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Space-Time Geometry and Some Applications of

Cliﬀord Algebra in Physics

Ying-Qiu Gu∗

School of Mathematical Science, Fudan University, Shanghai 200433, China

(Dated: 1st August 2018)

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 faithfully and exactly reﬂects intrinsic symmetry of space-time and 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 diﬀerential 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 gravity 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; gravity

I. INTRODUCTION

Cliﬀord algebra sometimes is also called space-time algebra or geometrical algebra. It naturally

combines geometrical concepts with algebraic ones, and endows them with simple and concise

algorithm[1]-[23]. In [24], we show that the following framework based on Cliﬀord algebra give a

uniﬁed view on physical ﬁelds:

A1. The element of space-time is described by

dx=γµdxµ=γµdxµ=γaδX a=γaδXa.(1.1)

in which γaand γµsatisfy the following C`(1,3) Cliﬀord algebra

γaγb+γbγa= 2ηab, γµγν+γνγµ= 2gµν .(1.2)

∗Electronic address: yqgu@fudan.edu.cn

2

A2. The dynamics for a deﬁnite physical system has the following canonical form

DΨ = F(Ψ), D ≡γµ∂µ,(1.3)

in which Ψ=(ψ1, ψ2,·· · , ψn)T, and F(Ψ) consists of some tensorial products of Ψ, so that the

total equation is covariant.

A3. Nature is consistent, i.e., for all solutions to (1.3)we have ∀ψk∈L∞.

We have seen the explaining power of the framework and the eﬀectiveness of Cliﬀord algebra in

[24]. Now we solve some other remaining problems such as Cliﬀord structure of equation, aﬃne

connection, torsion and so on.

At ﬁrst, we recall some concepts and relations of Cliﬀord algebra. In curved space-time, we

deﬁne the frame as

γµ=hµ

aγa, γµ=la

µγa.(1.4)

In which the frame coeﬃcients hµ

aand la

µare algebraically determined by

hµ

ahνbηab =gµν, l a

µlb

νηab =gµν .(1.5)

Cliﬀord algebra endows the space-time with invariant distance ds and directional area and volumes

which deﬁnes the ndimensional Minkowski space-time Mp,q ,

ds2=dx2=gµν dxµdxν=ηabδXaδXb.(1.6)

dx∧dy=γµ∧γνdxµdyν,

dx∧dy∧dz=γµ∧γν∧γωdxµdyνdzω,···

(1.7)

We can also deﬁned covariant diﬀerential operators

D=γµ∇µ.(1.8)

Cliﬀord algebra is graded, we have

I∈Λ0, γa∈Λ1, γab =γa∧γb∈Λ2, γabc =γa∧γb∧γc∈Λ3,··· (1.9)

In all these deﬁnitions, the frame γµcarry all geometrical information of the space-time, but

dxµ, dyµare only real numbers stand for length elements of tangents. Cliﬀord algebra deﬁnes

the calculus of γµaccording to (1.2) and (1.9) which reﬂect intrinsic geometrical relations of the

space-time. Since all γµhave simple representation of matrix, geometry in Mp,q can be converted

into clear algebraic operations, and then it can be easily understood.

3

In this paper, we mainly discuss problems and applications in 1+3 dimensional Minkowski

space-time. Denote the Minkowski metric by (ηab) = diag(1,−1,−1,−1), the Pauli matrices by

σa≡

1 0

0 1

,

0 1

1 0

,

0−i

i0

,

1 0

0−1

,(1.10)

σ0=eσ0=I, eσk=−σk,(k= 1,2,3).(1.11)

We use Greek characters to stand for curved space-time indices, which go from 0 to 3, Latin

characters to stand for Minkowski indices and k, l, j for spatial indices. Deﬁne γaby

γa=

0e

ϑa

ϑa0

, ϑa= diag(

n

z }| {

σa, σa,·· · , σa),e

ϑa= diag(

n

z }| {

eσa,eσa,··· ,eσa).(1.12)

which forms a grade-1 basis of Cliﬀord algebra C`(1,3) satisfying (1.2). In equivalent sense, the

representation (1.12) is unique. By γ-matrix (1.12), we have the complete bases of C`(1,3) as

follows,

I, γa, γab =i

2²abcdγcd γ5, γabc =i²abcdγdγ5, γ5=iγ0123 = diag(I , −I).(1.13)

The above γaand γaformally constitute the orthogonal bases of Minkowski space-time, but

they have a property diﬀerent from the usual frame eaand coframe eadeﬁned in geometry. γaare

constant matrices, and the direction of γain curved space-time is implicitly deﬁned by coeﬃcients

(1.5). Under any proper Lorentz transformation, γaall keep invariant formally, but the vectors

such as coordinates all have been transformed. Such feature makes algebraic calculations very

convenient, but the diﬀerential calculus should be realized via connection. Although in diﬀerential

geometry absolute diﬀerential is also realized via connection, but the explanation is a little diﬀerent

from the opinion of physics. We will discuss it in the next section.

The deﬁnition of vector, tensor and spinor in diﬀerential geometry involving a number of reﬁned

concepts such as vector bundle and dual bundle, which are too complicated for readers in other

specialty. Here we inherit the traditional deﬁnitions based on the bases γaand γµsimilar to (1.1).

For vector Aat any point x, it is directly described by

A(x) = γaAa=γaAa=γµAµ=γµAµ,(1.14)

where Aa(xα) and Aµ(xα) are coordinates of the vector, which are only numbers. The geometrical

and algebraic properties of Aare realized via bases γa.

The inner product of two vectors is deﬁned as

dx·dy=γµ·γνdxµdyν=γµ·γνdxµdyν=γa·γbδXaδY b=γa·γbδXaδYb

=gµν dxµdyν=gµν dxµdyν=ηabδXaδY b=ηabδXaδYb=dxµdyµ,(1.15)

4

in which γµ·γν=1

2(γµγν+γνγµ) = gµν . The exterior product is deﬁned by

dx∧dy=γµ∧γνdxµdyν=1

2(γµγν−γνγµ)dxµdyν=γµν dxµdyν.(1.16)

The exterior products can deﬁne continued multiplication. Assuming A∈Λpand B∈Λq, we have

A∧B= (−1)pqB∧A.(1.17)

The geometric product or Cliﬀord product is deﬁned as

AB =γaγbAaBb= (ηab +γab)AaBb=A·B+A∧B.(1.18)

Obviously, for geometric product, we have grade-ntimes grade-1 leads to grade-(n±1) algebra.

For example, we have

γαγµν =gαµγν−gαν γµ+γαµν , γµν γα=gαν γµ−gαµγν+γαµν .(1.19)

By Cliﬀord algebra, the geometrical concepts such as length and volume can be converted into

algebra calculus.

In physics, basis of tensors is deﬁned by direct products of grade-1 bases γµ. For metric we

have

g=gµν γµ⊗γν=gµν γµ⊗γν=δν

µγµ⊗γν

=ηabγa⊗γb=ηab γa⊗γb=δb

aγa⊗γb.(1.20)

For simplicity we denote tensor basis by

⊗γµ1µ2···µn=γµ1⊗γµ2⊗ · ·· γµn,⊗γµ2µ3···µn

µ1=γµ1⊗γµ2⊗ · ·· γµn,··· (1.21)

In general, a tensor of rank nis given by

T=Tµ1µ2···µn⊗γµ1µ2···µn=Tµ1

µ2···µn⊗γµ2µ3···µn

µ1=··· (1.22)

The geometrical messages of the tensor such as transformation law and diﬀerential connection are

all recorded by tetrad bases, and all representations of rank (r, s) tensor denote the same one

practical entity T(x). Tν···

µ··· is just a quantity table, but the physical and geometrical meanings of

the tensor Tare represented by basis γµ. Cliﬀord algebra is a special kind of tensor with exterior

product. Its algebraic calculus exactly reﬂects the intrinsic property of space-time and makes

physical calculation simple and clear.

5

In diﬀerential geometry, we use the natural bases (dxµ, ∂µ). Usually they are clear, but some-

times they are confusing. For example, dxµ=γµ·dxalways denotes increment of coordinate

along tangent, which is a variable quantity. But γµis an algebraic operator. gµν ≡γµ·γνis

natural, but the expression dxµ·dxνor dxµ⊗dxνlooks strange and contradicts the convention

ds2=gµν dxµdxν. The following discussion shows the above deﬁnitions and treatments make the

corresponding subtle and fallible concepts in diﬀerential geometry much simpler.

The relations between tetrad matrices and metric, i.e., the relations la

µ=F(gµν , L b

a), hµ

a=

H(gµν , L b

a) are given by the following theorems[25], which can be determined to an arbitrary

Lorentz transformation Lb

a. Here γαis just matrix, which can not be regarded as tetrad when

make the following diﬀerential operation. The absolute derivatives of tetrad involving the concept

of connection will be discussed in the next section. This is a subtle problem to which careful

attention should be paid.

Theorem 1 Deﬁne a spinor coeﬃcient tensor by

Sµν

ab ≡1

2(hµ

ahνb+hνahµ

b)sgn(a−b) = −h{µ

[ahν}

b],(1.23)

for any solution of tetrad la

µand hµ

a, we have

∂l n

α

∂gµν

=1

4(δµ

αhνm+δν

αhµ

m)ηnm +1

2Sµν

ab la

αηnb.(1.24)

∂hα

a

∂gµν

=−1

4(hµ

agαν +hνagµα)−1

2Sµν

ab hα

nηnb.(1.25)

Or equivalently,

δγα=1

2γβ(δgαβ +Sµν

ab la

αlb

βδgµν ),(1.26)

δγλ=−1

2gλβγα(δgαβ +Sµν

ab la

αlb

βδgµν ) = −gλα δγα.(1.27)

δγαonly involves the length change of tetrad, but the absolute derivatives also includes direction

change of tetrad. Sµν

ab is symmetry for indices (µ, ν) but antisymmetry for indices (a, b). It appears

in dynamical equation and energy momentum tensor of spinor[25,26]. Its property is quite strange

and unusual.

The materials are organized as follows. In the next section we deﬁne the aﬃne connection,

absolute diﬀerentials for ﬁelds and then derive their dynamics in curved space-time according to

Cliﬀord algebra. In section III, we discuss gravity and torsion. In the last section, we give some

discussion and a summary.

6

II. CONNECTION AND COVARIANT DERIVATIVE OF FIELDS

The algebraic calculus of bases is deﬁned at a ﬁxed point in the manifold. But the diﬀerential

operator has to compare values of variables at diﬀerent points, which involves the local trans-

formation of bases γµ. For example, in Minkowski space-time with spherical coordinate system

(t, r, θ, ϕ), we have tetrad coeﬃcients and elements of coordinates[26,27]

(hµ

a) = diag µ1,1,1

r,1

rsin θ¶,(la

µ) = diag (1,1, r, r sin θ).(2.1)

∂a=µ∂t, ∂r,1

r∂θ,1

rsin θ∂ϕ¶, δX a= (dt, dr, rdθ, r sin θdϕ).(2.2)

The direction of the tetrad basis turns into: γ0along dt,γ1along dr,γ2along dθ, and γ3along

dϕ direction. The directions of γ1, γ2and γ3are all varying with the coordinates (θ, ϕ), but these

γamatrices are still constant Dirac matrices. This has not inﬂuences on the algebraic calculation

at ﬁx point, but for derivatives, we should introduce aﬃne connection to describe the change of

direction of tetrad.

At ﬁrst, we examine the absolute diﬀerential of vector ﬁeld A, we have

dA≡lim

∆x→0[A(x+ ∆x)−A(x)]

= (∂αAµγµ+Aµdαγµ)dxα= (∂αAµγµ+Aµdαγµ)dxα.(2.3)

We call dαconnection operator. According to its geometrical meanings, connection operator should

satisfy the following conditions: 1◦it is a real linear transformation of basis γµ, 2◦it satisﬁes metric

consistent condition dg= 0. Then the diﬀerential connection can be generally expressed as

dαγµ=−(Πµ

αβ +Tµ

αβ)γβ,Πµ

αβ = Πµ

βα,Tµ

αβ =−T µ

βα.(2.4)

The split of symmetrical part and anti-symmetrical part has important geometrical meanings,

because Πµ

αβ is determined geometrically by metric, but Tµ

αβ should be determined by dynamical

equation of ﬁeld.

For metric g=gµν γµ⊗γν, by metric consistent condition we have

0 = dg=d(gµν γµ⊗γν)

= [(∂αgµν )γµ⊗γν+gµν (dαγµ)⊗γν+gµν γµ⊗dαγν]dxα

= [(∂αgµν −gνβ Πβ

αµ −gµβΠβ

αν )−(gνβ Tβ

αµ +gµβTβ

αν )]γµ⊗γνdxα.(2.5)

Solving (2.5) we get the usual Christoﬀel symbols and torsion respectively

Πα

µν =1

2gαβ(∂µgβν +∂νgµβ −∂βgµν ) = Γα

µν ,(2.6)

gνβ Tβ

αµ =gµβTβ

να .(2.7)

7

Γα

µν is called Levi-Civita connection, and Tβ

αµ torsion.

Substituting (2.4) into

0 = dg=δµ

ν[(dαγµ)⊗γν+γµ⊗dαγν]dxα,(2.8)

we get

dαγµ= Γβ

αµγβ+Tβ

αµγβ,(2.9)

Clearly, the basis γµhas diﬀerent geometrical meaning from the coordinate increment dxα=γα·dx.

Deﬁne two 1-forms by

dγλ=dαγλdxα−δγλ,

θλ

β=−1

2gλκ(∂κgαβ −∂βgακ +Sµν

ab la

βlb

κ∂αgµν )dxα,

(2.10)

in which δγλdeﬁned by (1.27) is only the length change of tetrad, but dγλdeﬁnes the rotational

change of tetrad. By (1.27), (2.4), (2.10) and calculating the exterior diﬀerential of θλ

β, we get

concrete Cartan’s structure equation in this tetrad as follows[28,29],

dγλ−γβ∧θλ

β=Tλ

αβdxα∧γβ,(2.11)

dθλ

β−θα

β∧θλ

α=Kλ

βµν dxµ∧dxν,(2.12)

where Kλ

βµν stands for curvature.

In [29], γµis used to simplify the diﬀerential forms, where only its algebraic property is used

but its geometrical meaning is ignored. From the above analysis we found γµmatrix itself can be

naturally used as tetrad. In [30], some concepts for Cliﬀord-valued diﬀerential forms are deﬁned in

language of diﬀerential geometry, but they are too complicated for readers of other specialty. The

above system is more accessible for physicists, and it is impossible that the above deﬁnitions are

insuﬃcient to describe physical theories. The pseudo-Riemannian geometry rewritten by Cliﬀord

algebra may be very simple and clear.

By (2.7) we ﬁnd torsion Tµνω ≡gµβ Tβ

νω is a skew-symmetrical tensor, which satisﬁes

Tµνα =Tαµν =Tν αµ =−Tµαν =−Tνµα =−Tανµ.(2.13)

Tµνα is equivalent to a pseudo vector in C`(1,3). This characteristic can be clariﬁed in spinor

equation as shown below. It can be also checked by Cliﬀord algebra calculus

T=Tµνω γµνω =Tabc γabc =Tabc²abcdγd(iγ5)≡iγdγ5Td=iγαγ5Tα,(2.14)

8

in which

Tα=hα

dTabc²abcd =Tµν ωhµ

ahνbhω

chα

d²abcd =1

√g²µνωαTµνω .(2.15)

So we get

Tµνω =√g²µνωα Tα,Tµνω Tω= 0,Tα

µν Tν= 0.(2.16)

Denote the absolute diﬀerential and usual covariant derivative respectively as usual,

dA=∇αAµγµdxα, Aµ

;α=∂αAµ+ Γµ

αβAβ.(2.17)

Aµ

;αis the common covariant derivative without torsion. By (2.9) and (2.17) we can easily calculate

the absolute diﬀerentials for vectors and tensors. For vector Awe have

dA= [(∂αAµ)γµ+Aµdαγµ]dxα= (Aβ

;α+AµTβ

αµ)γβdxα

= [(∂αAµ)γµ+Aµdαγµ]dxα= (Aβ;α−AµTµ

αβ)γβdxα.(2.18)

we get

∇αAµ=Aµ

;α+Tµ

αβAβ,∇αAµ=Aµ;α− T β

αµAβ.(2.19)

Especially, by (2.16) we get the derivatives for torsion

∇αTµ=Tµ

;α+Tµ

αβTβ=Tµ

;α.(2.20)

The absolute derivatives of torsion are usual covariant derivatives.

Now we examine the connection in orthogonal basis γa, by γµ=la

µγawe have

dA= [∂α(Aµla

µ)γa+Aadαγa]dxα= (Aβ

;α+AµTβ

αµ)la

βγadxα.(2.21)

Notice the arbitrary of Aµ, we get equation for connection dαγaas

dαγa=−hµ

a[∂αγµ−(Γν

αµ +Tν

αµ)γν] = −hµ

a∇αγµ,(2.22)

in which ∇αγµjust a notation without special geometrical meanings. By (2.22) we get

∇αγµ=−la

µdαγa.(2.23)

Similarly we have

dαγa=−la

µ[∂αγµ+ (Γµ

αν +Tµ

αν )γν]

=−la

µ∇αγµ=−gµν la

µ∇αγν=ηabdαγb.(2.24)

9

By (2.24) we ﬁnd dαγais similar to a Lorentz vector.

(2.18) and (2.21) clearly show how the connections convert the derivatives of tetrad into the

absolute derivatives of ﬁelds. For algebraic calculus, the basis γµand matrix γµare identical. As for

diﬀerential operation, we can only deﬁne connection dαfor basis γµor γa, which is equivalent to a

linear transformation of basis. When we calculate the absolute derivatives of ﬁelds, we simply take

metric gµν and tetrad γµas constants, because their variation has been included in the absolute

derivatives of ﬁelds.

We take Maxwell equation system as example to show the relations. By absolute derivatives

and Cliﬀord algebra, we can derive the dynamical equations for a vector Aαas follows. We omit

torsion for simplicity. Denote D=γα∇αand A=γβAβ, we have

DA =γαγβ∇αAβ= (gαβ +γαβ)∇αAβ

=∇µAµ+1

2γµν (∂µAν−∂νAµ)≡H+1

2γµν Fµν .(2.25)

Clearly DA ∈Λ0∪Λ2⊂C`(1,3). By relation (1.19) we get

D2A=γα∂αH+1

2γαγµν ∇αFµν

=γα∂αH+gαµγν∇αFµν +1

2γαµν ∇αFµν

=γα(∂αH− ∇µFαµ) + iγωγ5

2√g(²αµνω ∇αFµν ).(2.26)

Therefore D2A∈Λ1∪Λ3, and then D3A∈Λ0∪Λ2∪Λ4. If the γαµν term does not vanish in

D2A, the dynamical equation should be closed at least in D4A. However, all ﬁeld equations in

physics are closed in D2[24,31]. That is to say, we have

D2A=−b2A+eq=γα(−b2Aα+eqα).(2.27)

In contrast the above equation with (2.26), we get the ﬁrst order torsion-free dynamical equation

for vector ﬁeld in curved space-time independent of γα,

∂µAν−∂νAµ=Fµν ,∇αAα=H

∂µH− ∇νFµν +b2Aµ=eqµ, ²αµνω∇αFµν = 0.

(2.28)

In which all variables are covariant ones with clear physical meanings. If Lorentz gauge condition

H= 0 holds, for long distance interaction b= 0, (2.28) gives the complete Maxwell equation system

in curved space-time with current conservation qµ

;µ= 0. The last equation is a kind of integrable

condition. The Lagrangian of Ais given by

LA=−1

2(∇αAβ∇αAβ−b2AαAα)−eqαAα.(2.29)

10

The above derivation shows how Cliﬀord algebra exactly reﬂects the intrinsic properties of

space-time and ﬁelds with no more or less contents. In this natural language, we derive complicated

physical equations and relations similarly to making arithmetic calculations. In what follows we

examine the absolute diﬀerentials of spinors. Although this problem has been solved in previous

literatures[30,32,33,34,35], we gain new insights through restudying old material, and also

investigate the interaction between spinor and torsion by the way. We start from Weyl spinors for

simplicity and clearness.

Theorem 2 For Weyl spinors (ψ, e

ψ), we have spinor connection as follows

Γµ=1

4e%ν%ν

;µ−1

8Tµνω (e%ν%ω−e%ω%ν) + iλµ,(2.30)

e

Γµ=1

4%νe%ν

;µ−1

8Tµνω (%νe%ω−%ωe%ν) + ie

λµ,(2.31)

where %α

;µ=∂µ%α+ Γα

µβ%βand e%α

;µ=∂µe%α+ Γα

µβ e%βare usual covariant derivatives for vector. λµ

is an arbitrary real ﬁelds. For bispinor φ= (ψ, e

ψ), we have e

λµ=λµand

∇µφ= (∂µ+¯

Γµ)φ, ¯

Γµ=1

4eγαeγα

;µ−1

4Tµνω γνω +iλµ.(2.32)

Proof. Since qµ=ψ+%µψis a contra-variant vector under general coordinate transformation,

we have the covariant derivative in the case of presence of torsion Tµ

νω

∇νqµ=∂νqµ+ (Γµ

νω +Tµ

νω )qω,

= (∂νψ)+%µψ+ψ+%µ(∂νψ) + ψ+%µ

;νψ+Tµ

νω qω,(2.33)

in which %µ

;ν=∂ν%µ+ Γµ

αν %αis just a convenient notation for matrix. On the other hand, we have

the deﬁnition of absolute derivative for spinor compatible with tetrad

∇νqµ= (∇νψ)+%µψ+ψ+%µ(∇νψ),

= (∂νψ)+%µψ+ψ+%µ(∂νψ) + ψ+(Γ+

ν%µ+%µΓν)ψ,

(2.34)

in which spinor connection Γνstands for the connection dνγµin spinor case. That is to say, the

changes of tetrad and the transformation of spinor have been implicitly merged into Γν. Similarly

to the case of tensors, the tetrad γµcan be regarded as constants when calculating covariant

derivatives of spinor. However, this treatment does not means we really have ∇α%µ≡0 or ∇αγµ≡0

as used in some previous literatures.

In contrast (2.33) with (2.34), and noticing the arbitrary ψ, we get

Γ+

ν%µ+%µΓν=%µ

;ν+Tµ

νω %ω.(2.35)

11

For any 2 ×2 matrix A, It is easy to check

e%µA%µ= 2tr(A),e%µ%µ= 4.(2.36)

e%µleft multiplies (2.35), we have

2tr(Γν)++ 4Γν=e%µ%µ

:ν+1

2Tµνω (e%µ%ω−e%ω%µ).(2.37)

Taking trace of (2.37), we have

4[tr(Γν)++ tr(Γν)] = tr(e%µ%µ

:ν) = tr(gµα e%α%µ

;ν) = gµαhα

ahµ

b;νηab =1

2gµαgµα

;ν= 0.(2.38)

The solution is given by tr(Γν) = i2λν, where ﬁeld λνis real and arbitrary. Then we solve spinor

connection from (2.37) as

Γµ=1

4e%ν%ν

;µ−1

8Tµνω (e%ν%ω−e%ω%ν) + iλµ.(2.39)

Similarly we have

e

Γµ=1

4%νe%ν

;µ−1

8Tµνω (%νe%ω−%ωe%ν) + ie

λµ.(2.40)

For bispinor φ= (ψ, e

ψ), ψ+e

ψis a mixed scalar, so we have

∂ν(ψ+e

ψ) = (∇νψ)+e

ψ+ψ+(∇νe

ψ),

= (∂νψ)+e

ψ+ψ+(∂νe

ψ) + 1

4ψ+¡(e%µ%µ

;ν)++%µe%µ

;ν¢e

ψ+i(e

λν−λν)ψ+e

ψ

=∂ν(ψ+e

ψ) + 1

4gµαψ+¡%α

;νe%µ+%αe%µ

;ν¢e

ψ+i(e

λν−λν)ψ+e

ψ

=∂ν(ψ+e

ψ) + 1

4gµαgµα

;νψ+e

ψ+i(e

λν−λν)ψ+e

ψ. (2.41)

By gµν

;ν= 0 we get e

λν=λν. Therefore for Dirac bispinor φ= (ψ, e

ψ)T, we have

∇µφ=∂µφ+iλµφ+1

4

e%ν%ν;µ0

0%νe%ν;µ

ψ

e

ψ

−1

4Tµνω

e%ν%ω0

0%νe%ω

ψ

e

ψ

=∂µφ+iλµφ+1

4

0e%ν

%ν0

0e%ν

;µ

%ν

;µ0

φ−1

4Tµνω

0e%ν

%ν0

0e%ω

%ω0

φ

= (∂µ+1

4γνγν

;µ−1

4Tµνω γνω +iλµ)φ. (2.42)

The proof is ﬁnished.

This theorem shows how the change of tetrad is converted into spinor connection. For covariant

derivatives of spinor, the tetrad can be regarded as constant. The arbitrary ﬁeld λµreﬂects the

12

gravity is compatible with vector potentials such as electromagnetic potential eAµ. There are some

other expression of spinor connection such as ωab

αγab. But detailed calculation shows this form is

more complicated than (2.32). In the form (2.32), some derivations similar to (2.38) become

simpler.

By (2.42), we get Dirac equation in curved space-time as[26,36]

γµi(∂µ+1

4γνγν

;µ−1

4Tµνω γνω +iλµ)φ=mφ. (2.43)

By (1.19) and Cliﬀord algebra, we ﬁnd the coeﬃcient operators of Dirac equation belong to Λ1∪Λ3.

Thus we have

γµ[i(∂µ+ Υµ)−eAµ+1

2γ5(Ωµ+1

2Tµ)]φ=mφ, (2.44)

in which the parameters are redeﬁned by

Υµ≡1

2(la

µ∂νhν

a+ Γν

µν ) = 1

2hν

a(∂µla

ν−∂νla

µ),Ωα=1

4²dabchα

dhβ

aSµν

bc ∂βgµν .(2.45)

(2.44) can be rewritten in Hermitian form,

αµˆpµφ+Sµ(Ωµ+1

2Tµ)φ=mγ0φ, (2.46)

in which αµis current operator, ˆpµmomentum and Sµspin. They are respectively deﬁned by

αµ= diag(%µ,e%µ),ˆpµ=i(∂µ+ Υµ)−eAµ, Sµ=1

2diag(%µ,−e%µ).(2.47)

Only in this representation we can properly deﬁne the classical concepts such as momentum and

spin of a spinor, because momentum pµ∈Λ1, but the operator i∇αincludes components of Λ3.

By Cliﬀord algebra we learn Υµ∈Λ1has diﬀerent physical meanings from that of Ωµ∈Λ3[25,26].

(∂µ+ Υµ)∈Λ1becomes a closed operator in logic for spinor. We ﬁnd the spin Sµ∈Λ3is also

diﬀerent from the angular momentum ~r ×~p ∈Λ2, although we have Lie algebra ~

S×~

S=i

2~

S.

The spin-gravity coupling term SµΩµmay be the origin of magnetic ﬁeld of celestial body. Ωµ

is proportional to angular momentum of gravitational source. For example, in Kerr-type metric

for a rotational ball

ds2=gttdt2+grr dr2+gθθ dθ2+ 2J dtdϕ +gϕϕdϕ2,(2.48)

in which gµν =gµν(r, θ), J =J(r, θ), but is independent of (t, ϕ). For Kerr metric J→2

rmω sin2θ.

By (2.45), we have

Ωµ=f(0, ∂θJ, −∂rJ, 0) →f1mω(0,2rcos θ, sin θ, 0),(2.49)

13

in which (f, f1) are two functions irrelevant to the following discussion. The ﬂux line

dxµ

ds = Ωµ→dr

dθ =2rcos θ

sin θ↔r=Rsin2θ(2.50)

is similar to that of a magnetic ﬁeld ~

Bgenerated by a magnetic dipole. The spin of particles will

be arranged along these lines and a macroscopic magnetic ﬁeld is induced.

III. LAGRANGIAN FOR GRAVITY AND TORSION

Now we derive the Lagrangian of gravity by Cliﬀord calculus. Since the Dirac bispinor can

automatically specify the geometrical meanings of tetrad, and the coeﬃcient of Dirac equation is

a Cliﬀord algebra itself. We examine the second order Dirac equation. Since torsion Tαtakes the

place of Ωαin Dirac equation (2.46), we omit Tαin the following derivation. When we consider

torsion, it can be simply employed by replacing Ωµby Ωµ+1

2Tµ.

In the case of Aµ=Tµ= 0, multiplying (2.44) by γµ[i(∂µ+ Υµ) + 1

2γ5Ωµ] we have

m2φ=−gµν µ∂µν + 2Υµ∂ν+ ΥµΥν+1

4ΩµΩν¶φ−γµ(∂µγν)∂νφ+

γµi∂µµγν(iΥν+1

2γ5Ων)¶φ−iγµν γ5Ωµ(∂ν+ Υν)φ. (3.1)

In (3.1) the calculation is just common diﬀerential operation irrelevant to the abstract derivatives.

So the γµshould be treated as matrix when making diﬀerential calculus. We have 3 kinds of

quantities and operators in second order Dirac equation belonging to {Λ0,Λ2,Λ4}respectively. In

(3.1), the ﬁrst term corresponds to a scalar, the last one to a bivector. However, they are still

incomplete terms in logic.

To get logically closed results, we simplify the following terms,

Y1=i

2γµγ5∂µ(γνΩν) = i

2(ηab +γab)γ5hµ

a∂µ(hνbΩν),(3.2)

Y2=−γµ(∂µγν)∂ν=−(ηab +γab)hµ

a∂µ(hνb)∂ν,(3.3)

Y3=−γµ∂µ(γνΥν) = −(ηab +γab)hµ

a∂µ(hνbΥν).(3.4)

In (3.2) the simplest and closed term is pseudo scalar P=ηabhµ

a∂µ(hνbΩν)∈Λ4. By (2.45) we

have

P=hµ

a∂µ(la

νΩν) = ∂µΩµ+ Ωνhµ

a∂µla

ν

=∂µΩµ+ Γν

µν Ωµ−2ΥµΩµ= Ωµ

;µ+ ΓµΩµ,(3.5)

14

where and hereafter Γµ=−2Υµ. Similarly, in (3.4) and (3.3) we have

ηabhµ

a∂µ(hνbΥν) = Υµ

;µ−2ΥµΥµ= Υµ

;µ+ ΓµΥµ.(3.6)

ηabhµ

a∂µ(hνb∂ν) = (∂µgµν +gανΓµ

µα −2Υν)∂ν

= (−gαβΓµ

αβ + Γµ)∂µ.(3.7)

Substituting (3.5), (3.6) and (3.7) into (3.1) we get second order Dirac equation as follows

µ−2+1

2Lg+i

2γ5P+γabYab ¶φ=m2φ, (3.8)

in which 2=gµν(∂µν −Γα

µν ∂α) is d’Alembertian for scalar ﬁeld which is a closed scalar operator,

Yab ∈Λ2is a bivectorial operator with complicated form in which (∂µ+ Υµ) is a closed operator,

Lg∈Λ0is an important scalar deﬁning the Lagrangian of gravity

Lg= Γµ

;µ+1

2(ΓµΓµ−ΩµΩµ) = 1

2R.(3.9)

Since Γµ

;µwill vanish in action due to integration, we get the equivalent Lagrangian of gravity with

quadratic form of ﬁrst order derivatives

Lg=1

2(ΓµΓµ−ΩµΩµ).(3.10)

Since the time component of Ωµalmost vanishes Ω0→0, so Lgis positive deﬁnite. This form is

not an exact scalar now, but it is much helpful to discuss the existence and regularity of solution

to Einstein’s ﬁeld equation.

In [37], a similar treatment was discussed as follows. A quadratic spinor action is deﬁned as

S[Ψ] = ZL=Z2D¯

Ψγ5DΨ(3.11)

where the gravitational variable is a spinor-valued 1-form ﬁeld Ψ = ϑψ, which includes an or-

thonormal frame 1-form ϑ≡γαϑαand a normalized spinor ﬁeld ψ. The covariant derivative

DΨ≡dΨ + ωΨ includes the matrix valued connection one-form ω≡1

4γαβωαβ. The new spinor-

curvature identity

2D¯

Ψγ5DΨ = −¯

ψψR ∗ I+d[(D¯

Ψ)γ5Ψ + ¯

Ψγ5(DΨ)] (3.12)

reveals that the quadratic-spinor Lagrangian diﬀers from the standard Einstein-Hilbert Lagrangian

only by a total diﬀerential term. Hence they yield the same ﬁeld equations. However, the new

quadratic spinor Lagrangian is asymptotically O(r−4) which guarantees ﬁnite action, an advantage

over the Einstein-Hilbert O(r−3).

15

In second order Dirac equation (3.8), the term iγ5Pis anti-Hermitian, Since Ωµis a ﬁeld similar

to magnetic ﬁeld ~

B, the pseudo scalar P= Ωµ

;µ+ ΓµΩµmay vanish like ∇ · ~

B= 0, so P= 0 may

be provable.

Now we examine Λ2terms in (3.1). The term include (∂µ+ Υµ) is a closed term, so we

do not consider it. We only need to combine the like terms in (3.2) and (3.4). Substituting

γab =1

2i²abcdγcd γ5into (3.4), we get the following γcd terms,

Y1+Y2+Y3=i

2ηacηbd γcdγ5hµ

a∂µ(hνbΩν)−i

2²abcdγcd γ5hµ

ahνb∂µΥν+ηabK0

ab.(3.13)

Collecting like terms, we get a new anti-symmetrical tensor

Ycd =²abcdhµ

ahνb∂µΓν+ηacηbd[hµ

a∂µ(hνbΩν)−hµ

b∂µ(hνaΩν)].(3.14)

Yαβ =hα

chβ

dYcd =−Yβα seems to be the Weyl tensor. Similarly, we can derive high order

variables and relations by analyzing high order Dirac equation.

Now we discuss torsion Tµ. As a vector, calculating DTand D2Tsimilarly to (2.25) and (2.26),

we get Maxwell-like dynamics for components Tαsimilar to (2.28), and the Lagrangian of torsion

is similar to (2.29),

LT=1

2τ¡gµν Tα;µTα

;ν−k2TαTα¢.(3.15)

The source of torsion should be determined by coupling system with other ﬁelds. According to the

above conclusion and [24], we have Lagrangian of all ﬁelds as

L=1

κLg+Lm+LT

=1

2κ[ΓµΓµ−(Ωµ+1

2Tµ)(Ωµ+1

2Tµ)−2Λ] + Lm+LT,(3.16)

in which

Lm=

N

X

k=1 µφ+

k[αµˆpµ

k+Sµ(Ωµ+1

2Tµ)]φk−µkcˇγk+Nk+1

2wˇαµ,k ˇαµ

k¶−1

2wQµQµ

−1

2∇µAν∇µAν+1

2(∇µΦν∇µΦν−b2ΦµΦµ), Qµ=

N

X

k=1

ˇαµ

k.(3.17)

For vectors the absolute derivative ∇µis given by (2.19). Variation with respect to Tαwe get

gµν Tα

;µ;ν+k2Tα=1

2τX

k

ˇ

Sα

k+τ√ggαβ²βµνω(Aω∇µAν−Φω∇µΦν)−τ

4κ(4Ωα+Tα).(3.18)

We ﬁnd Tµsatisﬁes a linear equation. The consistence of torsion with non-Abelian gauge ﬁeld is

discussed in [38]. The source terms in (3.18) such as spin of fermions ˇ

Sµ

kare unstable variables, so

Tαmay be absent in Nature.

16

IV. THE SPINOR-LIKE DYNAMICS WITH TETRAD

The above discussion shows Cliﬀord algebra is a very convenient algorithm for spinor and vector

ﬁelds. But how to generalize it to calculate high rank tensors is a diﬃcult problem, because the

tensor algebra and Cliﬀord algebra are two diﬀerent logical system which cannot couple directly.

In [24,31] we derived the canonical formalism of dynamics for all ﬁelds in Minkowski space-time,

which shows the intrinsic symmetries and relations of ﬁelds. In what follows, we derive the spinor-

like dynamics for vector and metric tensor in curved space-time by Cliﬀord algebra calculus. We

only consider the torsion-free cases for simplicity.

Denote the constant quaternion-like matrices βaby β0=I[24,31],

β1=

0 1 0 0

1 0 0 0

0 0 0 i

0 0 −i0

, β2=

0 0 1 0

0 0 0 −i

1 0 0 0

0i0 0

, β3=

0 0 0 1

0 0 i0

0−i0 0

1 0 0 0

,(4.1)

and its conjugation e

β0=β0,e

βk=−βk. For contravariant real vector A= (A0, A1, A2, A3)Tand

its complex ﬁeld intensity H, we have the canonical dynamics in Minkowski space-time

βa∂aA=−H, e

βa∂aH=b2A−eq. (4.2)

In what follows we discuss the canonical covariant derivatives ∇µfor vector pair (A, H). In order

to use Cliﬀord algebra and the above results of spinor equation, we assume b > 0, e = 0 and make

transformation H=−ib e

A. Denote Ψ = (A, e

A)T. Similarly to the case of spinor, assume the left

hand coeﬃcient belonging to Λ1∪Λ3, Then in equivalent sense we have symmetrical dynamics

similar to (2.44) in curved space-time as

γµ(∂µ+Kµ+Lµγ5)Ψ = ibΨ, γµ=

0e

βµ

βµ0

,(4.3)

in which βα=hα

aβa. (Kµ, Lµ) are vectors to be determined, which are related to spinor connection

and Christoﬀel symbols. Multiplying (4.3) by γµ(∂µ+Kµ+Lµγ5), we get the second order equation

similar to (3.1),

[2+ (2Kµ−Γµ)∂µ+C0+γµν Yµν ] Ψ = −b2Ψ,(4.4)

in which Yµν stands for cross terms, C0is a scalar connection

C0= (KµKµ−LµLµ)+(Kµ

;µ+ ΓµKµ) + γ5(Lµ

;µ+ ΓµLµ).(4.5)

17

Noticing the above derivation is actually independent of (b, e), then the second order equation for

real potential Ais given by

[2+ (2Kµ−Γµ)∂µ+C1]A+1

2(e

βµβν−e

βνβµ)Yµν A=−b2A+eq, (4.6)

where

C1= (KµKµ−LµLµ)+(Kµ

;µ+ ΓµKµ)+(Lµ

;µ+ ΓµLµ).(4.7)

In contrast (4.6) with the trace equation of gµνAα

;µ;ν=−b2Aα+eqα, we get

Kµ= Γα

µα +1

2Γµ, Lµ=1

2Ωµ.(4.8)

Substituting (4.8) into (4.2) we get the spinor-like equation for vector ﬁeld

βµ[∂µ+ Γα

µα +1

2(Γµ+ Ωµ)]A=−H,

e

βµ[∂µ+ Γα

µα +1

2(Γµ−Ωµ)]H=b2A−eq.

(4.9)

For metric in matrix form G= (gµν ), by derivations similar to (4.9) we have

e

βµ³[∂µ−1

2(gµν gαβΓν

αβ + Γµ+ Ωµ)]G − GNµ´=−e

G,

βµ³[∂µ−1

2(gµν gαβΓν

αβ + Γµ−Ωµ)] e

G+e

GNµ´=λG − κT ,

(4.10)

where Mαis a vector to be determined and Nαis matrix to be determined. We take e

βαfor G

is because Gis a (0,2) type covariant tensor. The tensor expressed in matrix form is a strong

constraint for Cliﬀord algebra, so we cannot represent the dynamical equation by standard Cliﬀord

algebra now. Contracting the second order equation of (4.10) we get

gθκ [2+ (2Mµ−Γµ)∂µ+C1]gθκ +··· ≡ −2R.(4.11)

Comparing (4.11) with scalar Einstein’s ﬁeld equation R= 4Λ + 8π GT µ

µ, we ﬁnd

Mα=−1

2gµν Γα

µν ,Nα= (Γµ

αν ), λ =−1

2R+ Λ, κ = 8πG. (4.12)

For matrix Nα= (Γµ

αν ), µis row index and νcolumn index.

From the above results we learn that, only spinor, vector and rank-2 tensor have dynamics

with spinor structure. For other kind ﬁelds, we can hardly deﬁne the aﬃne connection by Cliﬀord

algebra. This feature also indicates the absence of these ﬁelds in Nature.

In Minkowski space-time, the transformation property of (A, H) is clariﬁed by quaternion rep-

resentation of ﬁelds and Lorentz transformation, so the geometrical meaning of (A, H) in (4.2) is

18

clear. In curved space-time, the geometrical meaning of Ain (2.25)-(2.28) is also clear. But the

geometrical meaning of (A, H) in (4.9) and (G,e

G) in (4.10) are implicitly determined by spinor-like

dynamics. Whether these representation of ﬁelds is the usual one or it is only a linear transfor-

mation of the usual one depending on tetrad is unknown. Analyzing (4.9) and (4.10) in detail, we

may ﬁnd some new interesting symmetries and phenomena of ﬁelds.

For free vector equation (4.3), whether the constraint of coeﬃcient in Λ0∪Λ1∪Λ3is necessary?

In general, we have canonical dynamics expressed by standard Cliﬀord algebra as

γµ∂µΨ = [U+V γ5+γµ(Kµ+Lµγ5) + γµν Nµν ]Ψ,(4.13)

in which (U, V, Kµ, Lµ, Nµν ) depend on (hµ

a, gµν ). By (3.7) we have second order equation

2Ψ = [C+Bγ5+γµ(Uµ+Vµγ5) + γµν Wµν ]Ψ +

[Cα+Bαγ5+γµ(Uα

µ+Vα

µγ5) + γµν Wα

µν ]∂αΨ.(4.14)

In general case we may have Nµν 6= 0, which can be determined by comparing (4.14) with usual

concrete equations. Each coeﬃcient has special geometrical and physical meanings. We discuss

the problem furthermore elsewhere.

V. DISCUSSION AND CONCLUSION

In this paper, we provided some typical applications of Cliﬀord algebra in physics. We estab-

lished the quadratic form of Lagrangian of gravity, dynamical equation of torsion and ﬁrst order

dynamical equation of vector and metric, which provide some new insights into physical ﬁelds.

Directly taking the γµmatrices as tetrad of space-time is much helpful to describe properties of

ﬁelds and space-time.

From the above discussion we ﬁnd that, Cliﬀord algebra is a natural and perfect language for

physical theory, which faithfully and exactly reﬂects the intrinsic properties of space-time and ﬁelds.

Cliﬀord algebra automatically displays the symmetry and graded structures of parameters with no

more or less contents. The representational matrix of its generator simultaneously has geometrical

and algebraic functions. Cliﬀord calculus is simple and clear like arithmetic operation. It can

be well understood by a common reader. If rewriting physical theories and pseudo-Riemannian

geometry in Cliﬀord algebra, the contents will be much accessible. By virtue of this excellent

language, the physical researches may be greatly promoted, and some long standing puzzling

problems may be easily solved. All in all, we sum up:

19

For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know

fully, even as I have been fully known.

Acknowledgments

The paper greatly beneﬁted from the enlightening discussions with Prof. James M. Nester.

Some contents have been modiﬁed according to the suggestions of a referee.

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