Pauli matrices

In mathematical physics and mathematics, the **Pauli matrices** are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. They are

σ
1

=

σ
x

=

(

0

1

1

0

)

σ
2

=

σ
y

=

(

0

−
i

i

0

)

σ
3

=

σ
z

=

(

1

0

0

−
1

)

.

{\displaystyle
\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}}

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 *σ*0), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices.
This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

Hermitian operators represent observables in quantum mechanics, 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 *iσ*1, *iσ*2, *iσ*3 form a basis for the real Lie algebra

s
u

(
2
)

{\displaystyle \mathfrak{su}(2)}

, which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices *σ*1, *σ*2, *σ*3 is isomorphic to the Clifford algebra of ℝ3 , and the algebra generated by *iσ*1, *iσ*2, *iσ*3 is isomorphic to the quaternions.

## Algebraic properties

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

σ
a

=

(

δ

a
3

δ

a
1

−
i

δ

a
2

δ

a
1

i

δ

a
2

−

δ

a
3

)

{\displaystyle
 \sigma\_a =
 \begin{pmatrix}
 \delta\_{a3} & \delta\_{a1} - i\delta\_{a2}\\
 \delta\_{a1} + i\delta\_{a2} & -\delta\_{a3}
 \end{pmatrix}
}

where *i* = √−1 is the imaginary unit, and *δab* 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:

σ
1
2

=

σ
2
2

=

σ
3
2

=
−
i

σ
1

σ
2

σ
3

=

(

1

0

0

1

)

=
I

{\displaystyle \sigma\_1^2 = \sigma\_2^2 = \sigma\_3^2 = -i\sigma\_1 \sigma\_2 \sigma\_3 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I}

where *I* is the identity matrix.

The determinants and traces of the Pauli matrices are:

det

σ
i

=
−
1
,

tr
⁡

σ
i

=
0
.

{\displaystyle \begin{align}
 \det \sigma\_i &= -1, \\
 \operatorname{tr} \sigma\_i &= 0 .
\end{align}}

From which, we can deduce that the eigenvalues of each *σ**i* are ±1.

With the inclusion of the identity matrix, *I* (sometimes denoted *σ*0), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the real Hilbert space of 2 × 2 complex Hermitian matrices,

H

2

(

C

)

{\displaystyle \mathcal{H}\_2(\mathbb{C})}

, and the complex Hilbert space of all 2 × 2 matrices,

M

2
,
2

(

C

)

{\displaystyle \mathcal{M}\_{2,2}(\mathbb{C})}

.

### Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. Using a convention in which prior to normalization, the 1 is placed into the top and bottom positions of the + and – wavefunctions respectively, the corresponding normalized eigenvectors are:

ψ

x

=

1

2

(

1

1

)

,

ψ

x
−

=

1

2

(

−
1

1

)

,

ψ

y

=

1

2

(

1

i

)

,

ψ

y
−

=

1

2

(

i

1

)

,

ψ

z

=

(

1

0

)

,

ψ

z
−

=

(

0

1

)

.

{\displaystyle \begin{align}
 \psi\_{x+} &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}, &
 \psi\_{x-} &= \frac{1}{\sqrt{2}} \begin{pmatrix} -1 \\ 1 \end{pmatrix}, \\
 \psi\_{y+} &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}, &
 \psi\_{y-} &= \frac{1}{\sqrt{2}} \begin{pmatrix} i \\ 1 \end{pmatrix}, \\
 \psi\_{z+} &= \begin{pmatrix} 1 \\ 0 \end{pmatrix}, &
 \psi\_{z-} &= \begin{pmatrix} 0 \\ 1 \end{pmatrix}.
\end{align}}

An advantage of using this convention is that the + and – wavefunctions may be related to one another, using the Pauli matrices themselves, by

ψ

x

=
i

σ

y

ψ

x
−

{\displaystyle {{\psi }\_{x+}}=i{{\sigma }\_{y}}{{\psi }\_{x-}}}

,

ψ

y

=

σ

z

ψ

y
−

{\displaystyle {{\psi }\_{y+}}={{\sigma }\_{z}}{{\psi }\_{y-}}}

 and

ψ

z

=

σ

x

ψ

z
−

{\displaystyle {{\psi }\_{z+}}={{\sigma }\_{x}}{{\psi }\_{z-}}}

.

### Pauli vector

The Pauli vector is defined by

σ
→

=

σ
1

x
^

σ
2

y
^

σ
3

z
^

{\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 as follows,

a
→

⋅

σ
→

=

(

a
i

x
^

i

)

⋅

(

σ
j

x
^

j

)

=

a
i

σ
j

x
^

i

⋅

x
^

j

=

a
i

σ
j

δ

i
j

=

a
i

σ
i

=

(

a
3

a
1

−
i

a
2

a
1

i

a
2

−

a
3

)

{\displaystyle \begin{align}
 \vec{a} \cdot \vec{\sigma}
 &= \left(a\_i \hat{x}\_i\right) \cdot \left(\sigma\_j \hat{x}\_j\right)
 = 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{align}}

using the summation convention. Further,

det

a
→

⋅

σ
→

=
−

a
→

⋅

a
→

=
−

a
→

2

,

{\displaystyle \det \vec{a} \cdot \vec{\sigma} = -\vec{a} \cdot \vec{a} = -\left|\vec{a}\right|^2,}

its eigenvalues being

±

a
→

{\displaystyle \pm |\vec{a}| }

, and moreover (see completeness, below)

1
2

tr
⁡

(

(

a
→

⋅

σ
→

)

σ
→

)

=

a
→

.

{\displaystyle \frac{1}{2} \operatorname{tr} \left(\left(\vec{a} \cdot \vec{\sigma}\right) \vec{\sigma}\right) = \vec{a} ~.}

Its (unnormalized) eigenvectors are

ψ

=

(

a
3

a
→

a
1

i

a
2

)

;

ψ
−

=

(

i

a
2

−

a
1

a
3

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:

[

σ
a

,

σ
b

# ]
2
i

ε

a
b
c

σ
c

,

{\displaystyle [\sigma\_a, \sigma\_b] = 2 i \varepsilon\_{a b c}\,\sigma\_c \, , }

and anticommutation relations:

{

σ
a

,

σ
b

# }
2

δ

a
b

I
.

{\displaystyle \{\sigma\_a, \sigma\_b\} = 2 \delta\_{a b}\,I.}

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

For example,

commutators

anticommutators

[

σ
1

,

σ
2

]

=
2
i

σ
3

{

σ
1

,

σ
1

}

=
2
I

[

σ
2

,

σ
3

]

=
2
i

σ
1

{

σ
1

,

σ
2

}

=
0

[

σ
3

,

σ
1

]

=
2
i

σ
2

{

σ
1

,

σ
3

}

=
0

[

σ
1

,

σ
1

]

=
0

{

σ
2

,

σ
2

}

=
2
I

.

{\displaystyle \begin{align}
 \text{commutators} & & \text{anticommutators} & \, \\
 \left[\sigma\_1, \sigma\_2\right] &= 2i\sigma\_3 \, & \left\{\sigma\_1, \sigma\_1\right\} &= 2I\, \\
 \left[\sigma\_2, \sigma\_3\right] &= 2i\sigma\_1 \, & \left\{\sigma\_1, \sigma\_2\right\} &= 0\, \\
 \left[\sigma\_3, \sigma\_1\right] &= 2i\sigma\_2 \, & \left\{\sigma\_1, \sigma\_3\right\} &= 0\, \\
 \left[\sigma\_1, \sigma\_1\right] &= 0 \, & \left\{\sigma\_2, \sigma\_2\right\} &= 2I\,. \\

\end{align}}

### 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

[

σ
a

,

σ
b

]

{

σ
a

,

σ
b

}

=
(

σ
a

σ
b

−

σ
b

σ
a

)
(

σ
a

σ
b

σ
b

σ
a

)

2
i

ε

a
b
c

σ
c

2

δ

a
b

I

=
2

σ
a

σ
b

{\displaystyle \begin{align}
 \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) \\
 2i\varepsilon\_{a b c}\,\sigma\_c + 2 \delta\_{a b}I &= 2\sigma\_a \sigma\_b
\end{align}}

so that,

σ
a

σ
b

=

δ

a
b

I
i

ε

a
b
c

σ
c

.

{\displaystyle \sigma\_a \sigma\_b = \delta\_{a b}I + i\varepsilon\_{a b c}\,\sigma\_c~ .}

Contracting each side of the equation with components of two 3-vectors *ap* and *bq* (which commute with the Pauli matrices, i.e., *apσq* = *σqap*) for each matrix *σq* and vector component *ap* (and likewise with *bq*), and relabeling indices *a*, *b*, *c* → *p*, *q*, *r*, to prevent notational conflicts, yields

a
p

b
q

σ
p

σ
q

=

a
p

b
q

(

i

ε

p
q
r

σ
r

δ

p
q

I

)

a
p

σ
p

b
q

σ
q

=
i

ε

p
q
r

a
p

b
q

σ
r

a
p

b
q

δ

p
q

I

.

{\displaystyle \begin{align}
 a\_p b\_q \sigma\_p \sigma\_q & = a\_p b\_q \left(i\varepsilon\_{pqr}\,\sigma\_r + \delta\_{pq}I\right) \\
 a\_p \sigma\_p b\_q \sigma\_q & = i\varepsilon\_{pqr}\,a\_p b\_q \sigma\_r + a\_p b\_q \delta\_{pq}I ~.
\end{align}}

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 \left(\vec{a} \cdot \vec{\sigma}\right)\left(\vec{b} \cdot \vec{\sigma}\right) = \left(\vec{a} \cdot \vec{b}\right) \, I + i \left(\vec{a} \times \vec{b}\right) \cdot \vec{\sigma}}

If

i

{\displaystyle i}

 is identified with the pseudoscalar

σ
x

σ
y

σ
z

{\displaystyle \sigma\_x \sigma\_y \sigma\_z }

 then the right hand side becomes

a
⋅
b
a
∧
b

{\displaystyle a \cdot b + a \wedge 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
⁡

(

σ
a

)

=
0

tr
⁡

(

σ
a

σ
b

)

=
2

δ

a
b

tr
⁡

(

σ
a

σ
b

σ
c

)

=
2
i

ε

a
b
c

tr
⁡

(

σ
a

σ
b

σ
c

σ
d

)

=
2

(

δ

a
b

δ

c
d

−

δ

a
c

δ

b
d

δ

a
d

δ

b
c

)

{\displaystyle \begin{align}
 \operatorname{tr}\left(\sigma\_a\right) &= 0 \\
 \operatorname{tr}\left(\sigma\_a \sigma\_b\right) &= 2\delta\_{ab} \\
 \operatorname{tr}\left(\sigma\_a \sigma\_b \sigma\_c\right) &= 2i\varepsilon\_{abc} \\
 \operatorname{tr}\left(\sigma\_a \sigma\_b \sigma\_c \sigma\_d\right) &= 2\left(\delta\_{ab}\delta\_{cd} - \delta\_{ac}\delta\_{bd} + \delta\_{ad}\delta\_{bc}\right)
\end{align}}

### Exponential of a Pauli vector

For

a
→

=
a

n
^

,

n
^

=
1
,

{\displaystyle \vec{a} = a\hat{n}, \quad |\hat{n}| = 1,}

one has, for even powers,

2
p
,

# p
0
,
1
,
2
,
3
,
…

{\displaystyle 2p, \ \ p = 0, 1, 2, 3, \ldots}

(

n
^

⋅

σ
→

)

2
p

=
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,

2
q
1
,

# q
0
,
1
,
2
,
3
,
…

{\displaystyle 2q + 1,\ \ q = 0, 1, 2, 3, \ldots}

(

n
^

⋅

σ
→

)

2
q
1

=

n
^

⋅

σ
→

.

{\displaystyle \left(\hat{n} \cdot \vec{\sigma}\right)^{2q+1} = \hat{n} \cdot \vec{\sigma} \, .}

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

e

i
a

(

n
^

⋅

σ
→

)

=

∑

# k
0

∞

i
k

[

a

(

n
^

⋅

σ
→

)

]

k

k
!

=

∑

# p
0

∞

(
−
1

)
p

(
a

n
^

⋅

σ
→

)

2
p

(
2
p
)
!

i

∑

# q
0

∞

(
−
1

)
q

(
a

n
^

⋅

σ
→

)

2
q
1

(
2
q
1
)
!

=
I

∑

# p
0

∞

(
−
1

)
p

a

2
p

(
2
p
)
!

i
(

n
^

⋅

σ
→

)

∑

# q
0

∞

(
−
1

)
q

a

2
q
1

(
2
q
1
)
!

{\displaystyle \begin{align}
 e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)}
 &= \sum\_{k=0}^\infty{\frac{i^k \left[a \left(\hat{n} \cdot \vec{\sigma}\right)\right]^k}{k!}} \\
 &= \sum\_{p=0}^\infty{\frac{(-1)^p (a\hat{n}\cdot \vec{\sigma})^{2p}}{(2p)!}} + i\sum\_{q=0}^\infty{\frac{(-1)^q (a\hat{n}\cdot \vec{\sigma})^{2q + 1}}{(2q + 1)!}} \\
 &= I\sum\_{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{align}}

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

e

i
a

(

n
^

⋅

σ
→

)

=
I
cos
⁡

a

i
(

n
^

⋅

σ
→

)
sin
⁡

a

{\displaystyle e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} = I\cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} }

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

Note that

det
[
i
a
(

n
^

⋅

σ
→

)
# ]

a
2

{\displaystyle \det[i a(\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,

f
(
a
(

n
^

⋅

σ
→

)
# )
I

f
(
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). One may directly solve for c in

e

i
a

(

n
^

⋅

σ
→

)

e

i
b

(

m
^

⋅

σ
→

)

=
I
(
cos
⁡
a
cos
⁡
b
−

n
^

⋅

m
^

sin
⁡
a
sin
⁡
b
)
i
(

n
^

sin
⁡
a
cos
⁡
b

m
^

sin
⁡
b
cos
⁡
a
−

n
^

×

m
^

sin
⁡
a
sin
⁡
b
)
⋅

σ
→

=
I
cos
⁡

c

i
(

k
^

⋅

σ
→

)
sin
⁡

c

=

e

i
c

(

k
^

⋅

σ
→

)

,

{\displaystyle \begin{align}
 e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} e^{i b\left(\hat{m} \cdot \vec{\sigma}\right)}
 &= I(\cos a \cos b - \hat{n} \cdot\hat{m} \sin a \sin b) + i(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin b )\cdot \vec{\sigma } \\
 &= I\cos{c} + i (\hat{k} \cdot \vec{\sigma}) \sin{c} \\
 &= e^{i c \left(\hat{k} \cdot \vec{\sigma}\right)},
\end{align}}

which specifies the generic group multiplication, where, manifestly,

cos
⁡
# c
cos
⁡
a
cos
⁡
b
−

n
^

⋅

m
^

sin
⁡
a
sin
⁡
b

,

{\displaystyle \cos c = \cos a \cos b - \hat{n} \cdot\hat{m} \sin a \sin b~,}

the spherical law of cosines. Given c, then,

k
^

=

1

sin
⁡
c

(

n
^

sin
⁡
a
cos
⁡
b

m
^

sin
⁡
b
cos
⁡
a
−

n
^

×

m
^

sin
⁡
a
sin
⁡
b

)

.

{\displaystyle \hat{k} = \frac{1}{\sin c}\left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} \sin a \sin b\right) ~.}

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

e

i
c

k
^

⋅

σ
→

=
exp
⁡

(

i

c

sin
⁡
c

(

n
^

sin
⁡
a
cos
⁡
b

m
^

sin
⁡
b
cos
⁡
a
−

n
^

×

m
^

sin
⁡
a
sin
⁡
b
)
⋅

σ
→

)

.

{\displaystyle e^{ic \hat{k}\cdot \vec{\sigma}}= \exp \left( i\frac{c}{\sin c} (\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin 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, 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,

e

i
a

(

n
^

⋅

σ
→

)

σ
→

e

−
i
a

(

n
^

⋅

σ
→

)

=

σ
→

cos
⁡
(
2
a
)

n
^

×

σ
→

sin
⁡
(
2
a
)

n
^

n
^

⋅

σ
→

(
1
−
cos
⁡
(
2
a
)
)

.

{\displaystyle
e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} ~ \vec{\sigma}~ e^{-i a\left(\hat{n} \cdot \vec{\sigma}\right)} = \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
→

α
β

⋅

σ
→

γ
δ

≡

∑

# i
1

3

σ

α
β

i

σ

γ
δ

i

=
2

δ

α
δ

δ

β
γ

−

δ

α
β

δ

γ
δ

.

{\displaystyle \vec{\sigma}\_{\alpha\beta}\cdot\vec{\sigma}\_{\gamma\delta}\equiv \sum\_{i=1}^3 \sigma^i\_{\alpha\beta}\sigma^i\_{\gamma\delta} = 2 \delta\_{\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
c
I

∑
i

a
i

σ
i

{\displaystyle M = c I + \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

tr

σ
i

σ
j

=
2

δ

i
j

{\displaystyle \operatorname{tr}\, \sigma^i\sigma^j = 2\delta\_{ij}}

where "tr" denotes the trace, and hence that

c

=

1
2

tr

M
,

a
i

=

1
2

tr

σ
i

M

.

∴

2
M

=
I
tr

M

∑
i

σ
i

tr

σ
i

M

,

{\displaystyle \begin{align}
 c &= \frac{1}{2}\operatorname{tr}\,M,\ \ a\_i = \frac{1}{2}\operatorname{tr}\,\sigma^i M ~. \\[3pt]
 \therefore ~~ 2M &= I \operatorname{tr}\, M + \sum\_i \sigma^i \operatorname{tr}\, \sigma^i M ~,
\end{align}}

which can be rewritten in terms of matrix indices as

2

M

α
β

=

δ

α
β

M

γ
γ

∑
i

σ

α
β

i

σ

γ
δ

i

M

δ
γ

,

{\displaystyle 2M\_{\alpha\beta} = \delta\_{\alpha\beta} M\_{\gamma\gamma} + \sum\_i \sigma^i\_{\alpha\beta} \sigma^i\_{\gamma\delta} 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 *σ*0, so *σ*0*αβ* = *δ**αβ*.
The completeness relation can alternatively be expressed as

∑

# i
0

3

σ

α
β

i

σ

γ
δ

i

=
2

δ

α
δ

δ

β
γ

.

{\displaystyle \sum\_{i=0}^3 \sigma^i\_{\alpha\beta}\sigma^i\_{\gamma\delta} = 2 \delta\_{\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 unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of {*σ*0, *σ*1, *σ*2, *σ*3} as above, and then imposing the positive-semidefinite and trace 1 conditions.

For a pure state, in polar coordinates,

a
→

=
(
sin
⁡
θ
cos
⁡
ϕ
,
sin
⁡
θ
sin
⁡
ϕ
,
cos
⁡
θ
)

{\displaystyle \vec a = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta )}

 , the idempotent density matrix

1
2

(
1

1

a
→

⋅

σ
→

# )

(

cos
2

⁡
θ

/

2

sin
⁡
θ

/

2

cos
⁡
θ

/

2

e

−
i
ϕ

sin
⁡
θ

/

2

cos
⁡
θ

/

2

e

i
ϕ

sin
2

⁡
θ

/

2

)

{\displaystyle \frac{1}{2}(1\!\! 1 + \vec a \cdot \vec \sigma) =
 \begin{pmatrix}
 \cos^2 \theta/2 & \sin \theta/2 ~ \cos\theta/2 ~e^{-i\phi} \\
 \sin \theta/2 ~ \cos\theta/2 ~e^{i\phi} & \sin^2 \theta/2
 \end{pmatrix}
}

acts on the state eigenvector

(
cos
⁡
θ

/

2
,

e

i
ϕ

sin
⁡
θ

/

2
)

{\displaystyle (\cos \theta/2 , e^{i\phi} \sin \theta/2) }

 with eigenvalue 1, so like a  projection operator for it.

### Relation with the permutation operator

Let *Pij* be the transposition (also known as a permutation) between two spins *σ**i* and *σ**j* living in the tensor product space ℂ2 ⊗ ℂ2,

P

i
j

σ
i

σ
j

# ⟩

σ
j

σ
i

⟩

.

{\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,

P

i
j

=

1
2

(

σ
→

i

⋅

σ
→

j

1
)

.

{\displaystyle P\_{ij} = \frac{1}{2}(\vec{\sigma}\_i\cdot\vec{\sigma}\_j + 1)\,.}

Its eigenvalues are therefore 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

s
u

2

{\displaystyle \mathfrak{su}\_2}

 is the 3-dimensional real algebra spanned by the set {*iσj*}. In compact notation,

s
u

(
2
# )
span
⁡
{
i

σ
1

,
i

σ
2

,
i

σ
3

}
.

{\displaystyle \mathfrak{su}(2) = \operatorname{span} \{ i \sigma\_1, i \sigma\_2 , i \sigma\_3 \}.}

As a result, each *iσj* 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* = 12, so that

s
u

(
2
# )
span
⁡

{

i

σ
1

2

,

i

σ
2

2

,

i

σ
3

2

}

.

{\displaystyle \mathfrak{su}(2) = \operatorname{span} \left\{\frac{i \sigma\_1}{2}, \frac{i \sigma\_2}{2}, \frac{i \sigma\_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 *iσj* 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

Main articles: Spinor#Three dimensions
The real linear span of {*I*, *iσ*1, *iσ*2, *iσ*3} 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

↦
−

σ
2

σ
3

=
−
i

σ
1

,

j

↦
−

σ
3

σ
1

=
−
i

σ
2

,

k

↦
−

σ
1

σ
2

=
−
i

σ
3

.

{\displaystyle
 1 \mapsto I, \quad
 \mathbf{i} \mapsto - \sigma\_2\sigma\_3 = - i \sigma\_1, \quad
 \mathbf{j} \mapsto - \sigma\_3\sigma\_1 = - i \sigma\_2, \quad
 \mathbf{k} \mapsto - \sigma\_1\sigma\_2 = - i \sigma\_3.
}

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

1
↦
I
,

i

↦
i

σ
3

,

j

↦
i

σ
2

,

k

↦
i

σ
1

.

{\displaystyle
 1 \mapsto I, \quad
 \mathbf{i} \mapsto i \sigma\_3, \quad
 \mathbf{j} \mapsto i \sigma\_2, \quad
 \mathbf{k} \mapsto i \sigma\_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

Main articles: Quaternions and spatial rotation
In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters. 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
→

=
x

σ
x

y

σ
y

z

σ
z

.

{\displaystyle P = \vec{x} \cdot \vec{\sigma} = x\sigma\_x + y\sigma\_y + z\sigma\_z ~.}

Consequently, the transformation matrix

Q
θ

{\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

Q
θ

=
1

1
cos
⁡

θ
2

i

σ
x

sin
⁡

θ
2

.

{\displaystyle Q\_\theta = 1\!\!1 \cos\frac{\theta}{2} + i\sigma\_x \sin\frac{\theta}{2} ~.}

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

## Real Clifford algebra **R**3,1

From Wikipedia:Clifford module

Developed by Ettore Majorana, this Clifford module enables the construction of a Dirac-like equation without complex numbers, and its elements are called Majorana spinors.

The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix. The signature is (+++−). For the signatures (+−−−) and (−−−+) often used in physics, 4×4 complex matrices or 8×8 real matrices are needed.