Pauli matrices

Wolfgang Pauli (1900–1958), ca. 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 2 × 2 complex matrices which are Hermitian and unitary.^[1] Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. They are

σ1=σx=(0110)σ2=σy=(0−ii0)σ3=σz=(100−1).{displaystyle {begin{aligned}sigma _{1}=sigma _{x}&={begin{pmatrix}0&1\1&0end{pmatrix}}\sigma _{2}=sigma _{y}&={begin{pmatrix}0&-i\i&0end{pmatrix}}\sigma _{3}=sigma _{z}&={begin{pmatrix}1&0\0&-1end{pmatrix}},.end{aligned}}}

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 of a particle with an external electromagnetic field.

Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ₀), the Pauli matrices (multiplied by real coefficients) form a basis for the vector space of 2 × 2 Hermitian matrices.

Hermitian operators represent observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work, σ_k represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space ℝ³.

The Pauli matrices (after multiplication by i to make them anti-Hermitian), also generate transformations in the sense of Lie algebras: the matrices iσ₁, iσ₂, iσ₃ form a basis for su(2){displaystyle {mathfrak {su}}(2)}, which exponentiates to the special unitary group SU(2).^{[nb 1]} The algebra generated by the three matrices σ₁, σ₂, σ₃ is isomorphic to the Clifford algebra of ℝ³.

Contents

• 
  1 Algebraic properties
  
  
  
  • 
    1.1 Eigenvectors and eigenvalues
    
  
  
  • 
    1.2 Pauli vector
    
  
  
  • 
    1.3 Commutation relations
    
  
  
  • 
    1.4 Relation to dot and cross product
    
  
  
  • 
    1.5 Some trace relations
    
  
  
  • 
    1.6 Exponential of a Pauli vector
    
    
    
    • 
      1.6.1 The group composition law of SU(2)
      
    
    
    • 
      1.6.2 Adjoint action
      
    
    
    
    
  
  
  • 
    1.7 Completeness relation
    
  
  
  • 
    1.8 Relation with the permutation operator
    
  
  
  
  


• 
  2 SU(2)
  
  
  
  • 
    2.1 SO(3)
    
  
  
  • 
    2.2 Quaternions
    
  
  
  
  


• 
  3 Physics
  
  
  
  • 
    3.1 Classical mechanics
    
  
  
  • 
    3.2 Quantum mechanics
    
  
  
  • 
    3.3 Relativistic quantum mechanics
    
  
  
  • 
    3.4 Quantum information
    
  
  
  
  


• 
  4 See also
  


• 
  5 Remarks
  


• 
  6 Notes
  


• 
  7 References
  

Algebraic properties

All three of the Pauli matrices can be compacted into a single expression:

σa=(δa3δa1−iδa2δa1+iδa2−δa3){displaystyle sigma _{a}={begin{pmatrix}delta _{a3}&delta _{a1}-idelta _{a2}\delta _{a1}+idelta _{a2}&-delta _{a3}end{pmatrix}}}

where i = √−1 is the imaginary unit, and δ_ab is the Kronecker delta, which equals +1 if a = b and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of a = 1, 2, 3, in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.

The matrices are involutory:

σ12=σ22=σ32=−iσ1σ2σ3=(1001)=I{displaystyle sigma _{1}^{2}=sigma _{2}^{2}=sigma _{3}^{2}=-isigma _{1}sigma _{2}sigma _{3}={begin{pmatrix}1&0\0&1end{pmatrix}}=I}

where I is the identity matrix.

• The determinants and traces of the Pauli matrices are:

detσi=−1,Tr⁡σi=0.{displaystyle {begin{aligned}det sigma _{i}&=-1,\operatorname {Tr} sigma _{i}&=0.end{aligned}}}

From above we can deduce that the eigenvalues of each σ_i are ±1.

• Together with the 2 × 2 identity matrix I (sometimes written as σ₀), the Pauli matrices form an orthogonal basis, in the sense of Hilbert–Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:

ψx+=12(11),ψx−=12(1−1),ψy+=12(1i),ψy−=12(1−i),ψz+=(10),ψz−=(01).{displaystyle {begin{aligned}&psi _{x+}={frac {1}{sqrt {2}}}{begin{pmatrix}1\1end{pmatrix}},&&psi _{x-}={frac {1}{sqrt {2}}}{begin{pmatrix}1\-1end{pmatrix}},\&psi _{y+}={frac {1}{sqrt {2}}}{begin{pmatrix}1\iend{pmatrix}},&&psi _{y-}={frac {1}{sqrt {2}}}{begin{pmatrix}1\-iend{pmatrix}},\&psi _{z+}={begin{pmatrix}1\0end{pmatrix}},&&psi _{z-}={begin{pmatrix}0\1end{pmatrix}}.end{aligned}}}

Pauli vector

The Pauli vector is defined by^{[nb 2]}

σ→=σ1x^+σ2y^+σ3z^{displaystyle {vec {sigma }}=sigma _{1}{hat {x}}+sigma _{2}{hat {y}}+sigma _{3}{hat {z}}}

and provides a mapping mechanism from a vector basis to a Pauli matrix basis^[2] as follows,

a→⋅σ→=(aix^i)⋅(σjx^j)=aiσjx^i⋅x^j=aiσjδij=aiσi=(a3a1−ia2a1+ia2−a3){displaystyle {begin{aligned}{vec {a}}cdot {vec {sigma }}&=(a_{i}{hat {x}}_{i})cdot (sigma _{j}{hat {x}}_{j})\&=a_{i}sigma _{j}{hat {x}}_{i}cdot {hat {x}}_{j}\&=a_{i}sigma _{j}delta _{ij}\&=a_{i}sigma _{i}={begin{pmatrix}a_{3}&a_{1}-ia_{2}\a_{1}+ia_{2}&-a_{3}end{pmatrix}}end{aligned}}}

using the summation convention. Further,

deta→⋅σ→=−a→⋅a→=−|a→|2,{displaystyle det {vec {a}}cdot {vec {sigma }}=-{vec {a}}cdot {vec {a}}=-|{vec {a}}|^{2},}

its eigenvalues being ±|a→|{displaystyle pm |{vec {a}}|}, and moreover (see completeness, below)

12tr[(a→⋅σ→)σ→]=a→ .{displaystyle {frac {1}{2}}mathrm {tr} [({vec {a}}cdot {vec {sigma }}){vec {sigma }}]={vec {a}}~.}

Its (unnormalized) eigenvectors are
ψ+=(a3+|a→|a1+ia2);ψ−=(ia2−a1a3+|a→|).{displaystyle psi _{+}={begin{pmatrix}a_{3}+|{vec {a}}|\a_{1}+ia_{2}end{pmatrix}};qquad psi _{-}={begin{pmatrix}ia_{2}-a_{1}\a_{3}+|{vec {a}}|end{pmatrix}}.}

Commutation relations

The Pauli matrices obey the following commutation relations:

[σa,σb]=2iεabcσc,{displaystyle [sigma _{a},sigma _{b}]=2ivarepsilon _{abc},sigma _{c},,}

and anticommutation relations:

{σa,σb}=2δabI.{displaystyle {sigma _{a},sigma _{b}}=2delta _{ab},I.}

where the structure constant ε_abc is the Levi-Civita symbol, Einstein summation notation is used, δ_ab is the Kronecker delta, and I is the 2 × 2 identity matrix.

For example,

[σ1,σ2]=2iσ3[σ2,σ3]=2iσ1[σ3,σ1]=2iσ2[σ1,σ1]=0{σ1,σ1}=2I{σ1,σ2}=0.{displaystyle {begin{aligned}left[sigma _{1},sigma _{2}right]&=2isigma _{3},\left[sigma _{2},sigma _{3}right]&=2isigma _{1},\left[sigma _{3},sigma _{1}right]&=2isigma _{2},\left[sigma _{1},sigma _{1}right]&=0,\left{sigma _{1},sigma _{1}right}&=2I,\left{sigma _{1},sigma _{2}right}&=0,.\end{aligned}}}

Relation to dot and cross product

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives

[σa,σb]+{σa,σb}=(σaσb−σbσa)+(σaσb+σbσa)2iεabcσc+2δabI=2σaσb{displaystyle {begin{aligned}left[sigma _{a},sigma _{b}right]+{sigma _{a},sigma _{b}}&=(sigma _{a}sigma _{b}-sigma _{b}sigma _{a})+(sigma _{a}sigma _{b}+sigma _{b}sigma _{a})\2ivarepsilon _{abc},sigma _{c}+2delta _{ab}I&=2sigma _{a}sigma _{b}end{aligned}}}

so that,

σaσb=δabI+iεabcσc .{displaystyle sigma _{a}sigma _{b}=delta _{ab}I+ivarepsilon _{abc},sigma _{c}~.}

Contracting each side of the equation with components of two 3-vectors a_p and b_q (which commute with the Pauli matrices, i.e., a_pσ_q = σ_qa_p) for each matrix σ_q and vector component a_p (and likewise with b_q), and relabeling indices a, b, c → p, q, r, to prevent notational conflicts, yields

apbqσpσq=apbq(iεpqrσr+δpqI)apσpbqσq=iεpqrapbqσr+apbqδpqI .{displaystyle {begin{aligned}a_{p}b_{q}sigma _{p}sigma _{q}&=a_{p}b_{q}left(ivarepsilon _{pqr},sigma _{r}+delta _{pq}Iright)\a_{p}sigma _{p}b_{q}sigma _{q}&=ivarepsilon _{pqr},a_{p}b_{q}sigma _{r}+a_{p}b_{q}delta _{pq}I~.end{aligned}}}

Finally, translating the index notation for the dot product and cross product results in

(a→⋅σ→)(b→⋅σ→)=(a→⋅b→)I+i(a→×b→)⋅σ→{displaystyle ({vec {a}}cdot {vec {sigma }})({vec {b}}cdot {vec {sigma }})=({vec {a}}cdot {vec {b}}),I+i({vec {a}}times {vec {b}})cdot {vec {sigma }}}

(1 )

If i{displaystyle i} is identified with the pseudoscalar σxσyσz{displaystyle sigma _{x}sigma _{y}sigma _{z}} then the right hand side becomes a⋅b+a∧b{displaystyle acdot b+awedge b} which is also the definition for the product of two vectors in geometric algebra.

Some trace relations

Following traces can be derived using the commutation and anticommutation relations.

tr(σa)=0{displaystyle tr(sigma _{a})=0}

tr(σaσb)=2δab{displaystyle tr(sigma _{a}sigma _{b})=2delta _{ab}}

tr(σaσbσc)=2iεabc{displaystyle tr(sigma _{a}sigma _{b}sigma _{c})=2ivarepsilon _{abc}}

tr(σaσbσcσd)=2(δabδcd−δacδbd+δadδbc){displaystyle tr(sigma _{a}sigma _{b}sigma _{c}sigma _{d})=2(delta _{ab}delta _{cd}-delta _{ac}delta _{bd}+delta _{ad}delta _{bc})}

Exponential of a Pauli vector

For

a→=an^,|n^|=1,{displaystyle {vec {a}}=a{hat {n}},quad |{hat {n}}|=1,}

one has, for even powers, 2p,  p=0,1,2,3,...{displaystyle 2p, p=0,1,2,3,...}

(n^⋅σ→)2p=I{displaystyle ({hat {n}}cdot {vec {sigma }})^{2p}=I}

which can be shown first for the p=1{displaystyle p=1} case using the anticommutation relations. For convenience, the case p=0{displaystyle p=0} is taken to be I{displaystyle I} by convention.

For odd powers, 2q+1,  q=0,1,2,3,...{displaystyle 2q+1, q=0,1,2,3,...}

(n^⋅σ→)2q+1=n^⋅σ→.{displaystyle ({hat {n}}cdot {vec {sigma }})^{2q+1}={hat {n}}cdot {vec {sigma }},.}

Matrix exponentiating, and using the Taylor series for sine and cosine,

eia(n^⋅σ→)=∑k=0∞ik[a(n^⋅σ→)]kk!=∑p=0∞(−1)p(an^⋅σ→)2p(2p)!+i∑q=0∞(−1)q(an^⋅σ→)2q+1(2q+1)!=I∑p=0∞(−1)pa2p(2p)!+i(n^⋅σ→)∑q=0∞(−1)qa2q+1(2q+1)!{displaystyle {begin{aligned}e^{ia({hat {n}}cdot {vec {sigma }})}&=sum _{k=0}^{infty }{frac {i^{k}left[a({hat {n}}cdot {vec {sigma }})right]^{k}}{k!}}\&=sum _{p=0}^{infty }{frac {(-1)^{p}(a{hat {n}}cdot {vec {sigma }})^{2p}}{(2p)!}}+isum _{q=0}^{infty }{frac {(-1)^{q}(a{hat {n}}cdot {vec {sigma }})^{2q+1}}{(2q+1)!}}\&=Isum _{p=0}^{infty }{frac {(-1)^{p}a^{2p}}{(2p)!}}+i({hat {n}}cdot {vec {sigma }})sum _{q=0}^{infty }{frac {(-1)^{q}a^{2q+1}}{(2q+1)!}}\end{aligned}}}.

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,

eia(n^⋅σ→)=Icos⁡a+i(n^⋅σ→)sin⁡a{displaystyle e^{ia({hat {n}}cdot {vec {sigma }})}=Icos {a}+i({hat {n}}cdot {vec {sigma }})sin {a}}

(2 )

which is analogous to Euler's formula, extended to quaternions.

Note that

det[ia(n^⋅σ→)]=a2{displaystyle det[ia({hat {n}}cdot {vec {sigma }})]=a^{2}},

while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).

A more abstract version of formula (2) for a general 2 × 2 matrix can be found in the article on matrix exponentials. A general version of (2) for an analytic (at a and −a) function is provided by application of Sylvester's formula,^[3]

f(a(n^⋅σ→))=If(a)+f(−a)2+n^⋅σ→f(a)−f(−a)2 .{displaystyle f(a({hat {n}}cdot {vec {sigma }}))=I{frac {f(a)+f(-a)}{2}}+{hat {n}}cdot {vec {sigma }}{frac {f(a)-f(-a)}{2}}~.}

The group composition law of SU(2)

A straightforward application of formula (2) provides a parameterization of the composition law of the group SU(2).^{[nb 3]} One may directly solve for c in

eia(n^⋅σ→)eib(m^⋅σ→)=I(cos⁡acos⁡b−n^⋅m^sin⁡asin⁡b)+i(n^sin⁡acos⁡b+m^sin⁡bcos⁡a−n^×m^ sin⁡asin⁡b)⋅σ→=Icos⁡c+i(k^⋅σ→)sin⁡c=eic(k^⋅σ→),{displaystyle {begin{aligned}e^{ia({hat {n}}cdot {vec {sigma }})}e^{ib({hat {m}}cdot {vec {sigma }})}&=I(cos acos b-{hat {n}}cdot {hat {m}}sin asin b)+i({hat {n}}sin acos b+{hat {m}}sin bcos a-{hat {n}}times {hat {m}}~sin asin b)cdot {vec {sigma }}\&=Icos {c}+i({hat {k}}cdot {vec {sigma }})sin {c}\&=e^{icleft({hat {k}}cdot {vec {sigma }}right)},end{aligned}}}

which specifies the generic group multiplication, where, manifestly,

cos⁡c=cos⁡acos⁡b−n^⋅m^sin⁡asin⁡b ,{displaystyle cos c=cos acos b-{hat {n}}cdot {hat {m}}sin asin b~,}

the spherical law of cosines. Given c, then,

k^=1sin⁡c(n^sin⁡acos⁡b+m^sin⁡bcos⁡a−n^×m^sin⁡asin⁡b) .{displaystyle {hat {k}}={frac {1}{sin c}}left({hat {n}}sin acos b+{hat {m}}sin bcos a-{hat {n}}times {hat {m}}sin asin bright)~.}

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to^[4]

eick^⋅σ→=exp⁡(icsin⁡c(n^sin⁡acos⁡b+m^sin⁡bcos⁡a−n^×m^ sin⁡asin⁡b)⋅σ→) .{displaystyle e^{ic{hat {k}}cdot {vec {sigma }}}=exp left(i{frac {c}{sin c}}({hat {n}}sin acos b+{hat {m}}sin bcos a-{hat {n}}times {hat {m}}~sin asin b)cdot {vec {sigma }}right)~.}

(Of course, when n̂ is parallel to m̂, so is k̂, and c = a + b.)

See also: Rotation formalisms in three dimensions § Rodrigues parameters and Gibbs representation, and Spinor § Three dimensions

Adjoint action

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation effectively by double the angle a,

eia(n^⋅σ→) σ→ e−ia(n^⋅σ→)=σ→cos⁡(2a)+n^×σ→ sin⁡(2a)+n^ n^⋅σ→ (1−cos⁡(2a)) .{displaystyle e^{ia({hat {n}}cdot {vec {sigma }})}~{vec {sigma }}~e^{-ia({hat {n}}cdot {vec {sigma }})}={vec {sigma }}cos(2a)+{hat {n}}times {vec {sigma }}~sin(2a)+{hat {n}}~{hat {n}}cdot {vec {sigma }}~(1-cos(2a))~.}

See also: Rodrigues' rotation formula

Completeness relation

An alternative notation that is commonly used for the Pauli matrices is to write the vector index i in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the i-th Pauli matrix is σ ⁱ_αβ.

In this notation, the completeness relation for the Pauli matrices can be written

σ→αβ⋅σ→γδ≡∑i=13σαβiσγδi=2δαδδβγ−δαβδγδ.{displaystyle {vec {sigma }}_{alpha beta }cdot {vec {sigma }}_{gamma delta }equiv sum _{i=1}^{3}sigma _{alpha beta }^{i}sigma _{gamma delta }^{i}=2delta _{alpha delta }delta _{beta gamma }-delta _{alpha beta }delta _{gamma delta }.}

Proof: The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the complex Hilbert space of all 2 × 2 matrices means that we can express any matrix M as

M=cI+∑iaiσi{displaystyle M=cI+sum _{i}a_{i}sigma ^{i}}

where c is a complex number, and a is a 3-component complex vector. It is straightforward to show, using the properties listed above, that

trσiσj=2δij{displaystyle mathrm {tr} ,sigma ^{i}sigma ^{j}=2delta _{ij}}

where "tr" denotes the trace, and hence that

c=12trM{displaystyle c={frac {1}{2}}mathrm {tr} ,M} and ai=12trσiM .{displaystyle a_{i}={frac {1}{2}}mathrm {tr} ,sigma ^{i}M~.}

∴  2M=ItrM+∑iσitrσiM ,{displaystyle therefore ~~2M=Imathrm {tr} ,M+sum _{i}sigma ^{i}mathrm {tr} ,sigma ^{i}M~,}

which can be rewritten in terms of matrix indices as

2Mαβ=δαβMγγ+∑iσαβiσγδiMδγ ,{displaystyle 2M_{alpha beta }=delta _{alpha beta }M_{gamma gamma }+sum _{i}sigma _{alpha beta }^{i}sigma _{gamma delta }^{i}M_{delta gamma }~,}

where summation is implied over the repeated indices γ and δ. Since this is true for any choice of the matrix M, the completeness relation follows as stated above.

As noted above, it is common to denote the 2 × 2 unit matrix by σ₀, so σ⁰_αβ = δ_αβ.
The completeness relation can alternatively be expressed as

∑i=03σαβiσγδi=2δαδδβγ .{displaystyle sum _{i=0}^{3}sigma _{alpha beta }^{i}sigma _{gamma delta }^{i}=2delta _{alpha delta }delta _{beta gamma }~.}

The fact that any 2 × 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states' density matrix, (2 × 2 positive semidefinite matrices with trace 1). This can be seen by simply first writing an arbitrary Hermitian matrix as a real linear combination of {σ₀, σ₁, σ₂, σ₃} as above, and then imposing the positive-semidefinite and trace 1 conditions.

Relation with the permutation operator

Let P_ij be the transposition (also known as a permutation) between two spins σ_i and σ_j living in the tensor product space ℂ² ⊗ ℂ²,

Pij|σiσj⟩=|σjσi⟩.{displaystyle P_{ij}|sigma _{i}sigma _{j}rangle =|sigma _{j}sigma _{i}rangle ,.}

This operator can also be written more explicitly as Dirac's spin exchange operator,

Pij=12(σ→i⋅σ→j+1).{displaystyle P_{ij}={tfrac {1}{2}}({vec {sigma }}_{i}cdot {vec {sigma }}_{j}+1),.}

Its eigenvalues are therefore^[5] 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.

SU(2)

The group SU(2) is the Lie group of unitary 2×2 matrices with unit determinant; its Lie algebra is the set of all 2×2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra su2{displaystyle {mathfrak {su}}_{2}} is the 3-dimensional real algebra spanned by the set {iσ_j}. In compact notation,

su(2)=span⁡{iσ1,iσ2,iσ3}.{displaystyle {mathfrak {su}}(2)=operatorname {span} {isigma _{1},isigma _{2},isigma _{3}}.}

As a result, each iσ_j can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is λ = 1/2, so that

su(2)=span⁡{iσ12,iσ22,iσ32}.{displaystyle {mathfrak {su}}(2)=operatorname {span} left{{frac {isigma _{1}}{2}},{frac {isigma _{2}}{2}},{frac {isigma _{3}}{2}}right}.}

As SU(2) is a compact group, its Cartan decomposition is trivial.

SO(3)

The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that the iσ_j are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though su(2) and so(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one group homomorphism from SU(2) to SO(3), see relationship between SO(3) and SU(2).

Quaternions

The real linear span of {I, iσ₁, iσ₂, iσ₃} is isomorphic to the real algebra of quaternions ℍ. The isomorphism from ℍ to this set is given by the following map (notice the reversed signs for the Pauli matrices):

1↦I,i↦−iσ1,j↦−iσ2,k↦−iσ3.{displaystyle 1mapsto I,quad imapsto -isigma _{1},quad jmapsto -isigma _{2},quad kmapsto -isigma _{3}.}

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,^[6]

1↦I,i↦iσ3,j↦iσ2,k↦iσ1.{displaystyle 1mapsto I,quad imapsto isigma _{3},quad jmapsto isigma _{2},quad kmapsto isigma _{1}.}

As the set of versors U ⊂ ℍ forms a group isomorphic to SU(2), U gives yet another way of describing SU(2). The two-to-one homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.

Quaternions form a division algebra—every non-zero element has an inverse—whereas Pauli matrices do not.

Physics

Classical mechanics

In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters.^[7] The matrix P corresponding to the position x→{displaystyle {vec {x}}} of a point in space is defined in terms of the above Pauli vector matrix,

P=x→⋅σ→=xσx+yσy+zσz .{displaystyle P={vec {x}}cdot {vec {sigma }}=xsigma _{x}+ysigma _{y}+zsigma _{z}~.}

Consequently, the transformation matrix Qθ{displaystyle Q_{theta }} for rotations about the x-axis through an angle θ may be written in terms of Pauli matrices and the unit matrix as ^[7]

Qθ=11cos⁡θ2+iσxsin⁡θ2 .{displaystyle Q_{theta }=1!!1cos {tfrac {theta }{2}}+isigma _{x}sin {tfrac {theta }{2}}~.}

Similar expressions follow for general Pauli vector rotations as detailed above.

Quantum mechanics

In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin ½ particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, iσ_j are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin ½. The states of the particles are represented as two-component spinors. In the same way, the Pauli matrices are related to the isospin operator.

An interesting property of spin ½ particles is that they must be rotated by an angle of 4π in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the 2-sphere S ², they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.

For a spin ½ particle, the spin operator is given by J=ħ/2σ, the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3)#A note on Lie algebra. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.^[8]

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_n is defined to consist of all n-fold tensor products of Pauli matrices.

Relativistic quantum mechanics

In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as

Σi=(σi00σi).{displaystyle {mathsf {Sigma }}_{i}={begin{pmatrix}{mathsf {sigma }}_{i}&0\0&{mathsf {sigma }}_{i}end{pmatrix}}.}

It follows from this definition that Σi{displaystyle {mathsf {Sigma }}_{i}} matrices have the same algebraic properties as σi{displaystyle {mathsf {sigma }}_{i}} matrices.

However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence Σi{displaystyle {mathsf {Sigma }}_{i}} needs to be replaced by Σμν{displaystyle Sigma _{mu nu }}, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Σμν{displaystyle Sigma _{mu nu }} are also antisymmetric. Hence there are only six independent matrices.

The first three are the Σjk≡ϵijkΣi.{displaystyle {mathsf {Sigma }}_{jk}equiv epsilon _{ijk}{mathsf {Sigma }}_{i}.} The remaining three, Σ0i≡αi{displaystyle {mathsf {Sigma }}_{0i}equiv {mathsf {alpha }}_{i}}, are the Dirac αi{displaystyle {mathsf {alpha }}_{i}} matrices defined as

αi=(0σiσi0).{displaystyle {mathsf {alpha }}_{i}={begin{pmatrix}0&{mathsf {sigma }}_{i}\{mathsf {sigma }}_{i}&0end{pmatrix}}.}

The relativistic spin matrices Σμν{displaystyle Sigma _{mu nu }} are written in compact form in terms of commutator of gamma matrices as

Σμν=i2[γμ,γν]{displaystyle Sigma _{mu nu }={tfrac {i}{2}}[gamma _{mu },gamma _{nu }]}.

Quantum information

• In quantum information, single-qubit quantum gates are 2 × 2 unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the Z–Y decomposition of a single-qubit gate. Choosing a different Cartan pair gives a similar X–Y decomposition of a single-qubit gate.

See also

• Spinors in three dimensions


• 
  Gamma matrices
  
  • § Dirac basis
  
  


• Angular momentum


• Gell-Mann matrices


• Poincaré group


• Generalizations of Pauli matrices


• Bloch sphere


• Euler's four-square identity


• For higher spin generalizations of the Pauli matrices, see spin (physics) § Higher spins
  

Remarks

1. 
   ^ This conforms to the mathematics convention for the matrix exponential, iσ ↦ exp(iσ). In the physics convention, σ ↦ exp(−iσ), hence in it no pre-multiplication by i is necessary to land in SU(2).
   


2. 
   ^ The Pauli vector is a formal device. It may be thought of as an element of M₂(ℂ) ⊗ ℝ³, where the tensor product space is endowed with a mapping ⋅: ℝ³ × M₂(ℂ) ⊗ ℝ³ → M₂(ℂ) induced by the dot product on ℝ³.
   


3. 
   ^ N.B. The relation among a, b, c, n, m, k derived here in the 2 × 2 representation holds for all representations of SU(2), being a group identity. Note that, by virtue of the standard normalization of that group's generators as half the Pauli matrices, the parameters a,b,c correspond to half the rotation angles of the rotation group.
   

Notes

1. 
   ^ "Pauli matrices". Planetmath website. 28 March 2008. Retrieved 28 May 2013..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
   


2. 
   ^ See the spinor map.
   


3. 
   ^
   Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-63235-5. OCLC 43641333.
   


4. 
   ^ cf. J W Gibbs (1884). Elements of Vector Analysis, New Haven, 1884, p. 67. In fact, however, the formula goes back to Olinde Rodrigues, 1840, replete with half-angle: "Des lois géometriques qui regissent les déplacements d' un systéme solide dans l' espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire", J. Math. Pures Appl. 5 (1840), 380–440;
   


5. 
   ^ Explicitly, in the convention of "right-space matrices into elements of left-space matrices", it is (1000001001000001) .{displaystyle quad left({begin{smallmatrix}1&0&0&0\0&0&1&0\0&1&0&0\0&0&0&1end{smallmatrix}}right)~.}
   


6. 
   ^ Nakahara, Mikio (2003). Geometry, topology, and physics (2nd ed.). CRC Press. ISBN 978-0-7503-0606-5, pp. xxii.
   


7. 
   ^ ^ab Goldstein, Herbert (1959). Classical Mechanics. Addison-Wesley. pp. 109–118.
   


8. 
   ^ Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10: 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084.
   

References

• 
  Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
  


• 
  Schiff, Leonard I. (1968). Quantum Mechanics. McGraw-Hill. ISBN 978-0070552876.
  


• 
  Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press. ISBN 0-521-14505-8.
  

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