Pauli matrices

The Pauli matrices (after Wolfgang Pauli ) are special complex Hermitian 2 × 2 matrices . Together with the 2 × 2 identity matrix , which is referred to in this context as , they form both a basis of the 4-dimensional real vector space of all complex Hermitian 2 × 2 matrices and a basis of the 4-dimensional complex vector space of all complex 2 × 2 matrices. ${\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}}$ ${\displaystyle \sigma _{0}}$

They were introduced by Wolfgang Pauli in 1927 to describe the spin, but were already known in mathematics.

definition

The Pauli matrices are originally:

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

Here denotes the imaginary unit . The matrices were originally introduced in quantum mechanics to meet the basic commutation rules of the components of the spin operator (see below). Often, especially in relativistic quantum mechanics, the identity matrix is ​​added as the zeroth Pauli matrix: ${\displaystyle \mathrm {i} }$

${\displaystyle \sigma _{0}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}.}$

multiplication

For the multiplication of a Pauli matrix with another Pauli matrix, the following table results from the calculation rules for matrix multiplication :

${\displaystyle \cdot }$	${\displaystyle \sigma _{0}}$	${\displaystyle \sigma _{1}}$	${\displaystyle \sigma _{2}}$	${\displaystyle \sigma _{3}}$
${\displaystyle \sigma _{0}}$	${\displaystyle \sigma _{0}}$	${\displaystyle \sigma _{1}}$	${\displaystyle \sigma _{2}}$	${\displaystyle \sigma _{3}}$
${\displaystyle \sigma _{1}}$	${\displaystyle \sigma _{1}}$	${\displaystyle \sigma _{0}}$	${\displaystyle \mathrm {i} \,\sigma _{3}}$	${\displaystyle -\mathrm {i} \,\sigma _{2}}$
${\displaystyle \sigma _{2}}$	${\displaystyle \sigma _{2}}$	${\displaystyle -\mathrm {i} \,\sigma _{3}}$	${\displaystyle \sigma _{0}}$	${\displaystyle \mathrm {i} \,\sigma _{1}}$
${\displaystyle \sigma _{3}}$	${\displaystyle \sigma _{3}}$	${\displaystyle \mathrm {i} \,\sigma _{2}}$	${\displaystyle -\mathrm {i} \,\sigma _{1}}$	${\displaystyle \sigma _{0}}$

The product is in the row marked with and the column marked with , for example . The 4 Pauli matrices do not form a group. ${\displaystyle \sigma _{i}\cdot \sigma _{j}}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle \sigma _{j}}$ ${\displaystyle \sigma _{2}\cdot \sigma _{1}=-\mathrm {i} \,\sigma _{3}}$

Those of them with the matrix multiplication as a shortcut created group is named . It contains the element which is in the center , i.e. which commutes with all elements. The group thus consists of the 16 elements so that your multiplication table can easily be derived from the above. It contains the quaternion group Q ₈ as a normal divisor (see The Quaternions as a subring of C ⁴ and list of small groups # Pauli matrices ), which results. The cycle graph is . ${\displaystyle G_{16}^{10}}$ ${\displaystyle \sigma _{1}\cdot \sigma _{2}\cdot \sigma _{3}=\mathrm {i} \,\sigma _{0}=\left({\begin{smallmatrix}\mathrm {i} &0\\0&\mathrm {i} \end{smallmatrix}}\right)}$ ${\displaystyle G_{16}^{10}}$ ${\displaystyle \mathrm {i} ^{j}\,\sigma _{k}\;\;(j,k=0,1,2,3),}$ ${\displaystyle G_{16}^{10}\cong Q_{8}\rtimes \{\sigma _{0},\sigma _{1}\}}$

Decomposition of matrices

A complex 2 × 2 matrix with the elements is given . Then complex numbers can be found for which the following applies: ${\displaystyle \mathbf {A} }$ ${\displaystyle \left\{a_{ij}\ |\ i,j\in \left\{0,1\right\},a_{ij}\in \mathbb {C} \right\}}$ ${\displaystyle \left\{z_{i}\ |\ i\in \left\{0,1,2,3\right\},z_{i}\in \mathbb {C} \right\}}$

${\displaystyle \mathbf {A} }$	${\displaystyle ={\begin{pmatrix}a_{00}&a_{01}\\a_{10}&a_{11}\end{pmatrix}}={\begin{pmatrix}z_{0}+z_{3}&z_{1}-\mathrm {i} \,z_{2}\\z_{1}+\mathrm {i} \,z_{2}&z_{0}-z_{3}\\\end{pmatrix}}}$
	${\displaystyle =z_{0}\,{\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$	${\displaystyle +\,z_{1}\,{\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$	${\displaystyle +\,z_{2}\,{\begin{pmatrix}0&-\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}}$	${\displaystyle +\,z_{3}\,{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$
	${\displaystyle =z_{0}\;\;\sigma _{0}}$	${\displaystyle +\,z_{1}\;\;\sigma _{1}}$	${\displaystyle +\,z_{2}\;\;\sigma _{2}}$	${\displaystyle +\,z_{3}\;\;\sigma _{3}\;.}$

The following conversions apply:

${\displaystyle a_{00}=z_{0}+z_{3},\quad a_{01}=z_{1}-\mathrm {i} \,z_{2},\quad a_{10}=z_{1}+\mathrm {i} \,z_{2},\quad a_{11}=z_{0}-z_{3},}$

or.:

${\displaystyle z_{0}={\frac {a_{00}+a_{11}}{2}},\quad z_{1}={\frac {a_{01}+a_{10}}{2}},\quad z_{2}=\mathrm {i} {\frac {a_{01}-a_{10}}{2}},\quad z_{3}={\frac {a_{00}-a_{11}}{2}}.}$

A complex 2 × 2 matrix can therefore be written as a linear combination of , and this representation is unambiguous. The Pauli matrices thus form a basis of the vector space (and matrix ring ) , and this basis is an orthogonal one under the Frobenius scalar product , which makes the latter a Hilbert space . ${\displaystyle \sigma _{i}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} ^{2\times 2}}$ ${\displaystyle \mathbb {C} ^{2\times 2}}$

The conversions define a ring isomorphism

${\displaystyle \mathbb {C} ^{2\times 2}\to \mathbb {C} ^{4}}$

with the usual vector addition , the usual - scalar multiplication and the vector multiplication ${\displaystyle \mathbb {C} }$

${\displaystyle (x_{0},x_{1},x_{2},x_{3})\star (y_{0},y_{1},y_{2},y_{3}):=(}$	${\displaystyle x_{0}y_{0}+\,x_{1}y_{1}+\,x_{2}y_{2}+\,x_{3}y_{3},}$
	${\displaystyle x_{0}y_{1}+\,x_{1}y_{0}+\mathrm {i} x_{2}y_{3}-\mathrm {i} x_{3}y_{2},}$
	${\displaystyle x_{0}y_{2}-\mathrm {i} x_{1}y_{3}+\,x_{2}y_{0}+\mathrm {i} x_{3}y_{1},}$
	${\displaystyle x_{0}y_{3}+\mathrm {i} x_{1}y_{2}-\mathrm {i} x_{2}y_{1}+\,x_{3}y_{0}\,)}$

in Two vectors are interchangeable if and only if ${\displaystyle \mathbb {C} ^{4}.}$

${\displaystyle {\begin{aligned}&x_{2}y_{3}-x_{3}y_{2}={\begin{vmatrix}x_{2}&x_{3}\\y_{2}&y_{3}\end{vmatrix}}=0\\&x_{3}y_{1}-x_{1}y_{3}={\begin{vmatrix}x_{3}&x_{1}\\y_{3}&y_{1}\end{vmatrix}}=0\\&x_{1}y_{2}-x_{2}y_{1}={\begin{vmatrix}x_{1}&x_{2}\\y_{1}&y_{2}\end{vmatrix}}=0,\end{aligned}}}$

if the vector parts and -linearly depend on each other . ${\displaystyle (x_{1},x_{2},x_{3})}$ ${\displaystyle (y_{1},y_{2},y_{3})}$ ${\displaystyle \mathbb {C} }$

The inverse matrix of is calculated from this in the case of ${\displaystyle \mathbf {A} =z_{0}\,\sigma _{0}+z_{1}\,\sigma _{1}+z_{2}\,\sigma _{2}+z_{3}\,\sigma _{3}}$ ${\displaystyle z_{0}^{2}-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}\neq 0}$

${\displaystyle \mathbf {A} ^{-1}={\frac {z_{0}\,\sigma _{0}-z_{1}\,\sigma _{1}-z_{2}\,\sigma _{2}-z_{3}\,\sigma _{3}}{z_{0}^{2}-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}}}.}$

Hermitian 2 × 2 matrices

The subset of the Hermitian 2 × 2 matrices, i.e. the matrices with ${\displaystyle \mathbf {A} }$

${\displaystyle \mathbf {A} ={\overline {\mathbf {A} }}^{\mathrm {T} },}$

is a subspace for which the Pauli matrices also form a basis, but the coefficients are real. In other words: in Hermitian 2 × 2 matrices there are four (real) free parameters, da and are real and . ${\displaystyle \mathbb {R} }$ ${\displaystyle z_{i}}$ ${\displaystyle a_{00}}$ ${\displaystyle a_{11}}$ ${\displaystyle a_{01}={\overline {a_{10}}}}$

The product of two Hermitian matrices is Hermitian if they commute. The subspace is therefore not a (sub) ring.

The quaternions as a subring of C ⁴

(Unter) ring is another subspace of , which can be spanned by the coefficients of . It is also compatible with scalar multiplication and is also closed with regard to multiplication . This subspace is isomorphic to the quaternions . ${\displaystyle \mathbb {C} ^{4}}$ ${\displaystyle z_{0}\in \mathbb {R} ,}$ ${\displaystyle z_{1}\in \mathrm {i} \mathbb {R} ,}$ ${\displaystyle z_{2}\in \mathrm {i} \mathbb {R} ,}$ ${\displaystyle z_{3}\in \mathrm {i} \mathbb {R} }$ ${\displaystyle (\sigma _{0},\sigma _{1},\sigma _{2},\sigma _{3})}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \star }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {H} }$

As a basis for real coefficients, one can take the Pauli matrices multiplied by the imaginary unit together with the identity matrix, i.e. the set with the isomorphic assignment: ${\displaystyle \{\sigma _{0},\mathrm {i} \sigma _{1},\mathrm {i} \sigma _{2},\mathrm {i} \sigma _{3}\}}$

${\displaystyle 1\mapsto \sigma _{0},\quad i_{\mathbb {H} }\mapsto -\mathrm {i} \sigma _{1},\quad j_{\mathbb {H} }\mapsto -\mathrm {i} \sigma _{2},\quad k_{\mathbb {H} }\mapsto -\mathrm {i} \sigma _{3},}$

with as the known unit quaternions. Each of the 24 automorphisms of the quaternion group Q _{8 can be} switched before this assignment . An isomorphism can also be built "in reverse order": ${\displaystyle i_{\mathbb {H} },j_{\mathbb {H} },k_{\mathbb {H} }}$

${\displaystyle 1\mapsto \sigma _{0},\quad i_{\mathbb {H} }\mapsto +\mathrm {i} \sigma _{3},\quad j_{\mathbb {H} }\mapsto +\mathrm {i} \sigma _{2},\quad k_{\mathbb {H} }\mapsto +\mathrm {i} \sigma _{1}.}$

application

In quantum physics, in which the physical observables correspond to Hermitian operators or matrices on the mathematical side, the angular momentum operator of spin -½ states, for example for electrons , is represented by the Pauli matrices: ${\displaystyle {\hat {S}}_{i},\ i\in \{1,2,3\}}$

${\displaystyle {\hat {S}}_{i}\doteq {\tfrac {\hbar }{2}}\sigma _{i}}$,

where "is represented by" means. ${\displaystyle \doteq }$

In relativistic quantum mechanics, where one has four space-time or energy-momentum variables according to the relativistic four-vector formalism, the unit matrix is ​​equal to the three Pauli matrices (as the "zeroth" Pauli matrix) and with its help it becomes the Dirac equation built with the Dirac matrices .

The Pauli matrices appear directly in the Pauli equation for the quantum mechanical description of particles with spin in the magnetic field, which results from the non-relativistic reduction of the Dirac equation, and in the description of Majorana fermions (Majorana equation).

presentation

The Pauli matrices can be represented as matrices using Dirac notation : Either the standard basic vectors or the eigenvectors of the Pauli matrices can be used for the linear combination.

Pauli matrix	matrix	Linear combination (standard basis vectors)	Linear combination (eigenvectors)
${\displaystyle \sigma _{1}=\sigma _{x}}$	${\displaystyle {\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$	${\displaystyle |0\rangle \langle 1|+|1\rangle \langle 0|}$	${\displaystyle |+\rangle \langle +|\,-\,|-\rangle \langle -|}$
${\displaystyle \sigma _{2}=\sigma _{y}}$	${\displaystyle {\begin{pmatrix}0&-\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}}$	${\displaystyle \mathrm {i} \left(|1\rangle \langle 0|-|0\rangle \langle 1|\right)}$	${\displaystyle |\phi ^{+}\rangle \langle \phi ^{+}|-|\phi ^{-}\rangle \langle \phi ^{-}|}$
${\displaystyle \sigma _{3}=\sigma _{z}}$	${\displaystyle {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$	${\displaystyle |0\rangle \langle 0|-|1\rangle \langle 1|}$	${\displaystyle |0\rangle \langle 0|-|1\rangle \langle 1|}$

The vectors used are defined as follows, with the kets used being represented by vectors of what is identified by " ": ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \doteq }$

${\displaystyle |0\rangle =|s_{z}+\rangle }$	${\displaystyle \doteq {\begin{pmatrix}1\\0\end{pmatrix}}}$	${\displaystyle |1\rangle =|s_{z}-\rangle }$	${\displaystyle \doteq {\begin{pmatrix}0\\1\end{pmatrix}}}$
${\displaystyle |+\rangle }$	${\displaystyle \doteq {\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}}$	${\displaystyle |-\rangle }$	${\displaystyle \doteq {\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\-1\end{pmatrix}}}$
${\displaystyle |\phi ^{+}\rangle }$	${\displaystyle \doteq {\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\\mathrm {i} \end{pmatrix}}}$	${\displaystyle |\phi ^{-}\rangle }$	${\displaystyle \doteq {\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\-\mathrm {i} \end{pmatrix}}}$

properties

The Pauli matrices are Hermitian and unitary . This follows with the fourth basic element defined by ${\displaystyle \sigma _{0}:={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$

${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=\sigma _{0}^{2}=\sigma _{0}.}$

The determinants and traces of the Pauli matrices are

${\displaystyle {\begin{matrix}\det \sigma _{i}&=&-1&\\[1ex]\operatorname {tr} \sigma _{i}&=&0&\end{matrix}}}$   For ${\displaystyle i=1,2,3.}$

From the above it follows that every Pauli matrix has the eigenvalues +1 and −1. ${\displaystyle \mathbf {\sigma } _{i}}$

Furthermore:

${\displaystyle \sigma _{1}\,\sigma _{2}\,\sigma _{3}=\mathrm {i} \,\sigma _{0}}$

The Pauli matrices satisfy the algebraic relation

${\displaystyle \sigma _{i}\,\sigma _{j}=\delta _{ij}\sigma _{0}+\mathrm {i} \,\sum _{k=1}^{3}\epsilon _{ijk}\;\sigma _{k}}$   For ${\displaystyle i,j=1,2,3\,}$

( is the Levi-Civita symbol ), i.e. in particular the same relations as the angular momentum algebra up to a factor of 2 ${\displaystyle \epsilon _{ijk}}$

${\displaystyle [\sigma _{i}\,,\sigma _{j}]=\sigma _{i}\,\sigma _{j}-\sigma _{j}\,\sigma _{i}=2\,\mathrm {i} \,\sum _{k=1}^{3}\epsilon _{ijk}\;\sigma _{k}}$   For ${\displaystyle i,j=1,2,3.}$

and the Clifford or Dirac algebra ${\displaystyle \mathrm {Cl} (0,3,\mathbb {R} )}$

${\displaystyle \{\sigma _{i}\,,\sigma _{j}\}=\sigma _{i}\,\sigma _{j}+\sigma _{j}\,\sigma _{i}=2\,\delta _{ij}\sigma _{0}}$   For ${\displaystyle i,j=1,2,3.}$

The Pauli matrices belong to the special case of angular momentum operators, which act on basis vectors of an angular momentum multiplet with quantum numbers in systems of measurement as follows: ${\displaystyle l=1/2}$ ${\displaystyle \Lambda _{m}}$ ${\displaystyle l}$ ${\displaystyle m}$ ${\displaystyle \hbar =1}$

${\displaystyle L_{3}\Lambda _{m}=m\Lambda _{m}\,,\ m\in \{-l,-l+1,\dots ,l\}\,,}$

${\displaystyle L_{+}\Lambda _{m}={\sqrt {(l-m)(l+m+1)}}\,\Lambda _{m+1}\,,}$

${\displaystyle L_{-}\Lambda _{m}={\sqrt {(l+m)(l-m+1)}}\,\Lambda _{m-1}\,.}$

Where is a natural number and the different quantum numbers occur for. For , the angular momentum operators act on the components of linear combinations of the two basis vectors and therefore through multiplication with the following matrices ${\displaystyle 2l+1}$ ${\displaystyle m}$ ${\displaystyle 2l+1}$ ${\displaystyle m=-l,-l+1,\dots ,l}$ ${\displaystyle l=1/2}$ ${\displaystyle \Lambda _{1/2}}$ ${\displaystyle \Lambda _{-1/2}}$

${\displaystyle L_{3}={\frac {1}{2}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,,L_{+}={\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,,L_{-}={\begin{pmatrix}0&0\\1&0\end{pmatrix}}\,.}$

With and we then see that the angular momentum operators act on the components of spin 1/2 states by multiplying them by half the Pauli matrices. ${\displaystyle L_{1}={\frac {1}{2}}(L_{+}+L_{-})}$ ${\displaystyle L_{2}={\frac {1}{2\mathrm {i} }}(L_{+}-L_{-})}$

Assigned rotation group, connection with spin-1/2 systems

The linear envelope of the Pauli matrices multiplied by is a Lie algebra with the usual matrix multiplication . Because of the identity valid for every unit vector ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} \,\sigma _{1},\,\mathrm {i} \,\sigma _{2},\,\mathrm {i} \,\sigma _{3}}$ ${\displaystyle {\vec {n}}\cdot {\vec {\sigma }}\,=n_{1}\sigma _{1}+n_{2}\sigma _{2}+n_{3}\sigma _{3}}$ ${\displaystyle {\vec {n}}\in \mathbb {R} ^{3}}$

${\displaystyle \exp \left(-\mathrm {i} \,{\tfrac {\alpha }{2}}{\vec {n}}\cdot {\vec {\sigma }}\right)=\sigma _{0}\,\cos {\tfrac {\alpha }{2}}-\mathrm {i} \,({\vec {n}}\cdot {\vec {\sigma }})\,\sin {\tfrac {\alpha }{2}}}$

these three matrices are the generators of the complex rotating group ${\displaystyle SU(2)}$ .

The factor 1/2 in the above equation is mathematically dispensable. However, the equation is often required in precisely this form in physical applications. Because (as mentioned in the introduction) in quantum physics the matrices represent the operators for the spin components of a spin 1/2 system (e.g. an electron ). On the other hand, the matrix given by the exponential expression describes the change in the spin state during a spatial rotation . is the angle of rotation, the axis of rotation. For results ; d. H. the state vector of a spin-1/2 system is converted into its negative by turning around the angle and only converted back into itself by turning around the angle (“ spinor rotations ”). ${\displaystyle S_{i}={\tfrac {\hbar }{2}}\sigma _{i}}$ ${\displaystyle \alpha }$ ${\displaystyle {\vec {n}}}$ ${\displaystyle \alpha =2\pi }$ ${\displaystyle \exp {\bigl (}\!\!-\mathrm {i} \,\pi \;{\vec {n}}\cdot {\vec {\sigma }}{\bigr )}=-\sigma _{0}}$ ${\displaystyle 2\pi }$ ${\displaystyle 4\pi }$

Eigenvectors

The matrix has the eigenvectors ${\displaystyle \sigma _{3}}$

${\displaystyle \chi _{31}={\begin{pmatrix}1\\0\end{pmatrix}},\quad \chi _{32}={\begin{pmatrix}0\\1\end{pmatrix}}}$

as you can easily see:

${\displaystyle \sigma _{3}\chi _{31}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}1\\0\end{pmatrix}}={\begin{pmatrix}1\\0\end{pmatrix}},\quad \sigma _{3}\chi _{32}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}0\\1\end{pmatrix}}={\begin{pmatrix}0\\-1\end{pmatrix}}=-1{\begin{pmatrix}0\\1\end{pmatrix}}}$

according to the eigenvalues . The eigenvectors of are ${\displaystyle \pm 1}$ ${\displaystyle \sigma _{1}}$

${\displaystyle \chi _{11}={\begin{pmatrix}1\\1\end{pmatrix}},\quad \chi _{12}={\begin{pmatrix}1\\-1\end{pmatrix}}:}$

${\displaystyle \sigma _{1}\chi _{11}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}1\\1\end{pmatrix}}={\begin{pmatrix}1\\1\end{pmatrix}},\quad \sigma _{1}\chi _{12}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}1\\-1\end{pmatrix}}={\begin{pmatrix}-1\\1\end{pmatrix}}=-1{\begin{pmatrix}1\\-1\end{pmatrix}}}$

and the eigenvectors of ${\displaystyle \sigma _{2}}$

${\displaystyle \chi _{21}={\begin{pmatrix}1\\\mathrm {i} \end{pmatrix}},\quad \chi _{22}={\begin{pmatrix}\mathrm {i} \\1\end{pmatrix}}:}$

${\displaystyle \sigma _{2}\chi _{21}={\begin{pmatrix}0&-\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}{\begin{pmatrix}1\\\mathrm {i} \end{pmatrix}}={\begin{pmatrix}1\\\mathrm {i} \end{pmatrix}},\quad \sigma _{2}\chi _{22}={\begin{pmatrix}0&-\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}{\begin{pmatrix}\mathrm {i} \\1\end{pmatrix}}={\begin{pmatrix}-\mathrm {i} \\-1\end{pmatrix}}=-1{\begin{pmatrix}\mathrm {i} \\1\end{pmatrix}}}$

Kronecker product from Pauli Matrizen

In mathematics, with the help of the tensor product (Kronecker product) of Pauli matrices (with identity matrix), the representations of the higher Clifford algebras can be constructed over the real numbers.

Pauli matrices can be used to represent Hamilton operators and to approximate the exponential function of such operators. If the four are Pauli matrices, then one can generate higher dimensional matrices with the help of the Kronecker product . ${\displaystyle \sigma _{0},\sigma _{1},\sigma _{2},\sigma _{3}}$

${\displaystyle p:=\sigma _{\mu _{1}}\otimes \sigma _{\mu _{2}}\otimes ...\otimes \sigma _{\mu _{n}}\quad ;\quad \mu _{1},\mu _{2},...,\mu _{n}\in \{0,1,2,3\}\quad ;\quad n\in \mathbb {N} }$

Properties of the Pauli matrices are inherited by these matrices. If and are two Kronecker products from Pauli-Matrizen, the following applies: ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$

• ${\displaystyle p_{1},p_{2}}$are matrices ${\displaystyle 2^{n}\times 2^{n}}$
• ${\displaystyle p_{1}^{2}=p_{2}^{2}=1\qquad }$(The identity matrix) ${\displaystyle 2^{n}\times 2^{n}}$
• ${\displaystyle p_{1}p_{2}=p_{2}p_{1}}$or ( commutativity ) ${\displaystyle p_{1}p_{2}=-p_{2}p_{1}\qquad }$
• ${\displaystyle \operatorname {Spur} \sigma _{\mu _{1}}\otimes \sigma _{\mu _{2}}\otimes ...\otimes \sigma _{\mu _{n}}=2^{n}\delta _{\mu _{1},0}\delta _{\mu _{2},0}...\delta _{\mu _{n},0}}$
• The Kronecker products of Pauli matrices are linearly independent and form a basis in the vector space of the matrices. Hamilton operators of many physical models can be expressed as the sum of such matrices due to the basic property ( linear combination ). In particular, producers and annihilators of fermions , which can assume a finite number of states, can simply be expressed by them. ${\displaystyle 2^{n}\times 2^{n}}$ ${\displaystyle H}$

${\displaystyle H=\sum _{k=0}^{N}h_{k}p_{k}\quad }$mit is a Kronecker product from Pauli Matrizen. ${\displaystyle \quad N\in \mathbb {N} ,h_{k}\in \mathbb {R} ,p_{k}}$

Examples of such models are the Hubbard model , Heisenberg model and Anderson model .

The Kronecker product of Pauli matrices occurs when describing spin 1/2 systems that are made up of several subsystems. The relationship is given by the fact that the tensor product of two operators in the associated matrix representation is given by the Kronecker product of the matrices (see Kronecker product # relationship with tensor products ).

Approximation of the exponential function of the Hamilton operator

Often one is interested in the exponential function of the Hamilton operator.

${\displaystyle \exp\{-\beta H\}=\sum _{l=0}^{\infty }{\frac {(-\beta )^{l}}{l!}}{\biggl (}\sum _{k=0}^{N}h_{k}p_{k}{\biggr )}^{l}}$   With   ${\displaystyle \beta \in \mathbb {R} }$

Due to the commutativity, the matrices in a product can be arranged as desired. If a permutation is , then: ${\displaystyle \pi }$

${\displaystyle p_{\pi _{1}}p_{\pi _{2}}...p_{\pi _{n}}=ap_{1}p_{2}...p_{n}}$   With   ${\displaystyle n\in \mathbb {N} ,a\in \{1,-1\}}$

Therefore there exist rational numbers with: ${\displaystyle E_{k_{1}k_{2}...k_{N}}}$

${\displaystyle \exp\{-\beta H\}=\sum _{k_{1}=0}^{\infty }\sum _{k_{2}=0}^{\infty }...\sum _{k_{N}=0}^{\infty }E_{k_{1}k_{2}...k_{N}}(-\beta h_{1})^{k_{1}}(-\beta h_{2})^{k_{2}}...(-\beta h_{N})^{k_{N}}p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{N}^{k_{N}}}$

With a few exceptions, these rational numbers are difficult to calculate.

A first approximation is obtained by only considering summands that consist of commuting matrices.

${\displaystyle E_{k_{1}k_{2}...k_{N}}=0}$if a pair with and exists ${\displaystyle 1\leq a,b\leq N}$ ${\displaystyle p_{a}p_{b}=-p_{b}p_{a}}$ ${\displaystyle k_{a},k_{b}>0}$

${\displaystyle E_{k_{1}k_{2}...k_{N}}={\frac {1}{k_{1}!}}{\frac {1}{k_{2}!}}...{\frac {1}{k_{N}!}}}$ otherwise

The approximation can be further improved by considering pairs, triples, ... of non-commuting matrices.

See also

• Gell-Mann matrices

literature

• Willi-Hans Steeb: Kronecker Product of Matrices and Applications . BI Wissenschaftsverlag, Mannheim 1991, ISBN 3-411-14811-X .

Web links

• Eric W. Weisstein : Pauli Matrices . In: MathWorld (English).

References and comments

1. ↑ Wolfgang Pauli: To the quantum mechanics of the magnetic electron . In: Zeitschrift für Physik , Volume 43, 1927, p. 601
2. ↑ Numbering according to The Small Groups library . quoted from RJ Mathar: cycle graph plots of finite groups up to order 36 . 2014.
3. ^ RJ Mathar: Cycle graph plots of finite groups up to order 36 . 2014.
4. ↑ Mikio Nakahara: Geometry, topology, and physics . CRC Press, 2003, pp. Xxii ff. ( Google Books ).
5. ↑ By multiplying by , Hermitian matrices are turned into skewed Hermitian matrices. A representation with the help of Hermitian operators and matrices is preferred by physicists, because in quantum mechanics measurable quantities (so-called observables ) are always described by Hermitian operators. ${\displaystyle \pm \mathrm {i} }$
6. ^ Charles Misner , Kip S. Thorne , John. A. Wheeler : Gravitation . WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0 , p. 1142