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

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

$${\displaystyle {\widehat {\Omega }}\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t')}$$

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

$${\displaystyle {\widehat {\Omega }}\left|\mathbf {r} (t)\right\rangle =\left|\mathbf {r} '(t')\right\rangle }$$

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

$${\displaystyle {\widehat {A}}\psi ={\widehat {\Omega }}^{\dagger }{\widehat {A}}{\widehat {\Omega }}\psi \quad \Rightarrow \quad {\widehat {\Omega }}{\widehat {A}}\psi ={\widehat {A}}{\widehat {\Omega }}\psi }$$

and thus:

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

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

Overview of Lie group theory

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]

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

• the dimension of the group, N, is the number of parameters it has.
• the group elements, g, in G are functions of the parameters:
  $${\displaystyle g=G(\xi _{1},\xi _{2},\dots )}$$

  
  and all parameters set to zero returns the identity element of the group:
  $${\displaystyle I=G(0,0,\dots )}$$

  
  Group elements are often matrices which act on vectors, or transformations acting on functions.
• The generators of the group are the partial derivatives of the group elements with respect to the group parameters with the result evaluated when the parameter is set to zero:
  $${\displaystyle X_{j}=\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}$$

  
  In the language of manifolds, the generators are the elements of the tangent space to G at the identity. The generators are also known as infinitesimal group elements or as the elements of the Lie algebra of G. (See the discussion below of the commutator.)
  One aspect of generators in theoretical physics is they can be constructed themselves as operators corresponding to symmetries, which may be written as matrices, or as differential operators. In quantum theory, for unitary representations of the group, the generators require a factor of i:
  $${\displaystyle X_{j}=i\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}$$

  
  The generators of the group form a vector space, which means linear combinations of generators also form a generator.
• The generators (whether matrices or differential operators) satisfy the commutation relations:
  $${\displaystyle \left[X_{a},X_{b}\right]=if_{abc}X_{c}}$$

  
  where f_abc are the (basis dependent) structure constants of the group. This makes, together with the vector space property, the set of all generators of a group a Lie algebra. Due to the antisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices.
• The representations of the group then describe the ways that the group G (or its Lie algebra) can act on a vector space. (The vector space might be, for example, the space of eigenvectors for a Hamiltonian having G as its symmetry group.) We denote the representations using a capital D. One can then differentiate D to obtain a representation of the Lie algebra, often also denoted by D. These two representations are related as follows:
  $${\displaystyle D[g(\xi _{j})]\equiv D(\xi _{j})=e^{i\xi _{j}D(X_{j})}}$$

  
  without summation on the repeated index j. Representations are linear operators that take in group elements and preserve the composition rule:
  $${\displaystyle D(\xi _{a})D(\xi _{b})=D(\xi _{a}\xi _{b}).}$$

  

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

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

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

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

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

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

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

Commutators

Operators	Generators
${\displaystyle \left[{\widehat {T}}(\mathbf {r} _{1}),{\widehat {T}}(\mathbf {r} _{2})\right]\psi (\mathbf {r} ,t)=0}$	${\displaystyle \left[{\widehat {p}}_{i},{\widehat {p}}_{j},\right]\psi (\mathbf {r} ,t)=0}$
${\displaystyle \left[{\widehat {U}}(t_{1}),{\widehat {U}}(t_{2})\right]\psi (\mathbf {r} ,t)=0}$	${\displaystyle \left[{\widehat {E}},{\widehat {p}}_{i},\right]\psi (\mathbf {r} ,t)=0}$

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

$${\displaystyle {\widehat {U}}(t)=\exp \left(-{\frac {i\Delta tE}{\hbar }}\right)}$$

and all stationary states have the form

$${\displaystyle \psi (\mathbf {r} ,t+t_{0})={\widehat {U}}(t-t_{0})\psi (\mathbf {r} ,t_{0})}$$

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

An alternative notation is ${\displaystyle {\widehat {U}}(t-t_{0})\equiv U(t,t_{0})}$.

Angular momentum as the generator of rotations

Orbital angular momentum

The rotation operator acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle Δθ:

$${\displaystyle {R}(\Delta \theta ,{\hat {\mathbf {a} }})\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t)}$$

where r′ are the rotated coordinates about an axis defined by a unit vector ${\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})}$ through an angular increment Δθ, given by:

$${\displaystyle \mathbf {r} '={\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\mathbf {r} \,.}$$

where ${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})}$ is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis ${\displaystyle \Delta \theta {\hat {\mathbf {a} }}=\Delta \theta (a_{1},a_{2},a_{3})}$ are the parameters, of the three-dimensional special orthogonal group, SO(3). The rotation matrices about the standard Cartesian basis vector ${\displaystyle {\hat {\mathbf {e} }}_{x},{\hat {\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}}$ through angle Δθ, and the corresponding generators of rotations J = (J_x, J_y, J_z), are:

${\displaystyle {\widehat {R}}_{x}\equiv {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{x})={\begin{pmatrix}1&0&0\\0&\cos \Delta \theta &-\sin \Delta \theta \\0&\sin \Delta \theta &\cos \Delta \theta \\\end{pmatrix}}\,,}$	${\displaystyle J_{x}\equiv J_{1}=i\left.{\frac {\partial {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{x})}{\partial \Delta \theta }}\right|_{\Delta \theta =0}=i{\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\\\end{pmatrix}}\,,}$
${\displaystyle {\widehat {R}}_{y}\equiv {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{y})={\begin{pmatrix}\cos \Delta \theta &0&\sin \Delta \theta \\0&1&0\\-\sin \Delta \theta &0&\cos \Delta \theta \\\end{pmatrix}}\,,}$	${\displaystyle J_{y}\equiv J_{2}=i\left.{\frac {\partial {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{y})}{\partial \Delta \theta }}\right|_{\Delta \theta =0}=i{\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\\\end{pmatrix}}\,,}$
${\displaystyle {\widehat {R}}_{z}\equiv {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{z})={\begin{pmatrix}\cos \Delta \theta &-\sin \Delta \theta &0\\\sin \Delta \theta &\cos \Delta \theta &0\\0&0&1\\\end{pmatrix}}\,,}$	${\displaystyle J_{z}\equiv J_{3}=i\left.{\frac {\partial {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{z})}{\partial \Delta \theta }}\right|_{\Delta \theta =0}=i{\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\\\end{pmatrix}}\,.}$

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

$${\displaystyle [{\widehat {R}}(\theta ,{\hat {\mathbf {a} }})]_{ij}=(\delta _{ij}-a_{i}a_{j})\cos \theta -\varepsilon _{ijk}a_{k}\sin \theta +a_{i}a_{j}}$$

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

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

	Rotation about ${\displaystyle {\hat {\mathbf {e} }}_{z}}$	Rotation about ${\displaystyle {\hat {\mathbf {a} }}}$
Action on wavefunction	${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{z})\psi (x,y,z,t)=\psi (x-\Delta \theta y,\Delta \theta x+y,z,t)}$	${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\psi (r_{i},t)=\psi (R_{ij}r_{j},t)=\psi (r_{i}-\varepsilon _{ijk}a_{k}\Delta \theta r_{j},t)}$
Infinitesimal operator	${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{z})=I-{\frac {i}{\hbar }}\Delta \theta {\widehat {L}}_{z}}$	${\displaystyle {\begin{aligned}{\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})&=I-(-\Delta \theta a_{k}\varepsilon _{kij}r_{j}){\frac {\partial }{\partial r_{i}}}\\&=I-(\Delta \theta a_{k}\varepsilon _{kji}r_{j}){\frac {\partial }{\partial r_{i}}}\\&=I-\Delta \theta {\hat {\mathbf {a} }}\cdot (\mathbf {r} \times \nabla )\\&=I-{\frac {i\Delta \theta }{\hbar }}{\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}\\\end{aligned}}}$
Infinitesimal rotations	${\displaystyle {\widehat {R}}=1-{\frac {i}{\hbar }}\Delta \theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}\,,\quad {\widehat {\mathbf {L} }}=i\hbar {\hat {\mathbf {a} }}{\frac {\partial }{\partial \theta }}}$	Same
Finite rotations	${\displaystyle \lim _{N\to \infty }\left(1-{\frac {i}{\hbar }}{\frac {\Delta \theta }{N}}{\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}\right)^{N}=\exp \left(-{\frac {i}{\hbar }}\Delta \theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}\right)={\widehat {R}}}$	Same
Generator	z-component of the angular momentum operator ${\displaystyle {\widehat {L}}_{z}=i\hbar {\frac {\partial }{\partial \theta }}}$	Full angular momentum operator ${\displaystyle {\widehat {\mathbf {L} }}}$.

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

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

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

$${\displaystyle R(\theta _{1}+\theta _{2},\mathbf {e} _{i})=R(\theta _{1}\mathbf {e} _{i})R(\theta _{2}\mathbf {e} _{i})\,,\quad [R(\theta _{1}\mathbf {e} _{i}),R(\theta _{2}\mathbf {e} _{i})]=0\,.}$$

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

$${\displaystyle [L_{i},L_{j}]=i\hbar \varepsilon _{ijk}L_{k}.}$$

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

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

Spin angular momentum

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

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

Spin rotation operator (finite)

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

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

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

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

$${\displaystyle {\widehat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }}}$$

where the Pauli matrices in the standard representation are:

$${\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\,,\quad \sigma _{2}=\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\,,\quad \sigma _{3}=\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$$

Total angular momentum

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

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

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:

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

Conserved quantities in the quantum harmonic oscillator

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

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

Pure rotations in spacetime

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

${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{x})={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos \Delta \theta &-\sin \Delta \theta \\0&0&\sin \Delta \theta &\cos \Delta \theta \\\end{pmatrix}}\,,}$	${\displaystyle J_{x}=J_{1}=i{\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{pmatrix}}\,,}$
${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{y})={\begin{pmatrix}1&0&0&0\\0&\cos \Delta \theta &0&\sin \Delta \theta \\0&0&1&0\\0&-\sin \Delta \theta &0&\cos \Delta \theta \\\end{pmatrix}}\,,}$	${\displaystyle J_{y}=J_{2}=i{\begin{pmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\\\end{pmatrix}}\,,}$
${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{z})={\begin{pmatrix}1&0&0&0\\0&\cos \Delta \theta &-\sin \Delta \theta &0\\0&\sin \Delta \theta &\cos \Delta \theta &0\\0&0&0&1\\\end{pmatrix}}\,,}$	${\displaystyle J_{z}=J_{3}=i{\begin{pmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\\\end{pmatrix}}\,.}$

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

$${\displaystyle \mathbf {A} '={\widehat {R}}(\Delta \theta ,{\hat {\mathbf {n} }})\mathbf {A} }$$

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

Pure boosts in spacetime

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

${\displaystyle {\widehat {B}}_{x}\equiv {\widehat {B}}(\varphi ,{\hat {\mathbf {e} }}_{x})={\begin{pmatrix}\cosh \varphi &\sinh \varphi &0&0\\\sinh \varphi &\cosh \varphi &0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}\,,}$	${\displaystyle K_{x}=K_{1}=i\left.{\frac {\partial {\widehat {B}}(\varphi ,{\hat {\mathbf {e} }}_{x})}{\partial \varphi }}\right|_{\varphi =0}=i{\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}}\,,}$
${\displaystyle {\widehat {B}}_{y}\equiv {\widehat {B}}(\varphi ,{\hat {\mathbf {e} }}_{y})={\begin{pmatrix}\cosh \varphi &0&\sinh \varphi &0\\0&1&0&0\\\sinh \varphi &0&\cosh \varphi &0\\0&0&0&1\\\end{pmatrix}}\,,}$	${\displaystyle K_{y}=K_{2}=i\left.{\frac {\partial {\widehat {B}}(\varphi ,{\hat {\mathbf {e} }}_{y})}{\partial \varphi }}\right|_{\varphi =0}=i{\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\\\end{pmatrix}}\,,}$
${\displaystyle {\widehat {B}}_{z}\equiv {\widehat {B}}(\varphi ,{\hat {\mathbf {e} }}_{z})={\begin{pmatrix}\cosh \varphi &0&0&\sinh \varphi \\0&1&0&0\\0&0&1&0\\\sinh \varphi &0&0&\cosh \varphi \\\end{pmatrix}}\,,}$	${\displaystyle K_{z}=K_{3}=i\left.{\frac {\partial {\widehat {B}}(\varphi ,{\hat {\mathbf {e} }}_{z})}{\partial \varphi }}\right|_{\varphi =0}=i{\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\\\end{pmatrix}}\,.}$

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

$${\displaystyle \mathbf {A} '={\widehat {B}}(\varphi ,{\hat {\mathbf {n} }})\mathbf {A} }$$

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

Combining boosts and rotations

Products of rotations give another rotation (a frequent exemplification of a subgroup), while products of boosts and boosts or of rotations and boosts cannot be expressed as pure boosts or pure rotations. In general, any Lorentz transformation can be expressed as a product of a pure rotation and a pure boost. For more background see (for example) B.R. Durney (2011)^[6] and H.L. Berk et al.^[7] and references therein.

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

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

Commutators

	Generators	Representations
Pure rotation	${\displaystyle \left[J_{a},J_{b}\right]=i\varepsilon _{abc}J_{c}}$	${\displaystyle \left[{D(J_{a})},{D(J_{b})}\right]=i\varepsilon _{abc}{D(J_{c})}}$
Pure boost	${\displaystyle \left[K_{a},K_{b}\right]=-i\varepsilon _{abc}J_{c}}$	${\displaystyle \left[{D(K_{a})},{D(K_{b})}\right]=-i\varepsilon _{abc}{D(J_{c})}}$
Lorentz transformation	${\displaystyle \left[J_{a},K_{b}\right]=i\varepsilon _{abc}K_{c}}$	${\displaystyle \left[{D(J_{a})},{D(K_{b})}\right]=i\varepsilon _{abc}{D(K_{c})}}$

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	${\displaystyle {\widehat {B}}(\varphi ,{\hat {\mathbf {n} }})=\exp \left(-{\frac {i}{\hbar }}\varphi {\hat {\mathbf {n} }}\cdot \mathbf {K} \right)}$	${\displaystyle D[{\widehat {B}}(\varphi ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i}{\hbar }}\varphi {\hat {\mathbf {n} }}\cdot D(\mathbf {K} )\right)}$
Pure rotation	${\displaystyle {\widehat {R}}(\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{\hbar }}\theta {\hat {\mathbf {a} }}\cdot \mathbf {J} \right)}$	${\displaystyle D[{\widehat {R}}(\theta ,{\hat {\mathbf {a} }})]=\exp \left(-{\frac {i}{\hbar }}\theta {\hat {\mathbf {a} }}\cdot D(\mathbf {J} )\right)}$
Lorentz transformation	${\displaystyle \Lambda (\varphi ,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf {a} }})=\exp \left[-{\frac {i}{\hbar }}\left(\varphi {\hat {\mathbf {n} }}\cdot \mathbf {K} +\theta {\hat {\mathbf {a} }}\cdot \mathbf {J} \right)\right]}$	${\displaystyle D[\Lambda (\theta ,{\hat {\mathbf {a} }},\varphi ,{\hat {\mathbf {n} }})]=\exp \left[-{\frac {i}{\hbar }}\left(\varphi {\hat {\mathbf {n} }}\cdot D(\mathbf {K} )+\theta {\hat {\mathbf {a} }}\cdot D(\mathbf {J} )\right)\right]}$

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

$${\displaystyle M^{0a}=-M^{a0}=K_{a}\,,\quad M^{ab}=\varepsilon _{abc}J_{c}\,.}$$

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

$${\displaystyle \omega _{0a}=-\omega _{a0}=\varphi n_{a}\,,\quad \omega _{ab}=\theta \varepsilon _{abc}a_{c}\,,}$$

The general Lorentz transformation is then:

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

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

$${\displaystyle \mathbf {A} '=\Lambda (\varphi ,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf {a} }})\mathbf {A} }$$

Transformations of spinor wavefunctions in relativistic quantum mechanics

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

Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψ_σ locally transform under some representation D of the Lorentz group:^{[8] [9]}

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

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

$${\displaystyle \psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{\sigma =s}(\mathbf {r} ,t)\\\psi _{\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }={\begin{bmatrix}{\psi _{\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{\sigma =-s+1}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}}$$

Real irreducible representations and spin

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

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

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

$${\displaystyle \left[A_{i},A_{j}\right]=\varepsilon _{ijk}A_{k}\,,\quad \left[B_{i},B_{j}\right]=\varepsilon _{ijk}B_{k}\,,\quad \left[A_{i},B_{j}\right]=0\,,}$$

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

$${\displaystyle \left(A_{x}\right)_{m'n',mn}=\delta _{n'n}\left(J_{x}^{(m)}\right)_{m'm}\,\quad \left(B_{x}\right)_{m'n',mn}=\delta _{m'm}\left(J_{x}^{(n)}\right)_{n'n}}$$

$${\displaystyle \left(A_{y}\right)_{m'n',mn}=\delta _{n'n}\left(J_{y}^{(m)}\right)_{m'm}\,\quad \left(B_{y}\right)_{m'n',mn}=\delta _{m'm}\left(J_{y}^{(n)}\right)_{n'n}}$$

$${\displaystyle \left(A_{z}\right)_{m'n',mn}=\delta _{n'n}\left(J_{z}^{(m)}\right)_{m'm}\,\quad \left(B_{z}\right)_{m'n',mn}=\delta _{m'm}\left(J_{z}^{(n)}\right)_{n'n}}$$

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

$${\displaystyle \left(J_{z}^{(m)}\right)_{m'm}=m\delta _{m'm}\,\quad \left(J_{x}^{(m)}\pm iJ_{y}^{(m)}\right)_{m'm}=m\delta _{a',a\pm 1}{\sqrt {(a\mp m)(a\pm m+1)}}}$$

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

Applying this to particles with spin s;

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

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

Relativistic wave equations

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

Unitary groups in quantum field theory

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

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

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

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

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

U(1)

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

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

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

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