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Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context ofquantum mechanics,relativistic quantum mechanics andquantum field theory, and with applications in themathematical formulation of the standard model andcondensed matter physics. In general,symmetry in physics,invariance, andconservation laws, are fundamentally important constraints for formulatingphysical 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 ofcontinuous symmetries as well as theirquantum operators, and relates them to theLie groups, and relativistic transformations in theLorentz group andPoincaré group.

The notational conventions used in this article are as follows. Boldface indicatesvectors,four vectors,matrices, andvectorial operators, whilequantum states usebra–ket notation. Wide hats are foroperators, narrow hats are forunit vectors (including their components intensor index notation). Thesummation convention on the repeatedtensor indices is used, unless stated otherwise. TheMinkowski metricsignature is (+−−−).

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

The form of the fundamental quantum operators, for example the energy operator as apartialtime derivative and momentum operator as a spatialgradient, 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 aunitary 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 of a unitary operator is itsHermitian conjugate${\displaystyle {\hat{\mathrm{\Omega }}}^{-1}={\hat{\mathrm{\Omega }}}^{†}}$. The results can be extended to many-particle wavefunctions. Written inDirac notation as standard, the transformations onquantum state vectors are:

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

Now, the action of${\displaystyle \hat{\mathrm{\Omega }}}$ changesψ(r,t) toψ(r′,t′), so the inverse${\displaystyle {\hat{\mathrm{\Omega }}}^{-1}={\hat{\mathrm{\Omega }}}^{†}}$ changesψ(r′,t′) back toψ(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 stateψ (i.e.${\displaystyle \hat{\mathrm{\Omega }}}$ and${\displaystyle \hat{A}}$ commute). Quantum operators representingobservables are also required to beHermitian so that theireigenvalues arereal numbers, i.e. the operator equals itsHermitian conjugate,${\displaystyle \hat{A}={\hat{A}}^{†}}$.

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^[1][2]

LetG be aLie group, which is a group that locally isparameterized by a finite numberN ofrealcontinuously varying parametersξ₁,ξ₂, ...,ξ_N. In more mathematical language, this means thatG is a smoothmanifold that is also a group, for which the group operations are smooth.

• thedimension of the group,N, is the number of parameters it has.
• thegroupelements,g, inG arefunctions of the parameters:$${\displaystyle g=G({\xi }_1,{\xi }_2,\dots )}$$ and all parameters set to zero returns theidentity element of the group:$${\displaystyle I=G(0,0,\dots )}$$ Group elements are often matrices which act on vectors, or transformations acting on functions.
• Thegenerators of the group are thepartial 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 toG at the identity. The generators are also known as infinitesimal group elements or as the elements of theLie algebra ofG. (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, forunitary representations of the group, the generators require a factor ofi:$${\displaystyle X_j=i{\frac{\mathrm{\partial }g}{\mathrm{\partial }{\xi }_j}|}_{{\xi }_j=0}}$$ The generators of the group form avector space, which meanslinear combinations of generators also form a generator.
• The generators (whether matrices or differential operators) satisfy thecommutation relations:$${\displaystyle \left[X_a,X_b\right]=i{f}_{abc}{X}_c}$$ wheref_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 aLie algebra. Due to theantisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices.
• Therepresentations of the group then describe the ways that the groupG (or its Lie algebra) can act on a vector space. (The vector space might be, for example, the space of eigenvectors for a Hamiltonian havingG as its symmetry group.) We denote the representations using a capitalD. One can then differentiateD to obtain a representation of the Lie algebra, often also denoted byD. These two representations are related as follows:$${\displaystyle D[g({\xi }_j)]\equiv D({\xi }_j)=e^{i{\xi }_{j}D(X_j)}}$$withoutsummation on the repeated indexj. 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 adirect sum of other representations, is calledirreducible. It is conventional to labelirreducible representations by a superscripted numbern in brackets, as inD⁽ⁿ⁾, or if there is more than one number, we writeD^{(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 aprojective representation, one that only satisfies the composition law up to a scalar. In the context of quantum mechanical spin, such representations are calledspinorial.

The spacetranslation operator${\displaystyle \hat{T}(\mathrm{\Delta }\mathbf{r})}$ acts on a wavefunction to shift the space coordinates by an infinitesimal displacementΔr. The explicit expression${\displaystyle \hat{T}}$ can be quickly determined by aTaylor expansion ofψ(r + Δr,t) aboutr, then (keeping the first order term and neglecting second and higher order terms), replace the space derivatives by themomentum operator${\displaystyle \hat{\mathbf{p}}}$. Similarly for thetime translation operator acting on the time parameter, the Taylor expansion ofψ(r,t + Δt) is aboutt, and the time derivative replaced by theenergy 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 toEuler, 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, replaceΔr byΔr/N andΔt byΔt/N, whereN is a positive non-zero integer. Then asN increases, the magnitude ofΔr andΔt become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunctionN times and taking the limit asN 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 arestationary states: the eigenstates of the Hamiltonian are the energy eigenvaluesE:

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

and all stationary states have the form

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

wheret₀ 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${\displaystyle \hat{U}(t-t_0)\equiv U(t,t_0)}$.

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

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

wherer′ are the rotated coordinates about an axis defined by aunit vector${\displaystyle \hat{\mathbf{a}}=(a_1,a_2,a_3)}$ through an angular incrementΔθ, 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 arotation 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-dimensionalspecial orthogonal group, SO(3). The rotation matrices about thestandardCartesian basis vector${\displaystyle {\hat{\mathbf{e}}}_x,{\hat{\mathbf{e}}}_y,{\hat{\mathbf{e}}}_z}$ through angleΔθ, and the corresponding generators of rotationsJ = (J_x,J_y,J_z), are:

More generally for rotations about an axis defined by${\displaystyle \hat{\mathbf{a}}}$, the rotation matrix elements are:^[3]

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

whereδ_ij is theKronecker delta, andε_ijk is theLevi-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 thex,y, orz-axis) then infer the general result, or use the general rotation matrix directly andtensor index notation withδ_ij andε_ijk. To derive the infinitesimal rotation operator, which corresponds to smallΔθ, we use thesmall angle approximationssin(Δθ) ≈ Δθ andcos(Δθ) ≈ 1, then Taylor expand aboutr orr_i, keep the first order term, and substitute theangular 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	${\displaystyle \hat{R}(\mathrm{\Delta }\theta ,{\hat{\mathbf{e}}}_z)=I-\frac{i}{\hslash }\mathrm{\Delta }\theta {\hat{L}}_z}$
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${\displaystyle \hat{\mathbf{L}}}$.

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

Again, a finite rotation can be made from many small rotations, replacingΔθ byΔθ/N and taking the limit asN tends to infinity gives the rotation operator for a finite rotation.

Rotations about thesame axis do commute, for example a rotation through anglesθ₁ andθ₂ about axisi 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 aboutdifferent 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.

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 spinvector 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 angleθ about the unit vector${\displaystyle \hat{a}}$ in space acting on a multicomponent wave function (spinor) at a point in space is represented by:

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

Evaluating the exponential for a givenz-projection spin quantum numbers gives a (2s + 1)-dimensional spin matrix. This can be used to define aspinor 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 ofs = 1/2, the spin operator is given by

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

where thePauli matrices in the standard representation are:

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:

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

The dynamical symmetry group of then 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.

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)^[4] and E. Abers (2004).^[5]

Lorentz transformations can be parametrized byrapidityφ for a boost in the direction of a three-dimensionalunit vector${\displaystyle \hat{\mathbf{n}}=(n_1,n_2,n_3)}$, and a rotation angleθ about a three-dimensionalunit 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.

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 generatorsJ = (J₁,J₂,J₃) for pure rotations are:

The rotation matrices act on anyfour vectorA = (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 acolumn vector.

A boost with velocityctanhφ in thex,y, orz directions given by thestandardCartesian 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 generatorsK = (K₁,K₂,K₃) are the remaining three group elements and generators of the Lorentz group:

The boost matrices act on any four vectorA = (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 thegenerator of translations, as explainedbelow.

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)^[6] and H.L. Berk et al.^[7] and references therein.

The boost and rotation generators have representations denotedD(K) andD(J) respectively, the capitalD in this context indicates agroup representation.

For the Lorentz group, the representationsD(K) andD(J) of the generatorsK andJ 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 generatorsK and rotation generatorsJ are sometimes combined into one generator for Lorentz transformationsM, 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 matrixω, with entries:

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

The general Lorentz transformation is then:

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

withsummation over repeated matrix indicesα andβ. The Λ matrices act on any four vectorA = (A₀,A₁,A₂,A₃) and mix the time-like and the space-like components, according to:

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

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

Under a properorthochronousLorentz transformation(r,t) → Λ(r,t) inMinkowski space, all one-particle quantum statesψ_σ locally transform under somerepresentationD of theLorentz group:^[8][9]

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

whereD(Λ) is a finite-dimensional representation, in other words a(2s + 1)×(2s + 1) dimensionalsquare matrix, andψ is thought of as acolumn vector containing components with the(2s + 1) allowed values ofσ:

Theirreducible representations ofD(K) andD(J), in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new operators:

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

soA andB are simplycomplex 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 andB form operator algebras analogous to angular momentum; sameladder 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 valuesm =a,a − 1, ... −a + 1, −a andn =b,b − 1, ... −b + 1, −b. The matrices satisfying the above commutation relations are the same as for spinsa andb have components given by multiplyingKronecker 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 numberm′n′ and column numbermn 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 forJ⁽ⁿ⁾.^{[note 1]} The threeJ^(m) matrices are each(2m + 1)×(2m + 1) square matrices, and the threeJ⁽ⁿ⁾ are each(2n + 1)×(2n + 1) square matrices. The integers or half-integersm andn 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 spins;

• left-handed(2s + 1)-component spinors transform under the real irrepsD^{(s, 0)},
• right-handed(2s + 1)-component spinors transform under the real irrepsD^(0,s),
• takingdirect sums symbolized by⊕ (seedirect sum of matrices for the simpler matrix concept), one obtains the representations under which2(2s + 1)-component spinors transform:D^(m,n) ⊕D^(n,m) wherem +n =s. These are also real irreps, but as shown above, they split into complex conjugates.

In these cases theD refers to any ofD(J),D(K), or a full Lorentz transformationD(Λ).

In the context of theDirac equation andWeyl 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 underD^{(1/2, 0)} and the 2-component right-handed Weyl spinor transforms underD^{(0, 1/2)}. Dirac spinors satisfying the Dirac equation transform under the representationD^{(1/2, 0)} ⊕D^{(0, 1/2)}, the direct sum of the irreps for the Weyl spinors.

Space translations,time translations,rotations, andboosts, all taken together, constitute thePoincaré 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.

Inspecial relativity, space and time can be collected into a four-position vectorX = (ct, −r), and in parallel so can energy and momentum which combine into afour-momentum vectorP = (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 displacementΔX = (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-momentumP (generators of spacetime translations), and angular momentumM (generators of Lorentz transformations), that define the Poincaré algebra:^[10][11]

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