Clifford algebras

A Clifford algebra is the unital associative algebra generated over an underlying vector space equipped with a quadratic form. Over the real numbers this is equivalent to being able to define a symmetric scalar product, u ⋅ v = 1/2(uv + vu) that can be used to orthogonalise the quadratic form, to give a basis {e₁, ..., e_k} such that:

$${\displaystyle {\frac {1}{2}}\left(e_{i}e_{j}+e_{j}e_{i}\right)={\begin{cases}-1,0,+1&i=j,\\0&i\not =j.\end{cases}}}$$

Imposing closure under multiplication generates a multivector space spanned by a basis of 2^k elements, {1, e₁, e₂, e₃, ..., e₁e₂, ..., e₁e₂e₃, ...}. These can be interpreted as the basis of a hypercomplex number system. Unlike the basis {e₁, ..., e_k}, the remaining basis elements need not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So e₁e₂ = −e₂e₁, but e₁(e₂e₃) = +(e₂e₃)e₁.

Putting aside the bases which contain an element e_i such that e_i² = 0 (i.e. directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Cl_p,q(${\displaystyle \mathbb {R} }$), indicating that the algebra is constructed from p simple basis elements with e_i² = +1, q with e_i² = −1, and where ${\displaystyle \mathbb {R} }$ indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.

These algebras, called geometric algebras, form a systematic set, which turn out to be very useful in physics problems which involve rotations, phases, or spins, notably in classical and quantum mechanics, electromagnetic theory and relativity.

Examples include: the complex numbers Cl_0,1(${\displaystyle \mathbb {R} }$), split-complex numbers Cl_1,0(${\displaystyle \mathbb {R} }$), quaternions Cl_0,2(${\displaystyle \mathbb {R} }$), split-biquaternions Cl_0,3(${\displaystyle \mathbb {R} }$), split-quaternions Cl_1,1(${\displaystyle \mathbb {R} }$) ≈ Cl_2,0(${\displaystyle \mathbb {R} }$) (the natural algebra of two-dimensional space); Cl_3,0(${\displaystyle \mathbb {R} }$) (the natural algebra of three-dimensional space, and the algebra of the Pauli matrices); and the spacetime algebra Cl_1,3(${\displaystyle \mathbb {R} }$).

The elements of the algebra Cl_p,q(${\displaystyle \mathbb {R} }$) form an even subalgebra Cl^[0]
_q+1,p(${\displaystyle \mathbb {R} }$) of the algebra Cl_q+1,p(${\displaystyle \mathbb {R} }$), which can be used to parametrise rotations in the larger algebra. There is thus a close connection between complex numbers and rotations in two-dimensional space; between quaternions and rotations in three-dimensional space; between split-complex numbers and (hyperbolic) rotations (Lorentz transformations) in 1+1-dimensional space, and so on.

Whereas Cayley–Dickson and split-complex constructs with eight or more dimensions are not associative with respect to multiplication, Clifford algebras retain associativity at any number of dimensions.

In 1995 Ian R. Porteous wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:^[17]

Let A be a real associative algebra with unit element 1. Then

• 1 generates ${\displaystyle \mathbb {R} }$ (algebra of real numbers),
• any two-dimensional subalgebra generated by an element e₀ of A such that e₀² = −1 is isomorphic to ${\displaystyle \mathbb {C} }$ (algebra of complex numbers),
• any two-dimensional subalgebra generated by an element e₀ of A such that e₀² = 1 is isomorphic to ${\displaystyle \mathbb {R} }$² (pairs of real numbers with component-wise product, isomorphic to the algebra of split-complex numbers),
• any four-dimensional subalgebra generated by a set {e₀, e₁} of mutually anti-commuting elements of A such that ${\displaystyle e_{0}^{2}=e_{1}^{2}=-1}$ is isomorphic to ${\displaystyle \mathbb {H} }$ (algebra of quaternions),
• any four-dimensional subalgebra generated by a set {e₀, e₁} of mutually anti-commuting elements of A such that ${\displaystyle e_{0}^{2}=e_{1}^{2}=1}$ is isomorphic to M₂(${\displaystyle \mathbb {R} }$) (2 × 2 real matrices, coquaternions),
• any eight-dimensional subalgebra generated by a set {e₀, e₁, e₂} of mutually anti-commuting elements of A such that ${\displaystyle e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=-1}$ is isomorphic to ²${\displaystyle \mathbb {H} }$ (split-biquaternions),
• any eight-dimensional subalgebra generated by a set {e₀, e₁, e₂} of mutually anti-commuting elements of A such that ${\displaystyle e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=1}$ is isomorphic to M₂(${\displaystyle \mathbb {C} }$) (2 × 2 complex matrices, biquaternions, Pauli algebra).

Cayley–Dickson construction

Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of i (red), j (green) and k (blue). In the SVG file, hover over or click a path to highlight it.

All of the Clifford algebras Cl_p,q(${\displaystyle \mathbb {R} }$) apart from the real numbers, complex numbers and the quaternions contain non-real elements that square to +1; and so cannot be division algebras. A different approach to extending the complex numbers is taken by the Cayley–Dickson construction. This generates number systems of dimension 2ⁿ, n = 2, 3, 4, ..., with bases ${\displaystyle \left\{1,i_{1},\dots ,i_{2^{n}-1}\right\}}$, where all the non-real basis elements anti-commute and satisfy ${\displaystyle i_{m}^{2}=-1}$. In 8 or more dimensions (n ≥ 3) these algebras are non-associative. In 16 or more dimensions (n ≥ 4) these algebras also have zero-divisors.

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. An algebraic symmetry is lost with each increase in dimensionality: quaternion multiplication is not commutative, octonion multiplication is non-associative, and the norm of sedenions is not multiplicative.

The Cayley–Dickson construction can be modified by inserting an extra sign at some stages. It then generates the "split algebras" in the collection of composition algebras instead of the division algebras:

split-complex numbers with basis ${\displaystyle \{1,\,i_{1}\}}$ satisfying ${\displaystyle \ i_{1}^{2}=+1}$,

split-quaternions with basis ${\displaystyle \{1,\,i_{1},\,i_{2},\,i_{3}\}}$ satisfying ${\displaystyle \ i_{1}^{2}=-1,\,i_{2}^{2}=i_{3}^{2}=+1}$, and

split-octonions with basis ${\displaystyle \{1,\,i_{1},\,\dots ,\,i_{7}\}}$ satisfying ${\displaystyle \ i_{1}^{2}=i_{2}^{2}=i_{3}^{2}=-1}$, ${\displaystyle \ i_{4}^{2}=i_{5}^{2}=i_{6}^{2}=i_{7}^{2}=+1.}$

Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain nontrivial zero divisors and nontrivial idempotents. As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the square matrices of dimension two. Split-octonions are non-associative and contain nilpotents.

Tensor products

The tensor product of any two algebras is another algebra, which can be used to produce many more examples of hypercomplex number systems.

In particular taking tensor products with the complex numbers (considered as algebras over the reals) leads to four-dimensional bicomplex numbers ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} }$ (isomorphic to tessarines ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }D}$), eight-dimensional biquaternions ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {H} }$, and 16-dimensional complex octonions ${\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {O} }$.

Further examples

• bicomplex numbers: a 4-dimensional vector space over the reals, 2-dimensional over the complex numbers, isomorphic to tessarines.
• multicomplex numbers: 2ⁿ-dimensional vector spaces over the reals, 2ⁿ⁻¹-dimensional over the complex numbers
• composition algebra: algebra with a quadratic form that composes with the product