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A Note on the Representation of Clifford Algebras

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In this note we construct explicit complex and real faithful matrix representations of the Clifford algebra $Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the 1+3 dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Therefore, we can expect that Clifford algebra can complete a large synthesis in science. Keywords: Clifford algebra, multi-inner product, gamma matrix, Pauli matrix, connection operator, torsion
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A Note on the Representation of
Clifford Algebra
Ying-Qiu Gu
School of Mathematical Science, Fudan University, Shanghai 200433, China
(Dated: 19th December 2021)
In this note we construct explicit complex and real faithful matrix representations of the
Clifford algebra C`p,q . The representation is based on Pauli matrices and has an elegant
structure similar to the fractal geometry. In the cases p+q= 4m, the representation is
unique in equivalent sense, and the 1+3 dimensional space-time corresponds to the simplest
and best case. Besides, the relation between the curvilinear coordinate frame and the local
orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives
of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent
space is calculated. These results are helpful for both theoretical analysis and practical
calculation. The basis matrices are the faithful representation of Clifford algebras in any p+q
dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the
complicated relations in geometry and physics into simple and concise algebraic operations.
Clifford numbers over any number field Fexpressed by this matrix basis form a well-defined
2ndimensional hypercomplex number system. Therefore, we can expect that Clifford algebra
will complete a large synthesis in science.
MSC: 15A66, 15A30, 15B99
Keywords: Clifford algebra, multi-inner product, gamma matrix, Pauli matrix, connection
operator, torsion, hypercomplex number
Contents
I. Introduction 2
II. Canonical Matrix Representation of Generators 4
III. About the Definitions of Clifford Algebra 10
IV. Some Applications in Physics and Geometry 15
Electronic address: yqgu@fudan.edu.cn
2
V. Discussion and Conclusion 21
Acknowledgments 23
References 23
I. INTRODUCTION
Clifford algebra was firstly defined by W. K. Clifford in 1878, which combines the concepts of
Hamilton’s quaternion(1843) and Grassmann’s exterior algebra(1844). The introduce of Dirac’s
spinor equation has greatly promoted the research on Clifford algebra. Further development of the
theory of Clifford algebras is associated with a number of famous mathematicians and physicists:
R. Lipschitz, T. Vahlen, E. Cartan, E. Witt, C. Chevalley, M. Riesz and others [1,2].
Matrix representations of geometric algebra carry additional information that cannot be ob-
tained from their pure algebra definition. In [3], by introduction of isometric transformation
T(an)=[T(a)]nand norm ||a|| =n
p|det(Mn×n(a))|, some isometric transformation such as rota-
tions, reflections, axial symmetries, duality, Clifford conjugation and Hermitian conjugation, can
be written in the canonical forms either similarity transformation T(a) = t1at or a combination
of reversion R(a) and similarity transformation S(a) = t1R(a)t. Moreover, the metric signature
of the vector space corresponds to the symmetry or skew-symmetry of the matrix representation
of the generators of Clifford algebra.
Due to its excellent properties, Clifford algebra has gradually become a unified language and
efficient tool of modern science, and is widely used in different branches of mathematics, physics and
engineering[4,5,6,7,8]. Theoretically we have some equivalent definitions for Clifford algebras.
For the present purpose, we introduce the original definition of Clifford, which is based on the
generators of basis[9].
Definition 1. Suppose Vis n-dimensional vector space over field R, and its basis
{e1, e2,··· , en}satisfies the following algebraic rules
eaeb+ebea= 2ηabIn, ηab = diag(Ip,Iq), n =p+q(1.1)
where the multiplication eaebis Clifford product of vectors. Denoting eab =eaeb,eabc =
eaebec,···, then the basis
ek∈ {I, ea, eab, eabc,·· · , e12···n; 1 a<b<cn}(1.2)
3
together with relation (1.1)and number multiplication C=Pkckek(ckR)form a 2n-
dimensional real unital associative algebra, which is called real universal Clifford algebra
C`p,q =Ln
k=0 kV, and C=Pkckekis called Clifford number.
For C`0,2, we have C=tI +xe1+ye2+ze12 with
e2
1=e2
2=e2
12 =1, e2e12 =e12e2=e1
e1e2=e2e1=e12, e12 e1=e1e12 =e2.
(1.3)
By (1.3) we find Cis equivalent to a quaternion, that is, we have isomorphism C`0,2
=H.
Similarly, for C`2,0we have C=tI +xe1+ye2+ze12 with
e2
1=e2
2=e2
12 = 1, e2e12 =e12e2=e1
e1e2=e2e1=e12, e12 e1=e1e12 =e2.
(1.4)
By (1.4), the basis is equivalent to
e1=
0 1
1 0
, e2=
1 0
01
, e12 =
01
1 0
.(1.5)
Thus (1.5) means C`2,0
=Mat(2,R). For a complete geometric algebra[3], it has a minimal 2n
order real matrix representation, e. g. C`2,0
=C`1,1
=Mat(2,R), C`3,1
=C`2,2
=Mat(4,R).
For general cases, the matrix representation of Clifford algebra is an old problem with a long
history. As early as in 1908, E. Cartan got the following periodicity of 8[9].
Theorem 1 For real universal Clifford algebra C`p,q , we have the following isomorphism
C`p,q
=
Mat(2n
2,R),if mod (pq, 8) = 0,2
Mat(2n1
2,R)Mat(2n1
2,R),if mod (pq, 8) = 1
Mat(2n1
2,C),if mod (pq, 8) = 3,7
Mat(2n2
2,H),if mod (pq, 8) = 4,6
Mat(2n3
2,H)Mat(2n3
2,H),if mod (pq, 8) = 5.
(1.6)
In contrast with the above representation for a whole Clifford algebra, we find the representation
of the generators (e1, e2···en) is more fundamental and important in some practical applications.
For example, C`0,2
=His miraculous in mathematics, but it is strange and incomprehensible
in geometry and physics, because the basis e12 ∈ ⊗2Vhas different geometrical dimensions from
that of e1and e2. How can e12 take the same place of e1and e2? Besides, C`2,0C`0,2is also
abnormal in physics, because the different signs of metric are simply caused by different conventions.
Similarly, the signature of space-time metric (,+,+,+) or (+,,,) is also a convention, but
4
C`3,1
=Mat(4,R) and C`1,3Mat(4,R). The following applications of Clifford algebra in physics
and geometry show the importance of the matrix representation of generators, and the convenience
in application by relaxing the constraints M(ek)Fand the entries of matrix Cor H.
For the generators in 1 + 3 dimensional space-time, Pauli got the following result[10].
Theorem 2 Consider two sets of 4×4complex matrices {γa, βa;a= 0,1,2,3}. The 2 sets
satisfy the following C`1,3
γaγb+γbγa=βaβb+βbβa= 2ηabI. (1.7)
Then there exists a unique (up to multiplication by a complex constant)complex matrix Tsuch
that
γa=T1βaT, a ∈ {0,1,2,3}.(1.8)
In this note we explicitly construct faithful complex and real matrix representations for the
generators of all finite dimensional Clifford algebra. The representations have a unified canonical
form which is convenient for programmers. The problem is aroused from the discussion on the
specificity of the 1 + 3 dimensional Minkowski space-time with Prof. Rafal Ablamowicz. He have
done a number of researches on general representation theory of Clifford algebra[11,12,13,14].
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[15,16,17,18], and it discloses that the
1+3 dimensional space-time is really special.
II. CANONICAL MATRIX REPRESENTATION OF GENERATORS
Denote Minkowski metric by (ηµν ) = (ηµν ) = diag(1,1,1,1), Pauli matrices σµby
σµ
1 0
0 1
,
0 1
1 0
,
0i
i 0
,
1 0
01
eσ0=σ0=I, eσk=σk, k = 1,2,3.
Define γµby
γµ=
0e
ϑµ
ϑµ0
Γµ(m)(2.9)
5
where m1,
ϑµ= diag(
m
z }| {
σµ, σµ,··· , σµ),e
ϑµ= diag(
m
z }| {
eσµ,eσµ,··· ,eσµ).(2.10)
γµforms the faithful matrix representation of generator or grade-1 basis of Clifford algebra C`1,3.
To denote γµby Γµ(m) is for the convenience of representation of high dimensional Clifford algebra.
For any matrices Cµsatisfying C`1,3Clifford algebra, we have[15,16]
Theorem 3 Assuming the matrices Cµsatisfy anti-commutative relation
CµCν+CνCµ= 2ηµν I(2.11)
then there is a natural number mand an invertible matrix K, such that K1CµK=γµ. This
means in equivalent sense, we have unique representation (2.9)of generators of C`1,3.
In this note, we derive faithful complex representation of C`p,q based on Theorem 3, and then
derived the real representations according to the complex representations. In the cases without
confusion, we omit the identity matrix I.
Theorem 4 Let
γ4= idiag(E, E), E diag(I2k,I2l),(kl 6= 0, k +l=n).(2.12)
Other {γµ, µ 3}are given by (2.9). Then the generators of Clifford algebra C`1,4are equivalent
to γµ, µ = 0,1,2,3,4.
Proof. Since we have gotten the unique generator γµfor C`1,3, so we only need to derive γ4
for C`1,4. Assuming 4n×4nmatrix
X=
A B
C D
(2.13)
satisfies γµX+Xγµ= 0, µ= 0,1,2,3. By γ0X+Xγ0= 0 we get D=A,C=B. By
γkX+Xγk= 0 we get
ϑkB+k= 0, ϑkAk= 0.(2.14)
By the first equation we get B= 0, and then X= diag(A, A). Assuming A= (Aab), where Aab
are 2×2 matrices. Then by the second equation in (2.14) we get block matrix A= (KabI2)KI2,
where Kis a n×nmatrix to be determined. Here, the direct product of matrices is defined as
Kronecker product.
6
For X2=I4nwe get A2=I2n, and then K2=In. Therefore, there exists an invertible n×n
matrix qsuch that q1Kq = diag(Ik,Il). Let 2n×2nblock matrix Q=qI2, we have
Q1AQ = diag(I2k,I2l)E, ϑkQ=k.(2.15)
Let γ4= idiag(E, E), then all {γµ, µ = 0,1,2,3,4}satisfy Clifford relation (1.1). Noticing the
complete Grassmann basis set of C`1,3
I, γa, γab =i
2²abcdγcd γ5, γabc = i²abcd γdγ5, γ0123 =iγ5(2.16)
in which γ5= diag(I, I) and ²0123 = 1. If k6= 0 and l6= 0, we prove γ4is linearly independent
of (2.16).
Assuming
4+x0I+xaγa+xabγab +yaγaγ5+y0γ5= 0,(2.17)
by γ4γa+γaγ4= 0 and Clifford calculus, we have
0 = 2x0γa+ 2xaI+xbc(γbc γa+γaγbc) + yb(γbγ5γa+γaγbγ5)
= 2x0γa+ 2xaI+ 2xbcγabc + 2ybγab γ5.
So we get x0=xa=ya= 0, and then equation (2.17) becomes
4+xabγab +y0γ5= 0.(2.18)
Expressing γab in matrix form, for a, b > 0 we have
γ0a= diag(ϑa,ϑa), γab =diag(ϑaϑb, ϑaϑb).(2.19)
Substituting (2.19) into (2.18) we get x12 =x23 =x13 = 0 and
izE +x0aϑa+y0I= 0,(a= 1,2,3).(2.20)
For E= diag(I2k,I2l) and kl 6= 0, the solution of (2.20) is z=x0a=y0= 0, so γ4is linearly inde-
pendent of (2.16). In this case, {γµ, µ = 0,1,2,3,4}constitute the complex matrix representation
for generators of C`1,4. We prove the theorem.
From the above results we find that, the square matrix representation γaof the generators of
any Clifford algebras derived above contains only 5 numbers (0,±1,±i), and the algebraic calculus
of the basis is essentially Clifford product that involves only 3 numbers (0,±1). Like the abstract
basis ea,γais actually an operator having little relation with the number field Fof the vector
7
space or the division algebras {R,C,H}. Therefore, in applications it is unnecessary to constraint
the elements of γain F. According to this point of view, the square matrices {γa, a = 0,1,2,3,4}
can generate a faithful matrix representation of total basis set of real Clifford algebra C`1,4.
Again assuming matrix X1satisfies γµX1+X1γµ= 0. By the above proof we learn that
X1= diag(A1,A1). Solving X1γ4+γ4X1= 0, we get X1= 0 if k6=l. In this cases we cannot
expand the derived γµas matrix representation for C`1,5. But in the case k=l, we find X2
1=I
have solution, and A1has a structure of iγ1. Then the construction of generators can proceed. In
this case, we have the following theorem.
Theorem 5 Suppose that 8n×8nmatrices Aµ=diag(Cµ,Cµ), µ ∈ {0,1,2,3}satisfy
AµAν+AνAµ= 2ηµν , Aµγν
2n+γν
2nAµ= 0,(2.21)
then there is an 8n×8nmatrix K, such that
K1AµK= diag(γµ
n,γµ
n)βµ
2n, Kγµ
2n=γµ
2nK. (2.22)
In which γµ
nmeans n σµin ϑµ. Then {γµ
2n, βµ
2n}constitute all generators of C`2,6.
Proof. By Kγµ
2n=γµ
2nKwe get K= diag(L, L) and L= (LabI2)e
LI2, where e
L= (Lab) is
a 2n×2nmatrix to be determined. By (2.21) we have Cµ= (Cµ
abI2)e
CµI2. Then e
Cµalso
satisfies C`1,3Clifford algebra. By Theorem 3, there is a matric e
Lsuch that e
L1e
Cµe
L=γµ. Then
this Kproves the theorem.
Since (iγµ)2=(γµ)2, instead of C`p,q we directly use C`p+qin some cases for complex represen-
tation. Similarly to the case C`4, in equivalent sense we have unique matrix representation for C`8.
For C`9, besides the generators constructed by the above Theorem 5, we need another generator γ9.
By calculation similar to (2.15), we find γ9= diag(E, E, E, E ) and E= diag(I2k,I2l), kl 6= 0.
For C`10, we also have two essentially different cases similar to C`6. If k6=l,γ9and the above
generators cannot be expanded as generators of C`10. We call this representation as normal rep-
resentation. Clearly k6=lis 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
αµ
4n= diag(γµ
n,γµ
n,γµ
n, γµ
n)I4.(2.23)
In order to express the general representation of generators, we introduce some simple notations.
Imstands for m×munit matrix. For any matrix A= (Aab), denote block matrix
AIm= (AabIm),[A, B, C, ···] = diag(A, B, C, ···).(2.24)
8
Obviously, we have I2I2=I4,I2I2I2=I8and so on. In what follows, we use Γµ(m) defined
in (2.9). For µ∈ {0,1,2,3}, Γµ(m) is 4m×4mmatrix, which constitute the generator of C`1,3.
Similarly to the above proofs, we can check the following theorem by method of induction.
Theorem 6 1In the case of neglecting an imaginary factor i, for the generators of C`4m,
there exists the following unique matrix representation in the equivalent sense
©Γa(n),£Γa¡n
22¢,Γa¡n
22¢¤I2,
£a¡n
24¢,Γa¡n
24¢],a¡n
24¢,Γa¡n
24¢]¤I22,
£a¡n
26¢,Γa¡n
26¢,Γa¡n
26¢,Γa¡n
26¢],a¡n
26¢,Γa¡n
26¢,
Γa¡n
26¢,Γa¡n
26¢]¤I23,···ª.
(2.25)
In which n= 2m1N,Nis any given positive integer. All matrices are 2m+1N×2m+1Ntype.
2For C`4m+1, besides (2.25)we have another real generator
γ4m+1 = [[[E, E],[E , E]],[[E, E],[E, E]] · · · ](2.26)
where E= [I2k,I2l],(kl 6= 0). If and only if k=l, this representation can be uniquely expanded
as generators of C`4m+4.
3The generators of C`4m+2 or C`4m+3 can be represented by 4m+ 2 or 4m+ 3 matrices from
the matrix representation of the C`4m+4 generators.
4For C`j,(j= 2,3), besides to select the matrices from the basis of C`4for the representation,
we also have the following matrix representation [15]
γa∈ {diag(ϑk,ξk), k = 1,2,3}(2.27)
where
ϑk= diag(
m
z }| {
σk, σk,··· , σk), ξk= diag(
nm
z }| {
σk, σk,··· , σk)(2.28)
and m(0 mn)is independent of k.
Then we obtain all complex matrix representations for generators of real C`p,q explicitly.
(2.25) clearly shows the specificity of 1+3 dimensional space-time. For the real matrix repre-
sentation of the Clifford algebra C`p,q generators, it is easily obtained based on the above complex
matrix representation theorem. To obtain the real matrix representation, we need to classify the
complex matrix representations of the generators derived above. For any set of complex matri-
ces representations of the C`ngenerators given in the Theorem 6, by multiplying each matrix by
9
a factor 1 or i, so that all matrices satisfy (γa)2=I; that is to say, the basis matrices is the
representation of C`n,0generators, we denote it as Gc+.
From the structure of complex matrix representation of generators, we have only two classes
of γamatrices. One has only real nonzero elements and the other has only imaginary nonzero
elements, because all non-zero elements of σ2are imaginary numbers ±i, but all other σa(a6= 2)
are real numbers. So we have
Gc+=GrGi,Gr={γa
r;γa
ris real},Gi={γa
i;γa
iis imaginary}.(2.29)
Denoting J2= iσ2, we have J2
2=I2.J2becomes the real matrix representation for imaginary
unit i. Using the direct products of complex generators with (I2, J2), we can easily construct the
real representation of all generators for C`p,q from Gc+as follows.
Theorem 7 1For C`n,0, we have real matrix representation of generators as
Gr+={γaI2(if γaGr),iγbJ2(if γbGi)}.(2.30)
2For C`0,n, we have real matrix representation of generators as
Gr={γaJ2(if γaGr),iγbI2(if γbGi)}.(2.31)
3For C`p,q, we have real matrix representation of generators as
Gr=
Γka
+,Γlb
;Γka
+=γkaGr+, a = 1,2,··· , p
Γlb
=γlbGr, b = 1,2,··· , q
(2.32)
where the complex matrix representation corresponding to Γlb
should be different from that corre-
sponding to Γka
+. In this way, for the real matrix representation of the generators of C`p,q , we have
Cp
ndifferent choices.
Proof. By calculating rules of block matrix, we have the following relations
(γaI2)(γbJ2)+(γbJ2)(γaI2) = (γaγb+γbγa)J2,(2.33)
(γaJ2)(γbJ2)+(γbJ2)(γaJ2) = (γaγb+γbγa)I2.(2.34)
From these relations we learn that, if a6=b, the above formulas are equal to 0, then the real
representation keeps all anti-commutative relations of the complex representation. Thus, Theorem
7becomes a direct result of Theorem 6.
10
For example, we have 4 ×4 real matrix representation for generators of C`3,0as follows,
{σ1, σ2, σ3}
={σ1I2,iσ2J2, σ3I2} ≡ {Σ1,Σ2,Σ3}
=
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
,
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
,
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
.(2.35)
{Σ23,Σ31 ,Σ12}={σ1J2,iσ2I2, σ3J2}=
=
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
,
0 0 1 0
0 0 0 1
1 0 0 0
01 0 0
,
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
(2.36)
and Σ123 = diag(iσ2,iσ2). For real representation of C`0,3we need higher order matrices, because
Σab is not a faithful representation for C`0,3.
By Theorem 6we learn that, the 1+3 dimensional Minkowski space-time is really special, and
the matrix representation of its basis is unique and minimum. This implies Nature has only used
the simplest but best mathematics.
The above derivation is constructive, so it can be used for both theoretical analysis and practical
calculation. From the results we find C`1,3has specificity and takes fundamental place in Clifford
algebra theory. The faithful representation matrices of Clifford algebra are bases of p+qdimensional
Minkowski space-time or pseudo-Riemann space as shown in (3.40)-(3.43), and Clifford algebra
converts the complicated relations in geometry into simple and concise algebraic calculus[16,17,19],
so the Riemann geometry expressed in Clifford algebra will be much simpler and clearer than the
current version.
III. ABOUT THE DEFINITIONS OF CLIFFORD ALGEBRA
For different purpose, Clifford algebra has several different definitions, and 5 kinds are listed in
references [9,20]. 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 Ais
a unital associative algebra. There is an injective mapping J:VAsuch that
i) I /J(V)
ii) (J(x))2=Q(x)I, xV
11
iii) J(V)generates A.
Then Atogether with mapping Jis called Clifford algebra C`(V, Q)over F.
The above definition includes the case of degenerate Clifford algebra C`p,q,r. For example, if
Q(x) = 0, the Clifford algebra C`(V , 0) becomes Grassmann algebra. In the non-degenerate case,
if the standard orthogonal basis of Vis introduced, we can derive Definition 1. The definition
based on the quotient algebra of tensor algebra in Vis introduced by Chevalley[9], but it is too
abstract for common readers. In the author’s opinion, the most efficient and convenient definition
of Clifford algebra should be as follows.
In the following discussion, we use Latin characters (a, b ∈ {0,1,2,3}) for the local Minkowski
indices, Greek characters (µ, ν ∈ {0,1,2,3}) for the curvilinear indices, and (j, k, l, m, n ∈ {1,2,3})
for spatial indices. Since the basis or frame eµof a space-time is isomorphic to the Dirac-γmatrices
γµ, and this isomorphism guarantees a one-to-one correspondence of vectors and their algebraic
products to the set of matrices (2.9) and their matrix algebra, so we need not distinguish basis eµ
with matrix γµin algebraic calculations. Define γµ=gµν γν, which is equivalent to the coframe eµ.
Therefore, in the sense of logical equivalence, the element of space-time can be uniquely described
as[16,17,18,19,21]
dx=γµdxµ=γaδXa, γµ=fµ
aγa, γµ=fa
µγa(3.37)
in which γaacts as an orthonormal basis in the tangent space-time of the manifold and the
coframe is γa=ηabγb. The relation between the local frame coefficients (fµ
a, f a
µR) and the
metric gµν is given by
fa
µfµ
b=δa
b, f a
µfν
a=δν
µ, fµ
afν
bηab =gµν , f a
µfb
νηab =gµν .(3.38)
The γ-matrices satisfies the following C`1,3Clifford relations,
γaγb+γbγa= 2ηab, γµγν+γνγµ= 2gµν ,(3.39)
which is related to the length of the line element as shown below. In (3.39) the associative product
γaγbis equivalent to the Clifford product or geometrical product of basis, which is isomorphic to
multiplication of matrices[9,16]. The notation systems (3.37) - (3.39) are not just formal relations,
but a logical system with invariance. For example, by (3.37) and (3.38) we have the transformation
rule between curvilinear coordinate element dxµin the manifold and the orthogonal coordinate
element δXain the tangent space as
dxγµdxµ= (γafa
µ)(fµ
bδXb) = γa(fa
µfµ
b)δXb=γaδX a.
12
Again by the Clifford relation (3.39) we have calculation
dx2=1
2(γµγν+γνγµ)dxµdxν=gµν dxµdxν
=1
2(γaγb+γbγa)δXaδX b=ηab δXaδX b.
These equations simultaneously display both geometric and algebraic messages, and the operations
are simple, intuitive and error-prevent.
In geometry and physics, the multiplication between quantities are Clifford products; but only
by projecting these quantities onto a Grassmann basis, are their geometrical and physical meanings
clarified. In an n-dimensional space-time with basis {γa|a= 1,2···n}, the Grassmann product or
exterior product is defined as
γa1γa2· ·· ∧ γak1
k!X
σ
σb1b2···bk
a1a2···akγb1γb2···γbk,(1 kn)
in which aj6=alif j6=l,σb1b2···bk
a1a2···akis the permutation tensor, if b1b2···bkis the even permutation
of a1a2···ak, it equals 1. Otherwise, it equals -1. The above formula is a summation for all
permutations; that is, it is antisymmetrized with respect to all indices.
Definition 3. Assume the element of an n=p+qdimensional space-time Mp,q over Ris
described by
dx=γµdxµ=γµdxµ=γaδXa=γaδXa(3.40)
where γais the local orthonormal frame and γais the coframe. The space-time is endowed with
distance ds=|dx|and oriented volumes dVkcalculated by
dx2=1
2(γµγν+γνγµ)dxµdxν=gµν dxµdxν=ηabδXaδXb(3.41)
dVk= dx1dx2∧ · ·· ∧ dxk=γµν···ωdxµ
1dxν
2···dxω
k,1kn(3.42)
in which Minkowski metric (ηab) = diag(Ip,Iq), and Grassmann basis γµν ···ω=γµγν···γω
Λk(Mp,q). Then the Clifford-Grassmann number
C=c0I+cµγµ+cµν γµν +· ·· +c12···nγ12···n, ckR(3.43)
together with multiplication rule of basis given in (3.41)and associativity define the 2n-dimensional
real universal Clifford algebra C`p,q .
The reason why this paper highlights the above definition is that, this definition corresponds
directly to the geometric concepts and can be used directly for geometric and physical computation.
13
This should also be the reason why Clifford himself called this algebra as geometric algebra.
(3.41) is just the quadratic form in the usual definitions. In (3.43), c0Iis a scalar, cµγµa true vector,
cµν γµν a bi-vector, cµ1µ2···µkγµ1µ2···µkak-vector, and each term is a skew-symmetrical tensor.
The geometrical meanings of the elements dx,dy,dxdyare shown in Fig.1. Fig.1shows that
Figure 1: Geometric meaning of the vectors dx,dyand dxdy
.
the exterior product is the oriented volume of parallel polyhedron of line element vectors, and
the Grassmann basis γab···cis just the orthonormal basis of k-dimensional volume[16,19]. Since
the length of a line element and the volumes of each grade constitute the fundamental contents
of geometry, the Grassmann basis set becomes units to represent various geometric and physical
quantities, which are special kinds of tensors.
The advantage of such a representation is that, as long as the geometric concepts, such as vector,
area and volume etc., correspond to the above matrix basis at the beginning, then the complicated
geometric relations can be transformed into mechanical algebraic relations, thus the learning and
researching geometry and physics become simpler and clearer, and the calculus becomes mechanical
manipulation as shown below. For example, in the spherical coordinate system, we have the element
dxand the directed area element dson the r=r0sphere as
dx=σ1dr+σ2rdθ+σ3rsin θdϕ,
ds= (σ2rdθ)(σ3rsin θdϕ) = (iσ1)r2sin θdθdϕ
where σ2σ3=σ23 = iσ1is the product of basis, which shows the operator nature of σa. The
14
total area of this sphere is calculated by
A=Ids= (iσ1)r2Isin θdθdϕ= (iσ1)4πr2.(3.44)
Obviously, the geometrical meanings of each factor in (3.44) is clear, iσ1is just a basis to clarify
the direction of the sphere, which does not mean we get an imaginary area AC.
In some sense, Definition 1 is for all scientists, Definition 2 is for mathematicians, and the
definition of Chevalley is for algebraists. However, the Definition 3 can be well understood by all
common readers including high school students[16,17,18]. From the geometric and physical point
of view, the definition of Clifford basis in Definition 1 is inconvenient, because in the case of non-
orthogonal basis, e12 =e1¯e2+e1e2Λ0Λ2is a mixture with different dimensions, and the
geometric meaning which represents is unclear. 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. To use the Definition 3, the transformation law between Clifford product and
Grassmann product is important. Here we discuss this issue briefly.
Theorem 8 For γµand γθ1θ2···θkΛk, we have
γµγθ1θ2···θk=γµ¯γθ1θ2···θk+γµθ1···θk(3.45)
γθ1θ2···θkγµ=γθ1θ2···θk¯γµ+γθ1···θkµ(3.46)
where the inner product of Clifford algebra ¯is defined as
γµ¯γθ1θ2···θk=gµθ1γθ2···θkgµθ2γθ1θ3···θk+· ·· + ()k+1gµθkγθ1···θk1,
γθ1θ2···θk¯γµ= ()k+1gµθ1γθ2···θk+ ()kgµθ2γθ1θ3···θk+· · · +gµθkγθ1···θk1.
Proof. Clearly γµγθ1θ2···θkΛk1Λk+1, so we have
γµγθ1θ2···θk=a1gµθ1γθ2···θk+a2gµθ2γθ1θ3···θk+···
· ·· +akgµθkγθ1···θk1+µθ1···θk.
Permuting the indices θ1and θ2, we find a2=a1. Let µ=θ1, we get a1= 1. Check the monomial
in exterior product, we get A= 1. Thus we prove (3.45). In like manner we prove (3.46).
In the case of multivectors γµ1µ2···µlγθ1θ2···θk, we can define multi-inner product A ¯kBas
follows[19]
γµν ¯γαβ =gµβγνα gµα γνβ +gν αγµβ gν β γµα
γµν ¯2γαβ =gµβgνα gµα gνβ ,···
15
For example, we have
γµν γαβ =γµν ¯2γαβ +γµν ¯γαβ +γµναβ .
IV. SOME APPLICATIONS IN PHYSICS AND GEOMETRY
Now we give some applications of the definition and matrix representation of Clifford algebra.
For a vector field A=γµAµin the space-time, we define its absolute differential as
dAlim
xdx
[A(x+ ∆x)A(x)]
= (αAµγµ+Aµdαγµ)dxα= (αAµγµ+Aµdαγµ)dxα.
We call dαconnection operator[17,18]. According to its geometrical meanings, connection
operator should satisfy the following conditions:
1It is a real linear transformation in the tangent space-time dα:T V T V .
2For any bilinear product of vector AB, it satisfies Leibniz formula
dα(AB) = (dαA)B+A(dαB),(4.47)
or in the form of basis
dα(γµγν) = (dαγµ)γν+γµ(dαγν).(4.48)
In condition 2, the bilinear product means for any a, b Fwe have
(µ+ν)γω=µγω+νγω,
γω(µ+ν) = ωγµ+ωγν.
Clearly, the inner product in real space-time as well as the exterior, Clifford and tensor products are
all bilinear products. However, the inner product in complex space-time is not a bilinear product,
because in this case we have
γω·(µ+ν) = ¯ω·γµ+¯
ω·γν.
By the definition we have
Theorem 9 For metric tensor g=gµνγµγν=γµγµ, where is tensor product, we have
metric consistent condition dg= 0.
16
Proof. By condition 1, we assume
dαγµ=Kν
αµγν,dαγµ=e
Kµ
αβγβ(4.49)
where (Kν
αµ,e
Kµ
αβ)Rare connection coefficients. For any vectors Aand B, by condition 2we
have
d(A·B) = α(AµBν)(γµ·γν)dxα+ (AµBν)[(dαγµ)·γν+γµ·(dαγν)]dxα
=³α(AµBµ)+(AµBν)[ e
Kµ
αβ(γβ·γν) + γµ·(Kβ
αν γβ)]´dxα(4.50)
=³α(AµBµ)+(AµBν)[ e
Kµ
αν +Kµ
αν ]´dxα.
On the other hand, A·Bis a scalar, so we directly have
d(A·B) = α(AµBµ)dxα.(4.51)
Comparing (4.51) and (4.50), by the arbitrariness of vectors (A,B,dx) we find
e
Kµ
αν =Kµ
αν ,dαγµ=Kµ
αβγβ.(4.52)
For metric tensor, we have
dg= d(γµγµ) = [(dαγµ)γµ+γµ(dαγµ)]dxα
= [Kν
αµγνγµγµ(Kµ
αν γν)]dxα= 0.
(4.53)
The proof is completed.
From the above derivation we find that dαis actually the partial derivatives of basis. By
gµν =γµ·γνwe have
dgµν =αgµν dxα
= d(γµ·γν) = [(dαγµ)·γν+γµ·(dαγν)]dxα
= [Kβ
αµγβ·γν+γµ·(Kβ
αν γβ)]dxα
= (gνβ Kβ
αµ +gµβKβ
αν )dxα.
(4.54)
By arbitrariness of dxα, (4.54) is equivalent to
αgµν =gνβ Kβ
αµ +gµβKβ
αν .(4.55)
(4.55) is a linear nonhomogeneous algebraic equation of Kµ
αβ. The solution is given by the following
theorem[19].
17
Theorem 10 For the connection coefficients
Kα
µν = Πα
µν +Tα
µν ,Πµ
αβ = Πµ
βα,Tµ
αβ =−T µ
βα (4.56)
we have solution Πα
µν = Γα
µν +πα
µν , where Γα
µν is the Christoffel symbol. For the contortion
πµ
αβ =πµ
βα and torsion Tµ
αβ =−T µ
βα, denoting
πµ|να =gµβ πβ
να ,Tµ|να =gµβ Tβ
να ,
we have the following relations
πµ|να =Tν|αµ +Tα|ν µ,(4.57)
Tµ|να =1
3(πα|µν πν|µα) + e
Tµνα ,(4.58)
and consistent condition
πµ|να +πα|µν +πν|αµ = 0.(4.59)
e
T=e
Tµνω γµνω Λ3is an arbitrary skew-symmetrical tensor.
For geodesics, we have
dv
ds =dvα
ds γα+vαdµγαvµ=µdvα
ds + (Γα
µν +πα
µν +Tα
µν )vµvνγα,
=µd
dsvα+ Γα
µν vµvνγα+πα
µν vµvνγα.
The symmetrical part πα
µν influences the geodesic, this means πα
µν 6= 0 violates Einstein’s equiv-
alence principle. So we set πα
µν = 0, and take skew-symmetrical torsion T=Tµνω γµν ω Λ3
as example to show the convenience of the above formalism and Clifford algebra. The absolute
differential of vector Acan be denoted as
dA=γµαAµdxα=γµαAµdxα,
αAµ=Aµ
;α+AβTµ
αβ,αAµ=Aµ;αAβTβ
αµ
where Aµ
;αand Aµ;αare the usual covariant derivatives
Aµ
;α=αAµ+AβΓµ
αβ, Aµ;α=αAµAβΓβ
αµ.
Now we examine the connection of orthonormal basis γa, by γµ=fa
µγawe have
dA= [α(Aµfa
µ)γa+Aadαγa]dxα= (Aβ
;α+AµTβ
αµ)fa
βγadxα.
18
Notice the arbitrariness of Aµand dxα, we get equation for connection dαγaas
dαγa=fµ
a[αγµν
αµ +Tν
αµ)γν] = fµ
aαγµ
in which αγµ=γaαfa
µand
αγµ=αγµν
αµ +Tν
αµ)γν
is just a notation without special geometrical meanings. Similarly we have
dαγa=fa
µ[αγµ+ (Γµ
αν +Tµ
αν )γν]
=fa
µαγµ=gµν fa
µαγν=ηabdαγb.
(4.60)
Although (γa, γa) are constant matrices, but as basis vectors, their connection operation is a linear
transformation.
Theorem 11 For the basis (2.16) of C`1,3Clifford algebra, we have
dαγ0123 =dαγ5= 0.(4.61)
dαγ123 = (dαγ0)γ0123,dαγ023 =(dαγ1)γ0123 ,
dαγ013 = (dαγ2)γ0123,dαγ012 =(dαγ3)γ0123 .
(4.62)
For skew-symmetrical tensor K=Kµνω γµνω =Kαγαγ0123 , we have
αK= (αKµ)γµγ0123,αKµ=αKβµ
αβ +Tµ
αβ)Kµ.(4.63)
In which αKµis the covariant derivative of a true vector. In the case of torsion itself, namely
K=T, we have αTµ=αKβΓµ
αβTµ.
Proof. For (4.61), by (4.48) we have calculation
dαγ0123 = (dαγ0)γ123 +γ0(dαγ1)γ23 +···
=f0
µ[αγµ+ (Γµ
αν +Tµ
αν )γν]γ123 +···
=f0
µ[αfµ
b+ (Γµ
αν +Tµ
αν )fν
b]γbγ123 +···
=fa
µ[αfµ
a+ (Γµ
αν +Tµ
αν )fν
a]γ0123
=(fµ
aαfa
µ+ Γµ
αµ +Tµ
αµ)γ0123 .
By Γµ
αµ =fµ
aαfa
µ,Tµ
αµ =gµν Tναµ = 0, we obtain dαγ0123 = 0. By γ5= iγ0123 , we have dαγ5= 0.
For (4.62), we have
dαγ123 =dα(γ0γ0123) = (dαγ0)γ0123 +γ0dαγ0123 = (dαγ0)γ0123 ,···
19
For skew-symmetrical tensor, by (2.16) we have Clifford calculus,
K=Kµνω γµνω =Kabcγabc =Kabci²abcdγdγ5
=−Kabc²abcd γdγ0123 =Kdγdγ0123 =Kαγαγ0123
in which Kd=−Kabc²abcd,Kα=fα
aKa. So the trivector Kµνω is equivalent to a pseudo-vector
Kα. we have calculus,
αK= (αKµ)γµγ0123 +Kµ(dαγµ)γ0123 +Kµγµdαγ0123
= (αKµ)γµγ0123 − Kµµ
αβ +Tµ
αβ)γβγ0123
=³αKβµ
αβ +Tµ
αβ)Kµ´γβγ0123 = (αKµ)γµγ0123.
If K=T, we have
Ta
bcTa=Tabc Ta=²abcdTdTa= 0.
The proof is completed.
The above theorem shows that, for absolute derivatives, we can treat a pseudo-scalar as a
true scalar, and a grade-3 skew-symmetrical tensor as a true vector. The covariant derivative for
torsion is a linear operator. Therefore, the dynamical equation of the torsion field is also Maxwell
equations[17].
Next, we consider Dirac equation. In the flat space-time, the Dirac equation for free bispinor φis
given by γaiaφ=. In curved space-time without torsion, we have Dirac equation[19,22,23,24],
γµ(iµeAµ)φ=mφ, µφ= (µ+ Γµ)φ(4.64)
in which the spinor connection is given by
Γµ1
4γνγν
;µ=1
4γνγν;µ=1
4γν(µγνΓα
µν γα).(4.65)
The Dirac operator connection γµΓµΛ1Λ3is a Clifford product, and its geometric and
physical significance needs to be clarified. By projecting it onto the Grassmann basis γaand γabc,
its geometric and physical meanings become clear. By Clifford calculus, the Dirac equation can be
rewritten in the following Hermitian form[16,17,18,24]
(αµˆpµSµµ)φ=0φ(4.66)
in which αµis the current operator, ˆpµis the momentum and Sµis the spin operator,
αµ= diag(σµ,eσµ),ˆpµ= i(µ+ Υµ)eAµ, Sµ=1
2diag(σµ,eσµ),(4.67)
20
where Υµis the Keller connection and Ωµis the Gu-Nester potential, and they are respectively
defined as
Υν1
2fµ
a(νfa
µµfa
ν) = 1
2[ν(lng)fµ
aµfa
ν],
α1
2fα
dfµ
afν
bµfe
ν²abcdηce =1
4g²αµνω ηabfa
ω(µfb
ννfb
µ).
In which the notation ²αµνω = 0,±1 is just a number, but sometimes 1
g²αµνω is defined as a tensor
in differential geometry. Υµis not a vector. For a diagonal metric we have Ωα0.
For a free spinor in curved space-time, (4.66) expressed in the standard Clifford algebra becomes
Dφ=mφ, D=γµ(ˆpµ1
2γ5µ),ˆpµ= i(µ+ Υµ).(4.68)
For a coordinate or tetrad transformation, by covariance we have transformation law[16,17,18],
φ0=Rφ, D0=RDR1,D02=RD2R1(4.69)
where RSpin(1,3). (4.69) shows DΛ1Λ3and D2Λ0Λ2Λ4are covariant Clifford
numbers. By Clifford calculus of (4.68), we obtain the second-order Dirac equation as follows,
D2φ=m2φ, D2= (2+1
2L)γabYab i
2γ5P
in which 2=gµν(µν Γα
µν α) is the d’Alembertian for scalar fields, which is a closed scalar
operator, Yab Λ2is a bivector of linear operator ˆpµ,LΛ0is the unique real scalar containing
the linear second order derivatives of tetrad,
L= Θµ
;µ+1
2µΘµµµ)(4.70)
where Θµ
;µ=µΘµ+ Γµ
µν Θν=1
gµµg) and
Θµ≡ −µ,Υµ=1
2Θµ.
P= Ωµ
;µ+ ΘµµΛ4is a pseudo scalar.
On the other hand, by (4.60) we find dµγais a Lorentz vector for the index abut a Riemannian
vector for the index µ. Therefore, gµν dµγadνγais also a covariant Clifford number, and its scalar
part is given by
C=gµν fa
α;µfα
a;ν=gµν (µfa
αΓβ
αµfa
β)(νfα
a+ Γα
λν fλ
a).
The Nester-Tung decomposition of Lagrangian of gravity [25,26] should be a linear combination
of (L, C) as
Lg= Θµ
;µ+1
2µΘµµµ)kC =1
2R
21
where kis a constant.
Since Θµ
;µwill vanish in action due to integration, by (4.70) we obtain the equivalent Lagrangian
of gravity with quadratic form of first order derivatives
e
Lg=1
2µΘµµµ)kC.
This form is not an exact scalar now, but it may be helpful to discuss the existence and regularity of
the solution to Einstein field equation. For an asymptotic flat space-time, we have asymptotically
fa
µca
µ+ O(r1) and αfa
µO(r2). As pointed in [25,26], the new quadratic spinor
Lagrangian is asymptotically O(r4), which guarantees a finite action. This is an advantage over
the Einstein-Hilbert Lagrangian O(r3).
In Dirac equation (4.66), we obtained a spin-gravity coupling energy Sµµ. If the gravitational
field is generated by a rotating ball, then the corresponding metric, like the Kerr metric, cannot
be diagonalized. In this case Sµµ6= 0, which has a non-zero coupling effect. Similar to the case
of charged particles in a magnetic field, the spins of spinors will be automatically arranged along
the force lines of Ωµ. If the spins of all charged particles are regularly arranged along these lines
of force, then a macroscopic magnetic field will be induced.
V. DISCUSSION AND CONCLUSION
In this note we construct explicit complex and real faithful matrix representations for the
generators of Clifford algebra C`p,q. The representation is based on Pauli matrices and has an
elegant structure similar to the fractal geometry. In the cases p+q= 4m, the representation is
unique in equivalent sense, and the 1+3 dimensional space-time corresponds to the simplest and
best case. These results are helpful for both theoretical analysis and practical calculation. The
generators of Clifford algebra are the faithful basis of p+qdimensional 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 is much
simple and clear.
From the calculation we find that Clifford algebra is indeed a unified language and an efficient
tool for geometry and physics. To represent physical and geometric quantities by Clifford algebra,
the formalism is neat and elegant, their meanings are clear, and the calculus and operation are
simple and standard. For geometric and physical quantities, not only the vector Aµγµis important,
but all Clifford-Grassmann numbers cµν···ωγµν···ωin (3.43) such as torsion T Λ3are coequally
22
fundamental. Therefore, we can expect that Clifford algebra will complete a big synthesis of
physics.
In Minkowski space-time M1,3over number field F, we have general Clifford numbers
K=sI4+Aaγa+Habγab +Qaγaγ0123 +0123,(s, p, Aa,· ·· ∈ F)(5.71)
where the basis set is given by (2.16). Denoting
~
E= (H01, H02 , H03),~
B= (H23, H31 , H12)(5.72)
computing the determinant of Kwe obtain
det(K) = (s2+p2A2Q2)2+ 4(AaQa)2(s2+p2)2
+(p2s2+~
E2~
B2)2+ 4( ~
E·~
Bsp)24A2Q2+ ∆
(5.73)
in which A2=AaAa,Q2=QaQaand
∆ = 2( ~
E2+~
B2)(A2+Q2) + 4( ~
E2+~
B2)(A2
0+Q2
0)
4[( ~
A·~
E)2+ ( ~
A·~
B)2+ ( ~
Q·~
E)2+ ( ~
Q·~
B)2]
+8[A0(~
A×~
B)·~
EQ0(~
Q×~
E)·~
B)]
+8s[( ~
Q×~
A)·~
E+ (Q0~
AA0~
Q)·~
B]
+8p[( ~
Q×~
A)·~
B(Q0~
AA0~
Q)·~
E]
(5.74)
where ~
A= (A1, A2, A3), ~
Q= (Q1, Q2, Q3). From the result we find det(K)Fis independent of
±i in the basis (2.16), this means that the matrix representation (2.16) is independent of F, and
its entries can violate the constraint of Theorem 1.
In the domain {det(K)6= 0}, we can define the inversion of Kas the matrix K1. Since K1(s)
can be expressed as Taylor series of s1at s=,K1is also a Clifford number with basis matrices
(2.16). This means that the basis (2.16) is closed for all algebraic calculation of matrix. Thus, we
can generally define analytic functions and equations of Clifford numbers, e. g.
Hp=NeW+YeArsin(ωT)MAnJm
where (Hp,N,Y,···) are all Clifford numbers with coefficients in field F. Therefore, the Clifford
algebra is actually a special matrix algebra or hypercomplex number system with basis matrices
(2.16) or (2.25). If the covariant differential operator D=γµµis introduced, then we can discuss
differential equations such as
DY=λGexp(kT)Yn+Qsin(ωT)Jm.(5.75)
By Fourier or Laplace transformation in natural coordinate system[16], (5.75) can be converted
into algebraic equation. Nature works in this way.
23
Acknowledgments
I am very grateful to the Professor Ivailo M. Mladenov and Professor James M. Nester for their
encouragement and help, and this paper has been modified according to their suggestions.
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... The Grassmann products have clear geometrical meaning, but the Clifford product is isomorphic to matrix algebra and the multiplication of physical variables is Clifford product. Therefore, representing geometrical and physical variables in the form of (2.4) will bring great convenience [20,21]. In this case, the relations among three products such as (2.6)-(2.9) ...
... which is the most important relation in Clifford algebra. Since Clifford algebra is isomorphic to some matrix algebra, by (2.17) γ a is equivalent to some special matrices [20]. In practical calculation, we need not distinguish the vector basis from its representation matrix. ...
... 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 [20,31] Theorem 6 Assuming the matrices C µ satisfy anti-commutative relation of C 1,3 ...
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In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area and volume, and unifies the calculus of scalar, spinor, vector and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion and vector algebra, converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.
... Clifford algebra is also called geometric algebra. If the definition is directly related to geometric concepts, it will bring great convenience to the study and research of geometry [17]. ...
... Since the Clifford algebra is isomorphic to matrix algebra [15,16,17], we need not distinguish matrix γ a with tetrad γ a . Thus we have ...
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Clifford algebra is a unified language and efficient tool for geometry and physics. In this paper, we derive the integrable conditions for Dirac and Pauli equations by Clifford algebra, which shows Clifford algebra has a standard operation procedure and deep insights into the structure of the equations. Here the integrable conditions are only regarded as algebraic conditions. The commutators expressed by Clifford algebra have a neat and covariant form, which are naturally valid in curvilinear coordinate system and curved space-time. For Pauli and Schr\"odinger equation, many solutions in axisymmetric potential and magnetic field are also integrable. By the virtue of Clifford algebra, the physical researches may be greatly promoted.
... Clifford algebra is also called geometric algebra. If the definition is directly related to geometric concepts, it will bring great convenience to the study and research of geometry [30,31]. ...
... By straightforward calculation we have [6,30,31] Theorem 1 For C 1,3 , we have the following useful relations I, γ a , γ ab = i 2 abcd γ cd γ 5 , γ abc = i abcd γ d γ 5 , γ 0123 = −iγ 5 . ...
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The interaction between spinors and gravity is the most complicated and subtle interaction in the universe, which involves the basic problem to unified quantum theory and general relativity. By means of Clifford Algebra, a unified language and tool to describe the rules of nature, this paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection, the classical approximation of Dirac equation, the energy momentum tensor of spinors and so on. To split spinor connection into Keller connection Υ µ ∈ Λ 1 and pseudo-vector potential Ω µ ∈ Λ 3 by Clifford algebra not only makes the calculation simpler, but also highlights their different physical meanings. The representation of the new spinor connection is dependent only on the metric, but not on the Dirac matrix. Keller connection only corresponds to geometric calculations, but the potential Ω µ has dynamical effects, which couples with the spin of a spinor and may be the origin of the celestial magnetic field. Only in the new form of connection can we clearly define the classical concepts for the spinor field and then derive its complete classical dynamics, that is, Newton's second law of particles. To study the interaction between space-time and fermion, we need an explicit form of the energy-momentum tensor of spinor fields. However, the energy-momentum tensor is closely related to the tetrad, and the tetrad cannot be uniquely determined by the metric. This uncertainty increases the difficulty of deriving rigorous expression. In this paper, through a specific representation of tetrad, we derive the concrete energy-momentum tensor and its classical approximation. In the derivation of energy-momentum tensor, we obtain a spinor coefficient table S µν ab , which plays an important role in the interaction between spinor and gravity. From this paper we find that, Clifford algebra has irreplaceable advantages in the study of geometry and physics.
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In this chapter, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation, and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area, and volume and unifies the calculus of scalar, spinor, vector, and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion, and vector algebra and converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So, Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.
Article
Full-text available
By means of Clifford Algebra, a unified language and tool to describe the rules of nature, this paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection, the classical approximation of the Dirac equation, the energy-momentum tensor of spinors and so on. To split the spinor connection into the Keller connection Υμ∈Λ1 and the pseudo-vector potential Ωμ∈Λ3 not only makes the calculation simpler, but also highlights their different physical meanings. The representation of the new spinor connection is dependent only on the metric, but not on the Dirac matrix. Only in the new form of connection can we clearly define the classical concepts for the spinor field and then derive its complete classical dynamics, that is, Newton’s second law of particles. To study the interaction between space-time and fermion, we need an explicit form of the energy-momentum tensor of spinor fields; however, the energy-momentum tensor is closely related to the tetrad, and the tetrad cannot be uniquely determined by the metric. This uncertainty increases the difficulty of deriving rigorous expression. In this paper, through a specific representation of tetrad, we derive the concrete energy-momentum tensor and its classical approximation. In the derivation of energy-momentum tensor, we obtain a spinor coefficient table Sabμν, which plays an important role in the interaction between spinor and gravity. From this paper we find that Clifford algebra has irreplaceable advantages in the study of geometry and physics.
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The paper presents a novel approach to the cosmological constant problem by the use of the Clifford algebras of space Cl(3,0) and anti-space Cl(0,3) with a particular focus on the paravector representation, emphasizing the fact that both algebras have an isomorphic center represented just by two coordinates. Since the paravector representation allows assigning the scalar element of grade 0 to the time coordinate, we consider the relativity in such two-dimensional spacetime for a uniformly accelerated frame with the constant acceleration 3Hc. Using the Rindler coordinate transformations in two-dimensional spacetime and then applying it to Minkowski coordinates, we obtain the FLRW metric, which in the case of the Clifford algebra of space Cl(3,0) corresponds to the anti-de Sitter (AdS) flat (k=0) case, the negative cosmological term and an oscillating model of the universe. The approach with anti-Euclidean Clifford algebra Cl(0,3) leads to the de Sitter model with the positive cosmological term and the exact form of the scale factor used in modern cosmology.
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In this chapter, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation, and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area, and volume and unifies the calculus of scalar, spinor, vector, and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion, and vector algebra and converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So, Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.
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In this paper, we provide some enlightening applications of Clifford algebra in physics. Directly taking the generators of Clifford algebra as tetrad of space-time, we redefine some concepts of field and then discuss the dynamical equation and symmetry by Clifford calculus. Clifford algebra exactly reflects intrinsic symmetry of fields with no more or less contents, and automatically classifies the parameters in field equation by grade, which is a definite guidance to set up dynamical equation and compatible constraints of fields. By insights of Clifford algebra, we discuss the connection of fields and torsion in details. The dynamical equation of torsion, the quadratic form of Lagrangian of gravity and first order dynamical equation of tensors in curved space-time are derived. By virtue of this excellent language, physical theories can be well understood by common readers, and some long standing puzzling problems may be easily solved. {Keywords:} {Clifford algebra, space-time geometry, tetrad, absolute derivative, connection, torsion, curvature}
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According to a framework based on Clifford algebra Cℓ(1,3), this paper gives a classification for elementary fields, and then derives their dynamical equations and transformation laws in detail. These results provide an outline on elementary fields and some new insights into their unusual properties. All elementary fields exist in pairs, and one part of the pair is a complex field. Some intrinsic symmetries and constraints such as Lorentz gauge condition are automatically included in the canonical equation. Clifford algebra Cℓ(1,3) is a natural language to describe the world. In this language, the representation formalism of dynamical equation is symmetrical and elegant with no more or less contents. This paper is also a summary of some previous problem-oriented researches. Solutions to some simple equations are given.
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In these lectures, we discuss some well-known facts about Clifford algebras: matrix representations, Cartan's periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in n dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.
Article
Representations of geometric (Clifford) algebras with real square matrices are reviewed by providing the general theorem as well as examples of lowest dimensions. New definitions for isometry and norm are proposed. Direct and indirect isometries are identified respectively with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every element is defined as the nth-root of the absolute value of the determinant of its matrix representation of order n. It is deduced in which geometric algebras direct isometries are inner automorphisms (similarity transformations of matrices). Indirect isometries need reversion too. Finally, the most common isometries are reviewed in order to write them in this way.
Article
The presentation of the theory of Clifford numbers and spinors given here has grown out of thirteen lectures delivered at the Institute for Fluid Dynamics and Applied Mathematics of the University of Maryland. The work is divided into six chapters which, for the convenience of those readers who are only interested in certain parts of the material treated, are largely independent of each other. This arrangement has, of course, certain disadvantages such as repetitions and over-lappings.
Article
A historical review of spinors is given together with a construction of spinor spaces as minimal left ideals of Clifford algebras. Spinor spaces of euclidean spaces over reals have a natural linear structure over reals, complex numbers or quaternions. Clifford algebras have involutions which induce bilinear forms or scalar products on spinor spaces. The automorphism groups of these scalar products of spinors are determined and also classified.