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{{Tone|article|date=December 2024}}
{{Short description|Properties underlying modern physics}}
{{quantum mechanics}}
{{quantum field theory}}
'''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 [[Standard Model (mathematical formulation)|mathematical formulation of the standard model]] and [[condensed matter physics]]. In general, [[symmetry in physics]], [[invariant (physics)|invariance]], and [[Conservation law (physics)|conservation law]]s, are fundamentally important constraints for formulating [[Theoretical physics|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 symmetry|continuous symmetries]] as well as their [[operator (physics)|quantum operators]], and relates them to the [[Lie group]]s, and relativistic transformations in the [[Lorentz group]] and [[Poincaré group]].

==Notation==

The notational conventions used in this article are as follows. Boldface indicates [[Euclidean vector|vector]]s, [[four vector]]s, [[matrix (mathematics)|matrices]], and [[tensor operator|vectorial operators]], while [[quantum state]]s use [[bra–ket notation]]. Wide hats are for [[operator (physics)|operators]], narrow hats are for [[unit vector]]s (including their components in [[tensor index notation]]). The [[Einstein notation|summation convention]] on the repeated [[tensor index notation|tensor indices]] is used, unless stated otherwise. The [[Minkowski metric]] [[metric signature|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 derivative|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:

<math display="block"> \widehat{\Omega}\psi(\mathbf{r},t) = \psi(\mathbf{r}',t') </math>

are considered, where <math> \widehat{\Omega} </math> 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]] <math> \widehat{\Omega}^{-1} = \widehat{\Omega}^\dagger </math>. The results can be extended to many-particle wavefunctions. Written in [[Dirac notation]] as standard, the transformations on [[quantum state]] vectors are:

<math display="block"> \widehat{\Omega}\left|\mathbf{r}(t)\right\rangle = \left|\mathbf{r}'(t')\right\rangle </math>

Now, the action of <math> \widehat{\Omega} </math> changes {{math|''ψ''('''r''', ''t'')}} to {{math|''ψ''('''r'''&prime;, ''t''&prime;)}}, so the inverse <math> \widehat{\Omega}^{-1} = \widehat{\Omega}^\dagger </math> changes {{math|''ψ''('''r'''&prime;, ''t''&prime;)}} back to {{math|''ψ''('''r''', ''t'')}}. Thus,  an operator <math> \widehat{A} </math> invariant under <math> \widehat{\Omega} </math> satisfies [I am sorry, but this is non-sequitor. You have not laid a foundation for this proposition]:

<math display="block"> \widehat{A}\psi = \widehat{\Omega}^\dagger\widehat{A}\widehat{\Omega}\psi \quad \Rightarrow \quad \widehat{\Omega}\widehat{A}\psi = \widehat{A}\widehat{\Omega}\psi. </math>

Concomitantly,

<math display="block"> [\widehat{\Omega},\widehat{A}]\psi = 0 </math>

for any state ''ψ''. Quantum operators representing [[observables]] are also required to be [[Hermitian operator|Hermitian]] so that their [[eigenvalue]]s are [[real number]]s, i.e. the operator equals its [[Hermitian conjugate]], <math> \widehat{A} = \widehat{A}^\dagger </math>.

===Overview of Lie group theory===

{{For|a full exposition and details|Lie group|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<ref>{{harvnb|Hall|2015}}</ref><ref>{{harvnb|Hall|2013 }}</ref>

Let {{math|''G''}} be a ''[[Lie group]]'', which is a group that locally is [[parameter]]ized by a finite number {{mvar|N}} of [[real number|real]] [[Continuous function|continuously varying]] parameters {{math|''ξ''<sub>1</sub>, ''ξ''<sub>2</sub>, ..., ''ξ''<sub>''N''</sub>}}. In more mathematical language, this means that {{math|''G''}} is a smooth [[manifold]] that is also a group, for which the group operations are smooth.
*the ''dimension of the group'', {{mvar|N}}, is the number of parameters it has.
*the ''group [[element (mathematics)|element]]s'', {{mvar|g}}, in {{math|''G''}} are [[function (mathematics)|function]]s of the parameters: <math display="block">g = G(\xi_1, \xi_2, \dots )</math> and all parameters set to zero returns the ''[[identity element]]'' of the group: <math display="block">I = G(0, 0,\dots )</math> Group elements are often matrices which act on vectors, or transformations acting on functions.
*The ''generators of the group'' are the [[partial derivative]]s of the group elements with respect to the group parameters with the result evaluated when the parameter is set to zero: <math display="block">X_j = \left. \frac{\partial g}{\partial \xi_j} \right|_{\xi_j = 0} </math> 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.) {{pb}} 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 representation]]s of the group, the generators require a factor of {{math|''i''}}: <math display="block">X_j = i \left. \frac{\partial g}{\partial \xi_j} \right|_{\xi_j = 0} </math> The generators of the group form a [[vector space]], which means [[linear combination]]s of generators also form a generator.
*The generators (whether matrices or differential operators) satisfy the ''[[commutator|commutation relations]]'': <math display="block">\left[X_a,X_b\right] = i f_{abc} X_c</math> where {{math|''f<sub>abc</sub>''}} are the (basis dependent) ''[[structure constant]]s'' 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 ''[[group representation|representations of the group]]'' then describe the ways that the group {{math|''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 {{math|''G''}} as its symmetry group.) We denote the representations using a capital {{math|''D''}}. One can then differentiate {{math|''D''}} to obtain a  representation of the Lie algebra, often also denoted by {{math|''D''}}. These two representations are related as follows: <math display="block">D[g(\xi_j)] \equiv D(\xi_j) = e^{ i \xi_j D(X_j)}</math> ''without'' [[Einstein notation|summation on the repeated index]] {{mvar|j}}. Representations are linear operators that take in group elements and preserve the composition rule: <math display="block"> D(\xi_a)D(\xi_b) = D(\xi_a \xi_b). </math>
A representation which cannot be decomposed into a [[direct sum]] of other representations, is called ''irreducible''. It is conventional to label [[irreducible representation]]s by a superscripted number {{mvar|n}} in brackets, as in {{math|''D''<sup>(''n'')</sup>}}, or if there is more than one number, we write {{math|''D''<sup>(''n'', ''m'', ...)</sup>}}.

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 [[spinor]]ial.

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

The space [[Translation operator (quantum mechanics)|translation operator]] <math>\widehat{T}(\Delta \mathbf{r})</math> acts on a wavefunction to shift the space coordinates by an infinitesimal displacement {{math|Δ'''r'''}}. The explicit expression <math>\widehat{T}</math> can be quickly determined by a [[Taylor expansion]] of {{math|''ψ''('''r''' + Δ'''r''', ''t'')}} about {{math|'''r'''}}, then (keeping the first order term and neglecting second and higher order terms), replace the space derivatives by the [[momentum operator]] <math>\widehat{\mathbf{p}}</math>. Similarly for the [[time translation]] operator acting on the time parameter, the Taylor expansion of {{math|''ψ''('''r''', ''t'' + Δ''t'')}} is about {{mvar|t}}, and the time derivative replaced by the [[energy operator]] <math>\widehat{E}</math>.

{|class="wikitable"
|-
!Name
!Translation operator <math>\widehat{T}</math>
!Time translation/evolution operator <math>\widehat{U}</math>
|-
!Action on wavefunction
| <math>\widehat{T}(\Delta\mathbf{r})\psi(\mathbf{r},t) = \psi(\mathbf{r} + \Delta\mathbf{r},t)</math>
| <math>\widehat{U}(\Delta t)\psi(\mathbf{r},t) = \psi(\mathbf{r}, t + \Delta t)</math>
|-
!Infinitesimal operator
| <math>\widehat{T}(\Delta\mathbf{r}) = I + \frac{i}{\hbar}\Delta\mathbf{r} \cdot \widehat{\mathbf{p}}</math>
| <math> \widehat{U}(\Delta t) = I - \frac{i}{\hbar}\Delta t \widehat{E} </math>
|-
!Finite operator
|<math> \lim_{N \to \infty} \left(I + \frac{i}{\hbar}\frac{\Delta\mathbf{r}}{N} \cdot \widehat{\mathbf{p}}\right)^N = \exp\left(\frac{i}{\hbar}\Delta\mathbf{r} \cdot \widehat{\mathbf{p}}\right) = \widehat{T}(\Delta\mathbf{r})</math>
|<math> \lim_{N \to \infty} \left(I - \frac{i}{\hbar}\frac{\Delta t}{N} \cdot \widehat{E}\right)^N = \exp\left(-\frac{i}{\hbar}\Delta t \widehat{E}\right) = \widehat{U}(\Delta t)</math>
|-
!Generator
|[[Momentum operator]] <math>\widehat{\mathbf{p}} = -i \hbar \nabla</math>
|[[Energy operator]] <math>\widehat{E} = i \hbar \frac{\partial }{\partial t}</math>
|-
|}

The exponential functions arise by definition as those limits, due to [[Leonhard Euler|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 {{math|Δ'''r'''}} by {{math|Δ'''r'''/''N''}} and {{math|Δ''t''}} by {{math|Δ''t''/''N''}}, where {{mvar|N}} is a positive non-zero integer. Then as {{mvar|N}} increases, the magnitude of {{math|Δ'''r'''}} and {{math|Δ''t''}} become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunction {{mvar|N}} times and taking the limit as {{mvar|N}} tends to infinity gives the finite operators.

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

{|class="wikitable"
|+Commutators
!Operators
!Generators
|-
|<math>\left[\widehat{T}(\mathbf{r}_1), \widehat{T}(\mathbf{r}_2) \right] \psi(\mathbf{r},t) = 0 </math>
|<math>\left[\widehat{p}_i,\widehat{p}_j,\right] \psi(\mathbf{r},t) = 0 </math>
|-
|<math>\left[\widehat{U}(t_1), \widehat{U}(t_2) \right]\psi(\mathbf{r},t) = 0 </math>
|<math>\left[\widehat{E},\widehat{p}_i,\right] \psi(\mathbf{r},t) = 0 </math>
|}

For a time-independent Hamiltonian, energy is conserved in time and quantum states are [[stationary state]]s: the eigenstates of the Hamiltonian are the energy eigenvalues {{mvar|E}}:

<math display="block">\widehat{U}(t) = \exp\left( - \frac{i \Delta t E}{\hbar}\right) </math>

and all stationary states have the form

<math display="block">\psi(\mathbf{r}, t + t_0) = \widehat{U}(t - t_0) \psi(\mathbf{r},t_0) </math>

where {{math|''t''<sub>0</sub>}} 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 <math>\widehat{U}(t - t_0) \equiv U(t, t_0)</math>.

===Angular momentum as the generator of rotations===

====Orbital angular momentum====

The rotation operator, <math>\widehat{R}</math>, acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle {{math|Δ''θ''}}:

<math display="block">{\widehat{R}}(\Delta\theta,\hat{\mathbf{a}})\psi(\mathbf{r},t) = \psi(\mathbf{r}',t)</math>

where {{math|'''r&prime;'''}} are the rotated coordinates about an axis defined by a [[unit vector]] <math>\hat{\mathbf{a}} = (a_1, a_2, a_3)</math> through an angular increment {{math|Δ''θ''}}, given by:

<math display="block">\mathbf{r}' = \widehat{R}(\Delta\theta,\hat{\mathbf{a}})\mathbf{r}\,.</math>

where <math>\widehat{R}(\Delta\theta,\hat{\mathbf{a}})</math> 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 <math>\Delta \theta \hat{\mathbf{a}} = \Delta\theta(a_1, a_2, a_3)</math> are the parameters, of the three-dimensional [[special orthogonal group]], SO(3). The rotation matrices about the [[standard basis|standard]] [[Cartesian coordinate system#Representing a vector in the standard basis|Cartesian basis vector]] <math>\hat{\mathbf{e}}_x, \hat{\mathbf{e}}_y, \hat{\mathbf{e}}_z</math> through angle {{math|Δ''θ''}}, and the corresponding generators of rotations {{math|1='''J''' = (''J<sub>x</sub>'', ''J<sub>y</sub>'', ''J<sub>z</sub>'')}}, are:

{|
|-
|<math>
\widehat{R}_x \equiv \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_x) = \begin{pmatrix}
1 & 0 & 0 \\
0 & \cos\Delta\theta & -\sin\Delta\theta \\
0 & \sin\Delta\theta & \cos\Delta\theta \\
\end{pmatrix} \,,
</math>
||<math> J_x \equiv J_1 = i\left.\frac{\partial \widehat{R}(\Delta \theta,\hat{\mathbf{e}}_x)}{\partial \Delta \theta}\right|_{\Delta\theta = 0} = i\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & -1 \\
0 & 1 & 0 \\
\end{pmatrix} \,, </math>
|-
|<math>
\widehat{R}_y \equiv \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_y) = \begin{pmatrix}
\cos\Delta\theta & 0 & \sin\Delta\theta \\
0 & 1 & 0 \\
-\sin\Delta\theta & 0 & \cos\Delta\theta \\
\end{pmatrix} \,,
</math>
||<math> J_y \equiv J_2 = i\left.\frac{\partial \widehat{R}(\Delta \theta,\hat{\mathbf{e}}_y)}{\partial \Delta \theta}\right|_{\Delta \theta = 0} = i\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0 \\
\end{pmatrix} \,, </math>
|-
|<math>
\widehat{R}_z \equiv \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_z) = \begin{pmatrix}
\cos\Delta\theta & -\sin\Delta\theta & 0 \\
\sin\Delta\theta & \cos\Delta\theta & 0 \\
0 & 0 & 1 \\
\end{pmatrix} \,,
</math>
||<math> J_z \equiv J_3 = i\left.\frac{\partial \widehat{R}(\Delta \theta,\hat{\mathbf{e}}_z)}{\partial \Delta \theta}\right|_{\Delta\theta = 0} = i\begin{pmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix} \,. </math>
|-
|}

More generally for rotations about an axis defined by <math>\hat{\mathbf{a}}</math>, the rotation matrix elements are:<ref>{{cite book| first=C.B. |last=Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| edition=2nd| page = [https://archive.org/details/mcgrawhillencycl1993park/page/1333 1333]| year=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park/page/1333}}</ref>

<math display="block">[\widehat{R}(\theta, \hat{\mathbf{a}})]_{ij} = (\delta_{ij} - a_i a_j) \cos\theta - \varepsilon_{ijk} a_k \sin\theta  + a_i a_j </math>

where {{math|''δ<sub>ij</sub>''}} is the [[Kronecker delta]], and {{math|''ε<sub>ijk</sub>''}} 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 {{math|''x''}}, {{math|''y''}}, or {{math|''z''}}-axis) then infer the general result, or use the general rotation matrix directly and [[tensor index notation]] with {{math|''δ<sub>ij</sub>''}} and {{math|''ε<sub>ijk</sub>''}}. To derive the infinitesimal rotation operator, which corresponds to small {{math|Δ''θ''}}, we use the [[small angle approximation]]s {{math|sin(Δ''θ'') ≈ Δ''θ''}} and {{math|cos(Δ''θ'') ≈ 1}}, then Taylor expand about {{math|'''r'''}} or {{math|''r<sub>i</sub>''}}, keep the first order term, and substitute the [[angular momentum operator]] components.

{| class="wikitable"
|-
!
! Rotation about <math>\hat{\mathbf{e}}_z</math>
! Rotation about <math>\hat{\mathbf{a}}</math>
|-
! Action on wavefunction
|| <math>\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_z)\psi(x,y,z,t) = \psi(x - \Delta\theta y, \Delta\theta x + y, z, t)</math>
|| <math>\widehat{R}(\Delta\theta,\hat{\mathbf{a}})\psi(r_i,t) = \psi(R_{ij} r_j , t) = \psi(r_i - \varepsilon_{ijk} a_k \Delta\theta r_j , t)</math>
|-
! Infinitesimal operator
|| <math> \widehat{R}(\Delta\theta,\hat{\mathbf{e}}_z) = I - \frac{i}{\hbar}\Delta\theta \widehat{L}_z</math>
|| <math> \begin{align}
\widehat{R}(\Delta\theta,\hat{\mathbf{a}}) 
& = I - (-\Delta\theta a_k\varepsilon_{kij} r_j ) \frac{\partial}{\partial r_i}\\
& = I - (\Delta\theta a_k\varepsilon_{kji} r_j ) \frac{\partial}{\partial r_i}\\
& = I - \Delta \theta \hat{\mathbf{a}} \cdot (\mathbf{r} \times \nabla ) \\
& = I - \frac{i\Delta \theta }{\hbar} \hat{\mathbf{a}} \cdot \widehat{\mathbf{L}} \\
\end{align}
</math>
|-
!Infinitesimal rotations
|<math> \widehat{R} = 1 - \frac{i}{\hbar}\Delta\theta \hat{\mathbf{a}} \cdot \widehat{\mathbf{L}} \,,\quad \widehat{\mathbf{L}} = i\hbar \hat{\mathbf{a}}\frac{\partial}{\partial \theta}</math>
| Same
|-
!Finite rotations
|<math> \lim_{N \to \infty} \left(1 - \frac{i}{\hbar}\frac{\Delta \theta}{N} \hat{\mathbf{a}} \cdot \widehat{\mathbf{L}} \right)^N = \exp\left( - \frac{i}{\hbar}\Delta \theta \hat{\mathbf{a}} \cdot \widehat{\mathbf{L}}\right) = \widehat{R} </math>
| Same
|-
! Generator
|| ''z''-component of the angular momentum operator <math>\widehat{L}_z = i\hbar\frac{\partial}{\partial \theta}</math>
|| Full angular momentum operator <math>\widehat{\mathbf{L}}</math>.
|-
|}

The {{math|''z''}}-component of angular momentum can be replaced by the component along the axis defined by <math>\hat{\mathbf{a}}</math>, using the [[dot product]] <math>\hat{\mathbf{a}}\cdot\widehat{\mathbf{L}}</math>.

Again, a finite rotation can be made from many small rotations, replacing {{math|Δ''θ''}} by {{math|Δ''θ''/''N''}} and taking the limit as {{mvar|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 {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}} about axis {{math|''i''}} can be written

<math display="block">R(\theta_1 + \theta_2 , \mathbf{e}_i) = R(\theta_1 \mathbf{e}_i)R(\theta_2 \mathbf{e}_i)\,,\quad [R(\theta_1 \mathbf{e}_i),R(\theta_2 \mathbf{e}_i)]=0\,.</math>

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

<math display="block"> [ L_i , L_j ] = i \hbar \varepsilon_{ijk} L_k. </math>

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 <math> \widehat{\mathbf{S}} = (\widehat{S_x}, \widehat{S_y}, \widehat{S_z}) </math>. The eigenvalues of its components are the possible outcomes (in units of <math>\hbar</math>) of a measurement of the spin projected onto one of the basis directions.

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

{{Equation box 1
|indent =:
|title='''Spin rotation operator''' (''finite'')
|equation = <math> \widehat{S}(\theta , \hat{\mathbf{a}}) = \exp\left( - \frac{i}{\hbar}\theta \hat{\mathbf{a}} \cdot \widehat{\mathbf{S}}\right) </math>
|cellpadding= 6
|border = 1
|border colour = black
|background colour=white}}

However, unlike orbital angular momentum in which the ''z''-projection quantum number ''{{ell}}'' 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 (2''s'' + 1)-dimensional spin matrix. This can be used to define a [[spinor]] as a column vector of 2''s'' + 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

<math display="block"> \widehat{\mathbf{S}} = \frac{\hbar}{2} \boldsymbol{\sigma} </math>

where the [[Pauli matrices]] in the standard representation are:

<math display="block">
\sigma_1 = \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\,,\quad
\sigma_2 = \sigma_y =
\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}
\,,\quad
\sigma_3 = \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
</math>

====Total angular momentum====

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

<math display="block"> \widehat{\mathbf{J}} = \widehat{\mathbf{L}} + \widehat{\mathbf{S}} </math>

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:

<math display="block"> \widehat{J}(\theta,\hat{\mathbf{a}}) = \exp\left( - \frac{i}{\hbar}\theta \hat{\mathbf{a}} \cdot \widehat{\mathbf{J}}\right) </math>

===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) [[Tommy Ohlsson|T. Ohlsson]] (2011)<ref>{{cite book|isbn=978-1-13950-4324|first=T. |last=Ohlsson|author-link= Tommy Ohlsson |title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory|publisher=Cambridge University Press|year=2011|pages=7–10|url=https://books.google.com/books?id=hRavtAW5EFcC&q=pauli-lubanski+pseudovector&pg=PA11}}
</ref> and E. Abers (2004).<ref>{{cite book|title=Quantum Mechanics|first=E. |last=Abers|publisher=Addison Wesley|year=2004|pages=11, 104, 105, 410–1|isbn=978-0-13-146100-0}}</ref>

Lorentz transformations can be parametrized by [[rapidity]] {{math|''φ''}} for a boost in the direction of a three-dimensional [[unit vector]] <math>\hat{\mathbf{n}} = (n_1, n_2, n_3)</math>, and a rotation angle {{mvar|θ}} about a three-dimensional [[unit vector]] <math>\hat{\mathbf{a}} = (a_1, a_2, a_3)</math> defining an axis, so <math>\varphi\hat{\mathbf{n}} = \varphi(n_1, n_2, n_3)</math> and <math>\theta\hat{\mathbf{a}} = \theta(a_1, a_2, a_3)</math> 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 <math>\widehat{R}_x, \widehat{R}_y, \widehat{R}_z </math> and generators {{math|1='''J''' = (''J''<sub>1</sub>, ''J''<sub>2</sub>, ''J''<sub>3</sub>)}} for pure rotations are:

{|
|-
|<math>
\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_x) = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \cos\Delta\theta & -\sin\Delta\theta \\
0 & 0 & \sin\Delta\theta & \cos\Delta\theta \\
\end{pmatrix} \,,
</math>
||<math> J_x = J_1 = i\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 \\
\end{pmatrix} \,, </math>
|-
|<math>
\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_y) = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & \cos\Delta\theta & 0 & \sin\Delta\theta \\
0 & 0 & 1 & 0 \\
0 & -\sin\Delta\theta & 0 & \cos\Delta\theta \\
\end{pmatrix} \,,
</math>
||<math> J_y = J_2 = i\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
\end{pmatrix}\,,</math>
|-
|<math>
\widehat{R}(\Delta\theta,\hat{\mathbf{e}}_z) = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & \cos\Delta\theta & -\sin\Delta\theta & 0 \\
0 & \sin\Delta\theta & \cos\Delta\theta & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix} \,,
</math>
||<math> J_z = J_3 = i\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}\,.</math>
|-
|}

The rotation matrices act on any [[four vector]] {{math|1='''A''' = (''A''<sub>0</sub>, ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>)}} and rotate the space-like components according to

<math display="block">\mathbf{A}' = \widehat{R}(\Delta\theta,\hat{\mathbf{n}})\mathbf{A}</math>

leaving the time-like coordinate unchanged. In matrix expressions, {{math|'''A'''}} is treated as a [[column vector]].

===Pure boosts in spacetime===

A boost with velocity {{math|''c''tanh''φ''}} in the ''x'', ''y'', or ''z'' directions given by the [[standard basis|standard]] [[Cartesian coordinate system#Representing a vector in the standard basis|Cartesian basis vector]] <math>\hat{\mathbf{e}}_x, \hat{\mathbf{e}}_y, \hat{\mathbf{e}}_z</math>, are the boost transformation matrices. These matrices <math> \widehat{B}_x, \widehat{B}_y, \widehat{B}_z</math> and the corresponding generators {{math|1='''K''' = (''K''<sub>1</sub>, ''K''<sub>2</sub>, ''K''<sub>3</sub>)}} are the remaining three group elements and generators of the Lorentz group:

{|
|-
|<math>
\widehat{B}_x \equiv \widehat{B}(\varphi,\hat{\mathbf{e}}_x) = \begin{pmatrix}
\cosh\varphi & \sinh\varphi & 0 & 0 \\
\sinh\varphi & \cosh\varphi & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}  \,,
</math>
|| <math> K_x = K_1 = i\left.\frac{\partial \widehat{B}(\varphi,\hat{\mathbf{e}}_x)}{\partial \varphi}\right|_{\varphi=0} = i \begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}  \,,</math>
|-
|<math>
\widehat{B}_y \equiv \widehat{B}(\varphi,\hat{\mathbf{e}}_y) = \begin{pmatrix}
\cosh\varphi & 0 & \sinh\varphi & 0 \\
0 & 1 & 0 & 0 \\
\sinh\varphi & 0 & \cosh\varphi & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}  \,,
</math>
|| <math> K_y = K_2 = i \left.\frac{\partial \widehat{B}(\varphi,\hat{\mathbf{e}}_y)}{\partial \varphi}\right|_{\varphi=0} = i \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix} \,,</math>
|-
|<math>
\widehat{B}_z \equiv \widehat{B}(\varphi,\hat{\mathbf{e}}_z) = \begin{pmatrix}
\cosh\varphi & 0 & 0 & \sinh\varphi \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\sinh\varphi & 0 & 0 & \cosh\varphi \\
\end{pmatrix} \,,
</math>
||<math> K_z = K_3 = i\left.\frac{\partial \widehat{B}(\varphi,\hat{\mathbf{e}}_z)}{\partial \varphi}\right|_{\varphi=0} = i \begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{pmatrix}  \,.</math>
|}

The boost matrices act on any four vector '''A''' = (''A''<sub>0</sub>, ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>) and mix the time-like and the space-like components, according to:

<math display="block">\mathbf{A}' = \widehat{B}(\varphi,\hat{\mathbf{n}}) \mathbf{A}</math>

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 [[#The Poincaré group in relativistic quantum mechanics and field theory|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)<ref>{{cite book |first= B.R. |last=Durney |title=Lorentz Transformations |year=2011 |url= https://archive.org/details/arxiv-1103.0156 |arxiv=1103.0156}}</ref> and H.L. Berk et al.<ref>{{cite news|title=The Proper Homogeneous Lorentz Transformation Operator ''e<sup>L</sup>'' = ''e''<sup>− ''ω''·''S'' − ''ξ''·''K''</sup>, Where's It Going, What's the Twist|first1=H.L. |last1=Berk |first2=K. |last2=Chaicherdsakul |first3=T. |last3=Udagawa |url=http://w3fusion.ph.utexas.edu/ifs/ifsreports/907_berk.pdf|location=Texas, Austin}}</ref> and references therein.

The boost and rotation generators have representations denoted {{math|''D''('''K''')}} and {{math|''D''('''J''')}} respectively, the capital {{math|''D''}} in this context indicates a [[group representation#Definitions|group representation]].

For the Lorentz group, the representations {{math|''D''('''K''')}} and {{math|''D''('''J''')}} of the generators {{math|'''K'''}} and {{math|'''J'''}} fulfill the following commutation rules.

{|class="wikitable"
|+Commutators
|-
!
!Generators
!Representations
|-
!Pure rotation
|<math>\left[J_a ,J_b\right] = i\varepsilon_{abc}J_c</math>
|<math>\left[{D(J_a)} ,{D(J_b)}\right] = i\varepsilon_{abc}{D(J_c)}</math>
|-
!Pure boost
|<math>\left[K_a ,K_b\right] = -i\varepsilon_{abc}J_c</math>
|<math>\left[{D(K_a)} ,{D(K_b)}\right] = -i\varepsilon_{abc}{D(J_c)}</math>
|-
!Lorentz transformation
|<math>\left[J_a ,K_b\right] = i\varepsilon_{abc}K_c</math>
|<math>\left[{D(J_a)} ,{D(K_b)}\right] = i\varepsilon_{abc}{D(K_c)}</math>
|}

In all commutators, the boost entities mixed with those for rotations, although rotations alone simply give another rotation. [[Exponential function#Lie algebras|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.

{|class="wikitable"
|+Transformation laws
!
!Transformations
!Representations
|-
!Pure boost
|<math>\widehat{B}(\varphi,\hat{\mathbf{n}}) = \exp\left(-\frac{i}{\hbar} \varphi\hat{\mathbf{n}} \cdot \mathbf{K}\right)</math>
|<math>D[\widehat{B}(\varphi,\hat{\mathbf{n}})] = \exp\left(-\frac{i}{\hbar} \varphi \hat{\mathbf{n}} \cdot D(\mathbf{K})\right)</math>
|-
!Pure rotation
|<math>\widehat{R}(\theta,\hat{\mathbf{a}}) = \exp\left(-\frac{i}{\hbar} \theta \hat{\mathbf{a}} \cdot \mathbf{J}\right) </math>
|<math>D[\widehat{R}(\theta,\hat{\mathbf{a}})] = \exp\left(-\frac{i}{\hbar}\theta \hat{\mathbf{a}} \cdot D(\mathbf{J})\right) </math>
|-
!Lorentz transformation
|<math>\Lambda(\varphi,\hat{\mathbf{n}},\theta,\hat{\mathbf{a}}) = \exp\left[-\frac{i}{\hbar} \left(\varphi\hat{\mathbf{n}} \cdot \mathbf{K} + \theta\hat{\mathbf{a}} \cdot \mathbf{J}\right)\right] </math>
|<math>D[\Lambda(\theta,\hat{\mathbf{a}},\varphi,\hat{\mathbf{n}})] = \exp\left[-\frac{i}{\hbar}\left( \varphi \hat{\mathbf{n}} \cdot D(\mathbf{K}) + \theta \hat{\mathbf{a}} \cdot D(\mathbf{J})\right)\right] </math>
|}

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

<math display="block">M^{0a} = -M^{a0} = K_a \,,\quad M^{ab} = \varepsilon_{abc} J_c \,.</math>

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

<math display="block">\omega_{0a} = - \omega_{a0} = \varphi n_a \,,\quad \omega_{ab} = \theta \varepsilon_{abc} a_c \,,</math>

The general Lorentz transformation is then:

<math display="block">\Lambda(\varphi,\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(\varphi \hat{\mathbf{n}} \cdot \mathbf{K} + \theta \hat{\mathbf{a}} \cdot \mathbf{J}\right)\right]</math>

with [[Einstein notation|summation over repeated matrix indices]] ''α'' and ''β''. The Λ matrices act on any four vector '''A''' = (''A''<sub>0</sub>, ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>) and mix the time-like and the space-like components, according to:

<math display="block">\mathbf{A}' = \Lambda(\varphi,\hat{\mathbf{n}}, \theta,\hat{\mathbf{a}}) \mathbf{A} </math>

===Transformations of spinor wavefunctions in relativistic quantum mechanics===

In [[relativistic quantum mechanics]], wavefunctions are no longer single-component scalar fields, but now 2(2''s'' + 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]] {{math|('''r''', ''t'') → Λ('''r''', ''t'')}} in [[Minkowski space]], all one-particle quantum states {{math|''ψ<sub>σ</sub>''}} locally transform under some [[Representation theory of the Lorentz group|representation]] {{math|''D''}} of the [[Lorentz group]]:<ref name="Weinberg">{{cite journal|last=Weinberg|first=S.|journal=Phys. Rev.|volume=133|pages=B1318–32|year=1964|doi=10.1103/PhysRev.133.B1318|title=Feynman Rules ''for Any'' spin|issue=5B|bibcode=1964PhRv..133.1318W|url=http://theory.fi.infn.it/becattini/files/weinberg1.pdf|access-date=2018-11-20|archive-date=2020-12-04|archive-url=https://web.archive.org/web/20201204053451/http://theory.fi.infn.it/becattini/files/weinberg1.pdf|url-status=dead}}<br/>{{cite journal|last=Weinberg|first=S.|journal=Phys. Rev.|volume=134|pages=B882–96|year=1964|doi=10.1103/PhysRev.134.B882|title=Feynman Rules ''for Any'' spin. II. Massless Particles|issue=4B|bibcode=1964PhRv..134..882W|url=http://theory.fi.infn.it/becattini/files/weinberg2.pdf|access-date=2013-06-05|archive-date=2022-03-09|archive-url=https://web.archive.org/web/20220309040610/http://theory.fi.infn.it/becattini/files/weinberg2.pdf|url-status=dead}}<br/>{{cite journal|last=Weinberg|first=S.|journal=Phys. Rev.|volume=181|pages=1893–9|year=1969|doi=10.1103/PhysRev.181.1893|title=Feynman Rules ''for Any'' spin. III|issue=5|bibcode=1969PhRv..181.1893W|url=http://theory.fi.infn.it/becattini/files/weinberg3.pdf|access-date=2013-06-05|archive-date=2022-03-25|archive-url=https://web.archive.org/web/20220325020742/http://theory.fi.infn.it/becattini/files/weinberg3.pdf|url-status=dead}}</ref>
<ref>{{cite arXiv
 | first = K. |last=Masakatsu
 | year = 2012
 | title = Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation
 |class=gr-qc
 | eprint = 1208.0644
}}</ref>

<math display="block">\psi_\sigma(\mathbf{r}, t) \rightarrow D(\Lambda) \psi_\sigma(\Lambda^{-1}(\mathbf{r}, t)) </math>

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

<math display="block">\psi(\mathbf{r},t) = \begin{bmatrix} \psi_{\sigma=s}(\mathbf{r},t) \\ \psi_{\sigma=s - 1}(\mathbf{r},t) \\ \vdots \\ \psi_{\sigma=-s + 1}(\mathbf{r},t) \\ \psi_{\sigma=-s}(\mathbf{r},t) \end{bmatrix}\quad\rightleftharpoons\quad {\psi(\mathbf{r},t)}^\dagger = \begin{bmatrix} {\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{bmatrix}</math>

===Real irreducible representations and spin===

The [[irreducible representation]]s of {{math|''D''('''K''')}} and {{math|''D''('''J''')}}, in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new operators:

<math display="block">\mathbf{A} = \frac{\mathbf{J} + i \mathbf{K}}{2}\,,\quad \mathbf{B} = \frac{\mathbf{J} - i \mathbf{K}}{2} \, ,</math>

so {{math|'''A'''}} and {{math|'''B'''}} are simply [[complex conjugate]]s of each other, it follows they satisfy the symmetrically formed commutators:

<math display="block">\left[A_i ,A_j\right] = \varepsilon_{ijk}A_k\,,\quad \left[B_i ,B_j\right] = \varepsilon_{ijk}B_k\,,\quad \left[A_i ,B_j\right] = 0\,,</math>

and these are essentially the commutators the orbital and spin angular momentum operators satisfy. Therefore, {{math|'''A'''}} and {{math|'''B'''}} form operator algebras analogous to angular momentum; same [[Ladder operator#Angular momentum|ladder operator]]s, ''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, {{math|''a,'' ''b''}}, with corresponding sets of values {{math|1=''m'' = ''a'', ''a'' − 1, ... −''a'' + 1, −''a''}} and {{math|1=''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:

<math display="block">\left(A_x\right)_{m'n',mn} = \delta_{n'n} \left(J_x^{(m)}\right)_{m'm}\,\quad \left(B_x\right)_{m'n',mn} = \delta_{m'm} \left(J_x^{(n)}\right)_{n'n}</math>
<math display="block">\left(A_y\right)_{m'n',mn} = \delta_{n'n} \left(J_y^{(m)}\right)_{m'm}\,\quad \left(B_y\right)_{m'n',mn} = \delta_{m'm} \left(J_y^{(n)}\right)_{n'n}</math>
<math display="block">\left(A_z\right)_{m'n',mn} = \delta_{n'n} \left(J_z^{(m)}\right)_{m'm}\,\quad \left(B_z\right)_{m'n',mn} = \delta_{m'm} \left(J_z^{(n)}\right)_{n'n}</math>

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

<math display="block">\left(J_z^{(m)}\right)_{m'm} = m\delta_{m'm} \,\quad \left(J_x^{(m)} \pm i J_y^{(m)}\right)_{m'm} = m\delta_{a',a\pm 1}\sqrt{(a \mp m)(a \pm m + 1)}</math>

and similarly for '''J'''<sup>(''n'')</sup>.<ref group="note">Sometimes the [[tuple]] abbreviations:

<math display="block">\left(\mathbf{A}\right)_{m'n',mn} \equiv \left[\left(A_x\right)_{m'n',mn}, \left(A_y\right)_{m'n',mn}, \left(A_z\right)_{m'n',mn}\right]</math> 
<math display="block">\left(\mathbf{B}\right)_{m'n',mn} \equiv \left[\left(B_x\right)_{m'n',mn}, \left(B_y\right)_{m'n',mn}, \left(B_z\right)_{m'n',mn}\right]</math> 
<math display="block">\left(\mathbf{J}^{(m)}\right)_{m'm} \equiv \left[\left(J_x^{(m)}\right)_{m'm}, \left(J_y^{(m)}\right)_{m'm}, \left(J_z^{(m)}\right)_{m'm}\right]</math>

are used.</ref> The three '''J'''<sup>(''m'')</sup> matrices are each {{math|(2''m'' + 1)×(2''m'' + 1)}} square matrices, and the three '''J'''<sup>(''n'')</sup> are each {{math|(2''n'' + 1)×(2''n'' + 1)}} square matrices. The integers or half-integers ''m'' and ''n'' numerate all the irreducible representations by, in equivalent notations used by authors: {{math|''D''<sup>(''m'', ''n'')</sup> ≡ (''m'', ''n'') ≡ ''D''<sup>(''m'')</sup> &otimes; ''D''<sup>(''n'')</sup>}}, which are each {{math|[(2''m'' + 1)(2''n'' + 1)]×[(2''m'' + 1)(2''n'' + 1)]}} square matrices.

Applying this to particles with spin {{mvar|s}};
*left-handed {{math|(2''s'' + 1)}}-component spinors transform under the real irreps {{math|''D''<sup>(''s'', 0)</sup>}},
*right-handed {{math|(2''s'' + 1)}}-component spinors transform under the real irreps {{math|''D''<sup>(0, ''s'')</sup>}},
*taking [[direct sum]]s symbolized by {{math|&oplus;}} (see [[direct sum of matrices]] for the simpler matrix concept), one obtains the representations under which {{math|2(2''s'' + 1)}}-component spinors transform: {{math|''D''<sup>(''m'', ''n'')</sup> &oplus; ''D''<sup>(''n'', ''m'')</sup>}} where {{math|1=''m'' + ''n'' = ''s''}}. These are also real irreps, but as shown above, they split into complex conjugates.

In these cases the {{math|''D''}} refers to any of {{math|''D''('''J''')}}, {{math|''D''('''K''')}}, or a full Lorentz transformation {{math|''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 {{math|''D''<sup>(1/2, 0)</sup>}} and the 2-component right-handed Weyl spinor transforms under {{math|''D''<sup>(0, 1/2)</sup>}}. Dirac spinors satisfying the Dirac equation transform under the representation {{math|''D''<sup>(1/2, 0)</sup> &oplus; ''D''<sup>(0, 1/2)</sup>}}, the direct sum of the irreps for the Weyl spinors.

==The Poincaré group in relativistic quantum mechanics and field theory==

[[Space translation symmetry|Space translations]], [[Time translation symmetry|time translations]], [[Rotational symmetry|rotations]], and [[Lorentz Boost|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 {{math|1='''X''' = (''ct'', −'''r''')}}, and in parallel so can energy and momentum which combine into a [[four-momentum]] vector {{math|1='''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 {{math|1=Δ'''X''' = (''c''Δ''t'', −Δ'''r''')}}, and the energy and momentum operators are inserted in the four-momentum to obtain a four-momentum operator,

<math display="block">\widehat{\mathbf{P}} = \left(\frac{\widehat{E}}{c},-\widehat{\mathbf{p}}\right)  = i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right) \,, </math>

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

<math display="block">\widehat{X}(\Delta \mathbf{X}) = \exp\left(-\frac{i}{\hbar}\Delta\mathbf{X}\cdot\widehat{\mathbf{P}}\right) = \exp\left[-\frac{i}{\hbar}\left(\Delta t\widehat{E} + \Delta \mathbf{r} \cdot\widehat{\mathbf{p}}\right)\right] \,. </math>

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:<ref>{{cite book|title=General Principles of Quantum Field Theory|first=N.N. |last=Bogolubov|publisher=Springer|edition=2nd|isbn=0-7923-0540-X|year=1989| page=272|url=https://books.google.com/books?id=7VLMj4AvvicC&q=pauli-lubanski+pseudovector&pg=PA273}}</ref><ref>{{harvnb|Ohlsson|2011|p=10}}</ref>

* <math>[P_\mu, P_\nu] = 0\,</math>
* <math>\frac{ 1 }{ i }[M_{\mu\nu}, P_\rho] = \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu\,</math>
* <math>\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}\,</math>

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 bracket]]s.

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

<math display="block">W_{\mu}=\frac{1}{2}\varepsilon_{\mu \nu \rho \sigma} J^{\nu \rho} P^\sigma ,</math>

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''':

<math display="block">\left[P^{\mu},W^{\nu}\right]=0 \,, </math>
<math display="block">\left[J^{\mu \nu},W^{\rho}\right]=i \left( \eta^{\rho \nu} W^{\mu} - \eta^{\rho \mu} W^{\nu}\right) \,, </math>
<math display="block">\left[W_{\mu},W_{\nu}\right]=-i \epsilon_{\mu \nu \rho \sigma} W^{\rho}  P^{\sigma} \,. </math>

Invariants constructed from '''W''', instances of [[Casimir invariant]]s 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 group]]s 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 <math> \widehat{U} </math> be a unitary operator, so the inverse is the [[Hermitian adjoint]] <math> \widehat{U}^{-1} = \widehat{U}^\dagger </math>, which commutes with the Hamiltonian:

<math display="block">\left[\widehat{U}, \widehat{H} \right]=0 </math>

then the observable corresponding to the operator <math> \widehat{U}</math> is conserved, and the Hamiltonian is invariant under the transformation <math> \widehat{U}</math>.

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:

<math display="block">U=e^{-i\theta}</math>

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:

<math display="block">U=e^{-ia\theta}</math>

====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'':

<math display="block">U = \begin{pmatrix}
a & b \\
-b^\star & a^\star \\
\end{pmatrix}</math>

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

<math display="block"> \det(U) = aa^\star + bb^\star = {|a|}^2 + {|b|}^2 = 1 </math>

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:

<math display="block"> [ \sigma_a , \sigma_b ] = 2i \hbar \varepsilon_{abc} \sigma_c </math>

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

<math display="block">U(\theta,\hat{\mathbf{e}}_j) = e^{i \theta \sigma_j /2}</math>

where ''σ<sub>j</sub>'' 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).<ref>{{cite arXiv|first=D. |last=Bonastos|title=Symmetry Algebra of the Planar Anisotropic Quantum Harmonic Oscillator with Rational Ratio of Frequencies|year=1994|eprint=hep-th/9402099|display-authors=etal}}</ref>

====U(3) and SU(3)====

The eight [[Gell-Mann matrices]] {{math|''λ<sub>n</sub>''}} (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):

<math display="block">U(\theta,\hat{\mathbf{e}}_j) = \exp\left(-\frac{i}{2} \sum_{n=1}^8 \theta_n \lambda_n \right) </math>

where {{math|''θ<sub>n</sub>''}} are eight independent parameters. The {{math|''λ<sub>n</sub>''}} matrices satisfy the commutator:

<math display="block">\left[\lambda_a, \lambda_b \right] = 2i f_{abc}\lambda_c</math>

where the indices {{mvar|a}}, {{mvar|b}}, {{mvar|c}} take the values 1, 2, 3, ..., 8. The structure constants ''f<sub>abc</sub>'' 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):

<math display="block">|r\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\,,\quad |g\rangle = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\,,\quad |b\rangle = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}</math>

the colour states are eigenstates of the {{math|''λ''<sub>3</sub>}} and {{math|''λ''<sub>8</sub>}} matrices, while the other matrices mix colour states together.

The eight [[gluon]]s states (8-dimensional column vectors) are simultaneous eigenstates of the [[adjoint representation]] of {{math|SU(3)}}, the 8-dimensional representation acting on its own Lie algebra {{math|su(3)}}, for the {{math|''λ''<sub>3</sub>}} and {{math|''λ''<sub>8</sub>}} 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 {{math|SU(3)}} of color. The representations of SU(3) can be described by a "theorem of the highest weight".<ref>{{harvnb|Hall|2015|loc=[https://link.springer.com/chapter/10.1007/978-3-319-13467-3_6 6. The Representations of sl(3;C)]}}</ref>

===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 (physics)|Parity]] mirrors the [[orientation (vector space)|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]].
*[[T-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===

*{{slink|Parity (physics)|Molecules}}
*[[CPT theorem]]
*[[CP violation]]
*[[Non-Hermitian quantum mechanics|PT symmetry]]
*[[Lorentz violation]]

===Gauge theory===

{{main|Gauge theory}}

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

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

The strong (color) interaction is mediated by [[gluon]]s, which can have eight [[color charge]]s. There are eight [[gluon field strength tensor]]s with corresponding [[gluon four potential]]s field, each possessing gauge symmetry.

===The strong (color) interaction===

====Color charge====

Analogous to the spin operator, there are [[color charge operator]]s in terms of the Gell-Mann matrices {{math|''λ<sub>j</sub>''}}:

<math display="block">\hat{F}_j = \frac{1}{2}\lambda_j </math>

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

<math display="block">\left[\hat{F}_j,\hat{H}\right] = 0 </math>

====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 [[Magnetic monopole#Duality transformation|duality transformation]].

====Electroweak symmetry====

*[[Electroweak symmetry]]
*[[Electroweak symmetry breaking]]

===Supersymmetry===

{{main|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 [[gravitino]]s, a spin 3/2 particle with mass, its supersymmetric partner being the [[graviton]].

==Exchange symmetry==
{{main|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 <math>\left| \psi \right|^2</math> for a system of identical particles, the [[wave function]] <math>\psi</math> must either remain the same or change sign upon such an exchange. More generally, for a system of ''n'' identical particles the wave function <math>\psi</math> must transform as an irreducible representation of the finite [[symmetric group]] '''S<sub>''n''</sub>'''. It turns out that, according to the [[spin-statistics theorem]], fermion states transform as the antisymmetric irreducible representation of '''S<sub>''n''</sub>''' 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),<ref>{{cite book|last=Feynman|first=Richard|title=The 1986 Dirac Memorial Lectures|date=13 July 1999|publisher=Cambridge University Press|isbn=978-0-521-65862-1|pages=57}}</ref> the symmetric nature of the wave function depends on the particle's [[Spin (physics)|spin]] after the [[Rotation operator (quantum mechanics)|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 [[boson]]s, or particles with a [[symmetric]] wave function. The particles for which the wave function of the system changes sign are called [[fermion]]s, or particles with an [[antisymmetric relation|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 [[Pauli exclusion principle|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).


==See also==
{{cols|colwidth=26em}}
* [[Symmetric group]]
* [[Spin-statistics theorem]]
* [[Projective representation]]
* [[Casimir operator]]
* [[Pauli–Lubanski pseudovector]]
* [[Symmetries in general relativity]]
* [[Renormalization group]]
* [[Representation of a Lie group]]
* [[Representation theory of the Poincaré group]]
* [[Representation theory of the Lorentz group]]
{{colend}}

==Footnotes==

{{Reflist|group="note"|1}}

==References==
{{reflist}}

==Further reading==
{{refbegin}}
*{{cite book |title=Group theory for the standard model and beyond |publisher=Taylor & Francis |first=K.J. |last=Barnes |isbn=978-142-007-874-9 |year=2010 |series=Series in high energy physics, cosmology, and gravitation |url=https://books.google.com/books?id=OG6BJAAACAAJ&q=Group+theory+for+the+standard+model+and+beyond}}
*{{cite book |title=Symmetry in quantum mechanics: From angular momentum to supersymmetry |publisher=Institute of physics (Bristol and Philadelphia) |first1=M. |last1=Chaichian |first2=R. |last2=Hagedorn |isbn=0-7503-0408-1 |year=1998 |series=Graduate student series in physics}}
*{{cite book |first=Brian C. |last=Hall |title=Quantum Theory for Mathematicians |series=Graduate Texts in Mathematics |volume=267 |publisher=Springer |year=2013 |isbn=978-1-4614-7115-8 |doi=10.1007/978-1-4614-7116-5 |bibcode=2013qtm..book.....H |s2cid=117837329}}
*{{cite book |first=Brian C. |last=Hall |title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction |edition=2nd |series=Graduate Texts in Mathematics |volume=222 |publisher=Springer |year=2015 |isbn=978-3-319-13466-6 |doi=10.1007/978-3-319-13467-3}}
*{{cite book |title=Symmetries and Conservation Laws in Particle Physics: An Introduction to Group Theory for Particle Physicists |publisher=World Scientific |year=2011 |isbn=978-184-816-703-2 |first=S. |last=Haywood |url={{GBurl |AyWcQ-9jdeUC |pg=PA131}}}}
*{{cite book |title=Symmetry in molecules and crystals |publisher=Ellis Horwood series in physical chemistry |first=M.F.C. |last=Ladd |isbn=0-85312-255-5 |year=1989 |series=Solid state science |url=https://books.google.com/books?id=cgfwAAAAMAAJ}}
*{{cite book |title=Symmetries in physics |publisher=Springer |first1=W. |last1=Ludwig |first2=C. |last2=Falter |isbn=3-540-60284-4 |edition=2nd |year=1996 |series=Solid state science}}
*{{cite book |first1=B.R. |last1=Martin |first2=G. |last2=Shaw |title=Particle Physics |date=2013 |edition=3rd |series=Manchester Physics Series |publisher=Wiley |pages=3 |isbn=978-0-470-03294-7 |url={{GBurl|7tsVcsm84rEC|pg=PT6}}}}
*{{cite book |title=Quantum Field Theory |first=D. |last=McMahon |publisher=McGraw Hill |year=2008 |isbn=978-0-07-154382-8}}
*{{cite book |first=V. |last=Moretti |title=Spectral Theory and Quantum Mechanics; Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation |series=Unitext |edition=2nd |publisher=Springer |year=2017 |volume=110 |url=https://link.springer.com/book/10.1007/978-3-319-70706-8 |isbn=978-3-319-70705-1 |doi=10.1007/978-3-319-70706-8 |bibcode=2017stqm.book.....M |s2cid=125121522}}
{{refend}}

==External links==
*''The molecular symmetry group''[https://physics.uwo.ca/~mhoude2/courses/astro9701/Hamiltonian_symmetry.pdf] @ The University of Western Ontario 
*[http://bohr.physics.berkeley.edu/classes/221/1011/notes/wigeck.pdf (2010) ''Irreducible Tensor Operators and the Wigner-Eckart Theorem''] {{Webarchive|url=https://web.archive.org/web/20140720064630/http://bohr.physics.berkeley.edu/classes/221/1011/notes/wigeck.pdf |date=2014-07-20 }}
*{{cite web |first=R.D. |last=Reece |date=2006 |title=A Derivation of the Quantum Mechanical Momentum Operator in the Position Representation |url=https://philpapers.org/rec/REEADO-4}}
*{{cite web |url=http://physics.uoregon.edu/~soper/QuantumMechanics/momentum.pdf |first=D.E. |last=Soper |date=2011 |title=Position and momentum in quantum mechanics}}
*[http://personalpages.to.infn.it/~billo/didatt/gruppi/liegroups.pdf ''Lie groups'']
*{{cite web |url=http://www.hep.caltech.edu/~fcp/math/groupTheory/Lie.pdf |first=F. |last=Porter |date=2009 |title=Lie Groups and Lie Algebras |access-date=2013-06-05 |archive-date=2017-03-29 |archive-url=https://web.archive.org/web/20170329071231/http://www.hep.caltech.edu/~fcp/math/groupTheory/Lie.pdf |url-status=dead }}
*[http://www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter7.pdf ''Continuous Groups, Lie Groups, and Lie Algebras''] {{Webarchive|url=https://web.archive.org/web/20160304000910/http://www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter7.pdf |date=2016-03-04 }}
*{{cite web |url=http://www.nat.vu.nl/~mulders/QFT-0.pdf |first=P.J. |last=Mulders |date=November 2011 |id=6.04 |title=Quantum field theory |publisher=Department of Theoretical Physics, VU University}}
*{{cite arXiv |eprint=math-ph/0005032 |first=B.C. |last=Hall |date=2000 |title=An Elementary Introduction to Groups and Representations}}

{{Quantum mechanics topics}}

[[Category:Pauli exclusion principle]]
[[Category:Special relativity]]
[[Category:Quantum field theory]]
[[Category:Group theory]]
[[Category:Theoretical physics]]