Editing Pauli matrices (section)

{{Short description|Matrices important in quantum mechanics and the study of spin}}
{{Use American English|date=March 2019}}
[[File:Wolfgang Pauli.jpg|thumb|right|[[Wolfgang Pauli]] (1900–1958), c. 1924. Pauli received the [[Nobel Prize in Physics]] in 1945, nominated by [[Albert Einstein]], for the [[Pauli exclusion principle]].]]
In [[mathematical physics]] and [[mathematics]], the '''Pauli matrices''' are a set of three {{math|2 × 2}} [[complex number|complex]] [[matrix (mathematics)|matrices]] that are [[traceless]], [[Hermitian matrix|Hermitian]], [[Involutory matrix|involutory]] and [[Unitary matrix|unitary]]. Usually indicated by the [[Greek (alphabet)|Greek]] letter [[sigma]] ({{mvar|σ}}), they are occasionally denoted by [[tau]] ({{mvar|τ}}) when used in connection with [[isospin]] symmetries.
<math display="block">
\begin{align}
  \sigma_1 = \sigma_x &=
    \begin{pmatrix}
      0&1\\
      1&0
    \end{pmatrix}, \\
  \sigma_2 = \sigma_y &=
    \begin{pmatrix}
      0& -i \\
      i&0
    \end{pmatrix}, \\
  \sigma_3 = \sigma_z &=
    \begin{pmatrix}
      1&0\\
      0&-1
    \end{pmatrix}. \\
\end{align}
</math>

These matrices are named after the physicist [[Wolfgang Pauli]]. In [[quantum mechanics]], they occur in the [[Pauli equation]], which takes into account the interaction of the [[Spin (physics)|spin]] of a particle with an external [[electromagnetic field]]. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45&nbsp;degree polarization (right/left), and circular polarization (right/left).

Each Pauli matrix is [[Hermitian matrix|Hermitian]], and together with the identity matrix {{mvar|I}} (sometimes considered as the zeroth Pauli matrix {{math|''σ''{{sub|0}} }}), the Pauli matrices form a [[Basis (linear algebra)|basis]] of the [[vector space]] of {{math|2 × 2}} Hermitian matrices over the [[real number|real numbers]], under addition. This means that any {{math|2 × 2}} [[Hermitian matrix]] can be written in a unique way as a [[linear combination]] of Pauli matrices, with all coefficients being real numbers.

The Pauli matrices satisfy the useful product relation:<math display="block">\begin{align}
 \sigma_i \sigma_j = \delta_{ij}+i\epsilon_{ijk}\sigma_k.
\end{align}</math>

[[Hermitian operator]]s represent [[observable]]s in quantum mechanics, so the Pauli matrices span the space of observables of the [[complex number|complex]] two-dimensional [[Hilbert space]]. In the context of Pauli's work, {{mvar|σ{{sub|k}}}} represents the observable corresponding to spin along the {{mvar|k}}th coordinate axis in three-dimensional [[Euclidean space]] <math>\mathbb{R}^3 .</math>

The Pauli matrices (after multiplication by {{mvar|i}} to make them [[skew-Hermitian|anti-Hermitian]]) also generate transformations in the sense of [[Lie algebra]]s: the matrices {{math|''iσ''{{sub|1}}, ''iσ''{{sub|2}}, ''iσ''{{sub|3}} }} form a basis for the real Lie algebra <math>\mathfrak{su}(2)</math>, which [[Exponential map (Lie theory)|exponentiates]] to the special unitary group [[SU(2)#n = 2|SU(2)]].{{efn|
This conforms to the convention ''in mathematics'' for the [[matrix exponential]], {{math|''iσ'' ⟼ exp(''iσ'')}}. In the convention ''in physics'', {{math|''σ'' ⟼ exp(−''iσ'')}}, hence in it no pre-multiplication by {{mvar|i}} is necessary to land in {{math|SU(2)}}.
}} The [[Algebra over a field|algebra]] generated by the three matrices {{math|''σ''{{sub|1}}, ''σ''{{sub|2}}, ''σ''{{sub|3}} }} is [[isomorphic]] to the [[Clifford algebra]] of <math> \mathbb{R}^3,</math><ref>
{{cite journal |author1=Gull, S. F. |author2=Lasenby, A. N. |author3=Doran, C. J. L. |date=January 1993 |title=Imaginary numbers are not Real – the geometric algebra of spacetime |via=geometry.mrao.cam.ac.uk |journal=Found. Phys. |volume=23 |issue=9 |pages=1175–1201 |doi=10.1007/BF01883676 |bibcode=1993FoPh...23.1175G |s2cid=14670523 |url=http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf |access-date=2023-05-05 |df=dmy-all}}
</ref> and the (unital) [[associative algebra]] generated by {{math|''iσ''{{sub|1}}, ''iσ''{{sub|2}}, ''iσ''{{sub|3}} }} functions identically ([[isomorphism|is isomorphic]]) to that of [[quaternion]]s (<math>\mathbb{H}</math>).