Advanced mathematics

This article is College level

## Commutator

In mathematics, the **commutator** gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

 The **commutator** of two elements *a* and *b* of a ring (including any associative algebra) is defined by

[
a
,
b
# ]
a
b
−
b
a
.

{\displaystyle [a,b]=ab-ba.}

It is zero if and only if *a* and *b* commute. In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.

The **anticommutator** of two elements a and b of a ring or an associative algebra is defined by

{
a
,
b
# }
a
b
b
a
.

{\displaystyle \{a,b\}=ab+ba.}

Sometimes

[
a
,
b

]

{\displaystyle [a,b]\_{+}}

 is used to denote anticommutator, while

[
a
,
b

]

−

{\displaystyle [a,b]\_{-}}

 is then used for commutator. The anticommutator is used less often, but can be used to define Clifford algebras and Jordan algebras, and in the derivation of the Dirac equation in particle physics.

The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödinger relation. In phase space, equivalent commutators of function star-products are called Moyal brackets, and are completely isomorphic to the Hilbert-space commutator structures mentioned.

The commutator has the following properties:

### Lie-algebra identities

1. [
A
B
,
C
# ]
[
A
,
C
]
[
B
,
C
]

{\displaystyle [A+B,C]=[A,C]+[B,C]}
2. [
A
,
A
# ]
0

{\displaystyle [A,A]=0}
3. [
A
,
B
# ]
−
[
B
,
A
]

{\displaystyle [A,B]=-[B,A]}
4. [
A
,
[
B
,
C
]
]
[
B
,
[
C
,
A
]
]
[
C
,
[
A
,
B
]
# ]
0

{\displaystyle [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0}

Relation (3) is called anticommutativity, while (4) is the Jacobi identity.

### Additional identities

1. [
A
,
B
C
# ]
[
A
,
B
]
C
B
[
A
,
C
]

{\displaystyle [A,BC]=[A,B]C+B[A,C]}
2. [
A
,
B
C
D
# ]
[
A
,
B
]
C
D
B
[
A
,
C
]
D
B
C
[
A
,
D
]

{\displaystyle [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]}
3. [
A
,
B
C
D
E
# ]
[
A
,
B
]
C
D
E
B
[
A
,
C
]
D
E
B
C
[
A
,
D
]
E
B
C
D
[
A
,
E
]

{\displaystyle [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]}
4. [
A
B
,
C
# ]
A
[
B
,
C
]
[
A
,
C
]
B

{\displaystyle [AB,C]=A[B,C]+[A,C]B}
5. [
A
B
C
,
D
# ]
A
B
[
C
,
D
]
A
[
B
,
D
]
C
[
A
,
D
]
B
C

{\displaystyle [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC}
6. [
A
B
C
D
,
E
# ]
A
B
C
[
D
,
E
]
A
B
[
C
,
E
]
D
A
[
B
,
E
]
C
D
[
A
,
E
]
B
C
D

{\displaystyle [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD}
7. [
A
,
B
C
# ]
[
A
,
B
]
[
A
,
C
]

{\displaystyle [A,B+C]=[A,B]+[A,C]}
8. [
A
B
,
C
D
# ]
[
A
,
C
]
[
A
,
D
]
[
B
,
C
]
[
B
,
D
]

{\displaystyle [A+B,C+D]=[A,C]+[A,D]+[B,C]+[B,D]}
9. [
A
B
,
C
D
# ]
A
[
B
,
C
]
D
[
A
,
C
]
B
D
C
A
[
B
,
D
]
C
[
A
,
D
]
B

{\displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B}
10. [
[
A
,
C
]
,
[
B
,
D
]
# ]
[
[
[
[
A
,
B
]
,
C
]
,
D
]
[
[
[
[
B
,
C
]
,
D
]
,
A
]
[
[
[
[
C
,
D
]
,
A
]
,
B
]
[
[
[
[
D
,
A
]
,
B
]
,
C
]

{\displaystyle [[A,C],[B,D]]=[[[[A,B],C],D]+[[[[B,C],D],A]+[[[[C,D],A],B]+[[[[D,A],B],C]}

If A is a fixed element of a ring *R*, identity (1) can be interpreted as a Leibniz rule for the map

ad

A

:
R
→
R

{\displaystyle \operatorname {ad} \_{A}:R\rightarrow R}

 given by

ad

A

⁡
(
B
# )
[
A
,
B
]

{\displaystyle \operatorname {ad} \_{A}(B)=[A,B]}

. In other words, the map ad*A* defines a derivation on the ring *R*. Identities (2), (3) represent Leibniz rules for more than two factors, and are valid for any derivation. Identities (4)–(6) can also be interpreted as Leibniz rules. Identities (7), (8) express **Z**-bilinearity.

Some of the above identities can be extended to the anticommutator using the above ± subscript notation.
For example:

1. [
A
B
,
C

]

−

=
A
[
B
,
C

]

∓

±
[
A
,
C

]

∓

B

{\displaystyle [AB,C]\_{-}=A[B,C]\_{\mp }\pm [A,C]\_{\mp }B}
2. [
A
B
,
C
D

]

−

=
A
[
B
,
C

]

∓

D
±
A
C
[
B
,
D

]

∓

[
A
,
C

]

∓

D
B
±
C
[
A
,
D

]

∓

B

{\displaystyle [AB,CD]\_{-}=A[B,C]\_{\mp }D\pm AC[B,D]\_{\mp }+[A,C]\_{\mp }DB\pm C[A,D]\_{\mp }B}
3. [

A
,
[
B
,
C

]

±

]

[

B
,
[
C
,
A

]

±

]

[

C
,
[
A
,
B

]

±

]

=
0

{\displaystyle \left[A,[B,C]\_{\pm }\right]+\left[B,[C,A]\_{\pm }\right]+\left[C,[A,B]\_{\pm }\right]=0}

## Cross product

From Wikipedia:Cross product

Cross product is a crude and primitive way of multiplying real vectors in real vector space but it is a brilliant way of multiplying imaginary vectors in complex vector space.

The cross product can be defined in terms of the exterior product. The cross product is then obtained by taking the Hodge star of the bivector *a* ∧ *b*:

a
×
# b
⋆
(
a
∧
b
)

.

{\displaystyle a\times b=\star (a\wedge b)\,.}

The cross product can also be described in terms of quaternions. The unit vectors **i**, **j**, **k** correspond to rotations about their respective axes.

In general, if a vector [*a*1, *a*2, *a*3] is represented as the quaternion *a*1*i* + *a*2*j* + *a*3*k*, the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors.

Alternatively, using the above identification of the 'purely imaginary' quaternions with **R**3, the cross product may be thought of as half of the (de)commutator of two quaternions. The commutator has the effect of deleting the scalar part (the commutative part).

A
B

=
(

a

1

e

12

a

2

e

23

a

3

e

31

)
(

b

1

e

12

b

2

e

23

b

3

e

31

)

=

a

1

b

1

e

1212

a

1

b

2

e

1223

a

1

b

3

e

1231

a

2

b

1

e

2312

a

2

b

2

e

2323

a

2

b

3

e

2331

a

3

b

1

e

3112

a

3

b

2

e

3123

a

3

b

3

e

3131

=
−

a

1

b

1

a

1

b

2

e

13

a

1

b

3

e

23

a

2

b

1

e

31

−

a

2

b

2

a

2

b

3

e

21

a

3

b

1

e

32

a

3

b

2

e

12

−

a

3

b

3

=
−

a

1

b

1

−

a

2

b

2

−

a

3

b

3

a

3

b

2

e

12

−

a

2

b

3

e

12

a

1

b

3

e

23

−

a

3

b

1

e

23

a

2

b

1

e

31

−

a

1

b

2

e

31

{\displaystyle {\begin{aligned}AB&=(a\_{1}e\_{12}+a\_{2}e\_{23}+a\_{3}e\_{31})(b\_{1}e\_{12}+b\_{2}e\_{23}+b\_{3}e\_{31})\\\\&=a\_{1}b\_{1}e\_{1212}+a\_{1}b\_{2}e\_{1223}+a\_{1}b\_{3}e\_{1231}\\&+a\_{2}b\_{1}e\_{2312}+a\_{2}b\_{2}e\_{2323}+a\_{2}b\_{3}e\_{2331}\\&+a\_{3}b\_{1}e\_{3112}+a\_{3}b\_{2}e\_{3123}+a\_{3}b\_{3}e\_{3131}\\\\&=-a\_{1}b\_{1}+a\_{1}b\_{2}e\_{13}+a\_{1}b\_{3}e\_{23}\\&+a\_{2}b\_{1}e\_{31}-a\_{2}b\_{2}+a\_{2}b\_{3}e\_{21}\\&+a\_{3}b\_{1}e\_{32}+a\_{3}b\_{2}e\_{12}-a\_{3}b\_{3}\\\\&=-a\_{1}b\_{1}-a\_{2}b\_{2}-a\_{3}b\_{3}\\&+a\_{3}b\_{2}e\_{12}-a\_{2}b\_{3}e\_{12}\\&+a\_{1}b\_{3}e\_{23}-a\_{3}b\_{1}e\_{23}\\&+a\_{2}b\_{1}e\_{31}-a\_{1}b\_{2}e\_{31}\end{aligned}}}

B
A

=
(

b

1

e

12

b

2

e

23

b

3

e

31

)
(

a

1

e

12

a

2

e

23

a

3

e

31

)

=

b

1

a

1

e

1212

b

1

a

2

e

1223

b

1

a

3

e

1231

b

2

a

1

e

2312

b

2

a

2

e

2323

b

2

a

3

e

2331

b

3

a

1

e

3112

b

3

a

2

e

3123

b

3

a

3

e

3131

=
−

b

1

a

1

b

1

a

2

e

13

b

1

a

3

e

23

b

2

a

1

e

31

−

b

2

a

2

b

2

a

3

e

21

b

3

a

1

e

32

b

3

a

2

e

12

−

b

3

a

3

=
−

b

1

a

1

−

b

2

a

2

−

b

3

a

3

b

3

a

2

e

12

−

b

2

a

3

e

12

b

1

a

3

e

23

−

b

3

a

1

e

23

b

2

a

1

e

31

−

b

1

a

2

e

31

{\displaystyle {\begin{aligned}BA&=(b\_{1}e\_{12}+b\_{2}e\_{23}+b\_{3}e\_{31})(a\_{1}e\_{12}+a\_{2}e\_{23}+a\_{3}e\_{31})\\\\&=b\_{1}a\_{1}e\_{1212}+b\_{1}a\_{2}e\_{1223}+b\_{1}a\_{3}e\_{1231}\\&+b\_{2}a\_{1}e\_{2312}+b\_{2}a\_{2}e\_{2323}+b\_{2}a\_{3}e\_{2331}\\&+b\_{3}a\_{1}e\_{3112}+b\_{3}a\_{2}e\_{3123}+b\_{3}a\_{3}e\_{3131}\\\\&=-b\_{1}a\_{1}+b\_{1}a\_{2}e\_{13}+b\_{1}a\_{3}e\_{23}\\&+b\_{2}a\_{1}e\_{31}-b\_{2}a\_{2}+b\_{2}a\_{3}e\_{21}\\&+b\_{3}a\_{1}e\_{32}+b\_{3}a\_{2}e\_{12}-b\_{3}a\_{3}\\\\&=-b\_{1}a\_{1}-b\_{2}a\_{2}-b\_{3}a\_{3}\\&+b\_{3}a\_{2}e\_{12}-b\_{2}a\_{3}e\_{12}\\&+b\_{1}a\_{3}e\_{23}-b\_{3}a\_{1}e\_{23}\\&+b\_{2}a\_{1}e\_{31}-b\_{1}a\_{2}e\_{31}\end{aligned}}}

A
B
−
B
A

=
2
⋅
(

a

3

b

2

−

a

2

b

3

)

e

12

2
⋅
(

a

1

b

3

−

a

3

b

1

)

e

23

2
⋅
(

a

2

b

1

−

a

1

b

2

)

e

31

{\displaystyle {\begin{aligned}AB-BA&=2\cdot (a\_{3}b\_{2}-a\_{2}b\_{3})e\_{12}\\&2\cdot (a\_{1}b\_{3}-a\_{3}b\_{1})e\_{23}\\&2\cdot (a\_{2}b\_{1}-a\_{1}b\_{2})e\_{31}\end{aligned}}}

The standard basis vectors **i**, **j**, and **k** satisfy the following equalities in a right hand coordinate system:

i

×

j

=

k

j

×

k

=

i

k

×

i

=

j

{\displaystyle {\begin{alignedat}{2}\mathbf {\color {blue}{i}} &\times \mathbf {\color {red}{j}} &&=\mathbf {\color {green}{k}} \\\mathbf {\color {red}{j}} &\times \mathbf {\color {green}{k}} &&=\mathbf {\color {blue}{i}} \\\mathbf {\color {green}{k}} &\times \mathbf {\color {blue}{i}} &&=\mathbf {\color {red}{j}} \end{alignedat}}}

which imply, by the anticommutativity of the cross product, that

j

×

i

=
−

k

k

×

j

=
−

i

i

×

k

=
−

j

{\displaystyle {\begin{alignedat}{2}\mathbf {\color {red}{j}} &\times \mathbf {\color {blue}{i}} &&=-\mathbf {\color {green}{k}} \\\mathbf {\color {green}{k}} &\times \mathbf {\color {red}{j}} &&=-\mathbf {\color {blue}{i}} \\\mathbf {\color {blue}{i}} &\times \mathbf {\color {green}{k}} &&=-\mathbf {\color {red}{j}} \end{alignedat}}}

The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that

i

×

i

=

j

×

j

=

k

×

k

=

0

{\displaystyle \mathbf {\color {blue}{i}} \times \mathbf {\color {blue}{i}} =\mathbf {\color {red}{j}} \times \mathbf {\color {red}{j}} =\mathbf {\color {green}{k}} \times \mathbf {\color {green}{k}} =\mathbf {0} }

 (the zero vector).
These equalities, together with the distributivity and linearity of the cross product (but both do not follow easily from the definition given above), are sufficient to determine the cross product of any two vectors **a** and **b**. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors:

a

=

a

1

i

a

2

j

a

3

k

b

=

b

1

i

b

2

j

b

3

k

{\displaystyle {\begin{alignedat}{3}\mathbf {a} &=a\_{1}\mathbf {\color {blue}{i}} &&+a\_{2}\mathbf {\color {red}{j}} &&+a\_{3}\mathbf {\color {green}{k}} \\\mathbf {b} &=b\_{1}\mathbf {\color {blue}{i}} &&+b\_{2}\mathbf {\color {red}{j}} &&+b\_{3}\mathbf {\color {green}{k}} \end{alignedat}}}

Their cross product **a** × **b** can be expanded using distributivity:

a

×

b

=

(

a

1

i

a

2

j

a

3

k

)
×
(

b

1

i

b

2

j

b

3

k

)

=

a

1

b

1

(

i

×

i

)

a

1

b

2

(

i

×

j

)

a

1

b

3

(

i

×

k

)

a

2

b

1

(

j

×

i

)

a

2

b

2

(

j

×

j

)

a

2

b

3

(

j

×

k

)

a

3

b

1

(

k

×

i

)

a

3

b

2

(

k

×

j

)

a

3

b

3

(

k

×

k

)

{\displaystyle {\begin{aligned}\mathbf {a} \times \mathbf {b} ={}&(a\_{1}\mathbf {\color {blue}{i}} +a\_{2}\mathbf {\color {red}{j}} +a\_{3}\mathbf {\color {green}{k}} )\times (b\_{1}\mathbf {\color {blue}{i}} +b\_{2}\mathbf {\color {red}{j}} +b\_{3}\mathbf {\color {green}{k}} )\\={}&a\_{1}b\_{1}(\mathbf {\color {blue}{i}} \times \mathbf {\color {blue}{i}} )+a\_{1}b\_{2}(\mathbf {\color {blue}{i}} \times \mathbf {\color {red}{j}} )+a\_{1}b\_{3}(\mathbf {\color {blue}{i}} \times \mathbf {\color {green}{k}} )+{}\\&a\_{2}b\_{1}(\mathbf {\color {red}{j}} \times \mathbf {\color {blue}{i}} )+a\_{2}b\_{2}(\mathbf {\color {red}{j}} \times \mathbf {\color {red}{j}} )+a\_{2}b\_{3}(\mathbf {\color {red}{j}} \times \mathbf {\color {green}{k}} )+{}\\&a\_{3}b\_{1}(\mathbf {\color {green}{k}} \times \mathbf {\color {blue}{i}} )+a\_{3}b\_{2}(\mathbf {\color {green}{k}} \times \mathbf {\color {red}{j}} )+a\_{3}b\_{3}(\mathbf {\color {green}{k}} \times \mathbf {\color {green}{k}} )\\\end{aligned}}}

Multiplying out and collecting similar terms, we obtain:

a

×

b

=

a

1

b

1

0

a

1

b

2

k

−

a

1

b

3

j

−

a

2

b

1

k

a

2

b

2

0

a

2

b

3

i

a

3

b

1

j

−

a

3

b

2

i

a

3

b

3

0

=

(

a

2

b

3

−

a

3

b

2

)

i

(

a

3

b

1

−

a

1

b

3

)

j

(

a

1

b

2

−

a

2

b

1

)

k

{\displaystyle {\begin{aligned}\mathbf {a} \times \mathbf {b} ={}&a\_{1}b\_{1}\mathbf {0} +a\_{1}b\_{2}\mathbf {\color {green}{k}} -a\_{1}b\_{3}\mathbf {\color {red}{j}} \\&-a\_{2}b\_{1}\mathbf {\color {green}{k}} +a\_{2}b\_{2}\mathbf {0} +a\_{2}b\_{3}\mathbf {\color {blue}{i}} \\&+a\_{3}b\_{1}\mathbf {\color {red}{j}} \ -a\_{3}b\_{2}\mathbf {\color {blue}{i}} \ +a\_{3}b\_{3}\mathbf {0} \\={}&(a\_{2}b\_{3}-a\_{3}b\_{2})\mathbf {\color {blue}{i}} +(a\_{3}b\_{1}-a\_{1}b\_{3})\mathbf {\color {red}{j}} +(a\_{1}b\_{2}-a\_{2}b\_{1})\mathbf {\color {green}{k}} \\\end{aligned}}}

meaning that the three scalar coefficients of the resulting vector **s** = *s*1**i** + *s*2**j** + *s*3**k** = **a** × **b** are

s

1

=

a

2

b

3

−

a

3

b

2

s

2

=

a

3

b

1

−

a

1

b

3

s

3

=

a

1

b

2

−

a

2

b

1

{\displaystyle {\begin{aligned}s\_{1}&=a\_{2}b\_{3}-a\_{3}b\_{2}\\s\_{2}&=a\_{3}b\_{1}-a\_{1}b\_{3}\\s\_{3}&=a\_{1}b\_{2}-a\_{2}b\_{1}\end{aligned}}}

Using column vectors, we can represent the same result as follows:

(

s

1

s

2

s

3

)

=

(

a

2

b

3

−

a

3

b

2

a

3

b

1

−

a

1

b

3

a

1

b

2

−

a

2

b

1

)

{\displaystyle {\begin{pmatrix}s\_{1}\\s\_{2}\\s\_{3}\end{pmatrix}}={\begin{pmatrix}a\_{2}b\_{3}-a\_{3}b\_{2}\\a\_{3}b\_{1}-a\_{1}b\_{3}\\a\_{1}b\_{2}-a\_{2}b\_{1}\end{pmatrix}}}

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### Algebraic properties

If the cross product of two vectors is the zero vector (i.e. **a** × **b** = **0**), then either one or both of the inputs is the zero vector, (**a** = **0** or **b** = **0**) or else they are parallel or antiparallel (**a** ∥ **b**) so that the sine of the angle between them is zero (*θ* = 0° or *θ* = 180° and sin*θ* = 0).

The self cross product of a vector is the zero vector:

a

×

a

=

0

{\displaystyle \mathbf {a} \times \mathbf {a} =\mathbf {0} }

The cross product is anticommutative,

a

×

b

=
−
(

b

×

a

)
,

{\displaystyle \mathbf {a} \times \mathbf {b} =-(\mathbf {b} \times \mathbf {a} ),}

is distributive over addition,

a

×
(

b

c

# )
(

a

×

b

)
(

a

×

c

)
,

{\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=(\mathbf {a} \times \mathbf {b} )+(\mathbf {a} \times \mathbf {c} ),}

is compatible with scalar multiplication so that

(
r

a

)
×

b

=

a

×
(
r

b

# )
r

(

a

×

b

)
.

{\displaystyle (r\,\mathbf {a} )\times \mathbf {b} =\mathbf {a} \times (r\,\mathbf {b} )=r\,(\mathbf {a} \times \mathbf {b} ).}

is not associative, but satisfies the Jacobi identity:

a

×
(

b

×

c

)

b

×
(

c

×

a

)

c

×
(

a

×

b

# )

0

.

{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )+\mathbf {b} \times (\mathbf {c} \times \mathbf {a} )+\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=\mathbf {0} .}

Distributivity, linearity and Jacobi identity show that the **R**3 vector space together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO(3).

The product rule of differential calculus applies to any bilinear operation, and therefore also to the cross product:

d

d
t

(

a

×

b

# )

d

a

d
t

×

b

a

×

d

b

d
t

,

{\displaystyle {\frac {d}{dt}}(\mathbf {a} \times \mathbf {b} )={\frac {d\mathbf {a} }{dt}}\times \mathbf {b} +\mathbf {a} \times {\frac {d\mathbf {b} }{dt}},}

where **a** and **b** are vectors that depend on the real variable *t*.

The cross product conveniently describes the infinitesimal generators of rotations in **R**3. Specifically, if **n** is a unit vector in **R**3 and *R*(*φ*, **n**) denotes a rotation about the axis through the origin specified by **n**, with angle φ (measured in radians, counterclockwise when viewed from the tip of **n**), then

d

d
ϕ

# ϕ
0

R
(
ϕ
,

n

)

x

=

n

×

x

{\displaystyle \left.{d \over d\phi }\right|\_{\phi =0}R(\phi ,{\boldsymbol {n}}){\boldsymbol {x}}={\boldsymbol {n}}\times {\boldsymbol {x}}}

for every vector **x** in **R**3. The cross product with **n** therefore describes the infinitesimal generator of the rotations about **n**. These infinitesimal generators form the Lie algebra **so**(3) of the rotation group SO(3), and we obtain the result that the Lie algebra **R**3 with cross product is isomorphic to the Lie algebra **so**(3).

## Lie algebra

From Wikipedia: Lie algebra

A **Lie algebra** is a vector space

g

{\displaystyle {\mathfrak {g}}}

 together with an binary operation called the **Lie bracket**, an alternating bilinear map

g

×

g

→

g

,

(
x
,
y
)
↦
[
x
,
y
]

{\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}},\ (x,y)\mapsto [x,y]}

, that satisfies the Jacobi identity.

The brackets [,] represent bilinear operation "×"; often, it is the commutator: [*x*,*y*] = *x* *y* − *y* *x*.

The vector space

g

{\displaystyle {\mathfrak {g}}}

 together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

On an associative algebra

A

{\displaystyle A}

 over a field

F

{\displaystyle F}

 with multiplication

(
x
,
y
)
↦
x
y

{\displaystyle (x,y)\mapsto xy}

, a Lie bracket may be defined by the commutator

[
x
,
y
# ]
x
y
−
y
x

{\displaystyle [x,y]=xy-yx}

. With this bracket,

A

{\displaystyle A}

 is a Lie algebra. The associative algebra *A* is called an *enveloping algebra* of the Lie algebra

(
A
,
[

⋅

,
⋅

]
)

{\displaystyle (A,[\,\cdot \,,\cdot \,])}

. Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see universal enveloping algebra.

Back to top

## Spinors

External links: The Construction of Spinors in Geometric Algebra, Spinor calculus, and An introduction to spinors
See also: Bispinor
Spinors generalize the idea of rotors (using reverse). They may be regarded as non-normalised rotors which transform single-sided.

Note: The (real) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

A spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°. This property characterizes spinors.

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### Spinors in two dimensions

A spinor in two dimensions is a complex number which represents a rotation. A complex number that lies on the unit circle represents a pure rotation.

Rotating a rotation results in a rotation just as multiplying a complex number by a complex number results in a complex number. Rotations therefore form a group called the spin group.

Back to top

### Spinors in three dimensions

A vector in three dimensions can be represented by Pauli matrices by

v

=

v

x

σ

x

v

y

σ

y

v

z

σ

z

{\displaystyle \mathbf {v} =v\_{x}\sigma \_{x}+v\_{y}\sigma \_{y}+v\_{z}\sigma \_{z}}

Spinor calculus differs by allowing the coefficients (*v*1, *v*2, *v*3) to be complex numbers. In other words it exists in complex vector space. (In complex vector space the dot product is replaced with an inner product.)

i

σ

x

=

σ

y

σ

z

=

(

0

i

i

0

)

=

k

i

σ

y

=

σ

z

σ

x

=

(

0

1

−
1

0

)

=

j

i

σ

z

=

σ

x

σ

y

=

(

i

0

0

−
i

)

=

i

{\displaystyle {\begin{aligned}i\sigma \_{x}=\sigma \_{y}\sigma \_{z}&={\begin{pmatrix}0&i\\i&0\end{pmatrix}}=\mathbf {k} \\i\sigma \_{y}=\sigma \_{z}\sigma \_{x}&={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}=\mathbf {j} \\i\sigma \_{z}=\sigma \_{x}\sigma \_{y}&={\begin{pmatrix}i&0\\0&-i\end{pmatrix}}=\mathbf {i} \end{aligned}}}

From Wikipedia:Spinors in three dimensions
Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:

* det *X* = – (length **x**)2.
* *X* 2 = (length **x**)2*I*, where *I* is the identity matrix.
* 1
2

(
X
Y
Y
X
# )
(

x

⋅

y

)
I

{\displaystyle {\frac {1}{2}}(XY+YX)=({\mathbf {x}}\cdot {\mathbf {y}})I}
* 1
2

(
X
Y
−
Y
X
# )
i
Z

{\displaystyle {\frac {1}{2}}(XY-YX)=iZ}

 where *Z* is the matrix associated to the cross product **z** = **x** × **y**.
* If **u** is a unit vector, then −*UXU* is the matrix associated to the vector obtained from **x** by reflection in the plane orthogonal to **u**.
* It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections. (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if *R* is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector **u**1 followed by the plane perpendicular to **u**2, then the matrix *U*2*U*1*XU*1*U*2 represents the rotation of the vector **x** through *R*.

From Wikipedia:Spinor:

> Quote from Elie Cartan: *The Theory of Spinors*, Hermann, Paris, 1966: "Spinors...provide a linear representation of the group of rotations in a space with any number
>
>
>
> n
>
>
> {\displaystyle n}
>
>  of dimensions, each spinor having
>
>
>
>
> 2
>
> ν
>
>
>
>
> {\displaystyle 2^{\nu }}
>
>  components where
>
>
>
> n
> =
> 2
> ν
> +
> 1
>
>
> {\displaystyle n=2\nu +1}
>
>  or
>
>
>
> 2
> ν
>
>
> {\displaystyle 2\nu }
>
> ." The star (\*) refers to Cartan 1913.
>
>
>

(Note:

ν

{\displaystyle \nu }

 is the number of simultaneous independent rotations an object can have in n dimensions.)

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act.

In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.

In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors (that is, the spinors).

Back to top

#### Column vectors

Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural (see Multivectors#Complex\_numbers) to ask what role, if any, the 2×1 matrices (i.e., the column vectors) play. Provisionally, a **spinor** is a column vector

# ξ

[

ξ

1

ξ

2

]

,

{\displaystyle \xi =\left[{\begin{matrix}\xi \_{1}\\\xi \_{2}\end{matrix}}\right],}

 with complex entries *ξ*1 and *ξ*2.
The space of spinors is evidently acted upon by complex 2×2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors.

Often, the first example of spinors that a student of physics
encounters are the 2×1 spinors used in Pauli's theory of electron spin.
The Pauli matrices are a vector of three 2×2 matrices
that are used as spin operators.

Given a unit vector in 3 dimensions, for example (*a*, *b*, *c*), one takes a
dot product with the Pauli spin matrices to obtain a spin matrix for
spin in the direction of the unit vector.

The eigenvectors of that spin matrix are the spinors for
spin-1/2 oriented in the direction given by the vector.

Example: *u* = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli
spin matrices gives the matrix:

S

u

=
(
0.8
,
−
0.6
,
0.0
)
⋅

σ
→

=
0.8

σ

1

−
0.6

σ

2

0.0

σ

3

=

[

0.0

0.8
0.6
i

0.8
−
0.6
i

0.0

]

{\displaystyle S\_{u}=(0.8,-0.6,0.0)\cdot {\vec {\sigma }}=0.8\sigma \_{1}-0.6\sigma \_{2}+0.0\sigma \_{3}={\begin{bmatrix}0.0&0.8+0.6i\\0.8-0.6i&0.0\end{bmatrix}}}

The eigenvectors may be found by the usual methods of
linear algebra, but a convenient trick
is to note that a Pauli spin matrix is an involutory matrix, that is, the squareof the above matrix is the identity matrix.

Thus a (matrix) solution to the eigenvector problem with eigenvalues of
±1 is simply 1 ± *Su*. That is,

S

u

(
1
±

S

u

# )
±
1
(
1
±

S

u

)

{\displaystyle S\_{u}(1\pm S\_{u})=\pm 1(1\pm S\_{u})}

One can then choose either of the columns of the eigenvector
matrix as the vector solution, provided that the column chosen
is not zero. Taking the first column of the above,
eigenvector solutions for the two eigenvalues are:

[

1.0
(
0.0
)

0.0
(
0.8
−
0.6
i
)

]

,

[

1.0
−
(
0.0
)

0.0
−
(
0.8
−
0.6
i
)

]

{\displaystyle {\begin{bmatrix}1.0+(0.0)\\0.0+(0.8-0.6i)\end{bmatrix}},{\begin{bmatrix}1.0-(0.0)\\0.0-(0.8-0.6i)\end{bmatrix}}}

The trick used to find the eigenvectors is related to the concept of
ideals, that is, the matrix eigenvectors (1 ± *Su*)/2 are projection operators or idempotents and therefore each generates an
ideal in the Pauli algebra. The same trick
works in any Clifford algebra, in particular
the Dirac algebra that are discussed below. These projection
operators are also seen in density matrix theory
where they are examples of pure density matrices.

More generally, the projection operator for spin in the (*a*, *b*, *c*) direction
is given by

1
2

[

1
c

a
−
i
b

a
i
b

1
−
c

]

{\displaystyle {\frac {1}{2}}{\begin{bmatrix}1+c&a-ib\\a+ib&1-c\end{bmatrix}}}

 and any non zero column can be taken as the projection operator. While the
two columns appear different, one can use *a*2 + *b*2 + *c*2 = 1 to show that they are multiples (possibly zero) of the same spinor.

Back to top

### Spinors in four dimensions

In four dimensions a vector is represented by the gamma matrices.

From Wikipedia:Gamma matrices

The gamma matrices are:

γ

0

=

(

I

2

0

0

−

I

2

)

,

γ

k

=

(

0

σ

k

−

σ

k

0

)

,

γ

5

=

(

0

I

2

I

2

0

)

.

{\displaystyle \gamma ^{0}={\begin{pmatrix}I\_{2}&0\\0&-I\_{2}\end{pmatrix}},\quad \gamma ^{k}={\begin{pmatrix}0&\sigma ^{k}\\-\sigma ^{k}&0\end{pmatrix}},\quad \gamma ^{5}={\begin{pmatrix}0&I\_{2}\\I\_{2}&0\end{pmatrix}}.}

where *k* runs from 1 to 3 and the σk are Pauli matrices.

The gamma matrices are appropriate for acting on Dirac spinors written in the *Dirac basis*; in fact, the Dirac basis is defined by the gamma matrices.

#### Weyl (chiral) basis

Another common choice is the *Weyl* or *chiral basis*, in which

γ

k

{\displaystyle \gamma ^{k}}

 remains the same but

γ

0

{\displaystyle \gamma ^{0}}

 is different, and so

γ

5

{\displaystyle \gamma ^{5}}

 is also different, and diagonal,

γ

0

=

(

0

I

2

I

2

0

)

,

γ

k

=

(

0

σ

k

−

σ

k

0

)

,

γ

5

=

(

−

I

2

0

0

I

2

)

,

{\displaystyle \gamma ^{0}={\begin{pmatrix}0&I\_{2}\\I\_{2}&0\end{pmatrix}},\quad \gamma ^{k}={\begin{pmatrix}0&\sigma ^{k}\\-\sigma ^{k}&0\end{pmatrix}},\quad \gamma ^{5}={\begin{pmatrix}-I\_{2}&0\\0&I\_{2}\end{pmatrix}},}

or in more compact notation:

γ

μ

=

(

0

σ

μ

σ
¯

μ

0

)

,

σ

μ

≡
(
1
,

σ

i

)
,

σ
¯

μ

≡
(
1
,
−

σ

i

)
.

{\displaystyle \gamma ^{\mu }={\begin{pmatrix}0&\sigma ^{\mu }\\{\bar {\sigma }}^{\mu }&0\end{pmatrix}},\quad \sigma ^{\mu }\equiv (1,\sigma ^{i}),\quad {\bar {\sigma }}^{\mu }\equiv (1,-\sigma ^{i}).}

The Weyl basis has the advantage that its chiral projections take a simple form,

ψ

L

=

1
2

(

1
−

γ

5

)

# ψ

(

I

2

0

0

0

)

ψ
,

ψ

R

=

1
2

(

1

γ

5

)

# ψ

(

0

0

0

I

2

)

ψ
.

{\displaystyle \psi \_{L}={\frac {1}{2}}\left(1-\gamma ^{5}\right)\psi ={\begin{pmatrix}I\_{2}&0\\0&0\end{pmatrix}}\psi ,\quad \psi \_{R}={\frac {1}{2}}\left(1+\gamma ^{5}\right)\psi ={\begin{pmatrix}0&0\\0&I\_{2}\end{pmatrix}}\psi .}

The idempotence of the chiral projections is manifest.
By slightly abusing the notation and reusing the symbols

ψ

L

/

R

{\displaystyle \psi \_{L/R}}

 we can then identify

# ψ

(

ψ

L

ψ

R

)

,

{\displaystyle \psi ={\begin{pmatrix}\psi \_{L}\\\psi \_{R}\end{pmatrix}},}

where now

ψ

L

{\displaystyle \psi \_{L}}

 and

ψ

R

{\displaystyle \psi \_{R}}

 are left-handed and right-handed two-component Weyl spinors. The Dirac basis can be obtained from the Weyl basis as

γ

D

μ

=
U

γ

W

μ

U

T

,

ψ

D

=
U

ψ

W

{\displaystyle \gamma \_{D}^{\mu }=U\gamma \_{W}^{\mu }U^{\mathsf {T}},\psi \_{D}=U\psi \_{W}}

 via the unitary transform

# U

1

2

(

I

I

−
I

I

)

.

{\displaystyle U={\frac {1}{\sqrt {2\,}}}{\begin{pmatrix}I&I\\-I&I\end{pmatrix}}.}

Another possible choice of the Weyl basis has

γ

0

=

(

0

−

I

2

−

I

2

0

)

,

γ

k

=

(

0

σ

k

−

σ

k

0

)

,

γ

5

=

(

I

2

0

0

−

I

2

)

.

{\displaystyle \gamma ^{0}={\begin{pmatrix}0&-I\_{2}\\-I\_{2}&0\end{pmatrix}},\quad \gamma ^{k}={\begin{pmatrix}0&\sigma ^{k}\\-\sigma ^{k}&0\end{pmatrix}},\quad \gamma ^{5}={\begin{pmatrix}I\_{2}&0\\0&-I\_{2}\end{pmatrix}}.}

The chiral projections take a slightly different form from the other Weyl choice,

ψ

R

=

(

I

2

0

0

0

)

ψ
,

ψ

L

=

(

0

0

0

I

2

)

ψ
.

{\displaystyle \psi \_{R}={\begin{pmatrix}I\_{2}&0\\0&0\end{pmatrix}}\psi ,\quad \psi \_{L}={\begin{pmatrix}0&0\\0&I\_{2}\end{pmatrix}}\psi .}

In other words,

# ψ

(

ψ

R

ψ

L

)

,

{\displaystyle \psi ={\begin{pmatrix}\psi \_{R}\\\psi \_{L}\end{pmatrix}},}

where

ψ

L

{\displaystyle \psi \_{L}}

 and

ψ

R

{\displaystyle \psi \_{R}}

 are the left-handed and right-handed two-component Weyl spinors, as before.

#### Majorana basis

There is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Regarding the Pauli matrices, the basis can be written as

γ

0

=

(

0

σ

2

σ

2

0

)

,

γ

1

=

(

i

σ

3

0

0

i

σ

3

)

,

γ

2

=

(

0

−

σ

2

σ

2

0

)

,

γ

3

=

(

−
i

σ

1

0

0

−
i

σ

1

)

,

γ

5

=

(

σ

2

0

0

−

σ

2

)

,

C

=

(

0

−
i

σ

2

−
i

σ

2

0

)

,

{\displaystyle {\begin{aligned}\gamma ^{0}&={\begin{pmatrix}0&\sigma ^{2}\\\sigma ^{2}&0\end{pmatrix}},&\gamma ^{1}&={\begin{pmatrix}i\sigma ^{3}&0\\0&i\sigma ^{3}\end{pmatrix}},&\gamma ^{2}&={\begin{pmatrix}0&-\sigma ^{2}\\\sigma ^{2}&0\end{pmatrix}},\\\gamma ^{3}&={\begin{pmatrix}-i\sigma ^{1}&0\\0&-i\sigma ^{1}\end{pmatrix}},&\gamma ^{5}&={\begin{pmatrix}\sigma ^{2}&0\\0&-\sigma ^{2}\end{pmatrix}},&C&={\begin{pmatrix}0&-i\sigma ^{2}\\-i\sigma ^{2}&0\end{pmatrix}},\end{aligned}}}

where

C

{\displaystyle C}

 is the charge conjugation matrix, defined to satisfy

C

†

γ

μ

# C
−
(

γ

μ

)

T

{\displaystyle C^{\dagger }\gamma ^{\mu }C=-(\gamma ^{\mu })^{T}}

.

(The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric (+, −, −, −), in which squared masses are positive. The Majorana representation, however, is real. One can factor out the i to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the

i

{\displaystyle i}

 is that the only possible metric with real gamma matrices is (−, +, +, +).)

The Majorana basis can be obtained from the Dirac basis above as

γ

M

μ

=
U

γ

D

μ

U

T

,

ψ

M

=
U

ψ

D

{\displaystyle \gamma \_{M}^{\mu }=U\gamma \_{D}^{\mu }U^{\mathsf {T}},~~\psi \_{M}=U\psi \_{D}}

 via the unitary transform

# U

1

2

(

I

2

σ

2

−

σ

2

I

2

)

.

{\displaystyle U={\frac {1}{\sqrt {2\,}}}{\begin{pmatrix}I\_{2}&\sigma ^{2}\\-\sigma ^{2}&I\_{2}\end{pmatrix}}.}

#### *C*ℓ1,3(C) and *C*ℓ1,3(R)

The Dirac algebra can be regarded as a complexification of the real algebra *C*ℓ1,3(**R**), called the space time algebra:

C

l

1
,
3

(

C

# )
C

l

1
,
3

(

R

)
⊗

C

{\displaystyle Cl\_{1,3}(\mathbb {C} )=Cl\_{1,3}(\mathbb {R} )\otimes \mathbb {C} }

*C*ℓ1,3(**R**) differs from *C*ℓ1,3(**C**): in *C*ℓ1,3(**R**) only *real* linear combinations of the gamma matrices and their products are allowed.

Two things deserve to be pointed out. As *Clifford algebras*, *C*ℓ1,3(**C**) and *C*ℓ4(**C**) are isomorphic, see classification of Clifford algebras. The reason is that the underlying signature of the spacetime metric loses its signature (3,1) upon passing to the complexification. However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" (at the very least impractical) since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.

Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to −1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.

However, in contemporary practice, the Dirac algebra rather than the space-time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.

Back to top

#### Electron spin

From Wikipedia:Spinor:
In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other spin-1/2 fermions,. This is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2 component spinors transforming under three-dimensional infinitesimal rotations. The relativistic Dirac equation for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists.

Back to top

## Linear groups

A square matrixw of order n is an n-by-n matrix. Any two square matrices of the same order can be added and multiplied. A matrix is invertible if and only if its determinant is nonzero.

GL*n*(*F*) or GL(*n*, *F*), or simply GL(*n*) is the Lie group of *n*×*n* invertible matrices with entries from the field *F*. The group operation is matrix multiplicationw. The group GL(n, F) and its subgroups are often called linear groups or matrix groups.

SL(*n*, *F*) or SL*n*(*F*), is the subgroup of GL(*n*, *F*) consisting of matrices with a determinantw of 1.
U(*n*), the Unitary group of degree *n* is the groupw of *n* × *n* unitary matricesw. The group operation is matrix multiplicationw. The determinant of a unitary matrix is a complex number with norm 1.
SU(*n*), the special unitary group of degree *n*, is the Lie group of *n*×*n* unitary matricesw with determinantw 1.
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## Symmetry groups

Affine group

Poincaré group: boosts, rotations, translations
Lorentz group: boosts, rotations
The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.)
Aff(n,K): the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

E(n): rotations, reflections, and translations.
O(n): rotations, reflections
SO(n)w: rotations
**so**(3) is the Lie algebra of SO(*3*) and consists of all skew-symmetricw *3* × *3* matrices.
Clifford group: The set of invertible elements *x* such that for all *v* in *V*

x
v
α
(
x

)

−
1

∈
V
.

{\displaystyle xv\alpha (x)^{-1}\in V.}

 The spinor norm *Q* is defined on the Clifford group by

Q
(
x
# )

x

t

x
.

{\displaystyle Q(x)=x^{\mathrm {t} }x.}

Pin*V*(*K*): The subgroup of elements of spinor norm 1. Maps 2-to-1 to the orthogonal group
Spin*V*(*K*): The subgroup of elements of Dickson invariant 0 in Pin*V*(*K*). When the characteristic is not 2, these are the elements of determinant 1. Maps 2-to-1 to the special orthogonal group. Elements of the spin group act as linear transformations on the space of spinors
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### Rotations

See also: Hypersphere of rotations, Rotation group SO(3), Special unitary group, Plate trick, Spin representation, Spin group, Pin group, Spinor, Clifford algebraw, Indefinite orthogonal group, Root system, Bivectorsw, Curlw
From Wikipedia:Rotation group SO(3):
Consider the solid ball in **R**3 of radius π. For every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The two rotations through π and through −π are the same. So we identify (or "glue together") antipodal points on the surface of the ball.

The ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. It is also diffeomorphic to the real 3-dimensional projective space **RP**3, so the latter can also serve as a topological model for the rotation group.

These identifications illustrate that SO(3) is connected but not simply connected. As to the latter, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". (In other words one full rotation is not equivalent to doing nothing.)

Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that φ runs from 0 to 4π, you get a closed loop which *can* be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The Balinese plate trick and similar tricks demonstrate this practically.

The same argument can be performed in general, and it shows that the fundamental group of SO(3) is cyclic groupw of order 2. In physics applications, the non-triviality of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spin-statistics theorem.

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#### Orientation entanglement

From Wikipedia:Orientation entanglement
In three dimensions the problem illustrated above corresponds to the fact that the Lie group SO(3) is not simply connected. Mathematically, one can tackle this problem by exhibiting the special unitary group SU(2)w, which is also the spin group in three Euclidean dimensions, as a double cover of SO(3).

SU(2) is the following group,

S
U

(
2
# )

{

(

α

−

β
¯

β

α
¯

)

:

α
,
β
∈

C

,

α

2

β

2

=
1

}

,

where the overline denotes complex conjugation.
For comparison: Using 2 × 2 complex matrices, the quaternion *a* + *bi* + *cj* + *dk* can be represented as

[

a
b
i

c
d
i

−
(
c
−
d
i
)

a
−
b
i

]

.

{\displaystyle {\begin{bmatrix}a+bi&c+di\\-(c-di)&a-bi\end{bmatrix}}.}

If *X* = (*x*1,*x*2,*x*3) is a vector in **R**3, then we identify *X* with the 2 × 2 matrix with complex entries

# X

(

x

1

x

2

−
i

x

3

x

2

i

x

3

−

x

1

)

{\displaystyle X=\left({\begin{matrix}x\_{1}&x\_{2}-ix\_{3}\\x\_{2}+ix\_{3}&-x\_{1}\end{matrix}}\right)}

Note that −det(*X*) gives the square of the Euclidean length of *X* regarded as a vector, and that *X* is a trace-free, or better, trace-zero Hermitian matrix.

The unitary group acts on *X* via

X
↦
M
X

M

{\displaystyle X\mapsto MXM^{+}}

where *M* ∈ SU(2). Note that, since *M* is unitary,

det
(
M
X

M

# )
det
(
X
)

{\displaystyle \det(MXM^{+})=\det(X)}

, and

M
X

M

{\displaystyle MXM^{+}}

 is trace-zero Hermitian.
Hence SU(2) acts via rotation on the vectors *X*. Conversely, since any change of basis which sends trace-zero Hermitian matrices to trace-zero Hermitian matrices must be unitary, it follows that every rotation also lifts to SU(2). However, each rotation is obtained from a pair of elements *M* and −*M* of SU(2). Hence SU(2) is a double-cover of SO(3). Furthermore, SU(2) is easily seen to be itself simply connected by realizing it as the group of unit quaternions, a space homeomorphic to the 3-sphere.

A unit quaternion has the cosine of half the rotation angle as its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis (here assumed fixed) as its pseudovector (or axial vector) part. If the initial orientation of a rigid body (with unentangled connections to its fixed surroundings) is identified with a unit quaternion having a zero pseudovector part and +1 for the scalar part, then after one complete rotation (2pi rad) the pseudovector part returns to zero and the scalar part has become -1 (entangled). After two complete rotations (4pi rad) the pseudovector part again returns to zero and the scalar part returns to +1 (unentangled), completing the cycle.

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## Clifford group

From Wikipedia:Clifford algebra

The class of Clifford groups (a.k.a. Clifford–Lipschitz groups) was discovered by Rudolf Lipschitz.

In this section we assume that *V* is finite-dimensional and the quadratic form *Q* is nondegenerate.

An action on the elements of a Clifford algebra by its group of units may be defined in terms of a twisted conjugation: twisted conjugation by *x* maps *y* ↦ *x y α*(*x*)−1, where *α* is the *main involution* defined above.

The Clifford group Γ is defined to be the set of invertible elements *x* that *stabilize the set of vectors* under this action, meaning that for all *v* in *V* we have:

α
(
x
)
v

x

−
1

∈
V
.

{\displaystyle \alpha (x)vx^{-1}\in V.}

This formula also defines an action of the Clifford group on the vector space *V* that preserves the quadratic form *Q*, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements *r* of *V* for which *Q*(*r*) is invertible in *K*, and these act on *V* by the corresponding reflections that take *v* to *v* − *2*⟨*v*,*r*⟩*r*/*Q*(*r*). (In characteristic 2 these are called orthogonal transvections rather than reflections.)

If *V* is a finite-dimensional real vector space with a non-degenerate quadratic form then the Clifford group maps onto the orthogonal group of *V* with respect to the form (by the Cartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field *K*. This leads to exact sequences

1
→

K

∗

→
Γ
→

O

V

(
K
)
→
1
,

{\displaystyle 1\rightarrow K^{\*}\rightarrow \Gamma \rightarrow {\mbox{O}}\_{V}(K)\rightarrow 1,\,}

1
→

K

∗

→

Γ

0

→

SO

V

(
K
)
→

{\displaystyle 1\rightarrow K^{\*}\rightarrow \Gamma ^{0}\rightarrow {\mbox{SO}}\_{V}(K)\rightarrow 1.\,}

Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.

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### Spinor norm

Further information: Spinor norm#Galois cohomology and orthogonal groups
From Wikipedia:Clifford algebra

In arbitrary characteristic, the spinor norm *Q* is defined on the Clifford group by

Q
(
x
# )

x

t

x
.

{\displaystyle Q(x)=x^{\mathrm {t} }x.}

It is a homomorphism from the Clifford group to the group *K*× of non-zero elements of *K*. It coincides with the quadratic form *Q* of *V* when *V* is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of −1, 2, or −2 on Γ1. The difference is not very important in characteristic other than 2.

The nonzero elements of *K* have spinor norm in the group (*K*×)2 of squares of nonzero elements of the field *K*. So when *V* is finite-dimensional and non-singular we get an induced map from the orthogonal group of *V* to the group *K*×/(*K*×)2, also called the spinor norm. The spinor norm of the reflection about *r*⊥, for any vector *r*, has image *Q*(*r*) in *K*×/(*K*×)2, and this property uniquely defines it on the orthogonal group. This gives exact sequences:

1
→
{
±
1
}
→

Pin

V

(
K
)
→

O

V

(
K
)
→

K

×

/

(

K

×

)

2

,

{\displaystyle 1\to \{\pm 1\}\to {\mbox{Pin}}\_{V}(K)\to {\mbox{O}}\_{V}(K)\to K^{\times }/(K^{\times })^{2},}

1
→
{
±
1
}
→

Spin

V

(
K
)
→

SO

V

(
K
)
→

K

×

/

(

K

×

)

2

.

{\displaystyle 1\to \{\pm 1\}\to {\mbox{Spin}}\_{V}(K)\to {\mbox{SO}}\_{V}(K)\to K^{\times }/(K^{\times })^{2}.}

Note that in characteristic 2 the group {±1} has just one element.

From the point of view of Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the algebraic group of square roots of 1 (over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action), the short exact sequence

1
→

μ

2

→

Pin

V

→

O

V

→
1

{\displaystyle 1\to \mu \_{2}\rightarrow {\mbox{Pin}}\_{V}\rightarrow {\mbox{O}}\_{V}\rightarrow 1}

yields a long exact sequence on cohomology, which begins

1
→

H

0

(

μ

2

;
K
)
→

H

0

(

Pin

V

;
K
)
→

H

0

(

O

V

;
K
)
→

H

1

(

μ

2

;
K
)
.

{\displaystyle 1\to H^{0}(\mu \_{2};K)\to H^{0}({\mbox{Pin}}\_{V};K)\to H^{0}({\mbox{O}}\_{V};K)\to H^{1}(\mu \_{2};K).}

The 0th Galois cohomology group of an algebraic group with coefficients in *K* is just the group of *K*-valued points: *H*0(*G*; *K*) = *G*(*K*), and *H*1(μ2; *K*) ≅ *K*×/(*K*×)2, which recovers the previous sequence

1
→
{
±
1
}
→

Pin

V

(
K
)
→

O

V

(
K
)
→

K

×

/

(

K

×

)

2

,

{\displaystyle 1\to \{\pm 1\}\to {\mbox{Pin}}\_{V}(K)\to {\mbox{O}}\_{V}(K)\to K^{\times }/(K^{\times })^{2},}

where the spinor norm is the connecting homomorphism *H*0(O*V*; *K*) → *H*1(μ2; *K*).

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## Spin and Pin groups

Further information: Spin group, Pin group, Spinor
From Wikipedia:Clifford algebra

In this section we assume that *V* is finite-dimensional and its bilinear form is non-singular. (If *K* has characteristic 2 this implies that the dimension of *V* is even.)

The Pin group Pin*V*(*K*) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin group Spin*V*(*K*) is the subgroup of elements of Dickson invariant 0 in Pin*V*(*K*). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group.

Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the special orthogonal group to be the image of Γ0. If *K* does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If *K* does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0.

There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm 1 ∈ *K*×/(*K*×)2. The kernel consists of the elements +1 and −1, and has order 2 unless *K* has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of *V*.

In the common case when *V* is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when *V* has dimension at least 3. Further the kernel of this homomorphism consists of 1 and −1. So in this case the spin group, Spin(*n*), is a double cover of SO(*n*). Please note, however, that the simple connectedness of the spin group is not true in general: if *V* is **R***p*,*q* for *p* and *q* both at least 2 then the spin group is not simply connected. In this case the algebraic group Spin*p*,*q* is simply connected as an algebraic group, even though its group of real valued points Spin*p*,*q*(**R**) is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.

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## Spinors

From Wikipedia:Clifford algebra

Clifford algebras Cℓ*p*,*q*(**C**), with *p* + *q* = 2*n* even, are matrix algebras which have a complex representation of dimension 2*n*. By restricting to the group Pin*p*,*q*(**R**) we get a complex representation of the Pin group of the same dimension, called the spin representation. If we restrict this to the spin group Spin*p*,*q*(**R**) then it splits as the sum of two *half spin representations* (or *Weyl representations*) of dimension 2*n*−1.

If *p* + *q* = 2*n* + 1 is odd then the Clifford algebra Cℓ*p*,*q*(**C**) is a sum of two matrix algebras, each of which has a representation of dimension 2*n*, and these are also both representations of the Pin group Pin*p*,*q*(**R**). On restriction to the spin group Spin*p*,*q*(**R**) these become isomorphic, so the spin group has a complex spinor representation of dimension 2*n*.

More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the structure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra.
For examples over the reals see the article on spinors.

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### Real spinors

Further information: spinor
From Wikipedia:Clifford algebra

To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The Pin group, Pin*p*,*q* is the set of invertible elements in Cℓ*p*,*q* that can be written as a product of unit vectors:

Pin

p
,
q

=
{

v

1

v

2

…

v

r

∀
i

‖

v

i

# ‖
±
1
}
.

{\displaystyle {\mbox{Pin}}\_{p,q}=\{v\_{1}v\_{2}\dots v\_{r}|\,\,\forall i\,\|v\_{i}\|=\pm 1\}.}

Comparing with the above concrete realizations of the Clifford algebras, the Pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group O(*p*, *q*). The Spin group consists of those elements of Pin*p*, *q* which are products of an even number of unit vectors. Thus by the Cartan-Dieudonné theorem Spin is a cover of the group of proper rotations SO(*p*, *q*).

Let *α* : Cℓ → Cℓ be the automorphism which is given by the mapping *v* ↦ −*v* acting on pure vectors. Then in particular, Spin*p*,*q* is the subgroup of Pin*p*,*q* whose elements are fixed by *α*. Let

C
ℓ

p
,
q

[
0
]

=
{
x
∈

C
ℓ

p
,
q

⁡

α
(
x
# )
x
}
.

{\displaystyle \operatorname {C\ell } \_{p,q}^{[0]}=\{x\in \operatorname {C\ell } \_{p,q}|\,\alpha (x)=x\}.}

(These are precisely the elements of even degree in Cℓ*p*,*q*.) Then the spin group lies within Cℓ[0]
*p*,*q*.

The irreducible representations of Cℓ*p*,*q* restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of Cℓ[0]
*p*,*q*.

To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)

C
ℓ

p
,
q

[
0
]

≈

C
ℓ

p
,
q
−
1

,

 for

q
>
0

{\displaystyle \operatorname {C\ell } \_{p,q}^{[0]}\approx \operatorname {C\ell } \_{p,q-1},{\text{ for }}q>0}

C
ℓ

p
,
q

[
0
]

≈

C
ℓ

q
,
p
−
1

,

 for

p
>
0

{\displaystyle \operatorname {C\ell } \_{p,q}^{[0]}\approx \operatorname {C\ell } \_{q,p-1},{\text{ for }}p>0}

and realize a spin representation in signature (*p*, *q*) as a pin representation in either signature (*p*, *q* − 1) or (*q*, *p* − 1).

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## The Lie derivative of a spinor field

From Wikipedia:Lie derivative

A definition for Lie derivatives of spinors along generic spacetime vector fields, not necessarily Killing ones, on a general (pseudo) Riemannian manifold was already proposed in 1972 by Yvette Kosmann. Later, it was provided a geometric framework which justifies her *ad hoc* prescription within the general framework of Lie derivatives on fiber bundles in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.

In a given spin manifold, that is in a Riemannian manifold

(
M
,
g
)

{\displaystyle (M,g)}

 admitting a spin structure, the Lie derivative of a spinor field

ψ

{\displaystyle \psi }

 can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the André Lichnerowicz's local expression given in 1963:

L

X

ψ
:=

X

a

∇

a

ψ
−

1
4

∇

a

X

b

γ

a

γ

b

ψ

,

{\displaystyle {\mathcal {L}}\_{X}\psi :=X^{a}\nabla \_{a}\psi -{\frac {1}{4}}\nabla \_{a}X\_{b}\gamma ^{a}\,\gamma ^{b}\psi \,,}

where

∇

a

X

b

=

∇

[
a

X

b
]

{\displaystyle \nabla \_{a}X\_{b}=\nabla \_{[a}X\_{b]}}

, as

# X

X

a

∂

a

{\displaystyle X=X^{a}\partial \_{a}}

 is assumed to be a Killing vector field, and

γ

a

{\displaystyle \gamma ^{a}}

 are Dirac matrices.

It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for a *generic* vector field

X

{\displaystyle X}

, but explicitly taking the antisymmetric part of

∇

a

X

b

{\displaystyle \nabla \_{a}X\_{b}}

 only. More explicitly, Kosmann's local expression given in 1972 is:

L

X

ψ
:=

X

a

∇

a

ψ
−

1
8

∇

[
a

X

b
]

[

γ

a

,

γ

b

]
ψ

=

∇

X

ψ
−

1
4

(
d

X

♭

)
⋅
ψ

,

{\displaystyle {\mathcal {L}}\_{X}\psi :=X^{a}\nabla \_{a}\psi -{\frac {1}{8}}\nabla \_{[a}X\_{b]}[\gamma ^{a},\gamma ^{b}]\psi \,=\nabla \_{X}\psi -{\frac {1}{4}}(dX^{\flat })\cdot \psi \,,}

where

[

γ

a

,

γ

b

# ]

γ

a

γ

b

−

γ

b

γ

a

{\displaystyle [\gamma ^{a},\gamma ^{b}]=\gamma ^{a}\gamma ^{b}-\gamma ^{b}\gamma ^{a}}

 is the commutator,

d

{\displaystyle d}

 is exterior derivative,

X

♭

=
g
(
X
,
−
)

{\displaystyle X^{\flat }=g(X,-)}

 is the dual 1 form corresponding to

X

{\displaystyle X}

 under the metric (i.e. with lowered indices) and

⋅

{\displaystyle \cdot }

 is Clifford multiplication.
It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the connection. This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on the spinor bundle. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel.

To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article, where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called the Kosmann lift.

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### 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, *iσj* 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*2***σ***, 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.

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

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### Pauli matrices in relativistic quantum mechanics

From Wikipedia:Pauli matrices

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

Σ

i

=

(

σ

i

0

0

σ

i

)

.

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

It follows from this definition that

Σ

i

{\displaystyle {\mathsf {\Sigma }}\_{i}}

 matrices have the same algebraic properties as

σ

i

{\displaystyle {\mathsf {\sigma }}\_{i}}

 matrices.

However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence

Σ

i

{\displaystyle {\mathsf {\Sigma }}\_{i}}

 needs to be replaced by

Σ

μ
ν

{\displaystyle \Sigma \_{\mu \nu }}

, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the

Σ

μ
ν

{\displaystyle \Sigma \_{\mu \nu }}

 are also antisymmetric. Hence there are only six independent matrices.

The first three are the

Σ

j
k

≡

ϵ

i
j
k

Σ

i

.

{\displaystyle \Sigma \_{jk}\equiv \epsilon \_{ijk}{\mathsf {\Sigma }}\_{i}.}

 The remaining three,

−
i

Σ

0
i

≡

α

i

{\displaystyle -i\Sigma \_{0i}\equiv {\mathsf {\alpha }}\_{i}}

, where the Dirac

α

i

{\displaystyle {\mathsf {\alpha }}\_{i}}

 matrices are defined as

α

i

=

(

0

σ

i

σ

i

0

)

.

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

The relativistic spin matrices

Σ

μ
ν

{\displaystyle \Sigma \_{\mu \nu }}

 are written in compact form in terms of commutator of gamma matrices as

Σ

μ
ν

=

i
2

[

γ

μ

,

γ

ν

]

{\displaystyle \Sigma \_{\mu \nu }={\frac {i}{2}}\left[\gamma \_{\mu },\gamma \_{\nu }\right]}

.
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## Coordinate-free

From Wikipedia:Coordinate-free:

A **coordinate-free**, or **component-free**, treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system.

Coordinate-free treatments generally allow for simpler systems of equations and inherently constrain certain types of inconsistency, allowing greater mathematical elegance at the cost of some abstraction from the detailed formulae needed to evaluate these equations within a particular system of coordinates.

Coordinate-free treatments were the only available approach to geometry (and are now known as synthetic geometry) before the development of analytic geometry by Descartes. After several centuries of generally coordinate-based exposition, the modern tendency is generally to introduce students to coordinate-free treatments early on, and then to derive the coordinate-based treatments from the coordinate-free treatment, rather than *vice versa*.

Fields that are now often introduced with coordinate-free treatments include vector calculus, tensors, differential geometry, and computer graphics.

In physics, the existence of coordinate-free treatments of physical theories is a corollary of the principle of general covariance.

From Wikipedia: General covariance

In theoretical physics, **general covariance**, also known as **diffeomorphism covariance** or **general invariance**, consists of the invariance of the *form* of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist *a priori* in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.

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## References

1. ↑ Wikipedia:Rotor (mathematics)
2. ↑ Wikipedia:Spinor#Three\_dimensions
3. ↑ Wikipedia:Spinor
4. ↑ Wikipedia:Special unitary group