Inhomogeneous cosmology

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

An inhomogeneous cosmology is a physical cosmological theory (an astronomical model of the physical universe's origin and evolution) which, unlike the dominant cosmological concordance model, postulates that inhomogeneities in the distribution of matter across the universe affect local gravitational forces (i.e., at the galactic level) enough to skew our view of the Universe. When the universe began, matter was distributed homogeneously, but over billions of years, galaxies, clusters of galaxies, and superclusters coalesced. Einstein's theory of general relativity states that they warp the space-time around them.

While the concordance model acknowledges this fact, it assumes that such inhomogeneities are not sufficient to affect large-scale averages of gravity observations. Two studies claimed in 1998-1999 that high redshift supernovae were further away than the distance predicted by calculations. It was suggested that the expansion of the universe was accelerating, and dark energy, a repulsive energy inherent in space, was proposed as an explanation. Dark energy became widely accepted, but remains unexplained. Inhomogeneous cosmology falls into the class of models that might not require dark energy.

Inhomogeneous cosmologies assume that the backreactions of denser structures and those of empty voids on space-time are significant. When not neglected, they distort understanding of time and observations of distant objects. Burchert's equations in 1997 and 2000 derive from general relativity, but allow for the inclusion of local gravitational variations. Alternative models were proposed under which the acceleration of the universe was a misinterpretation of astronomical observations and in which dark energy is unnecessary. For example, in 2007, David Wiltshire proposed a model (timescape cosmology) in which backreactions caused time to run more slowly or, in voids, more quickly, thus leading supernovae observed in 1998 to be thought to be further away than they were. Timescape cosmology may also imply that the expansion of the universe is in fact slowing.

History

Standard cosmological model

Main article: Lambda-CDM model

The conflict between the two cosmologies derives from the inflexibility of Einstein's theory of general relativity, which shows how gravity is formed by the interaction of matter, space, and time. Physicist John Wheeler famously summed up the theory's essence as "Matter tells space how to curve; space tells matter how to move." However, in order to build a workable cosmological model, all of the terms on both sides of Einstein's equations must be balanced: on one side, matter (i.e., all the things that warp time and space); on the other, the curvature of the universe and the speed at which space-time is expanding. In short, a model requires a particular amount of matter in order to produce particular curvatures and expansion rates.

In terms of matter, all modern cosmologies are founded on the cosmological principle, which states that whichever direction we look from Earth, the universe is basically the same: homogeneous and isotropic (uniform in all dimensions). This principle grew out of Copernicus's assertion that there were no special observers in the universe and nothing special about the Earth's location in the universe (i.e., Earth was not the center of the universe, as previously thought). Since the publication of general relativity in 1915, this homogeneity and isotropy have greatly simplified the process of devising cosmological models.

Possible shapes of the universe

In terms of the curvature of space-time and the shape of the universe, it can theoretically be closed (positive curvature, or space-time folding in itself as though on a four-dimensional sphere's surface), open (negative curvature, with space-time folding outward), or flat (zero curvature, like the surface of a "flat" four-dimensional piece of paper).

The first real difficulty came with regards to expansion, for in 1915, as previously, the universe was assumed to be static, neither expanding nor contracting. All of Einstein's solutions to his equations in general relativity, however, predicted a dynamic universe. Therefore, in order to make his equations consistent with the apparently static universe, he added a cosmological constant, a term representing some unexplained extra energy. But when in the late 1920s Georges Lemaître's and Edwin Hubble's observations proved Alexander Friedmann's notion (derived from general relativity) that the universe was expanding, the cosmological constant became unnecessary, Einstein calling it "my greatest blunder."

With this term gone from the equation, others derived the Friedmann-Lemaître–Robertson–Walker (FLRW) solution to describe such an expanding universe — a solution built on the assumption of a flat, isotropic, homogeneous universe. The FLRW model became the foundation of the standard model of a universe created by the Big Bang, and further observational evidence has helped to refine it. For example, a smooth, mostly homogeneous, and (at least when it was almost 400,000 years old) flat universe seemed to be confirmed by data from the cosmic microwave background (CMB). And after galaxies and clusters of galaxies were found in the 1970s, mainly by Vera Rubin, to be rotating faster than they should without flying apart, the existence of dark matter seemed also proven, confirming its inference by Jacobus Kapteyn, Jan Oort, and Fritz Zwicky in the 1920s and 1930s and demonstrating the flexibility of the standard model. Dark matter is believed to make up roughly 23% of the energy density of the universe.

Dark energy

Main article: Dark energy

Timeline of the universe according to the CMB

Another observation in 1998 seemed to complicate the situation further: two separate studies found distant supernovae to be fainter than expected in a steadily expanding universe; that is, they were not merely moving away from the earth but accelerating. The universe's expansion was calculated to have been accelerating since approximately 5 billion years ago. Given the gravitation braking effect that all the matter of the universe should have had on this expansion, a variation of Einstein's cosmological constant was reintroduced to represent an energy inherent in space, balancing the equations for a flat, accelerating universe. It also gave Einstein's cosmological constant new meaning, for by reintroducing it into the equation to represent dark energy, a flat universe expanding ever faster can be reproduced.

Although the nature of this energy has yet to be adequately explained, it makes up almost 70% of the energy density of the universe in the concordance model. And thus, when including dark matter, almost 95% of the universe's energy density is explained by phenomena that have been inferred but not entirely explained nor directly observed. Most cosmologists still accept the concordance model, although science journalist Anil Ananthaswamy calls this agreement a "wobbly orthodoxy."

Inhomogeneous universe

All-sky mollweide map of the CMB, created from 9 years of WMAP data. Tiny residual variations are visible, but they show a very specific pattern consistent with a hot gas that is mostly uniformly distributed.

While the universe began with homogeneously distributed matter, enormous structures have since coalesced over billions of years: hundreds of billions of stars inside of galaxies, clusters of galaxies, superclusters, and vast filaments of matter. These denser regions and the voids between them must, under general relativity, have some effect, as matter dictates how space-time curves. So the extra mass of galaxies and galaxy clusters (and dark matter, should particles of it ever be directly detected) must cause nearby space-time to curve more positively, and voids should have the opposite effect, causing space-time around them to take on negative curvatures. The question is whether these effects, called backreactions, are negligible or together comprise enough to change the universe's geometry. Most scientists have assumed that they are negligible, but this has partly been because there has been no way to average space-time geometry in Einstein's equations.

In 2000, a set of new equations—now referred to as the set of Buchert equations—based on general relativity was published by cosmologist Thomas Buchert of the École Normale Supérieure in Lyon, France, which allow the effects of a non-uniform distribution of matter to be taken into account but still allow the behavior of the universe to be averaged. Thus, models based on a lumpy, inhomogeneous distribution of matter could now be devised. "There is no dark energy, as far as I'm concerned," Buchert told New Scientist in 2016. "In ten years' time, dark energy is gone." In the same article, cosmologist Syksy Räsänen said, "It’s not been established beyond reasonable doubt that dark energy exists. But I’d never say that it has been established that dark energy does not exist." He also told the magazine that the question of whether backreactions are negligible in cosmology "has not been satisfactorily answered."

Inhomogeneous cosmology

Inhomogeneous cosmology in the most general sense (assuming a totally inhomogeneous universe) is modeling the universe as a whole with the spacetime which does not possess any spacetime symmetries. Typically considered cosmological spacetimes have either the maximal symmetry, which comprises three translational symmetries and three rotational symmetries (homogeneity and isotropy with respect to every point of spacetime), the translational symmetry only (homogeneous models), or the rotational symmetry only (spherically symmetric models). Models with fewer symmetries (e.g. axisymmetric) are also considered as symmetric. However, it is common to call spherically symmetric models or non-homogeneous models as inhomogeneous. In inhomogeneous cosmology, the large-scale structure of the universe is modeled by exact solutions of the Einstein field equations (i.e. non-perturbatively), unlike cosmological perturbation theory, which is study of the universe that takes structure formation (galaxies, galaxy clusters, the cosmic web) into account but in a perturbative way.

Inhomogeneous cosmology usually includes the study of structure in the Universe by means of exact solutions of Einstein's field equations (i.e. metrics) or by spatial or spacetime averaging methods. Such models are not homogeneous, but may allow effects which can be interpreted as dark energy, or can lead to cosmological structures such as voids or galaxy clusters.

Perturbative approach

Perturbation theory, which deals with small perturbations from e.g. a homogeneous metric, only holds as long as the perturbations are not too large, and N-body simulations use Newtonian gravity which is only a good approximation when speeds are low and gravitational fields are weak.

Non-perturbative approach

Work towards a non-perturbative approach includes the Relativistic Zel'dovich Approximation. As of 2016, Thomas Buchert, George Ellis, Edward Kolb, and their colleagues judged that if the universe is described by cosmic variables in a backreaction scheme that includes coarse-graining and averaging, then whether dark energy is an artifact of the traditional way of using the Einstein equation remains an unanswered question.

Exact solutions

The first historical examples of inhomogeneous (though spherically symmetric) solutions are the Lemaître–Tolman metric (or LTB model - Lemaître–Tolman-Bondi). The Stephani metric can be spherically symmetric or totally inhomogeneous. Other examples are the Szekeres metric, Szafron metric, Barnes metric, Kustaanheimo-Qvist metric, and Senovilla metric. The Bianchi metrics as given in the Bianchi classification and Kantowski-Sachs metrics are homogeneous.

Averaging methods

The simplest averaging approach is the scalar averaging approach, leading to the kinematical backreaction and mean 3-Ricci curvature functionals. Buchert's equations are the most commonly used equations of such averaging methods. The simplest averaging kernels include spheres (cylinders, when viewed with a time component), Gaussians, and hard-momentum cutoffs. The former work well for non-relativistic fluids (dust); the later are more convenient for relativistic fluid calculations (photons and pre-recombination universes).

Timescape cosmology

In 2007, David L Wiltshire, a professor of theoretical physics at the University of Canterbury in New Zealand, argued in the New Journal of Physics that quasilocal variations in gravitational energy had in 1998 given the false conclusion that the expansion of the universe is accelerating. Moreover, due to the equivalence principle, which holds that gravitational and inertial energy are equivalent and thus prevents aspects of gravitational energy from being differentiated at a local level, scientists thus misidentified these aspects as dark energy. This misidentification was the result of presuming an essentially homogeneous universe, as the standard cosmological model does, and not accounting for temporal differences between matter-dense areas and voids. Wiltshire and others argued that if the universe is not only assumed not to be homogeneous but also not flat, models could be devised in which the apparent acceleration of the universe's expansion could be explained otherwise.

One more important step being left out of the standard model, Wiltshire claimed, was the fact that as proven by observation, gravity slows time. Thus, from the perspective of the same observer, a clock will move faster in empty space, which possesses low gravitation, than inside a galaxy, which has much more gravity, and he argued that as large as a 38% difference between the time on clocks in the Milky Way galaxy and those floating in a void exists. Thus, unless we can correct for that—timescapes each with different times—our observations of the expansion of space will be, and are, incorrect. Wiltshire claims that the 1998 supernovae observations that led to the conclusion of an expanding universe and dark energy can instead be explained by Buchert's equations if certain strange aspects of general relativity are taken into account. The arguments of Wiltshire have been contested by Ethan Siegel.

Observational evidence

A 2024 study examining the Pantheon+ Type Ia Supernova dataset conducted a significant test of the Timescape cosmology. By employing a model-independent statistical approach, the researchers found that the Timescape model could account for the observed cosmic acceleration without the need for dark energy. This result suggested that inhomogeneous cosmological models may offer viable alternatives to the standard ΛCDM framework and warranted further exploration to assess their ability to explain other key cosmological phenomena.

Invariant (mathematics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Invariant_(mathematics)

A wallpaper is invariant under some transformations. This one is invariant under horizontal and vertical translation, as well as rotation by 180° (but not under reflection).

In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.

Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.

Examples

A simple example of invariance is expressed in our ability to count. For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in which we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting.

An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change.

The distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand, multiplication does not have this same property, as distance is not invariant under multiplication.

Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which is the basis of trigonometry. In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. As another example, all circles are similar: they can be transformed into each other and the ratio of the circumference to the diameter is invariant (denoted by the Greek letter π (pi)).

Some more complicated examples:

• The real part and the absolute value of a complex number are invariant under complex conjugation.
• The tricolorability of knots.
• The degree of a polynomial is invariant under a linear change of variables.
• The dimension and homology groups of a topological object are invariant under homeomorphism.
• The number of fixed points of a dynamical system is invariant under many mathematical operations.
• Euclidean distance is invariant under orthogonal transformations.
• Area is invariant under linear maps which have determinant ±1 (see Equiareal map § Linear transformations).
• Some invariants of projective transformations include collinearity of three or more points, concurrency of three or more lines, conic sections, and the cross-ratio.^[6]
• The determinant, trace, eigenvectors, and eigenvalues of a linear endomorphism are invariant under a change of basis. In other words, the spectrum of a matrix is invariant under a change of basis.
• The principal invariants of tensors do not change with rotation of the coordinate system (see Invariants of tensors).
• The singular values of a matrix are invariant under orthogonal transformations.
• Lebesgue measure is invariant under translations.
• The variance of a probability distribution is invariant under translations of the real line. Hence the variance of a random variable is unchanged after the addition of a constant.
• The fixed points of a transformation are the elements in the domain that are invariant under the transformation. They may, depending on the application, be called symmetric with respect to that transformation. For example, objects with translational symmetry are invariant under certain translations.
• The integral ${\int }_{M}Kd\mu$ of the Gaussian curvature ${\displaystyle K}$ of a two-dimensional Riemannian manifold ${\displaystyle (M,g)}$ is invariant under changes of the Riemannian metric ${\displaystyle g}$. This is the Gauss–Bonnet theorem.

MU puzzle

The MU puzzle is a good example of a logical problem where determining an invariant is of use for an impossibility proof. The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:

1. If a string ends with an I, a U may be appended (xI → xIU)
2. The string after the M may be completely duplicated (Mx → Mxx)
3. Any three consecutive I's (III) may be replaced with a single U (xIIIy → xUy)
4. Any two consecutive U's may be removed (xUUy → xy)

An example derivation (with superscripts indicating the applied rules) is

MI →² MII →² MIIII →³ MUI →² MUIUI →¹ MUIUIU →² MUIUIUUIUIU →⁴ MUIUIIUIU → ...

In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a property that is invariant to all rules (that is, not changed by any of them), and that demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider:

The number of I's in the string is not a multiple of 3.

This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules:

Rule	#I's	#U's	Effect on invariant
1	+0	+1	Number of I's is unchanged. If the invariant held, it still does.
2	×2	×2	If n is not a multiple of 3, then 2×n is not either. The invariant still holds.
3	−3	+1	If n is not a multiple of 3, n−3 is not either. The invariant still holds.
4	+0	−2	Number of I's is unchanged. If the invariant held, it still does.

The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either.

Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).

Invariant set

A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when ${\displaystyle x\in S⟺T(x)\in S.}$ The elements of S are not necessarily fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant, vs. pointwise invariant, to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under a homothety of space.

An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group. In linear algebra, if a linear transformation T has an eigenvector v, then the line through 0 and v is an invariant set under T, in which case the eigenvectors span an invariant subspace which is stable under T.

When T is a screw displacement, the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points.

In probability theory and ergodic theory, invariant sets are usually defined via the stronger property ${\displaystyle x\in S\iff T(x)\in S.}$ When the map ${\displaystyle T}$ is measurable, invariant sets form a sigma-algebra, the invariant sigma-algebra.

Formal statement

The notion of invariance is formalized in three different ways in mathematics: via group actions, presentations, and deformation.

Unchanged under group action

Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group.

Frequently one will have a group acting on a set X, which leaves one to determine which objects in an associated set F(X) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L(P); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.

More importantly, one may define a function on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions.

Dual to the notion of invariants are coinvariants, also known as orbits, which formalizes the notion of congruence: objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the perimeter of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.

These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In classification problems, one might seek to find a complete set of invariants, such that if two objects have the same values for this set of invariants, then they are congruent.

For example, triangles such that all three sides are equal are congruent under rigid motions, via SSS congruence, and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the AAA similarity criterion shows that this is a complete set of invariants.

Independent of presentation

Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying topological space (the manifold) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.

The most common examples are:

• The presentation of a manifold in terms of coordinate charts – invariants must be unchanged under change of coordinates.
• Various manifold decompositions, as discussed for Euler characteristic.
• Invariants of a presentation of a group.

Unchanged under perturbation

Thirdly, if one is studying an object which varies in a family, as is common in algebraic geometry and differential geometry, one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric).

Invariants in computer science

For other uses of the word "invariant" in computer science, see invariant (disambiguation).

In computer science, an invariant is a logical assertion that is always held to be true during a certain phase of execution of a computer program. For example, a loop invariant is a condition that is true at the beginning and the end of every iteration of a loop.

Invariants are especially useful when reasoning about the correctness of a computer program. The theory of optimizing compilers, the methodology of design by contract, and formal methods for determining program correctness, all rely heavily on invariants.

Programmers often use assertions in their code to make invariants explicit. Some object oriented programming languages have a special syntax for specifying class invariants.

Automatic invariant detection in imperative programs

Abstract interpretation tools can compute simple invariants of given imperative computer programs. The kind of properties that can be found depend on the abstract domains used. Typical example properties are single integer variable ranges like 0<=x<1024, relations between several variables like 0<=i-j<2*n-1, and modulus information like y%4==0. Academic research prototypes also consider simple properties of pointer structures.

More sophisticated invariants generally have to be provided manually. In particular, when verifying an imperative program using the Hoare calculus, a loop invariant has to be provided manually for each loop in the program, which is one of the reasons that this approach is generally impractical for most programs.

In the context of the above MU puzzle example, there is currently no general automated tool that can detect that a derivation from MI to MU is impossible using only the rules 1–4. However, once the abstraction from the string to the number of its "I"s has been made by hand, leading, for example, to the following C program, an abstract interpretation tool will be able to detect that ICount%3 cannot be 0, and hence the "while"-loop will never terminate.

void MUPuzzle(void) {
    volatile int RandomRule;
    int ICount = 1, UCount = 0;
    while (ICount % 3 != 0)                         // non-terminating loop
        switch(RandomRule) {
        case 1:                  UCount += 1;   break;
        case 2:   ICount *= 2;   UCount *= 2;   break;
        case 3:   ICount -= 3;   UCount += 1;   break;
        case 4:                  UCount -= 2;   break;
        }                                          // computed invariant: ICount % 3 == 1 || ICount % 3 == 2
}

Symmetry in quantum mechanics

From Wikipedia, the free encyclopedia

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

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.

This article outlines the connection between the classical form of continuous symmetries as well as their quantum operators, and relates them to the Lie groups, and relativistic transformations in the Lorentz group and Poincaré group.

Notation

The notational conventions used in this article are as follows. Boldface indicates vectors, four vectors, matrices, and vectorial operators, while quantum states use bra–ket notation. Wide hats are for operators, narrow hats are for unit vectors (including their components in tensor index notation). The summation convention on the repeated tensor indices is used, unless stated otherwise. The Minkowski metric signature is (+−−−).

Symmetry transformations on the wavefunction in non-relativistic quantum mechanics

Continuous symmetries

Generally, the correspondence between continuous symmetries and conservation laws is given by Noether's theorem.

The form of the fundamental quantum operators, for example the energy operator as a partial time derivative and momentum operator as a spatial gradient, becomes clear when one considers the initial state, then changes one parameter of it slightly. This can be done for displacements (lengths), durations (time), and angles (rotations). Additionally, the invariance of certain quantities can be seen by making such changes in lengths and angles, illustrating conservation of these quantities.

In what follows, transformations on only one-particle wavefunctions in the form:

$${\displaystyle \hat{\mathrm{\Omega }}\psi (\mathbf{r},t)=\psi ({\mathbf{r}}^{\prime },t^{\prime })}$$

are considered, where ${\displaystyle \hat{\mathrm{\Omega }}}$ denotes a unitary operator. Unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state (representing the total probability of finding the particle somewhere with some spin) must be invariant under these transformations. The inverse is the Hermitian conjugate ${\displaystyle {\hat{\mathrm{\Omega }}}^{-1}={\hat{\mathrm{\Omega }}}^{†}}$. The results can be extended to many-particle wavefunctions. Written in Dirac notation as standard, the transformations on quantum state vectors are:

$${\displaystyle \hat{\mathrm{\Omega }}|\mathbf{r}(t)⟩=|{\mathbf{r}}^{\prime }(t^{\prime })⟩}$$

Now, the action of ${\displaystyle \hat{\mathrm{\Omega }}}$ changes ψ(r, t) to ψ(r′, t′), so the inverse ${\displaystyle {\hat{\mathrm{\Omega }}}^{-1}={\hat{\mathrm{\Omega }}}^{†}}$ changes ψ(r′, t′) back to ψ(r, t). Thus, an operator ${\displaystyle \hat{A}}$ invariant under ${\displaystyle \hat{\mathrm{\Omega }}}$ satisfies:

$${\displaystyle \hat{A}\psi ={\hat{\mathrm{\Omega }}}^{†}\hat{A}\hat{\mathrm{\Omega }}\psi \Rightarrow \hat{\mathrm{\Omega }}\hat{A}\psi =\hat{A}\hat{\mathrm{\Omega }}\psi .}$$

Concomitantly,

$${\displaystyle [\hat{\mathrm{\Omega }},\hat{A}]\psi =0}$$

for any state ψ. Quantum operators representing observables are also required to be Hermitian so that their eigenvalues are real numbers, i.e. the operator equals its Hermitian conjugate, ${\displaystyle \hat{A}={\hat{A}}^{†}}$.

Overview of Lie group theory

For a full exposition and details, see Lie group and Generator (mathematics).

Following are the key points of group theory relevant to quantum theory, examples are given throughout the article. For an alternative approach using matrix groups, see the books of Hall

Let G be a Lie group, which is a group that locally is parameterized by a finite number N of real continuously varying parameters ξ₁, ξ₂, ..., ξ_N. In more mathematical language, this means that G is a smooth manifold that is also a group, for which the group operations are smooth.

• the dimension of the group, N, is the number of parameters it has.
• the group elements, g, in G are functions of the parameters: $${\displaystyle g=G({\xi }_1,{\xi }_2,\dots )}$$ and all parameters set to zero returns the identity element of the group: $${\displaystyle I=G(0,0,\dots )}$$ Group elements are often matrices which act on vectors, or transformations acting on functions.
• The generators of the group are the partial derivatives of the group elements with respect to the group parameters with the result evaluated when the parameter is set to zero: $${\displaystyle X_j={\frac{\mathrm{\partial }g}{\mathrm{\partial }{\xi }_j}|}_{{\xi }_j=0}}$$ In the language of manifolds, the generators are the elements of the tangent space to G at the identity. The generators are also known as infinitesimal group elements or as the elements of the Lie algebra of G. (See the discussion below of the commutator.)
  One aspect of generators in theoretical physics is they can be constructed themselves as operators corresponding to symmetries, which may be written as matrices, or as differential operators. In quantum theory, for unitary representations of the group, the generators require a factor of i: $${\displaystyle X_j=i{\frac{\mathrm{\partial }g}{\mathrm{\partial }{\xi }_j}|}_{{\xi }_j=0}}$$ The generators of the group form a vector space, which means linear combinations of generators also form a generator.
• The generators (whether matrices or differential operators) satisfy the commutation relations: $${\displaystyle \left[X_a,X_b\right]=i{f}_{abc}{X}_c}$$ where f_abc are the (basis dependent) structure constants of the group. This makes, together with the vector space property, the set of all generators of a group a Lie algebra. Due to the antisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices.
• The representations of the group then describe the ways that the group G (or its Lie algebra) can act on a vector space. (The vector space might be, for example, the space of eigenvectors for a Hamiltonian having G as its symmetry group.) We denote the representations using a capital D. One can then differentiate D to obtain a representation of the Lie algebra, often also denoted by D. These two representations are related as follows: $${\displaystyle D[g({\xi }_j)]\equiv D({\xi }_j)=e^{i{\xi }_{j}D(X_j)}}$$ without summation on the repeated index j. Representations are linear operators that take in group elements and preserve the composition rule: $${\displaystyle D({\xi }_a)D({\xi }_b)=D({\xi }_a{\xi }_b).}$$

A representation which cannot be decomposed into a direct sum of other representations, is called irreducible. It is conventional to label irreducible representations by a superscripted number n in brackets, as in D⁽ⁿ⁾, or if there is more than one number, we write D^{(n, m, ...)}.

There is an additional subtlety that arises in quantum theory, where two vectors that differ by multiplication by a scalar represent the same physical state. Here, the pertinent notion of representation is a projective representation, one that only satisfies the composition law up to a scalar. In the context of quantum mechanical spin, such representations are called spinorial.

Momentum and energy as generators of translation and time evolution, and rotation

The space translation operator ${\displaystyle \hat{T}(\mathrm{\Delta }\mathbf{r})}$ acts on a wavefunction to shift the space coordinates by an infinitesimal displacement Δr. The explicit expression ${\displaystyle \hat{T}}$ can be quickly determined by a Taylor expansion of ψ(r + Δr, t) about r, then (keeping the first order term and neglecting second and higher order terms), replace the space derivatives by the momentum operator ${\displaystyle \hat{\mathbf{p}}}$. Similarly for the time translation operator acting on the time parameter, the Taylor expansion of ψ(r, t + Δt) is about t, and the time derivative replaced by the energy operator ${\displaystyle \hat{E}}$.

Name	Translation operator ${\displaystyle \hat{T}}$	Time translation/evolution operator ${\displaystyle \hat{U}}$
Action on wavefunction	${\displaystyle \hat{T}(\mathrm{\Delta }\mathbf{r})\psi (\mathbf{r},t)=\psi (\mathbf{r}+\mathrm{\Delta }\mathbf{r},t)}$	${\displaystyle \hat{U}(\mathrm{\Delta }t)\psi (\mathbf{r},t)=\psi (\mathbf{r},t+\mathrm{\Delta }t)}$
Infinitesimal operator	${\displaystyle \hat{T}(\mathrm{\Delta }\mathbf{r})=I+\frac{i}{\hslash }\mathrm{\Delta }\mathbf{r}\cdot \hat{\mathbf{p}}}$	${\displaystyle \hat{U}(\mathrm{\Delta }t)=I-\frac{i}{\hslash }\mathrm{\Delta }t\hat{E}}$
Finite operator	${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\left(I+\frac{i}{\hslash }\frac{\mathrm{\Delta }\mathbf{r}}{N}\cdot \hat{\mathbf{p}}\right)}^N=\exp \left(\frac{i}{\hslash }\mathrm{\Delta }\mathbf{r}\cdot \hat{\mathbf{p}}\right)=\hat{T}(\mathrm{\Delta }\mathbf{r})}$	${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\left(I-\frac{i}{\hslash }\frac{\mathrm{\Delta }t}{N}\cdot \hat{E}\right)}^N=\exp \left(-\frac{i}{\hslash }\mathrm{\Delta }t\hat{E}\right)=\hat{U}(\mathrm{\Delta }t)}$
Generator	Momentum operator ${\displaystyle \hat{\mathbf{p}}=-i\hslash \mathrm{\nabla }}$	Energy operator ${\displaystyle \hat{E}=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}}$

The exponential functions arise by definition as those limits, due to Euler, and can be understood physically and mathematically as follows. A net translation can be composed of many small translations, so to obtain the translation operator for a finite increment, replace Δr by Δr/N and Δt by Δt/N, where N is a positive non-zero integer. Then as N increases, the magnitude of Δr and Δt become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunction N times and taking the limit as N tends to infinity gives the finite operators.

Space and time translations commute, which means the operators and generators commute.

Commutators

Operators	Generators
${\displaystyle \left[\hat{T}({\mathbf{r}}_1),\hat{T}({\mathbf{r}}_2)\right]\psi (\mathbf{r},t)=0}$	${\displaystyle \left[{\hat{p}}_i,{\hat{p}}_j,\right]\psi (\mathbf{r},t)=0}$
${\displaystyle \left[\hat{U}(t_1),\hat{U}(t_2)\right]\psi (\mathbf{r},t)=0}$	${\displaystyle \left[\hat{E},{\hat{p}}_i,\right]\psi (\mathbf{r},t)=0}$

For a time-independent Hamiltonian, energy is conserved in time and quantum states are stationary states: the eigenstates of the Hamiltonian are the energy eigenvalues E:

$${\displaystyle \hat{U}(t)=\exp \left(-\frac{i\mathrm{\Delta }tE}{\hslash }\right)}$$

and all stationary states have the form

$${\displaystyle \psi (\mathbf{r},t+t_0)=\hat{U}(t-t_0)\psi (\mathbf{r},t_0)}$$

where t₀ is the initial time, usually set to zero since there is no loss of continuity when the initial time is set.

An alternative notation is ${\displaystyle \hat{U}(t-t_0)\equiv U(t,t_0)}$.

Angular momentum as the generator of rotations

Orbital angular momentum

The rotation operator, ${\displaystyle \hat{R}}$, acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle Δθ:

$${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,\hat{\mathbf{a}})\psi (\mathbf{r},t)=\psi ({\mathbf{r}}^{\prime },t)}$$

where r′ are the rotated coordinates about an axis defined by a unit vector ${\displaystyle \hat{\mathbf{a}}=(a_1,a_2,a_3)}$ through an angular increment Δθ, given by:

$${\displaystyle {\mathbf{r}}^{\prime }=\hat{R}(\mathrm{\Delta }\theta ,\hat{\mathbf{a}})\mathbf{r}.}$$

where ${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,\hat{\mathbf{a}})}$ is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis ${\displaystyle \mathrm{\Delta }\theta \hat{\mathbf{a}}=\mathrm{\Delta }\theta (a_1,a_2,a_3)}$ are the parameters, of the three-dimensional special orthogonal group, SO(3). The rotation matrices about the standard Cartesian basis vector ${\displaystyle {\hat{\mathbf{e}}}_x,{\hat{\mathbf{e}}}_y,{\hat{\mathbf{e}}}_z}$ through angle Δθ, and the corresponding generators of rotations J = (J_x, J_y, J_z), are:

${\displaystyle {\hat{R}}_x\equiv \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_x)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \mathrm{\Delta }\theta & -\sin \mathrm{\Delta }\theta \\ 0& \sin \mathrm{\Delta }\theta & \cos \mathrm{\Delta }\theta \end{array}\right),}$	${\displaystyle J_x\equiv J_1=i{\frac{\mathrm{\partial }\hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_x)}{\mathrm{\partial }\mathrm{\Delta }\theta }|}_{\mathrm{\Delta }\theta =0}=i\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right),}$
${\displaystyle {\hat{R}}_y\equiv \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_y)=\left(\begin{array}{ccc}\cos \mathrm{\Delta }\theta & 0& \sin \mathrm{\Delta }\theta \\ 0& 1& 0\\ -\sin \mathrm{\Delta }\theta & 0& \cos \mathrm{\Delta }\theta \end{array}\right),}$	${\displaystyle J_y\equiv J_2=i{\frac{\mathrm{\partial }\hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_y)}{\mathrm{\partial }\mathrm{\Delta }\theta }|}_{\mathrm{\Delta }\theta =0}=i\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right),}$
${\displaystyle {\hat{R}}_z\equiv \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_z)=\left(\begin{array}{ccc}\cos \mathrm{\Delta }\theta & -\sin \mathrm{\Delta }\theta & 0\\ \sin \mathrm{\Delta }\theta & \cos \mathrm{\Delta }\theta & 0\\ 0& 0& 1\end{array}\right),}$	${\displaystyle J_z\equiv J_3=i{\frac{\mathrm{\partial }\hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_z)}{\mathrm{\partial }\mathrm{\Delta }\theta }|}_{\mathrm{\Delta }\theta =0}=i\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right).}$

More generally for rotations about an axis defined by ${\displaystyle \hat{\mathbf{a}}}$, the rotation matrix elements are:

$${\displaystyle [\hat{R}(\theta ,\hat{\mathbf{a}}){]}_{ij}=({\delta }_{ij}-a_i{a}_j)\cos \theta -{\epsilon }_{ijk}{a}_k\sin \theta +a_i{a}_j}$$

where δ_ij is the Kronecker delta, and ε_ijk is the Levi-Civita symbol.

It is not as obvious how to determine the rotational operator compared to space and time translations. We may consider a special case (rotations about the x, y, or z-axis) then infer the general result, or use the general rotation matrix directly and tensor index notation with δ_ij and ε_ijk. To derive the infinitesimal rotation operator, which corresponds to small Δθ, we use the small angle approximations sin(Δθ) ≈ Δθ and cos(Δθ) ≈ 1, then Taylor expand about r or r_i, keep the first order term, and substitute the angular momentum operator components.

	Rotation about ${\displaystyle {\hat{\mathbf{e}}}_z}$	Rotation about ${\displaystyle \hat{\mathbf{a}}}$
Action on wavefunction	${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_z)\psi (x,y,z,t)=\psi (x-\mathrm{\Delta }\theta y,\mathrm{\Delta }\theta x+y,z,t)}$	${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,\hat{\mathbf{a}})\psi (r_i,t)=\psi (R_{ij}{r}_j,t)=\psi (r_i-{\epsilon }_{ijk}{a}_k\mathrm{\Delta }\theta r_j,t)}$
Infinitesimal operator	${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_z)=I-\frac{i}{\hslash }\mathrm{\Delta }\theta {\hat{L}}_z}$	${\displaystyle \begin{array}{rl}\hat{R}(\mathrm{\Delta }\theta ,\hat{\mathbf{a}})& =I-(-\mathrm{\Delta }\theta a_k{\epsilon }_{kij}{r}_j)\frac{\mathrm{\partial }}{\mathrm{\partial }{r}_i}\\ & =I-(\mathrm{\Delta }\theta a_k{\epsilon }_{kji}{r}_j)\frac{\mathrm{\partial }}{\mathrm{\partial }{r}_i}\\ & =I-\mathrm{\Delta }\theta \hat{\mathbf{a}}\cdot (\mathbf{r}\times \mathrm{\nabla })\\ & =I-\frac{i\mathrm{\Delta }\theta }{\hslash }\hat{\mathbf{a}}\cdot \hat{\mathbf{L}}\end{array}}$
Infinitesimal rotations	${\displaystyle \hat{R}=1-\frac{i}{\hslash }\mathrm{\Delta }\theta \hat{\mathbf{a}}\cdot \hat{\mathbf{L}},\hat{\mathbf{L}}=i\hslash \hat{\mathbf{a}}\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}$	Same
Finite rotations	${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\left(1-\frac{i}{\hslash }\frac{\mathrm{\Delta }\theta }{N}\hat{\mathbf{a}}\cdot \hat{\mathbf{L}}\right)}^N=\exp \left(-\frac{i}{\hslash }\mathrm{\Delta }\theta \hat{\mathbf{a}}\cdot \hat{\mathbf{L}}\right)=\hat{R}}$	Same
Generator	z-component of the angular momentum operator ${\displaystyle {\hat{L}}_z=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}$	Full angular momentum operator ${\displaystyle \hat{\mathbf{L}}}$.

The z-component of angular momentum can be replaced by the component along the axis defined by ${\displaystyle \hat{\mathbf{a}}}$, using the dot product ${\displaystyle \hat{\mathbf{a}}\cdot \hat{\mathbf{L}}}$.

Again, a finite rotation can be made from many small rotations, replacing Δθ by Δθ/N and taking the limit as N tends to infinity gives the rotation operator for a finite rotation.

Rotations about the same axis do commute, for example a rotation through angles θ₁ and θ₂ about axis i can be written

$${\displaystyle R({\theta }_1+{\theta }_2,{\mathbf{e}}_i)=R({\theta }_1{\mathbf{e}}_i)R({\theta }_2{\mathbf{e}}_i),[R({\theta }_1{\mathbf{e}}_i),R({\theta }_2{\mathbf{e}}_i)]=0.}$$

However, rotations about different axes do not commute. The general commutation rules are summarized by

$${\displaystyle [L_i,L_j]=i\hslash {\epsilon }_{ijk}{L}_k.}$$

In this sense, orbital angular momentum has the common sense properties of rotations. Each of the above commutators can be easily demonstrated by holding an everyday object and rotating it through the same angle about any two different axes in both possible orderings; the final configurations are different.

In quantum mechanics, there is another form of rotation which mathematically appears similar to the orbital case, but has different properties, described next.

Spin angular momentum

All previous quantities have classical definitions. Spin is a quantity possessed by particles in quantum mechanics without any classical analogue, having the units of angular momentum. The spin vector operator is denoted ${\displaystyle \hat{\mathbf{S}}=(\widehat{S_x},\widehat{S_y},\widehat{S_z})}$. The eigenvalues of its components are the possible outcomes (in units of ${\displaystyle \hslash }$) of a measurement of the spin projected onto one of the basis directions.

Rotations (of ordinary space) about an axis ${\displaystyle \hat{\mathbf{a}}}$ through angle θ about the unit vector ${\displaystyle \hat{a}}$ in space acting on a multicomponent wave function (spinor) at a point in space is represented by:

Spin rotation operator (finite)

${\displaystyle \hat{S}(\theta ,\hat{\mathbf{a}})=\exp \left(-\frac{i}{\hslash }\theta \hat{\mathbf{a}}\cdot \hat{\mathbf{S}}\right)}$

However, unlike orbital angular momentum in which the z-projection quantum number ℓ can only take positive or negative integer values (including zero), the z-projection spin quantum number s can take all positive and negative half-integer values. There are rotational matrices for each spin quantum number.

Evaluating the exponential for a given z-projection spin quantum number s gives a (2s + 1)-dimensional spin matrix. This can be used to define a spinor as a column vector of 2s + 1 components which transforms to a rotated coordinate system according to the spin matrix at a fixed point in space.

For the simplest non-trivial case of s = 1/2, the spin operator is given by

$${\displaystyle \hat{\mathbf{S}}=\frac{\hslash }{2}\mathit{\sigma }}$$

where the Pauli matrices in the standard representation are:

$${\displaystyle {\sigma }_1={\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2={\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_3={\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)}$$

Total angular momentum

The total angular momentum operator is the sum of the orbital and spin

$${\displaystyle \hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}}$$

and is an important quantity for multi-particle systems, especially in nuclear physics and the quantum chemistry of multi-electron atoms and molecules.

We have a similar rotation matrix:

$${\displaystyle \hat{J}(\theta ,\hat{\mathbf{a}})=\exp \left(-\frac{i}{\hslash }\theta \hat{\mathbf{a}}\cdot \hat{\mathbf{J}}\right)}$$

Conserved quantities in the quantum harmonic oscillator

The dynamical symmetry group of the n dimensional quantum harmonic oscillator is the special unitary group SU(n). As an example, the number of infinitesimal generators of the corresponding Lie algebras of SU(2) and SU(3) are three and eight respectively. This leads to exactly three and eight independent conserved quantities (other than the Hamiltonian) in these systems.

The two dimensional quantum harmonic oscillator has the expected conserved quantities of the Hamiltonian and the angular momentum, but has additional hidden conserved quantities of energy level difference and another form of angular momentum.

Lorentz group in relativistic quantum mechanics

Following is an overview of the Lorentz group; a treatment of boosts and rotations in spacetime. Throughout this section, see (for example) T. Ohlsson (2011) and E. Abers (2004).

Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector ${\displaystyle \hat{\mathbf{n}}=(n_1,n_2,n_3)}$, and a rotation angle θ about a three-dimensional unit vector ${\displaystyle \hat{\mathbf{a}}=(a_1,a_2,a_3)}$ defining an axis, so ${\displaystyle \phi \hat{\mathbf{n}}=\phi (n_1,n_2,n_3)}$ and ${\displaystyle \theta \hat{\mathbf{a}}=\theta (a_1,a_2,a_3)}$ are together six parameters of the Lorentz group (three for rotations and three for boosts). The Lorentz group is 6-dimensional.

Pure rotations in spacetime

The rotation matrices and rotation generators considered above form the spacelike part of a four-dimensional matrix, representing pure-rotation Lorentz transformations. Three of the Lorentz group elements ${\displaystyle {\hat{R}}_x,{\hat{R}}_y,{\hat{R}}_z}$ and generators J = (J₁, J₂, J₃) for pure rotations are:

${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_x)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \cos \mathrm{\Delta }\theta & -\sin \mathrm{\Delta }\theta \\ 0& 0& \sin \mathrm{\Delta }\theta & \cos \mathrm{\Delta }\theta \end{array}\right),}$	${\displaystyle J_x=J_1=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right),}$
${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_y)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \mathrm{\Delta }\theta & 0& \sin \mathrm{\Delta }\theta \\ 0& 0& 1& 0\\ 0& -\sin \mathrm{\Delta }\theta & 0& \cos \mathrm{\Delta }\theta \end{array}\right),}$	${\displaystyle J_y=J_2=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right),}$
${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_z)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \mathrm{\Delta }\theta & -\sin \mathrm{\Delta }\theta & 0\\ 0& \sin \mathrm{\Delta }\theta & \cos \mathrm{\Delta }\theta & 0\\ 0& 0& 0& 1\end{array}\right),}$	${\displaystyle J_z=J_3=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right).}$

The rotation matrices act on any four vector A = (A₀, A₁, A₂, A₃) and rotate the space-like components according to

$${\displaystyle {\mathbf{A}}^{\prime }=\hat{R}(\mathrm{\Delta }\theta ,\hat{\mathbf{n}})\mathbf{A}}$$

leaving the time-like coordinate unchanged. In matrix expressions, A is treated as a column vector.

Pure boosts in spacetime

A boost with velocity ctanhφ in the x, y, or z directions given by the standard Cartesian basis vector ${\displaystyle {\hat{\mathbf{e}}}_x,{\hat{\mathbf{e}}}_y,{\hat{\mathbf{e}}}_z}$, are the boost transformation matrices. These matrices ${\displaystyle {\hat{B}}_x,{\hat{B}}_y,{\hat{B}}_z}$ and the corresponding generators K = (K₁, K₂, K₃) are the remaining three group elements and generators of the Lorentz group:

${\displaystyle {\hat{B}}_x\equiv \hat{B}(\phi ,{\hat{\mathbf{e}}}_x)=\left(\begin{array}{cccc}\cosh \phi & \sinh \phi & 0& 0\\ \sinh \phi & \cosh \phi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),}$	${\displaystyle K_x=K_1=i{\frac{\mathrm{\partial }\hat{B}(\phi ,{\hat{\mathbf{e}}}_x)}{\mathrm{\partial }\phi }|}_{\phi =0}=i\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),}$
${\displaystyle {\hat{B}}_y\equiv \hat{B}(\phi ,{\hat{\mathbf{e}}}_y)=\left(\begin{array}{cccc}\cosh \phi & 0& \sinh \phi & 0\\ 0& 1& 0& 0\\ \sinh \phi & 0& \cosh \phi & 0\\ 0& 0& 0& 1\end{array}\right),}$	${\displaystyle K_y=K_2=i{\frac{\mathrm{\partial }\hat{B}(\phi ,{\hat{\mathbf{e}}}_y)}{\mathrm{\partial }\phi }|}_{\phi =0}=i\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),}$
${\displaystyle {\hat{B}}_z\equiv \hat{B}(\phi ,{\hat{\mathbf{e}}}_z)=\left(\begin{array}{cccc}\cosh \phi & 0& 0& \sinh \phi \\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \sinh \phi & 0& 0& \cosh \phi \end{array}\right),}$	${\displaystyle K_z=K_3=i{\frac{\mathrm{\partial }\hat{B}(\phi ,{\hat{\mathbf{e}}}_z)}{\mathrm{\partial }\phi }|}_{\phi =0}=i\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right).}$

The boost matrices act on any four vector A = (A₀, A₁, A₂, A₃) and mix the time-like and the space-like components, according to:

$${\displaystyle {\mathbf{A}}^{\prime }=\hat{B}(\phi ,\hat{\mathbf{n}})\mathbf{A}}$$

The term "boost" refers to the relative velocity between two frames, and is not to be conflated with momentum as the generator of translations, as explained below.

Combining boosts and rotations

Products of rotations give another rotation (a frequent exemplification of a subgroup), while products of boosts and boosts or of rotations and boosts cannot be expressed as pure boosts or pure rotations. In general, any Lorentz transformation can be expressed as a product of a pure rotation and a pure boost. For more background see (for example) B.R. Durney (2011) and H.L. Berk et al. and references therein.

The boost and rotation generators have representations denoted D(K) and D(J) respectively, the capital D in this context indicates a group representation.

For the Lorentz group, the representations D(K) and D(J) of the generators K and J fulfill the following commutation rules.

Commutators

	Generators	Representations
Pure rotation	${\displaystyle \left[J_a,J_b\right]=i{\epsilon }_{abc}{J}_c}$	${\displaystyle \left[D(J_a),D(J_b)\right]=i{\epsilon }_{abc}D(J_c)}$
Pure boost	${\displaystyle \left[K_a,K_b\right]=-i{\epsilon }_{abc}{J}_c}$	${\displaystyle \left[D(K_a),D(K_b)\right]=-i{\epsilon }_{abc}D(J_c)}$
Lorentz transformation	${\displaystyle \left[J_a,K_b\right]=i{\epsilon }_{abc}{K}_c}$	${\displaystyle \left[D(J_a),D(K_b)\right]=i{\epsilon }_{abc}D(K_c)}$

In all commutators, the boost entities mixed with those for rotations, although rotations alone simply give another rotation. Exponentiating the generators gives the boost and rotation operators which combine into the general Lorentz transformation, under which the spacetime coordinates transform from one rest frame to another boosted and/or rotating frame. Likewise, exponentiating the representations of the generators gives the representations of the boost and rotation operators, under which a particle's spinor field transforms.

Transformation laws

	Transformations	Representations
Pure boost	${\displaystyle \hat{B}(\phi ,\hat{\mathbf{n}})=\exp \left(-\frac{i}{\hslash }\phi \hat{\mathbf{n}}\cdot \mathbf{K}\right)}$	${\displaystyle D[\hat{B}(\phi ,\hat{\mathbf{n}})]=\exp \left(-\frac{i}{\hslash }\phi \hat{\mathbf{n}}\cdot D(\mathbf{K})\right)}$
Pure rotation	${\displaystyle \hat{R}(\theta ,\hat{\mathbf{a}})=\exp \left(-\frac{i}{\hslash }\theta \hat{\mathbf{a}}\cdot \mathbf{J}\right)}$	${\displaystyle D[\hat{R}(\theta ,\hat{\mathbf{a}})]=\exp \left(-\frac{i}{\hslash }\theta \hat{\mathbf{a}}\cdot D(\mathbf{J})\right)}$
Lorentz transformation	${\displaystyle \mathrm{\Lambda }(\phi ,\hat{\mathbf{n}},\theta ,\hat{\mathbf{a}})=\exp \left[-\frac{i}{\hslash }\left(\phi \hat{\mathbf{n}}\cdot \mathbf{K}+\theta \hat{\mathbf{a}}\cdot \mathbf{J}\right)\right]}$	${\displaystyle D[\mathrm{\Lambda }(\theta ,\hat{\mathbf{a}},\phi ,\hat{\mathbf{n}})]=\exp \left[-\frac{i}{\hslash }\left(\phi \hat{\mathbf{n}}\cdot D(\mathbf{K})+\theta \hat{\mathbf{a}}\cdot D(\mathbf{J})\right)\right]}$

In the literature, the boost generators K and rotation generators J are sometimes combined into one generator for Lorentz transformations M, an antisymmetric four-dimensional matrix with entries:

$${\displaystyle M^{0a}=-M^{a0}=K_a,M^{ab}={\epsilon }_{abc}{J}_c.}$$

and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:

$${\displaystyle {\omega }_{0a}=-{\omega }_{a0}=\phi n_a,{\omega }_{ab}=\theta {\epsilon }_{abc}{a}_c,}$$

The general Lorentz transformation is then:

$${\displaystyle \mathrm{\Lambda }(\phi ,\hat{\mathbf{n}},\theta ,\hat{\mathbf{a}})=\exp \left(-\frac{i}{2}{\omega }_{\alpha \beta }{M}^{\alpha \beta }\right)=\exp \left[-\frac{i}{2}\left(\phi \hat{\mathbf{n}}\cdot \mathbf{K}+\theta \hat{\mathbf{a}}\cdot \mathbf{J}\right)\right]}$$

with summation over repeated matrix indices α and β. The Λ matrices act on any four vector A = (A₀, A₁, A₂, A₃) and mix the time-like and the space-like components, according to:

$${\displaystyle {\mathbf{A}}^{\prime }=\mathrm{\Lambda }(\phi ,\hat{\mathbf{n}},\theta ,\hat{\mathbf{a}})\mathbf{A}}$$

Transformations of spinor wavefunctions in relativistic quantum mechanics

In relativistic quantum mechanics, wavefunctions are no longer single-component scalar fields, but now 2(2s + 1) component spinor fields, where s is the spin of the particle. The transformations of these functions in spacetime are given below.

Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψ_σ locally transform under some representation D of the Lorentz group:

$${\displaystyle {\psi }_{\sigma }(\mathbf{r},t)\to D(\mathrm{\Lambda }){\psi }_{\sigma }({\mathrm{\Lambda }}^{-1}(\mathbf{r},t))}$$

where D(Λ) is a finite-dimensional representation, in other words a (2s + 1)×(2s + 1) dimensional square matrix, and ψ is thought of as a column vector containing components with the (2s + 1) allowed values of σ:

$${\displaystyle \psi (\mathbf{r},t)=\left[\begin{array}{c}{\psi }_{\sigma =s}(\mathbf{r},t)\\ {\psi }_{\sigma =s-1}(\mathbf{r},t)\\ ⋮\\ {\psi }_{\sigma =-s+1}(\mathbf{r},t)\\ {\psi }_{\sigma =-s}(\mathbf{r},t)\end{array}\right]\rightleftharpoons {\psi (\mathbf{r},t)}^{†}=\left[\begin{array}{ccccc}{{\psi }_{\sigma =s}(\mathbf{r},t)}^{\star }& {{\psi }_{\sigma =s-1}(\mathbf{r},t)}^{\star }& \cdots & {{\psi }_{\sigma =-s+1}(\mathbf{r},t)}^{\star }& {{\psi }_{\sigma =-s}(\mathbf{r},t)}^{\star }\end{array}\right]}$$

Real irreducible representations and spin

The irreducible representations of D(K) and D(J), in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new operators:

$${\displaystyle \mathbf{A}=\frac{\mathbf{J}+i\mathbf{K}}{2},\mathbf{B}=\frac{\mathbf{J}-i\mathbf{K}}{2},}$$

so A and B are simply complex conjugates of each other, it follows they satisfy the symmetrically formed commutators:

$${\displaystyle \left[A_i,A_j\right]={\epsilon }_{ijk}{A}_k,\left[B_i,B_j\right]={\epsilon }_{ijk}{B}_k,\left[A_i,B_j\right]=0,}$$

and these are essentially the commutators the orbital and spin angular momentum operators satisfy. Therefore, A and B form operator algebras analogous to angular momentum; same ladder operators, z-projections, etc., independently of each other as each of their components mutually commute. By the analogy to the spin quantum number, we can introduce positive integers or half integers, a, b, with corresponding sets of values m = a, a − 1, ... −a + 1, −a and n = b, b − 1, ... −b + 1, −b. The matrices satisfying the above commutation relations are the same as for spins a and b have components given by multiplying Kronecker delta values with angular momentum matrix elements:

$${\displaystyle {\left(A_x\right)}_{m^{\prime }{n}^{\prime },mn}={\delta }_{n^{\prime }n}{\left(J_x^{(m)}\right)}_{m^{\prime }m}{\left(B_x\right)}_{m^{\prime }{n}^{\prime },mn}={\delta }_{m^{\prime }m}{\left(J_x^{(n)}\right)}_{n^{\prime }n}}$$ $${\displaystyle {\left(A_y\right)}_{m^{\prime }{n}^{\prime },mn}={\delta }_{n^{\prime }n}{\left(J_y^{(m)}\right)}_{m^{\prime }m}{\left(B_y\right)}_{m^{\prime }{n}^{\prime },mn}={\delta }_{m^{\prime }m}{\left(J_y^{(n)}\right)}_{n^{\prime }n}}$$ $${\displaystyle {\left(A_z\right)}_{m^{\prime }{n}^{\prime },mn}={\delta }_{n^{\prime }n}{\left(J_z^{(m)}\right)}_{m^{\prime }m}{\left(B_z\right)}_{m^{\prime }{n}^{\prime },mn}={\delta }_{m^{\prime }m}{\left(J_z^{(n)}\right)}_{n^{\prime }n}}$$

where in each case the row number m′n′ and column number mn are separated by a comma, and in turn:

$${\displaystyle {\left(J_z^{(m)}\right)}_{m^{\prime }m}=m{\delta }_{m^{\prime }m}{\left(J_x^{(m)}\pm i{J}_y^{(m)}\right)}_{m^{\prime }m}=m{\delta }_{a^{\prime },a\pm 1}\sqrt{(a\mp m)(a\pm m+1)}}$$

and similarly for J⁽ⁿ⁾. The three J^(m) matrices are each (2m + 1)×(2m + 1) square matrices, and the three J⁽ⁿ⁾ are each (2n + 1)×(2n + 1) square matrices. The integers or half-integers m and n numerate all the irreducible representations by, in equivalent notations used by authors: D^{(m, n)} ≡ (m, n) ≡ D^(m) ⊗ D⁽ⁿ⁾, which are each [(2m + 1)(2n + 1)]×[(2m + 1)(2n + 1)] square matrices.

Applying this to particles with spin s;

• left-handed (2s + 1)-component spinors transform under the real irreps D^{(s, 0)},
• right-handed (2s + 1)-component spinors transform under the real irreps D^{(0, s)},
• taking direct sums symbolized by ⊕ (see direct sum of matrices for the simpler matrix concept), one obtains the representations under which 2(2s + 1)-component spinors transform: D^{(m, n)} ⊕ D^{(n, m)} where m + n = s. These are also real irreps, but as shown above, they split into complex conjugates.

In these cases the D refers to any of D(J), D(K), or a full Lorentz transformation D(Λ).

Relativistic wave equations

In the context of the Dirac equation and Weyl equation, the Weyl spinors satisfying the Weyl equation transform under the simplest irreducible spin representations of the Lorentz group, since the spin quantum number in this case is the smallest non-zero number allowed: 1/2. The 2-component left-handed Weyl spinor transforms under D^{(1/2, 0)} and the 2-component right-handed Weyl spinor transforms under D^{(0, 1/2)}. Dirac spinors satisfying the Dirac equation transform under the representation D^{(1/2, 0)} ⊕ D^{(0, 1/2)}, the direct sum of the irreps for the Weyl spinors.

The Poincaré group in relativistic quantum mechanics and field theory

Space translations, time translations, rotations, and boosts, all taken together, constitute the Poincaré group. The group elements are the three rotation matrices and three boost matrices (as in the Lorentz group), and one for time translations and three for space translations in spacetime. There is a generator for each. Therefore, the Poincaré group is 10-dimensional.

In special relativity, space and time can be collected into a four-position vector X = (ct, −r), and in parallel so can energy and momentum which combine into a four-momentum vector P = (E/c, −p). With relativistic quantum mechanics in mind, the time duration and spatial displacement parameters (four in total, one for time and three for space) combine into a spacetime displacement ΔX = (cΔt, −Δr), and the energy and momentum operators are inserted in the four-momentum to obtain a four-momentum operator,

$${\displaystyle \hat{\mathbf{P}}=\left(\frac{\hat{E}}{c},-\hat{\mathbf{p}}\right)=i\hslash \left(\frac{1}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t},\mathrm{\nabla }\right),}$$

which are the generators of spacetime translations (four in total, one time and three space):

$${\displaystyle \hat{X}(\mathrm{\Delta }\mathbf{X})=\exp \left(-\frac{i}{\hslash }\mathrm{\Delta }\mathbf{X}\cdot \hat{\mathbf{P}}\right)=\exp \left[-\frac{i}{\hslash }\left(\mathrm{\Delta }t\hat{E}+\mathrm{\Delta }\mathbf{r}\cdot \hat{\mathbf{p}}\right)\right].}$$

There are commutation relations between the components four-momentum P (generators of spacetime translations), and angular momentum M (generators of Lorentz transformations), that define the Poincaré algebra:

• ${\displaystyle [P_{\mu },P_{\nu }]=0}$
• ${\displaystyle \frac{1}{i}[M_{\mu \nu },P_{\rho }]={\eta }_{\mu \rho }{P}_{\nu }-{\eta }_{\nu \rho }{P}_{\mu }}$
• ${\displaystyle \frac{1}{i}[M_{\mu \nu },M_{\rho \sigma }]={\eta }_{\mu \rho }{M}_{\nu \sigma }-{\eta }_{\mu \sigma }{M}_{\nu \rho }-{\eta }_{\nu \rho }{M}_{\mu \sigma }+{\eta }_{\nu \sigma }{M}_{\mu \rho }}$

where η is the Minkowski metric tensor. (It is common to drop any hats for the four-momentum operators in the commutation relations). These equations are an expression of the fundamental properties of space and time as far as they are known today. They have a classical counterpart where the commutators are replaced by Poisson brackets.

To describe spin in relativistic quantum mechanics, the Pauli–Lubanski pseudovector

$${\displaystyle W_{\mu }=\frac{1}{2}{\epsilon }_{\mu \nu \rho \sigma }{J}^{\nu \rho }{P}^{\sigma },}$$

a Casimir operator, is the constant spin contribution to the total angular momentum, and there are commutation relations between P and W and between M and W:

$${\displaystyle \left[P^{\mu },W^{\nu }\right]=0,}$$ $${\displaystyle \left[J^{\mu \nu },W^{\rho }\right]=i\left({\eta }^{\rho \nu }{W}^{\mu }-{\eta }^{\rho \mu }{W}^{\nu }\right),}$$ $${\displaystyle \left[W_{\mu },W_{\nu }\right]=-i{ϵ}_{\mu \nu \rho \sigma }{W}^{\rho }{P}^{\sigma }.}$$

Invariants constructed from W, instances of Casimir invariants can be used to classify irreducible representations of the Lorentz group.

Symmetries in quantum field theory and particle physics

Unitary groups in quantum field theory

Group theory is an abstract way of mathematically analyzing symmetries. Unitary operators are paramount to quantum theory, so unitary groups are important in particle physics. The group of N dimensional unitary square matrices is denoted U(N). Unitary operators preserve inner products which means probabilities are also preserved, so the quantum mechanics of the system is invariant under unitary transformations. Let ${\displaystyle \hat{U}}$ be a unitary operator, so the inverse is the Hermitian adjoint ${\displaystyle {\hat{U}}^{-1}={\hat{U}}^{†}}$, which commutes with the Hamiltonian:

$${\displaystyle \left[\hat{U},\hat{H}\right]=0}$$

then the observable corresponding to the operator ${\displaystyle \hat{U}}$ is conserved, and the Hamiltonian is invariant under the transformation ${\displaystyle \hat{U}}$.

Since the predictions of quantum mechanics should be invariant under the action of a group, physicists look for unitary transformations to represent the group.

Important subgroups of each U(N) are those unitary matrices which have unit determinant (or are "unimodular"): these are called the special unitary groups and are denoted SU(N).

U(1)

The simplest unitary group is U(1), which is just the complex numbers of modulus 1. This one-dimensional matrix entry is of the form:

$${\displaystyle U=e^{-i\theta }}$$

in which θ is the parameter of the group, and the group is Abelian since one-dimensional matrices always commute under matrix multiplication. Lagrangians in quantum field theory for complex scalar fields are often invariant under U(1) transformations. If there is a quantum number a associated with the U(1) symmetry, for example baryon and the three lepton numbers in electromagnetic interactions, we have:

$${\displaystyle U=e^{-ia\theta }}$$

U(2) and SU(2)

The general form of an element of a U(2) element is parametrized by two complex numbers a and b:

$${\displaystyle U=\left(\begin{array}{cc}a& b\\ -b^{\star }& a^{\star }\end{array}\right)}$$

and for SU(2), the determinant is restricted to 1:

$${\displaystyle det(U)=a{a}^{\star }+b{b}^{\star }={|a|}^2+{|b|}^2=1}$$

In group theoretic language, the Pauli matrices are the generators of the special unitary group in two dimensions, denoted SU(2). Their commutation relation is the same as for orbital angular momentum, aside from a factor of 2:

$${\displaystyle [{\sigma }_a,{\sigma }_b]=2i\hslash {\epsilon }_{abc}{\sigma }_c}$$

A group element of SU(2) can be written:

$${\displaystyle U(\theta ,{\hat{\mathbf{e}}}_j)=e^{i\theta {\sigma }_j/2}}$$

where σ_j is a Pauli matrix, and the group parameters are the angles turned through about an axis.

The two-dimensional isotropic quantum harmonic oscillator has symmetry group SU(2), while the symmetry algebra of the rational anisotropic oscillator is a nonlinear extension of u(2).

U(3) and SU(3)

The eight Gell-Mann matrices λ_n (see article for them and the structure constants) are important for quantum chromodynamics. They originally arose in the theory SU(3) of flavor which is still of practical importance in nuclear physics. They are the generators for the SU(3) group, so an element of SU(3) can be written analogously to an element of SU(2):

$${\displaystyle U(\theta ,{\hat{\mathbf{e}}}_j)=\exp \left(-\frac{i}{2}\sum _{n=1}^8{\theta }_n{\lambda }_n\right)}$$

where θ_n are eight independent parameters. The λ_n matrices satisfy the commutator:

$${\displaystyle \left[{\lambda }_a,{\lambda }_b\right]=2i{f}_{abc}{\lambda }_c}$$

where the indices a, b, c take the values 1, 2, 3, ..., 8. The structure constants f_abc are totally antisymmetric in all indices analogous to those of SU(2). In the standard colour charge basis (r for red, g for green, b for blue):

$${\displaystyle |r⟩=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),|g⟩=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),|b⟩=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)}$$

the colour states are eigenstates of the λ₃ and λ₈ matrices, while the other matrices mix colour states together.

The eight gluons states (8-dimensional column vectors) are simultaneous eigenstates of the adjoint representation of SU(3), the 8-dimensional representation acting on its own Lie algebra su(3), for the λ₃ and λ₈ matrices. By forming tensor products of representations (the standard representation and its dual) and taking appropriate quotients, protons and neutrons, and other hadrons are eigenstates of various representations of SU(3) of color. The representations of SU(3) can be described by a "theorem of the highest weight".

Matter and antimatter

In relativistic quantum mechanics, relativistic wave equations predict a remarkable symmetry of nature: that every particle has a corresponding antiparticle. This is mathematically contained in the spinor fields which are the solutions of the relativistic wave equations.

Charge conjugation switches particles and antiparticles. Physical laws and interactions unchanged by this operation have C symmetry.

Discrete spacetime symmetries

• Parity mirrors the orientation of the spatial coordinates from left-handed to right-handed. Informally, space is "reflected" into its mirror image. Physical laws and interactions unchanged by this operation have P symmetry.
• Time reversal flips the time coordinate, which amounts to time running from future to past. A curious property of time, which space does not have, is that it is unidirectional: particles traveling forwards in time are equivalent to antiparticles traveling back in time. Physical laws and interactions unchanged by this operation have T symmetry.

C, P, T symmetries

• Parity (physics) § Molecules
• CPT theorem
• CP violation
• PT symmetry
• Lorentz violation

Gauge theory

Main article: Gauge theory

In quantum electrodynamics, the local symmetry group is U(1) and is abelian. In quantum chromodynamics, the local symmetry group is SU(3) and is non-abelian.

The electromagnetic interaction is mediated by photons, which have no electric charge. The electromagnetic tensor has an electromagnetic four-potential field possessing gauge symmetry.

The strong (color) interaction is mediated by gluons, which can have eight color charges. There are eight gluon field strength tensors with corresponding gluon four potentials field, each possessing gauge symmetry.

The strong (color) interaction

Color charge

Analogous to the spin operator, there are color charge operators in terms of the Gell-Mann matrices λ_j:

$${\displaystyle {\hat{F}}_j=\frac{1}{2}{\lambda }_j}$$

and since color charge is a conserved charge, all color charge operators must commute with the Hamiltonian:

$${\displaystyle \left[{\hat{F}}_j,\hat{H}\right]=0}$$

Isospin

Isospin is conserved in strong interactions.

The weak and electromagnetic interactions

Duality transformation

Magnetic monopoles can be theoretically realized, although current observations and theory are consistent with them existing or not existing. Electric and magnetic charges can effectively be "rotated into one another" by a duality transformation.

Electroweak symmetry

• Electroweak symmetry
• Electroweak symmetry breaking

Supersymmetry

Main article: Supersymmetry

A Lie superalgebra is an algebra in which (suitable) basis elements either have a commutation relation or have an anticommutation relation. Symmetries have been proposed to the effect that all fermionic particles have bosonic analogues, and vice versa. These symmetry have theoretical appeal in that no extra assumptions (such as existence of strings) barring symmetries are made. In addition, by assuming supersymmetry, a number of puzzling issues can be resolved. These symmetries, which are represented by Lie superalgebras, have not been confirmed experimentally. It is now believed that they are broken symmetries, if they exist. But it has been speculated that dark matter is constitutes gravitinos, a spin 3/2 particle with mass, its supersymmetric partner being the graviton.

Exchange symmetry

Main article: Indistinguishable particles

The concept of exchange symmetry is derived from a fundamental postulate of quantum statistics, which states that no observable physical quantity should change after exchanging two identical particles. It states that because all observables are proportional to ${\displaystyle {\left|\psi \right|}^2}$ for a system of identical particles, the wave function ${\displaystyle \psi }$ must either remain the same or change sign upon such an exchange. More generally, for a system of n identical particles the wave function ${\displaystyle \psi }$ must transform as an irreducible representation of the finite symmetric group S_n. It turns out that, according to the spin-statistics theorem, fermion states transform as the antisymmetric irreducible representation of S_n and boson states as the symmetric irreducible representation.

Because the exchange of two identical particles is mathematically equivalent to the rotation of each particle by 180 degrees (and so to the rotation of one particle's frame by 360 degrees), the symmetric nature of the wave function depends on the particle's spin after the rotation operator is applied to it. Integer spin particles do not change the sign of their wave function upon a 360 degree rotation—therefore the sign of the wave function of the entire system does not change. Semi-integer spin particles change the sign of their wave function upon a 360 degree rotation (see more in spin–statistics theorem).

Particles for which the wave function does not change sign upon exchange are called bosons, or particles with a symmetric wave function. The particles for which the wave function of the system changes sign are called fermions, or particles with an antisymmetric wave function.

Fermions therefore obey different statistics (called Fermi–Dirac statistics) than bosons (which obey Bose–Einstein statistics). One of the consequences of Fermi–Dirac statistics is the exclusion principle for fermions—no two identical fermions can share the same quantum state (in other words, the wave function of two identical fermions in the same state is zero). This in turn results in degeneracy pressure for fermions—the strong resistance of fermions to compression into smaller volume. This resistance gives rise to the “stiffness” or “rigidity” of ordinary atomic matter (as atoms contain electrons which are fermions).