Clock and shift matrices

In mathematics, a **Generalized Clifford algebra** (GCA) is an associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these **clock-and-shift** operators introduced by J. J. Sylvester (1882), and organized by Cartan (1898) and Schwinger.

Clock and shift matrices find routine applications in numerous areas of mathematical physics, providing the cornerstone of **quantum mechanical dynamics in finite-dimensional vector spaces**. The concept of a spinor can further be linked to these algebras.

The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.

## Definition and properties

### Abstract definition

The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by

e

j

e

k

=

ω

j
k

e

k

e

j

{\displaystyle e\_{j}e\_{k}=\omega \_{jk}e\_{k}e\_{j}\,}

ω

j
k

e

l

=

e

l

ω

j
k

{\displaystyle \omega \_{jk}e\_{l}=e\_{l}\omega \_{jk}\,}

ω

j
k

ω

l
m

=

ω

l
m

ω

j
k

{\displaystyle \omega \_{jk}\omega \_{lm}=\omega \_{lm}\omega \_{jk}\,}

and

e

j

N

j

=
# 1

ω

j
k

N

j

=

ω

j
k

N

k

{\displaystyle e\_{j}^{N\_{j}}=1=\omega \_{jk}^{N\_{j}}=\omega \_{jk}^{N\_{k}}\,}

∀ *j*,*k*,*l*,*m* = 1,...,*n*.

Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that

ω

j
k

=

ω

k
j

−
1

=

e

2
π
i

ν

k
j

/

N

k
j

{\displaystyle \omega \_{jk}=\omega \_{kj}^{-1}=e^{2\pi i\nu \_{kj}/N\_{kj}}}

∀ *j*,*k* = 1,...,*n*,   and

N

k
j

=

{\displaystyle N\_{kj}=}

gcd

(

N

j

,

N

k

)

{\displaystyle (N\_{j},N\_{k})}

. The field F is usually taken to be the complex numbers **C**.

### More specific definition

Main articles: Generalizations of Pauli matrices
In the more common cases of GCA, the n-dimensional generalized Clifford algebra of order p has the property  *ωkj* = *ω*,

N

k

=
p

{\displaystyle N\_{k}=p}

   for all *j*,*k*, and

ν

k
j

=
1

{\displaystyle \nu \_{kj}=1}

. It follows that

e

j

e

k

=
ω

e

k

e

j

{\displaystyle e\_{j}e\_{k}=\omega \,e\_{k}e\_{j}\,}

ω

e

l

=

e

l

ω

{\displaystyle \omega e\_{l}=e\_{l}\omega \,}

and

e

j

p

=
# 1

ω

p

{\displaystyle e\_{j}^{p}=1=\omega ^{p}\,}

for all *j*,*k*,l = 1,...,*n*, and

# ω

ω

−
1

=

e

2
π
i

/

p

{\displaystyle \omega =\omega ^{-1}=e^{2\pi i/p}}

is the pth root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.

Clifford algebra
In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with *ω* = −1, and *p* = 2.

## Matrix representation

Main articles: Generalizations of Pauli matrices#Construction: The clock and shift matrices
The Clock and Shift matrices can be represented by *n×n* matrices in Schwinger's canonical notation as

# V

(

0

1

0

⋯

0

0

0

1

⋯

0

0

0

⋯

1

0

⋯

⋯

⋯

⋯

⋯

1

0

0

⋯

0

)

{\displaystyle V={\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\cdots &0\\0&0&\cdots &1&0\\\cdots &\cdots &\cdots &\cdots &\cdots \\1&0&0&\cdots &0\end{pmatrix}}}

 ,

# U

(

1

0

0

⋯

0

0

ω

0

⋯

0

0

0

ω

2

⋯

0

⋯

⋯

⋯

⋯

⋯

0

0

0

⋯

ω

(
n
−
1
)

)

{\displaystyle U={\begin{pmatrix}1&0&0&\cdots &0\\0&\omega &0&\cdots &0\\0&0&\omega ^{2}&\cdots &0\\\cdots &\cdots &\cdots &\cdots &\cdots \\0&0&0&\cdots &\omega ^{(n-1)}\end{pmatrix}}}

 ,

# W

(

1

1

1

⋯

1

1

ω

ω

2

⋯

ω

n
−
1

1

ω

2

(

ω

2

)

2

⋯

ω

2
(
n
−
1
)

⋯

⋯

⋯

⋯

⋯

1

ω

n
−
1

ω

2
(
n
−
1
)

⋯

ω

(
n
−
1

)

2

)

{\displaystyle W={\begin{pmatrix}1&1&1&\cdots &1\\1&\omega &\omega ^{2}&\cdots &\omega ^{n-1}\\1&\omega ^{2}&(\omega ^{2})^{2}&\cdots &\omega ^{2(n-1)}\\\cdots &\cdots &\cdots &\cdots &\cdots \\1&\omega ^{n-1}&\omega ^{2(n-1)}&\cdots &\omega ^{(n-1)^{2}}\end{pmatrix}}}

 .
Notably, *Vn* = 1, *VU* = *ωUV* (the Weyl braiding relations), and  *W−1VW* = *U* (the Discrete Fourier transform).
With *e*1 = *V* , *e*2 = *VU*, and *e*3 = *U*, one has three basis elements which, together with ω, fulfil the above conditions of the Generalized Clifford Algebra (GCA).

These matrices, V and U, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s. (Note that the matrices V are cyclic permutation matrices that perform a circular shift; *they* *are not to be confused* with upper and lower shift matrices which have ones only either above or below the diagonal, respectively).

### Specific examples

Case *n* = *p* = 2.
In this case, we have ω = −1, and

# V

(

0

1

1

0

)

{\displaystyle V={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}

 ,

# U

(

1

0

0

−
1

)

{\displaystyle U={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}

 ,

# W

(

1

1

1

−
1

)

{\displaystyle W={\begin{pmatrix}1&1\\1&-1\end{pmatrix}}}

thus

e

1

=

(

0

1

1

0

)

{\displaystyle e\_{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}

 ,

e

2

=

(

0

−
1

1

0

)

{\displaystyle e\_{2}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}

 ,

e

3

=

(

1

0

0

−
1

)

{\displaystyle e\_{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}

 ,
which constitute the Pauli matrices.

Case *n* = *p* = 4,
In this case we have ω = i, and

# V

(

0

1

0

0

0

0

1

0

0

0

0

1

1

0

0

0

)

{\displaystyle V={\begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\end{pmatrix}}}

 ,

# U

(

1

0

0

0

0

i

0

0

0

0

−
1

0

0

0

0

−
i

)

{\displaystyle U={\begin{pmatrix}1&0&0&0\\0&i&0&0\\0&0&-1&0\\0&0&0&-i\end{pmatrix}}}

 ,

# W

(

1

1

1

1

1

i

−
1

−
i

1

−
1

1

−
1

1

−
i

−
1

i

)

{\displaystyle W={\begin{pmatrix}1&1&1&1\\1&i&-1&-i\\1&-1&1&-1\\1&-i&-1&i\end{pmatrix}}}

and *e*1, *e*2, *e*3 may be determined accordingly.