,z.


These 3+2 equations are the Vasiliev equations for the four-dimensional bosonic higher-spin theory. Several comments are in order.


To prove that the linearized Vasiliev equations do describe free massless higher-spin fields we need to consider the linearised fluctuations over the anti-de Sitter vacuum. First of all we take the exact solution where formula_150 is a flat connection of the anti-de Sitter algebra, formula_149 and formula_152 and add fluctuations


Above it was used several times that formula_155, i.e. the vacuum value of the S-field acts as the derivative under the commutator. It is convenient to split the four-component Y,Z into two-component variables as formula_156. Another trick that was used in the fourth equation is the invertibility of the Klein operators:


The fifth of the Vasiliev equations is now split into the last three equation above.


The analysis of the linearized fluctuations is in solving the equations one by one in the right order. Recall that one expects to find unfolded equations for two fields: one-form formula_158 and zero-form formula_104. From the fourth equation it follows that


formula_160 does not depend on the auxiliary Z-direction. Therefore, one can identify formula_161.


The second equation then immediately leads to


where formula_163 is the Lorentz covariant derivative


where ... denote the term with formula_165 that is similar to the first one. The Lorentz covariant derivative comes from the usual commutator action of the spin-connection part of formula_143. The term with the vierbein results from the formula_112-automorphism that flips the sign of the AdS-translations and produces anti-commutator formula_168.


To read off the content of the C-equation one needs to expand it in Y and analyze the C-equation component-wise


Then various components can be seen to have the following interpretation:


The last three equations can be recognized to be the equations of the form formula_179 where formula_180 is the exterior derivative on the space of differential forms in the Z-space. Such equations can be solved with the help of the Poincare Lemma. In addition one needs to know how to multiply by the Klein operator from the right, which is easy to derive from the integral formula for the star-product:


I.e. the result is to exchange the half of the Y and Z variables and to flip the sign. The solution to the last three equations can be written as


where a similar formula exists for formula_183.


Here the last term is the gauge ambiguity, i.e. the freedom to add exact forms in the Z-space, and formula_184.


One can gauge fix it to have formula_185. Then, one plugs the solution to the third equation, which of the same type, i.e. a differential equation of the first order in the Z-space. Its general solution is again given by the Poincare Lemma


where formula_187 is the integration constant in the Z-space, i.e. the de-Rham cohomology. It is this integration constant that is to be identified with the one-form formula_188 as the name suggests. After some algebra one finds


where we again dropped a term with dotted and undotted indices exchanged. The last step is to plug the solution into the first equation to find


and again the second term on the right is omitted. It is important that formula_191 is not a flat connection, while formula_192 is a flat connection. To analyze the formula_191-equations it is useful to expand formula_191 in Y


The content of the formula_191-equation is as follows:


To conclude, anti-de Sitter space is an exact solution of the Vasiliev equations and upon linearization over it one finds unfolded equations that are equivalent to the Fronsdal equations for fields with s=0,1,2,3... .


so that the fields are now function of formula_208 and space-time coordinates. The components of the fields are required to have the right spin-statistic. The equations need to be slightly modified.


There also exist Vasiliev's equations in other dimensions:


The equations are very similar to the four-dimensional ones, but there are some important modifications in the definition of the algebra that the fields take values in and there are further constraints in the d-dimensional case.


Discrepancies between Vasiliev equations and Higher Spin Theories.


Most of the studies concern with the four-dimensional Vasiliev equations.


The correction to the free spin-2 equations due to the scalar field stress-tensor was extracted out of the four-dimensional Vasiliev equations and found to be


where formula_211 are symmetrized derivatives with traces subtracted. The most important information is in the coefficients formula_212 and in the prefactor formula_213, where formula_214 is a free parameter that the equations have, see Other dimensions, extensions, and generalisations. It is important to note that the usual stress-tensor has no more than two derivative and the terms formula_215 are not independent (for example, they contribute to the same formula_216 AdS/CFT three-point function). This is a general property of field theories that one can perform nonlinear (and also higher derivative) field redefinitions and therefore there exist infinitely many ways to write the same interaction vertex at the classical level. The canonical stress-tensor has two derivatives and the terms with contracted derivatives can be related to it via such redefinitions.


A surprising fact that had been noticed before its inconsistency with the AdS/CFT was realized is that the stress-tensor can change sign and, in particular, vanishes for formula_217. This would imply that the corresponding correlation function in the Chern-Simons matter theories vanishes, formula_218, which is not the case.


The most important and detailed tests were performed much later. It was first shown that some of the three-point AdS/CFT functions, as obtained from the Vasiliev equations, turn out to be infinite or inconsistent with AdS/CFT, while some other do agree. Those that agree, in the language of Unfolded equations correspond to formula_219 and the infinities/inconsistencies resulted from formula_220. The terms of the first type are local and are fixed by the higher spin algebra. The terms of the second type can be non-local (when solved perturbatively the master field formula_221 is a generating functions of infinitely many derivatives of higher spin fields). These non-localities are not present in higher spin theories as can be seen from the explicit cubic action.


As is briefly mentioned in Other dimensions, extensions, and generalisations there is an option to introduce infinitely many additional coupling constants that enter via phase factor formula_223. As was noted, the second such coefficient formula_224 will affect five-point AdS/CFT correlation functions, but not the three-point ones, which seems to be in tension with the results obtained directly from imposing higher spin symmetry on the correlation functions. Later, it was shown that the terms in the equations that result from


formula_225 are too non-local and lead to an infinite result for the AdS/CFT correlation functions.


In three dimensions the Prokushkin-Vasiliev equations, which are supposed to describe interactions of matter fields with higher spin fields in three dimensions, are also affected by the aforementioned locality problem. For example, the perturbative corrections at the second order to the stress-tensors of the matter fields lead to infinite correlation functions. There is, however, another discrepancy: the spectrum of the Prokushkin-Vasiliev equations has, in addition to the matter fields (scalar and spinor) and higher spin fields, a set of unphysical fields that do not have any field theory interpretation, but interact with the physical fields.


Since the Vasiliev equations are quite complicated there are few exact solutions known


In general relativity, the quadrupole formula describes the rate at which gravitational waves are emitted from a system of masses based on the change of the (mass) quadrupole moment. The formula reads


where formula_2 is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave. formula_3 is the gravitational constant, formula_4 the speed of light in vacuum, and formula_5 is the mass quadrupole moment.


The formula was first obtained by Albert Einstein in 1918. After a long history of debate on its physical correctness, observations of energy loss due to gravitational radiation in the Hulse–Taylor binary discovered in 1974 confirmed the result, with agreement up to 0.2 percent (by 2005).


In physics, Born reciprocity, also called reciprocal relativity or Born–Green reciprocity, is a principle set up by theoretical physicist Max Born that calls for a duality-symmetry among space and momentum. Born and his co-workers expanded his principle to a framework that is also known as reciprocity theory.


Born noticed a symmetry among configuration space and momentum space representations of a free particle, in that its wave function description is invariant to a change of variables "x" → "p" and "p" → −"x". (It can also be worded such as to include scale factors, e.g. invariance to "x" → "ap" and "p" → −"bx" where "a", "b" are constants.) Born hypothesized that such symmetry should apply to the four-vectors of special relativity, that is, to the four-vector space coordinates


Both in classical and in quantum mechanics, the Born reciprocity conjecture postulates that the transformation "x" → "p" and "p" → −"x" leaves invariant the Hamilton equations:


From his reciprocity approach, Max Born conjectured the invariance of a space-time-momentum-energy line element. Born and H.S. Green similarly introduced the notion an invariant (quantum) metric operator formula_5 as extension of the Minkowski metric of special relativity to an invariant metric on phase space coordinates. The metric is invariant under the group of quaplectic transformations.


Such a reciprocity as called for by Born can be observed in much, but not all, of the formalism of classical and quantum physics. Born's reciprocity theory was not developed much further for reason of difficulties in the mathematical foundations of the theory.


However Born's idea of a quantum metric operator was later taken up by Hideki Yukawa when developing his nonlocal quantum theory in the 1950s. In 1981, Eduardo R. Caianiello proposed a "maximal acceleration", similarly as there is a minimal length at Planck scale, and this concept of maximal acceleration has been expanded upon by others. It has also been suggested that Born reciprocity may be the underlying physical reason for the T-duality symmetry in string theory, and that Born reciprocity may be of relevance to developing a quantum geometry.


Born chose the term "reciprocity" for the reason that in a crystal lattice, the motion of a particle can be described in "p"-space by means of the reciprocal lattice.


This article summarizes equations used in optics, including geometric optics, physical optics, radiometry, diffraction, and interferometry.


There are different forms of the Poynting vector, the most common are in terms of the E and B or E and H fields.


For spectral quantities two definitions are in use to refer to the same quantity, in terms of frequency or wavelength.


In astrophysics, "L" is used for "luminosity" (energy per unit time, equivalent to "power") and "F" is used for "energy flux" (energy per unit time per unit area, equivalent to "intensity" in terms of area, not solid angle). They are not new quantities, simply different names.


The capstan equation or belt friction equation, also known as Eytelwein's formula (after Johann Albert Eytelwein), relates the hold-force to the load-force if a flexible line is wound around a cylinder (a bollard, a winch or a capstan).


Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small "holding" force exerted on one side can carry a much larger "loading" force on the other side; this is the principle by which a capstan-type device operates.


A holding capstan is a ratchet device that can turn only in one direction; once a load is pulled into place in that direction, it can be held with a much smaller force. A powered capstan, also called a winch, rotates so that the applied tension is multiplied by the friction between rope and capstan. On a tall ship a holding capstan and a powered capstan are used in tandem so that a small force can be used to raise a heavy sail and then the rope can be easily removed from the powered capstan and tied off.


In rock climbing with so-called top-roping, a lighter person can hold (belay) a heavier person due to this effect.


where formula_2 is the applied tension on the line, formula_3 is the resulting force exerted at the other side of the capstan, formula_4 is the coefficient of friction between the rope and capstan materials, and formula_5 is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle formula_6).


Several assumptions must be true for the formula to be valid:


It can be observed that the force gain increases exponentially with the coefficient of friction, the number of turns around the cylinder, and the angle of contact. Note that "the radius of the cylinder has no influence on the force gain".


The table below lists values of the factor formula_9 based on the number of turns and coefficient of friction "μ".


From the table it is evident why one seldom sees a sheet (a rope to the loose side of a sail) wound more than three turns around a winch. The force gain would be extreme besides being counter-productive since there is risk of a riding turn, result being that the sheet will foul, form a knot and not run out when eased (by slacking grip on the tail (free end)).


It is both ancient and modern practice for anchor capstans and jib winches to be slightly flared out at the base, rather than cylindrical, to prevent the rope (anchor warp or sail sheet) from sliding down. The rope wound several times around the winch can slip upwards gradually, with little risk of a riding turn, provided it is tailed (loose end is pulled clear), by hand or a self-tailer.


For instance, the factor "153,552,935" (5 turns around a capstan with a coefficient of friction of 0.6) means, in theory, that a newborn baby would be capable of holding (not moving) the weight of two supercarriers (97,000 tons each, but for the baby it would be only a little more than 1 kg). The large number of turns around the capstan combined with such a high friction coefficient mean that very little additional force is necessary to hold such heavy weight in place. The cables necessary to support this weight, as well as the capstan's ability to withstand the crushing force of those cables, are separate considerations.


Generalization of the capstan equation for a V-belt.


The belt friction equation for a v-belt is:


where formula_11 is the angle (in radians) between the two flat sides of the pulley that the v-belt presses against. A flat belt has an effective angle of formula_12.


The material of a V-belt or multi-V serpentine belt tends to wedge into the mating groove in a pulley as the load increases, improving torque transmission.


For the same power transmission, a V-belt requires less tension than a flat belt, increasing bearing life.


Generalization of the capstan equation for a rope lying on an arbitrary orthotropic surface.


If a rope is lying in equilibrium under tangential forces on a rough orthotropic surface then all three following conditions are satisfied:


This generalization has been obtained by Konyukhov.


In general relativity and tensor calculus, the Palatini identity is:


where formula_2 denotes the variation of Christoffel symbols and formula_3 indicates covariant differentiation.


A proof can be found in the entry Einstein–Hilbert action.


The "same" identity holds for the Lie derivative formula_4. In fact, one has:


where formula_6 denotes any vector field on the spacetime manifold formula_7.


The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and retardation effects to the order of "1/c2". When other quantum electrodynamic effects are negligible, this equation has been shown to give results in good agreement with experiment. It was originally derived from the Darwin Lagrangian but later vindicated by the Wheeler–Feynman absorber theory and eventually quantum electrodynamics.


The Breit equation is not only an approximation in terms of quantum mechanics, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation. Just as does the Dirac equation, it treats nuclei as point sources of an external field for the particles it describes. For "N" particles, the Breit equation has the form ("rij" is the distance between particle "i" and "j"):


is the Dirac Hamiltonian (see Dirac equation) for particle "i" at position r"i" and "φ"(r"i") is the scalar potential at that position; "qi" is the charge of the particle, thus for electrons "qi" = −"e".


The one-electron Dirac Hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/"rij", form the "Dirac-Coulomb" operator. To this, Breit added the operator (now known as the (frequency-independent) Breit operator):


where the Dirac matrices for electron "i": a("i") = [αx("i"),αy("i"),αz("i")]. The two terms in the Breit operator account for retardation effects to the first order.


The wave function Ψ in the Breit equation is a spinor with 4"N" elements, since each electron is described by a Dirac bispinor with 4 elements as in the Dirac equation, and the total wave function is the tensor product of these.


The total Hamiltonian of the Breit equation, sometimes called the Dirac-Coulomb-Breit Hamiltonian ("HDCB") can be decomposed into the following practical energy operators for electrons in electric and magnetic fields (also called the Breit-Pauli Hamiltonian), which have well-defined meanings in the interaction of molecules with magnetic fields (for instance for nuclear magnetic resonance):


in which the consecutive partial operators are:


Following is a list of the frequently occurring equations in the theory of special relativity.


To derive the equations of special relativity, one must start with two postulates:


In this context, "speed of light" really refers to the speed supremum of information transmission or of the movement of ordinary (nonnegative mass) matter, locally, as in a classical vacuum. Thus, a more accurate description would refer to formula_1 rather than the speed of light per se. However, light and other massless particles do theoretically travel at formula_1 under vacuum conditions and experiment has nonfalsified this notion with fairly high precision. Regardless of whether light itself does travel at formula_1, though formula_1 does act as such a supremum, and that is the assumption which matters for Relativity.


From these two postulates, all of special relativity follows.


In the following, the relative velocity "v" between two inertial frames is restricted fully to the "x"-direction, of a Cartesian coordinate system.


The following notations are used very often in special relativity:


where β = formula_7 and "v" is the relative velocity between two inertial frames.


For two frames at rest, γ = 1, and increases with relative velocity between the two inertial frames. As the relative velocity approaches the speed of light, γ → ∞.


In this example the time measured in the frame on the vehicle, "t", is known as the proper time. The proper time between two events - such as the event of light being emitted on the vehicle and the event of light being received on the vehicle - is the time between the two events in a frame where the events occur at the same location. So, above, the emission and reception of the light both took place in the vehicle's frame, making the time that an observer in the vehicle's frame would measure the proper time.


This is the formula for length contraction. As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is "". The proper length of an object is the length of the object in the frame in which the object is at rest. Also, this contraction only affects the dimensions of the object which are parallel to the relative velocity between the object and observer. Thus, lengths perpendicular to the direction of motion are unaffected by length contraction.


In what follows, bold sans serif is used for 4-vectors while normal bold roman is used for ordinary 3-vectors.


where formula_18 is known as the metric tensor. In special relativity, the metric tensor is the Minkowski metric:


In the above, "ds"2 is known as the spacetime interval. This inner product is invariant under the Lorentz transformation, that is,


The sign of the metric and the placement of the "ct", "ct"', "cdt", and "cdt′" time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes "η" is replaced with −"η", making the spatial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution. These differences can be used in any combination, so long as the choice of standards is followed completely throughout the computations performed.


It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace "t", "t′", "dt", and "dt′" with "ct", "ct"', "cdt", and "cdt′", which has the dimensions of distance. So:


The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows:


In the above, formula_28 and formula_29 are the four-vector and the transformed four-vector, respectively, and Λ is the transformation matrix, which, for a given transformation is the same for all four-vectors one might want to transform. So formula_28 can be a four-vector representing position, velocity, or momentum, and the same Λ can be used when transforming between the same two frames. The most general Lorentz transformation includes boosts and rotations; the components are complicated and the transformation requires spinors.


Invariance and unification of physical quantities both arise from four-vectors. The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also.


Doppler shift for emitter and observer moving right towards each other (or directly away):


Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them:


The Elliott formula describes analytically, or with few adjustable parameters such as the dephasing constant, the light absorption or emission spectra of solids. It was originally derived by Roger James Elliott to describe linear absorption based on properties of a single electron–hole pair. The analysis can be extended to a many-body investigation with full predictive powers when all parameters are computed microscopically using, e.g., the semiconductor Bloch equations (abbreviated as SBEs) or the semiconductor luminescence equations (abbreviated as SLEs).


One of the most accurate theories of semiconductor absorption and photoluminescence is provided by the SBEs and SLEs, respectively. Both of them are systematically derived starting from the many-body/quantum-optical system Hamiltonian and fully describe the resulting quantum dynamics of optical and quantum-optical observables such as optical polarization (SBEs) and photoluminescence intensity (SLEs). All relevant many-body effects can be systematically included by using various techniques such as the cluster-expansion approach.


These exciton eigenstates provide valuable insight to SBEs and SLEs, especially, when one analyses the linear semiconductor absorption spectrum or photoluminescence at steady-state conditions. One simply uses the constructed eigenstates to diagonalize the homogeneous parts of the SBEs and SLEs. Under the steady-state conditions, the resulting equations can be solved analytically when one further approximates dephasing due to higher-order many-body effects. When such effects are fully included, one must resort to a numeric approach. After the exciton states are obtained, one can eventually express the linear absorption and steady-state photoluminescence analytically.


The same approach can be applied to compute absorption spectrum for fields that are in the terahertz (abbreviated as THz) range of electromagnetic radiation. Since the THz-photon energy lies within the meV range, it is mostly resonant with the many-body states, not the interband transitions that are typically in the eV range. Technically, the THz investigations are an extension of the ordinary SBEs and/or involve solving the dynamics of two-particle correlations explicitly. Like for the optical absorption and emission problem, one can diagonalize the homogeneous parts that emerge analytically with the help of the exciton eigenstates. Once the diagonalization is completed, one can then compute the THz absorption analytically.


All of these derivations rely on the steady-state conditions and the analytic knowledge of the exciton states. Furthermore, the effect of further many-body contributions, such as the excitation-induced dephasing, can be included microscopically to the Wannier solver, which removes the need to introduce phenomenological dephasing constant, energy shifts, or screening of the Coulomb interaction.


Linear absorption of broadband weak optical probe can then be expressed as


where formula_5 is the probe-photon energy, formula_6 is the oscillator strength of the exciton state formula_2, and formula_8 is the dephasing constant associated with the exciton state formula_2. For a phenomenological description, formula_8 can be used as a single fit parameter, i.e., formula_11. However, a full microscopic computation generally produces formula_12 that depends on both exciton index formula_2 and photon frequency. As a general tendency, formula_12 increases for elevated formula_3 while the formula_16 dependence is often weak.


Each of the exciton resonances can produce a peak to the absorption spectrum when the photon energy matches with formula_3. For direct-gap semiconductors, the oscillator strength is proportional to the product of dipole-matrix element squared and formula_18 that vanishes for all states except for those that are spherically symmetric. In other words, formula_6 is nonvanishing only for the formula_20-like states, following the quantum-number convention of the hydrogen problem. Therefore, optical spectrum of direct-gap semiconductors produces an absorption resonance only for the formula_20-like state. The width of the resonance is determined by the corresponding dephasing constant.


In general, the exciton eigen energies consist of a series of bound states that emerge energetically well below the fundamental bandgap energy and a continuum of unbound states that appear for energies above the bandgap. Therefore, a typical semiconductor's low-density absorption spectrum shows a series of exciton resonances and then a continuum-absorption tail. For realistic situations, formula_8 increases more rapidly than the exciton-state spacing so that one typically resolves only few lowest exciton resonances in actual experiments.


The concentration of charge carriers influence the shape of the absorption spectrum considerably. For high enough densities, all formula_3 energies correspond to continuum states and some of the oscillators strengths may become negative-valued due to the Pauli-blocking effect. Physically, this can be understood as the elementary property of Fermions; if a given electronic state is already excited it cannot be excited a second time due to the Pauli exclusion among Fermions. Therefore, the corresponding electronic states can produce only photon emission that is seen as negative absorption, i.e., gain that is the prerequisite to realizing semiconductor lasers.


Even though one can understand the principal behavior of semiconductor absorption on the basis of the Elliott formula, detailed predictions of the exact formula_3, formula_6, and formula_12 requires a full many-body computation already for moderate carrier densities.


After the semiconductor becomes electronically excited, the carrier system relaxes into a quasiequilibrium. At the same time, vacuum-field fluctuations trigger spontaneous recombination of electrons and holes (electronic vacancies) via spontaneous emission of photons. At quasiequilibrium, this yields a steady-state photon flux emitted by the semiconductor. By starting from the SLEs, the steady-state photoluminescence (abbreviated as PL) can be cast into the form


that is very similar to the Elliott formula for the optical absorption. As a major difference, the numerator has a new contribution – the spontaneous-emission source


that contains electron and hole distributions formula_28 and formula_29, respectively, where formula_30 is the carrier momentum. Additionally, formula_31 contains also a direct contribution from exciton populations formula_32 that describes truly bound electron–hole pairs.


The formula_33 term defines the probability to find an electron and a hole with same formula_34. Such a form is expected for a probability of two uncorrelated events to occur simultaneously at a desired formula_34 value. Therefore, formula_33 is the spontaneous-emission source originating from uncorrelated electron–hole plasma. The possibility to have truly correlated electron–hole pairs is defined by a two-particle exciton correlation formula_32; the corresponding probability is directly proportional to the correlation. Nevertheless, both the presence of electron–hole plasma and excitons can equivalently induce the spontaneous emission. A further discussion of the relative weight and nature of plasma vs. exciton sources is presented in connection with the SLEs.


Like for the absorption, a direct-gap semiconductor emits light only at the resonances corresponding to the formula_20-like states. As a typical trend, a quasiequilibrium emission is strongly peaked around the 1"s" resonance because formula_31 is usually largest for the formula_40 ground state. This emission peak often remains well below the fundamental bandgap energy even at the high excitations where all states are continuum states. This demonstrates that semiconductors are often subjects to massive Coulomb-induced renormalizations even when the system appears to have only electron–hole plasma states as emission resonances. To make an accurate prediction of the exact position and shape at elevated carrier densities, one must resort to the full SLEs.


As discussed above, it is often meaningful to tune the electromagnetic field to be resonant with the transitions between two many-body states. For example, one can follow how a bound exciton is excited from its 1"s" ground state to a 2"p" state. In several semiconductor systems, one needs THz fields to induce such transitions. By starting from a steady-state configuration of electron–hole correlations, the diagonalization of THz-induced dynamics yields a THz absorption spectrum


(\omega) = \mathrm{Im}\left[ \frac{\sum_{\nu, \lambda} S^{\nu, \lambda} (\omega) \Delta N_{\nu,\lambda} - \left[ S^{\nu, \lambda}(-\omega) \Delta N_{\nu,\lambda}\right]^{\star} }{ \omega (\hbar \omega + \mathrm{i} \gamma(\omega))} \right]\;.


In this notation, the diagonal contributions formula_41 determine the population of formula_2 excitons. The off-diagonal formula_43 elements formally determine transition amplitudes between two exciton states formula_44 and formula_45. For elevated densities, formula_43 build up spontaneously and they describe correlated electron–hole plasma that is a state where electrons and holes move with respect to each other without forming bound pairs.


In contrast to optical absorption and photoluminescence, THz absorption may involve all exciton states. This can be seen from the spectral response function


that contains the current-matrix elements formula_48 between two exciton states. The unit vector formula_49 is determined by the direction of the THz field. This leads to dipole selection rules among exciton states, in full analog to the atomic dipole selection rules. Each allowed transition produces a resonance in formula_50 and the resonance width is determined by a dephasing constant formula_51 that generally depends on exciton states involved and the THz frequency formula_16. The THz response also contains formula_53 that stems from the decay constant of macroscopic THz currents.


In contrast to optical and photoluminescence spectroscopy, THz absorption can directly measure the presence of exciton populations in full analogy to atomic spectroscopy. For example, the presence of a pronounced 1"s"-to-2"p" resonance in THz absorption uniquely identifies the presence of excitons as detected experimentally in Ref. As a major difference to atomic spectroscopy, semiconductor resonances contain a strong excitation-induced dephasing that produces much broader resonances than in atomic spectroscopy. In fact, one typically can resolve only one 1"s"-to-2"p" resonance because the dephasing constant formula_54 is broader than energetic spacing of n-"p" and (n+1)-"p" states making 1"s"-to-n-"p" and 1"s"-to-(n+1)"p" resonances merge into one asymmetric tail.


The sigma model was introduced by ; the name σ-model comes from a field in their model corresponding to a spinless meson called , a scalar meson introduced earlier by Julian Schwinger. The model served as the dominant prototype of spontaneous symmetry breaking of O(4) down to O(3): the three axial generators broken are the simplest manifestation of chiral symmetry breaking, the surviving unbroken O(3) representing isospin.


In conventional particle physics settings, the field is generally taken to be SU(N), or the vector subspace of quotient formula_1 of the product of left and right chiral fields. In condensed matter theories, the field is taken to be O(N). For the rotation group O(3), the sigma model describes the isotropic ferromagnet; more generally, the O(N) model shows up in the quantum Hall effect, superfluid Helium-3 and spin chains.


In supergravity models, the field is taken to be a symmetric space. Since symmetric spaces are defined in terms of their involution, their tangent space naturally splits into even and odd parity subspaces. This splitting helps propel the dimensional reduction of Kaluza–Klein theories.


In its most basic form, the sigma model can be taken as being purely the kinetic energy of a point particle; as a field, this is just the Dirichlet energy in Euclidean space.


In two spatial dimensions, the O(3) model is completely integrable.


The Lagrangian density of the sigma model can be written in a variety of different ways, each suitable to a particular type of application. The simplest, most generic definition writes the Lagrangian as the metric trace of the pullback of the metric tensor on a Riemannian manifold. For formula_2 a field over a spacetime formula_3, this may be written as


where the formula_5 is the metric tensor on the field space formula_6, and the formula_7 are the derivatives on the underlying spacetime manifold.


This expression can be unpacked a bit. The field space formula_8 can be chosen to be any Riemannian manifold. Historically, this is the "sigma" of the sigma model; the historically-appropriate symbol formula_9 is avoided here to prevent clashes with many other common usages of formula_9 in geometry. Riemannian manifolds always come with a metric tensor formula_11. Given an atlas of charts on formula_8, the field space can always be locally trivialized, in that given formula_13 in the atlas, one may write a map formula_14 giving explicit local coordinates formula_15 on that patch. The metric tensor on that patch is a matrix having components formula_16


The base manifold formula_3 must be a differentiable manifold; by convention, it is either Minkowski space in particle physics applications, flat two-dimensional Euclidean space for condensed matter applications, or a Riemann surface, the worldsheet in string theory. The formula_18 is just the plain-old covariant derivative on the base spacetime manifold formula_19 When formula_3 is flat, formula_21 is just the ordinary gradient of a scalar function (as formula_22 is a scalar field, from the point of view of formula_3 itself.) In more precise language, formula_24 is a section of the jet bundle of formula_25.


Taking formula_26 the Kronecker delta, "i.e." the scalar dot product in Euclidean space, one gets the formula_27 non-linear sigma model. That is, write formula_28 to be the unit vector in formula_29, so that formula_30, with formula_31 the ordinary Euclidean dot product. Then formula_32 the formula_33-sphere, the isometries of which are the rotation group formula_27. The Lagrangian can then be written as


For formula_36, this is the continuum limit of the isotropic ferromagnet on a lattice, i.e. of the classical Heisenberg model. For formula_37, this is the continuum limit of the classical XY model. See also the n-vector model and the Potts model for reviews of the lattice model equivalents. The continuum limit is taken by writing


as the finite difference on neighboring lattice locations formula_39 Then formula_40 in the limit formula_41, and formula_42 after dropping the constant terms formula_43 (the "bulk magnetization").


The sigma model can also be written in a more fully geometric notation, as a fiber bundle with fibers formula_8 over a differentiable manifold formula_3. Given a section formula_2, fix a point formula_47 The pushforward at formula_48 is a map of tangent bundles


where formula_51 is taken to be an orthonormal vector space basis on formula_52 and formula_53 the vector space basis on formula_54. The formula_55 is a differential form. The sigma model action is then just the conventional inner product on vector-valued "k"-forms


where the formula_57 is the wedge product, and the formula_58 is the Hodge star. This is an inner product in two different ways. In the first way, given "any" two differentiable forms formula_59 in formula_3, the Hodge dual defines an invariant inner product on the space of differential forms, commonly written as


The above is an inner product on the space of square-integrable forms, conventionally taken to be the Sobolev space formula_62 In this way, one may write


This makes it explicit and plainly evident that the sigma model is just the kinetic energy of a point particle. From the point of view of the manifold formula_3, the field formula_22 is a scalar, and so formula_55 can be recognized just the ordinary gradient of a scalar function. The Hodge star is merely a fancy device for keeping track of the volume form when integrating on curved spacetime. In the case that formula_3 is flat, it can be completely ignored, and so the action is


which is the Dirichlet energy of formula_22. Classical extrema of the action (the solutions to the Lagrange equations) are then those field configurations that minimize the Dirichlet energy of formula_22. Another way to convert this expression into a more easily-recognizable form is to observe that, for a scalar function formula_71 one has formula_72 and so one may also write


where formula_74 is the Laplace–Beltrami operator, "i.e." the ordinary Laplacian when formula_3 is flat.


That there is "another", second inner product in play simply requires not forgetting that formula_55 is a vector from the point of view of formula_8 itself. That is, given "any" two vectors formula_78, the Riemannian metric formula_79 defines an inner product


Since formula_55 is vector-valued formula_82 on local charts, one also takes the inner product there as well. More verbosely,


The tension between these two inner products can be made even more explicit by noting that


is a bilinear form; it is a pullback of the Riemann metric formula_79. The individual formula_86 can be taken as vielbeins. The Lagrangian density of the sigma model is then


for formula_88 the metric on formula_19 Given this gluing-together, the formula_55 can be interpreted as a solder form; this is articulated more fully, below.


Several interpretational and foundational remarks can be made about the classical (non-quantized) sigma model. The first of these is that the classical sigma model can be interpreted as a model of non-interacting quantum mechanics. The second concerns the interpretation of energy.


given above. Taking formula_92, the function formula_93 can be interpreted as a wave function, and its Laplacian the kinetic energy of that wave function. The formula_94 is just some geometric machinery reminding one to integrate over all space. The corresponding quantum mechanical notation is formula_95 In flat space, the Laplacian is conventionally written as formula_96. Assembling all these pieces together, the sigma model action is equivalent to


which is just the grand-total kinetic energy of the wave-function formula_98, up to a factor of formula_99. To conclude, the classical sigma model on formula_100 can be interpreted as the quantum mechanics of a free, non-interacting quantum particle. Obviously, adding a term of formula_101 to the Lagrangian results in the quantum mechanics of a wave-function in a potential. Taking formula_102 is not enough to describe the formula_33-particle system, in that formula_33 particles require formula_33 distinct coordinates, which are not provided by the base manifold. This can be solved by taking formula_33 copies of the base manifold.


It is very well-known that the geodesic structure of a Riemannian manifold is described by the Hamilton–Jacobi equations. In thumbnail form, the construction is as follows. "Both" formula_3 and formula_8 are Riemannian manifolds; the below is written for formula_8, the same can be done for formula_3. The cotangent bundle formula_111, supplied with coordinate charts, can always be locally trivialized, "i.e."


The trivialization supplies canonical coordinates formula_113 on the cotangent bundle. Given the metric tensor formula_79 on formula_8, define the Hamiltonian function


where, as always, one is careful to note that the inverse of the mertric is used in this definition: formula_117 Famously, the geodesic flow on formula_8 is given by the Hamilton–Jacobi equations


The geodesic flow is the Hamiltonian flow; the solutions to the above are the geodesics of the manifold. Note, incidentally, that formula_121 along geodesics; the time parameter formula_122 is the distance along the geodesic.


The sigma model takes the momenta in the two manifolds formula_111 and formula_124 and solders them together, in that formula_55 is a solder form. In this sense, the interpretation of the sigma model as an energy functional is not surprising; it is in fact the gluing together of "two" energy functionals. Caution: the precise definition of a solder form requires it to be an isomorphism; this can only happen if formula_3 and formula_8 have the same real dimension. Furthermore, the conventional definition of a solder form takes formula_8 to be a Lie group. Both conditions are satisfied in various applications.


The space formula_8 is often taken to be a Lie group, usually SU(N), in the conventional particle physics models, O(N) in condensed matter theories, or as a symmetric space in supergravity models. Since symmetric spaces are defined in terms of their involution, their tangent space (i.e. the place where formula_55 lives) naturally splits into even and odd parity subspaces. This splitting helps propel the dimensional reduction of Kaluza–Klein theories.


For the special case of formula_8 being a Lie group, the formula_79 is the metric tensor on the Lie group, formally called the Cartan tensor or the Killing form. The Lagrangian can then be written as the pullback of the Killing form. Note that the Killing form can be written as a trace over two matrices from the corresponding Lie algebra; thus, the Lagrangian can also be written in a form involving the trace. With slight re-arrangements, it can also be written as the pullback of the Maurer-Cartan form.


A common variation of the sigma model is to present it on a symmetric space. The prototypical example is the chiral model, which takes the product


of the "left" and "right" chiral fields, and then constructs the sigma model on the "diagonal"


Such a quotient space is a symmetric space, and so one can generically take formula_135 where formula_136 is the maximal subgroup of formula_137 that is invariant under the Cartan involution. The Lagrangian is still written exactly as the above, either in terms of the pullback of the metric on formula_137 to a metric on formula_139 or as a pullback of the Maurer-Cartan form.


In physics, the most common and conventional statement of the sigma model begins with the definition


Here, the formula_141 is the pullback of the Maurer-Cartan form, for formula_142, onto the spacetime manifold. The formula_143 is a projection onto the odd-parity piece of the Cartan involution. That is, given the Lie algebra formula_144 of formula_137, the involution decompses the space into odd and even parity components formula_146 corresponding to the two eigenstates of the involution. The sigma model Lagrangian can then be written as


This is instantly recognizable as the first term of the Skyrme model.


The equivalent metric form of this is to write a group element formula_142 as the geodesic formula_149 of an element formula_150 of the Lie algebra formula_144. The formula_152 are the basis elements for the Lie algebra; the formula_153 are the structure constants of formula_144.


Plugging this directly into the above and applying the infinitesimal form of the Baker–Campbell–Hausdorff formula promptly leads to the equivalent expression


where formula_156 is now obviously (proportional to) the Killing form, and the formula_157 are the vielbeins that express the "curved" metric formula_79 in terms of the "flat" metric formula_156. The article on the Baker–Campbell–Hausdorff formula provides an explicit expression for the vielbeins. They can be written as


where formula_3 is a matrix whose matrix elements are formula_162.


For the sigma model on a symmetric space, as opposed to a Lie group, the formula_163 are limited to span the subspace formula_164 instead of all of formula_146. The Lie commutator on formula_164 will "not" be within formula_164; indeed, one has formula_168 and so a projection is still needed.


The model can be extended in a variety of ways. Besides the aforementioned Skyrme model, which introduces quartic terms, the model may be augmented by a torsion term to yield the Wess–Zumino–Witten model.


Another possibility is frequently seen in supergravity models. Here, one notes that the Maurer-Cartan form formula_169 looks like "pure gauge". In the construction above for symmetric spaces, one can also consider the other projection


where, as before, the symmetric space corresponded to the split formula_171. This extra term can be interpreted as a connection on the fiber bundle formula_172 (it transforms as a gauge field). It is what is "left over" from the connection on formula_137. It can be endowed with its own dynamics, by writing


with formula_175. Note that the differential here is just "d", and not a covariant derivative; this is "not" the Yang-Mills stress-energy tensor. This term is not gauge invariant by itself; it must be taken together with the part of the connection that embeds into formula_176, so that taken together, the formula_176, now with the connection as a part of it, together with this term, forms a complete gauge invariant Lagrangian (which does have the Yang-Mills terms in it, when expanded out).


In physics, there are equations in every field to relate physical quantities to each other and perform calculations. Entire handbooks of equations can only summarize most of the full subject, else are highly specialized within a certain field. Physics is derived of formulae only.


In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined by a transition rate matrix. The equations are a set of differential equations – over time – of the probabilities that the system occupies each of the different states.


A master equation is a phenomenological set of first-order differential equations describing the time evolution of (usually) the probability of a system to occupy each one of a discrete set of states with regard to a continuous time variable "t". The most familiar form of a master equation is a matrix form:


where formula_2 is a column vector (where element "i" represents state "i"), and formula_3 is the matrix of connections. The way connections among states are made determines the dimension of the problem; it is either


When the connections are time-independent rate constants, the master equation represents a kinetic scheme, and the process is Markovian (any jumping time probability density function for state "i" is an exponential, with a rate equal to the value of the connection). When the connections depend on the actual time (i.e. matrix formula_3 depends on the time, formula_5 ), the process is not stationary and the master equation reads


When the connections represent multi exponential jumping time probability density functions, the process is semi-Markovian, and the equation of motion is an integro-differential equation termed the generalized master equation:


The matrix formula_3 can also represent birth and death, meaning that probability is injected (birth) or taken from (death) the system, where then, the process is not in equilibrium.


Detailed description of the matrix and properties of the system.


Let formula_3 be the matrix describing the transition rates (also known as kinetic rates or reaction rates). As always, the first subscript represents the row, the second subscript the column. That is, the source is given by the second subscript, and the destination by the first subscript. This is the opposite of what one might expect, but it is technically convenient.


For each state "k", the increase in occupation probability depends on the contribution from all other states to "k", and is given by:


where formula_11 is the probability for the system to be in the state formula_12, while the matrix formula_3 is filled with a grid of transition-rate constants. Similarly, formula_14 contributes to the occupation of all other states formula_15


In probability theory, this identifies the evolution as a continuous-time Markov process, with the integrated master equation obeying a Chapman–Kolmogorov equation.


The master equation can be simplified so that the terms with "ℓ" = "k" do not appear in the summation. This allows calculations even if the main diagonal of the formula_3 is not defined or has been assigned an arbitrary value.


The final equality arises from the fact that


because the summation over the probabilities formula_20 yields one, a constant function. Since this has to hold for any probability formula_2 (and in particular for any probability of the form formula_22 for some k) we get


Using this we can write the diagonal elements as


The master equation exhibits detailed balance if each of the terms of the summation disappears separately at equilibrium—i.e. if, for all states "k" and "ℓ" having equilibrium probabilities formula_25 and formula_26,


These symmetry relations were proved on the basis of the time reversibility of microscopic dynamics (microscopic reversibility) as Onsager reciprocal relations.


Many physical problems in classical, quantum mechanics and problems in other sciences, can be reduced to the form of a "master equation", thereby performing a great simplification of the problem (see mathematical model).


The Lindblad equation in quantum mechanics is a generalization of the master equation describing the time evolution of a density matrix. Though the Lindblad equation is often referred to as a "master equation", it is not one in the usual sense, as it governs not only the time evolution of probabilities (diagonal elements of the density matrix), but also of variables containing information about quantum coherence between the states of the system (non-diagonal elements of the density matrix).


Another special case of the master equation is the Fokker–Planck equation which describes the time evolution of a continuous probability distribution. Complicated master equations which resist analytic treatment can be cast into this form (under various approximations), by using approximation techniques such as the system size expansion.


Stochastic chemical kinetics are yet another example of the Master equation. A chemical Master equation is used to model a set of chemical reactions when the number of molecules of one or more species is small (of the order of 100 or 1000 molecules).


A quantum master equation is a generalization of the idea of a master equation. Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements. A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.


The Redfield equation and Lindblad equation are examples of approximate quantum master equations assumed to be Markovian. More accurate quantum master equations for certain applications include the polaron transformed quantum master equation, and the VPQME (variational polaron transformed quantum master equation).