A Note on the Representation of Cliﬀord Algebra

: In this note we construct explicit complex and real matrix representations for the generators of real Cliﬀord algebra C(cid:96) p,q . The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We ﬁnd two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into 4 m +1 dimension, but the exceptional representation can be expanded as generators of the next period. In the cases p + q = 4 m , the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Cliﬀord algebra are the faithful basis of p + q dimensional Minkowski space-time or Riemann space, and Cliﬀord algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Cliﬀord algebra will be much simple and clear.

and engineering [8,9,10,11,12,13,14,15]. Theoretically we have some equivalent definitions for Clifford algebras [16,17]. For the present purpose, we use the original definition of Clifford, which is based on the generators of basis.
Definition 1 Suppose V is n-dimensional vector space over field R, and its basis {e 1 , e 2 , · · · , e n } satisfies the following algebraic rules e a e b + e b e a = 2η ab I, η ab = diag(I p , −I q ), n = p + q. (1.1) Then the basis e k ∈ {I, e a , e ab = e a e b , e abc = e a e b e c , · · · , e 12···n = e 1 e 2 · · · e n |1 ≤ a < b < c ≤ n} (1.2) together with relation (1.1) and number multiplication C = k c k e k (∀c k ∈ R) form a 2 ndimensional real unital associative algebra, which is called real universal Clifford algebra C p,q = n k=0 ⊗ k V , and C = k c k e k is called Clifford number. For C 0,2 , we have C = tI + xe 1 + ye 2 + ze 12 with By (1.3) we find C is equivalent to a quaternion, that is we have isomorphic relation C 0,2 ∼ = H.
Similarly, for C 2,0 we have C = tI + xe 1 + ye 2 + ze 12 with For general cases, the matrix representation of Clifford algebra is an old problem with a long history. As early as in 1908, Cartan got the following periodicity of 8 [16,17].
Theorem 1 For real universal Clifford algebra C p,q , we have the following isomorphism Mat(2 (1.6) In contrast with the above representation for a whole Clifford algebra, we find the representation of the generators (e 1 , e 2 · · · e n ) is more fundamental and important in the practical applications.
For example, C 0,2 ∼ = H is miraculous in mathematics, but it is strange and incomprehensible in geometry and physics, because the basis e 12 ∈ ⊗ 2 V has different geometrical dimensions from that of e 1 and e 2 . How can e 12 take the same place of e 1 and e 2 ? Besides, C 2,0 C 0,2 is also abnormal in physics, because the different signs of metric are simply caused by different conventions.
Theorem 2 Consider two sets of 4 × 4 complex matrices γ µ , β µ , (µ = 0, 1, 2, 3). The 2 sets satisfy the following C 1,3 Then there exists a unique (up to multiplication by a complex constant) complex matrix T such that This theorem is generalized to the cases of real and complex Clifford algebras of even and odd dimensions in [19,20].
In this note we construct explicit complex and real matrix representations for the generators of Clifford algebra. The problem is aroused from the discussion on the specificity of the 1 + 3 dimensional Minkowski space-time with Prof. Rafal Ablamowicz and Prof. Dmitry Shirokov.
They have done a number of researches on general representation theory of Clifford algebra [16,17,19,20,21,22,23,24]. Many isomorphic or equivalent relations between Clifford algebra and matrices were provided. However, the representation of generators provides some new insights into the specific properties of the Minkowski space-time and the dynamics of fields [25,26,27], and it discloses that the 1+3 dimensional space-time is really special.
which forms the generator or grade-1 basis of Clifford algebra C 1,3 . To denote γ µ by Γ µ (m) is for the convenience of representation of high dimensional Clifford algebra. For any matrices C µ satisfying C 1, 3 Clifford algebra, we have [25,26] Theorem 3 Assuming the matrices C µ satisfy anti-commutative relation then there is a natural number m and an invertible matrix K, such that In this note, we derive complex representation of C (p, q) based on Thm.3, and then derived the real representations according to the complex representations.
Proof. Since we have gotten the unique generator γ µ for C 1,3 , so we only need to derive γ 5 By the first equation we get B = 0, and then X = diag(A, −A). Assuming A = (A ab ), where ∀A ab are 2×2 matrices. Then by the second equation in (2.7) we get block matrix where K is a n × n matrix to be determined. In this paper, the direct product ⊗ of matrices is defined as Kronecker product.
Again assuming matrix X 1 satisfies γ µ X 1 + X 1 γ µ = 0. By the above proof we learn that In this cases we can not expand the derived γ µ as matrix representation for C 1,5 . But in the case k = l, we find X 2 1 = −I have solution, and A 1 has a structure of iγ 1 . Then the construction of generators can proceed. In this case, we have the following theorem.
Since (iγ µ ) 2 = −(γ µ ) 2 , instead of C p,q we directly use C p+q in some cases for complex representation. Similarly to the case C 4 , in equivalent sense we have unique matrix representation for C 8 .
For C 10 , we also have two essentially different cases similar to C 6 . If k = l, γ 9 and the above generators cannot be expanded as generators of C 10 . We call this representation as normal representation. Clearly k = l is a large class of representations which are not definitely equivalent.
In the case of k = l, the above generators can be uniquely expanded as generators for C 12 . We call this representation as exceptional representation. The other generators are given by In order to express the general representation of generators, we introduce some simple notations. Similarly to the above proofs, we can check the following theorem by method of induction.
Theorem 6 1 • In equivalent sense, for C 4m , the matrix representation of generators is uniquely given by In which n = 2 m−1 N , N is any given positive integer. All matrices are 2 m+1 N × 2 m+1 N type.
2 • For C 4m+1 , besides (2.13) we have another real generator (2.14) If and only if k = l, this representation can be uniquely expanded as generators of C 4m + 4. Then we get all complex matrix representations for generators of real C p,q explicitly.
The real representation of C p,q can be easily constructed from the above complex representation. In order to get the real representation, we should classify the generators derived above.
Let G c (n) stand for any one set of all complex generators of C n given in Thm.6, exceptional representation or normal one, and set the coefficients before all σ µ and σ µ as 1 or i. Denote G c+ stands for the set of complex generators of C n,0 and G c− for the set of complex generators of C 0,n . Then we have By the construction of generators, we have only two kinds of γ µ matrices. One is the matrix with real nonzero elements, and the other is that with imaginary nonzero elements. This is because all nonzero elements of σ 2 are imaginary but all other σ µ (∀µ = 2) are real. Again assume Denote J 2 = iσ 2 , we have J 2 2 = −I 2 . J 2 becomes the real matrix representation for imaginary unit i. Using the direct products of complex generators with (I 2 , J 2 ), we can easily construct the real representation of all generators for C p,q from G c+ as follows.
Theorem 7 1 • For C n,0 , we have real matrix representation of generators as (2.17) 2 • For C 0,n , we have real matrix representation of generators as

we have real matrix representation of generators as
Obviously we have C p n C q n = (C p n ) 2 choices for the real generators of C p,q from each complex representation.
Proof. By calculating rules of block matrix, it is easy to check the following relations (2.21) By these relations, Thm.7 becomes a direct result of Thm.6.
For example, we have 4 × 4 real matrix representation for generators of C 0,3 as It is easy to check (2.23)

III. DISCUSSION AND CONCLUSION
For different purpose, Clifford algebra has several different definitions, and 5 kinds are listed in references [16,17,28]. The following definition is the most commonly used in theoretical analysis.
Definition 2 Suppose (V, Q) is an n < ∞ dimensional quadratic space over field F, and A is a unital associative algebra. There is an injective mapping J : V → A such that Then the set A together with mapping J is called Clifford algebra C (V, Q) over F.

1)
where γ a is the local orthogonal frame and γ a the coframe. The space-time is endowed with distance ds = |dx| and oriented volumes dV k calculated by
In some sense, Definition 1 is for all scientists, Definition 2 is for mathematicians, and the definition of Chevalley is only for algebraists. However, the Definition 3 can be well understood by all common readers including high school students, which directly connects normal intelligence with the deepest wisdom of Nature [26,27]. From the geometric and physical point of view, the definition of Clifford basis in Definition 1 is inappropriate, because in the case of non-orthogonal basis, e 12 = e 1 e 2 + e 1 ∧ e 2 ∈ Λ 0 ∪ Λ 2 is a mixture with different dimensions, and the geometric meaning which represents is not clear. But the Grassmann basis in Definition 3 is not the case, where each term has a specific geometric meaning and has covariant form under coordinate transformation.
The coefficients in (3.4) are all tensors with clear geometric and physical meanings.
To use the Definition 3, the transformation law of Grassmann basis under Clifford product is important. Now we discuss it briefly.
In the case of multivectors γ µ 1 µ 2 ···µ l γ θ 1 θ 2 ···θ k , we can define multi-inner product A k B as follows [29] γ µν γ αβ = g µβ γ να − g µα γ νβ + g να γ µβ − g νβ γ µα , (3.8) γ µν 2 γ αβ = g µβ g να − g µα g νβ , · · · (3.9) For example, we have γ µν γ αβ = γ µν 2 γ αβ + γ µν γ αβ + γ µναβ . (3.10) The derivation of the paper is constructive, so it can be used for both theoretical analysis and practical calculation. From the results we find C 1,3 has specificity and takes fundamental place in Clifford algebra theory. By the above representations of generators, we can easily get the relations between bases such as γ abc = i abcd γ d γ 5 in C 1,3 . The generators of Clifford algebra are the faithful bases of p + q dimensional Minkowski space-time or pseudo-Riemann space as shown in (3.1)-(3.4), and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic calculus [27], so the Riemann geometry expressed in Clifford algebra will be much simpler and clearer than current version.