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Space-Time Geometry and Some Applications of Clifford Algebra in Physics

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Abstract

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}
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 γµµ,(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 ψkL.
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)
dxdy=γµγνdxµdyν,
dxdydz=γµγνγωdxµdyνdzω,···
(1.7)
We can also defined covariant differential operators
D=γµµ.(1.8)
Clifford algebra is graded, we have
IΛ0, γaΛ1, γab =γaγbΛ2, γabc =γaγbγcΛ3,··· (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
,
0i
i0
,
1 0
01
,(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 =abcdγdγ5, γ5=0123 = 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
dxdy=γµγνdxµdyν=1
2(γµγνγνγµ)dxµdyν=γµν dxµdyν.(1.16)
The exterior products can define continued multiplication. Assuming AΛpand BΛq, we have
AB= (1)pqBA.(1.17)
The geometric product or Clifford product is defined as
AB =γaγbAaBb= (ηab +γab)AaBb=A·B+AB.(1.18)
Obviously, for geometric product, we have grade-ntimes grade-1 leads to grade-(n±1) algebra.
For example, we have
γαγµν =gαµγνgαν γµ+γαµν , γµν γα=gαν γµgαµγν+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 γµγν=gµν γµγν=δν
µγµγν
=ηabγaγb=ηab γaγb=δb
aγaγb.(1.20)
For simplicity we denote tensor basis by
γµ1µ2···µn=γµ1γµ2⊗ · ·· γµn,γµ2µ3···µn
µ1=γµ1γµ2⊗ · ·· γµn,··· (1.21)
In general, a tensor of rank nis given by
T=Tµ1µ2···µnγµ1µ2···µn=Tµ1
µ2···µnγµ2µ3···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 dxµdxνlooks 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(ab) = 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=µt, ∂r,1
rθ,1
rsin θϕ, δX a= (dt, dr, rdθ, r sin θ).(2.2)
The direction of the tetrad basis turns into: γ0along dt,γ1along dr,γ2along , and γ3along
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
dAlim
x0[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: 1it is a real linear transformation of basis γµ, 2it satisfies metric
consistent condition dg= 0. Then the differential connection can be generally expressed as
dαγµ=µ
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 γµγν, by metric consistent condition we have
0 = dg=d(gµν γµγν)
= [(αgµν )γµγν+gµν (dαγµ)γν+gµν γµdαγν]dxα
= [(αgµν gνβ Πβ
αµ gµβΠβ
αν )(gνβ Tβ
αµ +gµβTβ
αν )]γµγνdxα.(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+γµdαγν]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
λ=dαγλdxαδγλ,
θλ
β=1
2gλκ(κgαβ βgακ +Sµν
ab la
βlb
καgµν )dxα,
(2.10)
in which δγλdefined by (1.27) is only the length change of tetrad, but λdefines 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],
λγβθλ
β=Tλ
αβdxαγβ,(2.11)
λ
βθα
βθλ
α=Kλ
βµν dxµdxν,(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(5)dγ5Td=αγ5Tα,(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;αAµTµ
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[αγµν
αµ +Tν
αµ)γν] = hµ
aαγµ,(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
µαγµ=gµν la
µαγν=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=γααand A=γβAβ, we have
DA =γαγβαAβ= (gαβ +γαβ)αAβ
=µAµ+1
2γµν (µAννAµ)H+1
2γµν Fµν .(2.25)
Clearly DA Λ0Λ2C`(1,3). By relation (1.19) we get
D2A=γααH+1
2γαγµν αFµν
=γααH+gαµγναFµν +1
2γαµν αFµν
=γα(αH− ∇µFαµ) + ωγ5
2g(²αµνω αFµν ).(2.26)
Therefore D2AΛ1Λ3, and then D3AΛ0Λ2Λ4. 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,
µAννAµ=Fµν ,αAα=H
µH− ∇νFµν +b2Aµ=eqµ, ²αµνωαFµν = 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(αAβαAβb2AαAα)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
;µ1
8Tµνω (e%ν%ωe%ω%ν) + µ,(2.30)
e
Γµ=1
4%νe%ν
;µ1
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
;µ1
4Tµνω γνω +µ.(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++
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%µ0 or αγµ0
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%ωe%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
;µ1
8Tµνω (e%ν%ωe%ω%ν) + µ.(2.39)
Similarly we have
e
Γµ=1
4%νe%ν
;µ1
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
λνλν)ψ+e
ψ
=ν(ψ+e
ψ) + 1
4gµαψ+¡%α
;νe%µ+%αe%µ
;ν¢e
ψ+i(e
λνλν)ψ+e
ψ
=ν(ψ+e
ψ) + 1
4gµαgµα
;νψ+e
ψ+i(e
λνλν)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+µφ+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+µφ+1
4
0e%ν
%ν0
0e%ν
;µ
%ν
;µ0
φ1
4Tµνω
0e%ν
%ν0
0e%ω
%ω0
φ
= (µ+1
4γνγν
;µ1
4Tµνω γνω +µ)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
;µ1
4Tµνω γνω +µ)φ=mφ. (2.43)
By (1.19) and Clifford algebra, we find the coefficient operators of Dirac equation belong to Λ1Λ3.
Thus we have
γµ[i(µ+ Υµ)eAµ+1
2γ5(Ωµ+1
2Tµ)]φ=mφ, (2.44)
in which the parameters are redefined by
Υµ1
2(la
µνhν
a+ Γν
µν ) = 1
2hν
a(µla
ννla
µ),α=1
4²dabchα
dhβ
aSµν
bc βgµν .(2.45)
(2.44) can be rewritten in Hermitian form,
αµˆpµφ+Sµ(Ωµ+1
2Tµ)φ=0φ, (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 pµΛ1, but the operator iαincludes components of Λ3.
By Clifford algebra we learn ΥµΛ1has different physical meanings from that of ΩµΛ3[25,26].
(µ+ Υµ)Λ1becomes a closed operator in logic for spinor. We find the spin SµΛ3is also
different from the angular momentum ~r ×~p Λ2, although we have Lie algebra ~
S×~
S=i
2~
S.
The spin-gravity coupling term Sµµmay be the origin of magnetic field of celestial body. µ
is proportional to angular momentum of gravitational source. For example, in Kerr-type metric
for a rotational ball
ds2=gttdt2+grr dr2+gθθ 2+ 2J dtdϕ +gϕϕ2,(2.48)
in which gµν =gµν(r, θ), J =J(r, θ), but is independent of (t, ϕ). For Kerr metric J2
rsin2θ.
By (2.45), we have
µ=f(0, ∂θJ, rJ, 0) f1(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 = Ωµdr
=2rcos θ
sin θr=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
2γ5µ] we have
m2φ=gµν µµν + 2Υµν+ ΥµΥν+1
4µνφγµ(µγν)νφ+
γµi∂µµγν(iΥν+1
2γ5ν)φµν γ5µ(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
2γµγ5µ(γνν) = i
2(ηab +γab)γ5hµ
aµ(hνbν),(3.2)
Y2=γµ(µγν)ν=(ηab +γab)hµ
aµ(hνb)ν,(3.3)
Y3=γµµ(γνΥν) = (ηab +γab)hµ
aµ(hνbΥν).(3.4)
In (3.2) the simplest and closed term is pseudo scalar P=ηabhµ
aµ(hνbν)Λ4. By (2.45) we
have
P=hµ
aµ(la
νν) = µµ+ Ωνhµ
aµla
ν
=µµ+ Γν
µν µµµ= Ωµ
;µ+ Γµµ,(3.5)
14
where and hereafter Γµ=µ. Similarly, in (3.4) and (3.3) we have
ηabhµ
aµ(hνbΥν) = Υµ
;µµΥµ= Υµ
;µ+ ΓµΥµ.(3.6)
ηabhµ
aµ(hνbν) = (µgµν +gανΓµ
µα ν)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
µ2+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,
LgΛ0is an important scalar defining the Lagrangian of gravity
Lg= Γµ
;µ+1
2µΓµµµ) = 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
2µΓµµµ).(3.10)
Since the time component of Ωµalmost vanishes Ω00, 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 ϑγαϑαand a normalized spinor field ψ. The covariant derivative
DΨdΨ + ωΨ includes the matrix valued connection one-form ω1
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(r4) which guarantees finite action, an advantage
over the Einstein-Hilbert O(r3).
15
In second order Dirac equation (3.8), the term 5Pis anti-Hermitian, Since Ωµis a field similar
to magnetic field ~
B, the pseudo scalar P= Ωµ
;µ+ Γµµmay 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
2abcdγcd γ5into (3.4), we get the following γcd terms,
Y1+Y2+Y3=i
2ηacηbd γcdγ5hµ
aµ(hνbν)i
2²abcdγcd γ5hµ
ahνbµΥν+ηabK0
ab.(3.13)
Collecting like terms, we get a new anti-symmetrical tensor
Ycd =²abcdhµ
ahνbµΓν+ηacηbd[hµ
aµ(hνbν)hµ
bµ(hνaν)].(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
;νk2TαTα¢.(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
2κµΓµ(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)]φkµkcˇγk+Nk+1
2wˇαµ,k ˇαµ
k1
2wQµQµ
1
2µAνµAν+1
2(µΦνµΦνb2ΦµΦµ), 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+τggαβ²βµνω(AωµAνΦωµΦν)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
0i0 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
βaaA=H, e
βaaH=b2Aeq. (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 Λ1Λ3, 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+ (2KµΓµ)µ+C0+γµν Yµν ] Ψ = b2Ψ,(4.4)
in which Yµν stands for cross terms, C0is a scalar connection
C0= (KµKµLµLµ)+(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+ (2KµΓµ)µ+C1]A+1
2(e
βµβνe
βνβµ)Yµν A=b2A+eq, (4.6)
where
C1= (KµKµLµLµ)+(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
2µ.(4.8)
Substituting (4.8) into (4.2) we get the spinor-like equation for vector field
βµ[µ+ Γα
µα +1
2µ+ Ωµ)]A=H,
e
βµ[µ+ Γα
µα +1
2µµ)]H=b2Aeq.
(4.9)
For metric in matrix form G= (gµν ), by derivations similar to (4.9) we have
e
βµ³[µ1
2(gµν gαβΓν
αβ + Γµ+ Ωµ)]G − GNµ´=e
G,
βµ³[µ1
2(gµν gαβΓν
αβ + Γµµ)] 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+ (2MµΓµ)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 Λ0Λ1Λ3is necessary?
In general, we have canonical dynamics expressed by standard Clifford algebra as
γµµΨ = [U+V γ5+γµ(Kµ+Lµγ5) + γµν Nµν ,(4.13)
in which (U, V, Kµ, Lµ, Nµν ) depend on (hµ
a, gµν ). By (3.7) we have second order equation
2Ψ = [C+5+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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... Cosmology contains a variety of contents, so it is necessary to clarify the conventions and notations frequently used in the paper at first. The line element of the space-time is given by [28] ...
... Υ µ ∈ Λ 1 is Keller connection and Ω α is Gu-Nester potential. They are calculated by [28][29][30][31]. ...
... By the definition of EMT (17), the variation δg : L m → T µν is a linear mapping, so (27) holds. By (28), the variables φ and ψ have decoupling dynamic equations. Since the dynamics of variables is sufficient condition of energy-momentum conservation law, we can derive T µν I;ν = 0 from dynamic equation of φ, and T µν I I;ν = 0 from dynamic equation of ψ independently, so (29) holds. ...
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By analyzing the energy-momentum tensor and equations of state of ideal gas, scalar, spinor and vector potential in detail, we find that the total mass density of all matter is always positive, and the initial total pressure is negative. Under these conditions, by qualitatively analyzing the global behavior of the dynamical equation of cosmological model, we get the following results: (i) K=1, namely, the global spatial structure of the universe should be a three-dimensional sphere S3; (ii) 0≤Λ<10−24ly−2, the cosmological constant should be zero or an infinitesimal; (iii) a(t)>0, the initial singularity of the universe is unreachable, and the evolution of the universe should be cyclic in time. Since the matter components considered are quite complete and the proof is very elementary and strict, these conclusions are quite reliable in logic and compatible with all observational data. Obviously, these conclusions will be very helpful to correct some popular misconceptions and bring great convenience to further research other problems in cosmology such as the properties of dark matter and dark energy. In addition, the macroscopic Lagrangian of fluid model is derived.
... Cosmology contains a variety of contents, so it is necessary to clarify the conventions and notations frequently used in the paper at first. The line element of the space-time is given by [28] ...
... Υ µ ∈ Λ 1 is Keller connection and Ω α is Gu-Nester potential. They are calculated by [28][29][30][31]. ...
... By the definition of EMT (17), the variation δg : L m → T µν is a linear mapping, so (27) holds. By (28), the variables φ and ψ have decoupling dynamic equations. Since the dynamics of variables is sufficient condition of energy-momentum conservation law, we can derive T µν I;ν = 0 from dynamic equation of φ, and T µν I I;ν = 0 from dynamic equation of ψ independently, so (29) holds. ...
Preprint
Full-text available
By analyzing the energy-momentum tensor and equations of state of ideal gas, scalar, spinor and vector potential in detail, we find that the total mass density of all matter is always positive, and the initial total pressure is negative. Under these conditions, by qualitatively analyzing the global behavior of the dynamical equation of cosmological model, we get the following results: (i) K = 1, namely, the global spatial structure of the universe should be a 3-dimensional sphere S 3. (ii) 0 ≤ Λ < 10 −24 ly −2 , the cosmological constant should be zero or an infinitesimal. (iii) a(t) > 0, the initial singularity of the universe is unreachable, and the evolution of universe should be cyclic in time. Since the matter components considered are quite complete and the proof is very elementary and strict, these conclusions are quite reliable in logic and compatible with all observational data. Obviously, these conclusions will be much helpful to correct some popular misconceptions and bring great convenience to further research other problems in cosmology such as property of dark matter and dark energy. In addition, the macroscopic Lagrangian of fluid model is derived. Keywords: Cosmological model; energy-momentum tensor; equation of state; cosmic curvature;cosmological constant; negative pressure; dynamic analysis; Lagrangian of fluid
... Again by γ a n γ a n ¼ 1 (not summation), we prove (23). Other equations can be proved by antisymmetrization of indices. ...
... In physics, basis of tensors is defined by direct products of grade-1 bases γ μ . For metric, we have [23]: ...
... We call d α connection operator [23]. According to its geometrical meanings, connection operator should satisfy the following conditions: ...
Chapter
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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.
... The interaction between spinors and gravity is the most complicated and subtle interaction in the universe, which involves the basic problem of a unified quantum theory and general relativity. The spinor connection has been constructed and researched in many works [1][2][3][4][5]. The spinor field is used to explain the accelerating expansion of the universe and dark matter and dark energy [6][7][8][9][10][11]. ...
... By straightforward calculation we have [5,12,29] 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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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.
... In order to eliminate the dependency of the scale factor on a central mass, the model (From η R = kη m , which is the definition for an arbitrary R given by (9), and η m = c βH 0 as given by (24) in [3].) sets ...
... Another notable property of Cl 3,0 is that it allows the quick extrapolation of the results to the spacetime algebra (STA) using the isomorphism with Cl 0 1,3 . As an example, for Cl 1,3 where the basis is the Dirac matrices, and the transformation is γ µ → γ a = γ µ h µ a [24]. Hence, in the case of the FLRW metric (10), it yields γ k = aγ k , (k = 1, 2, 3). ...
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The paper briefly reviews 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.
... In order to eliminate the dependency of the scale factor on a central mass, the model (From η R = kη m , which is the definition for an arbitrary R given by (9), and η m = c βH 0 as given by (24) in [3].) sets ...
... Another notable property of Cl 3,0 is that it allows the quick extrapolation of the results to the spacetime algebra (STA) using the isomorphism with Cl 0 1,3 . As an example, for Cl 1,3 where the basis is the Dirac matrices, and the transformation is γ µ → γ a = γ µ h µ a [24]. Hence, in the case of the FLRW metric (10), it yields γ k = aγ k , (k = 1, 2, 3). ...
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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.
... Now we make some general considerations. We express the main contents in the formalism of Clifford algebra, because this algebra faithfully and exactly reflects the intrinsic symmetry and property of space-time and fields [10,37]. In this formalism, the calculations such as (3.3)-(3.6) ...
... Clifford algebra is the perfect language. Then Dirac equation (4.4) becomes [10,37,38] ( ) ( ) , ...
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By constructing the commutative operators chain, we derive the integrable conditions for solving the eigenfunctions of Dirac equation and Schrödinger equation. These commutative relations correspond to the intrinsic symmetry of the physical system, which are equivalent to the original partial differential equation can be solved by separation of variables. Detailed calculation shows that, only a few cases can be completely solved by separation of variables. In general cases, we have to solve the Dirac equation and the Schrödinger equation by effective perturbation or approximation methods, especially in the cases including nonlinear potential or self-interactive potentials.
... They correspond to scalar, vector, pseudo vector and pseudo scalar respectively. Clifford algebra naturally converts geometric calculation into algebra operation, according to which all 3-d Euclidean geometry and field theory can be derived, and the conclusion can be simply extended to the case of curved space [27] [28] [29]. ...
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The description of the microscopic world in quantum mechanics is very different from that in classical physics, and there are some points of view that are contrary to intuition and logic. The first is the problem of reality; quantum mechanics believes the behavior of micro particles is random and jumping. The second is the loss of certainty; the conjugate physical variables of a system cannot be determined synchronously, they satisfy the Heisenberg uncertainty principle. The third is the non-local correlation. The measurement of one particle in the quantum entanglement pair will influence the state of the other entangled particle simultaneously. In this paper, some concepts related to quantum entanglement, such as EPR correlation, quantum entanglement correlation function, Bell’s inequality and so on, are analyzed in detail. Analysis shows that the mystery and confusion in quantum theory may be caused by the logical problems in its basic framework. Bell’s inequality is only a mathematical theorem, but its physical meaning is actually unclear. The Bell state of quantum entangled pair may not satisfy the dynamic equation of quantum theory, so it cannot describe the true state of microscopic particles. In this paper, the correct correlation functions of spin entanglement pair and photonic entanglement pair are strictly derived according to normal logic. Quantum theory is a more fundamental theory than classical mechanics, and they are not equal relation in logic. However, there are still some unreasonable contents in the framework of quantum theory, which need to be improved. In order to disclose the real relationship between quantum theory and classical mechanics, we propose some experiments which provide intuitionistic teaching materials for the new interpretation of quantum theory.
... In [13,14], by straightforward calculation we have the following results. ...
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The magnetic field of the earth plays an important role in the ecosystem, and the magnetic field of celestial bodies is also important in the formation of cosmic large-scale structures, but the origin and evolution of the celestial magnetic field is still an unresolved mystery. Many hypotheses to explain the origin have been proposed, but there are some insurmountable difficulties for each one. At present, the theory widely accepted in scientific society is the dynamo model, it says that, the movement of magnetofluid inside celestial bodies, which can overcome the Ohmic dissipative effect and generate persistent weak electric current and macroscopic magnetic field. However, this model needs an initial seed magnetic field, and the true values of many physical parameters inside the celestial body are difficult to obtain, and there is no stable solution to the large range of fluid motion. These are all difficulties for the dynamo model. Furthermore, it is difficult for the dynamo to explain the correlation between the dipole magnetic field and angular momentum of a celestial body. In this paper, by calculating the interaction between spin of particles and gravity of celestial body according to Clifford algebra, we find that a rotational celestial body provides a field $\Omega^\alpha$ for spins, which is similar to the magnetic field of a dipole, and the spins of charged particles within the celestial body are arranged along the flux line of $\Omega^\alpha$, then a macroscopic magnetic field is induced. The calculation shows that the strength of $\Omega^\alpha$ is proportional to the angular momentum of the celestial body, which explains the correlation between the magnetic intensity and angular momentum. The results of this paper suggest that further study of the effects of internal variables such as density, velocity, pressure and temperature of a celestial body on $\Omega^\alpha$ may provide some new insights into the origin and evolution of celestial magnetic field.
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本文是一篇介绍Clifford代数基本性质和应用的文章。Clifford代数是实数、复数、四元数和向量代数的统一和推广,它准确、忠实地刻画了时空的内在性质,为众多数学和物理理论提供了统一、标准、优雅和开放的语言和工具,值得在本科物理和数学的教学中普及。回顾数系理论的发展和完善的漫长历史,我们发现包括克利福德代数在内的超复数系,在自然科学中占据核心位置。超复数的零模集是具有特殊几何意义的低维集合,就像现实时空中的光锥,对代数运算没有实质性影响。克利福德代数以统一的方式描述复数和四元数,并且可以直接地推广到2n元结合代数。在这个推广中,时空中标准正交基的矩阵表示起了重要作用。矩阵表示具有抽象概念难以表达的内容,例如行列式和逆元的定义。没有矩阵表示,关于超复数的一些讨论将很难展开。由于克利福德代数与矩阵代数的同构关系,由矩阵代数得到的结论对相应的超复数也必然成立。克利福德代数直接定义于基本时空之上,描述的几何和物理问题时,内容不多不少刚刚好,并且形式简洁、结构对称、推导规范,看上去让人赏心悦目。因此,我们有望超复数系能够完成现代科学的一次新的大综合。 关键词 : 四元数, Clifford代数, 循环数, 超复数系, 几何代数, Maxwell方程组
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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.
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We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field.
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1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5. Geometric Algebras of PseudoEuclidean Spaces.- 2 / Differentiation.- 2-1. Differentiation by Vectors.- 2-2. Multivector Derivative, Differential and Adjoints.- 2-3. Factorization and Simplicial Derivatives.- 3 / Linear and Multilinear Functions.- 3-1. Linear Transformations and Outermorphisms.- 3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem.- 3-3. Eigenblades and Invariant Spaces.- 3-4. Symmetric and Skew-symmetric Transformations.- 3-5. Normal and Orthogonal Transformations.- 3-6. Canonical Forms for General Linear Transformations.- 3-7. Metric Tensors and Isometries.- 3-8. Isometries and Spinors of PseudoEuclidean Spaces.- 3-9. Linear Multivector Functions.- 3-10. Tensors.- 4 / Calculus on Vector Manifolds.- 4-1. Vector Manifolds.- 4-2. Projection, Shape and Curl.- 4-3. Intrinsic Derivatives and Lie Brackets.- 4-4. Curl and Pseudoscalar.- 4-5. Transformations of Vector Manifolds.- 4-6. Computation of Induced Transformations.- 4-7. Complex Numbers and Conformal Transformations.- 5 / Differential Geometry of Vector Manifolds.- 5-1. Curl and Curvature.- 5-2. Hypersurfaces in Euclidean Space.- 5-3. Related Geometries.- 5-4. Parallelism and Projectively Related Geometries.- 5-5. Conformally Related Geometries.- 5-6. Induced Geometries.- 6 / The Method of Mobiles.- 6-1. Frames and Coordinates.- 6-2. Mobiles and Curvature 230.- 6-3. Curves and Comoving Frames.- 6-4. The Calculus of Differential Forms.- 7 / Directed Integration Theory.- 7-1. Directed Integrals.- 7-2. Derivatives from Integrals.- 7-3. The Fundamental Theorem of Calculus.- 7-4. Antiderivatives, Analytic Functions and Complex Variables.- 7-5. Changing Integration Variables.- 7-6. Inverse and Implicit Functions.- 7-7. Winding Numbers.- 7-8. The Gauss-Bonnet Theorem.- 8 / Lie Groups and Lie Algebras.- 8-1. General Theory.- 8-2. Computation.- 8-3. Classification.- References.
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The potential conflict between torsion and gauge symmetry in the Riemann-Cartan curved space-time was noted by Kibble in his 1961 pioneering paper, and has since been discussed by many authors. Kibble suggested that, to preserve gauge symmetry, one should forgo the covariant derivative in favor of the ordinary derivative in the definition of the field strength F_{\mu}{\nu} for massless gauge theories, while for massive vector fields covariant derivatives should be adopted. This view was further emphasized by Hehl and collaborators in their influential 1976 review paper. We address the question of whether this deviation from normal procedure of forgoing covariant derivatives in curved spacetime with torsion could give rise to inconsistencies in the theory, such as the quantum renormalizability of a realistic interacting theory. We demonstrate in this note the one-loop renormalizability of a realistic gauge theory of gauge bosons interacting with Dirac spinors, such as the SU(3) chromodynamics, for the case of a curved Riemann-Cartan spacetime with totally anti-symmetric torsion. This affirmative confirmation is one step towards providing justification for the assertion that the flat-space definition of the gauge field strength should be adopted as the proper definition.
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The STA is a mathematical system, rather than of itself containing new physics. However, when we employ it in the description of physical phenomena, we usually find that some fresh insight is obtained, often on old questions. The oldest question of 20th century physics is the interpretation of quantum mechanics, and in this lecture we aim to discuss some of the light that an STA approach can throw upon this issue. This will be undertaken in the context of specific examples, and so in the main these lecture notes contain the details of these, rather than the comments that will be made about issues of interpretation. We think these examples stand as interesting and important applications in their own right, irrespective of whether one thinks there is a problem to be solved in the interpretation of quantum theory. This is particularly true for the beginnings of a multiparticle STA approach which we describe here, since it seems undeniable that there is a need for a more coherent and justifiable rationale for the set of recipes and operational procedures that currently constitute relativistic multiparticle theory.
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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.
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We investigate using Clifford algebra methods the theory of algebraic dotted and undotted spinor fields over a Lorentzian spacetime and their realizations as matrix spinor fields, which are the usual dotted and undotted two component spinor fields. We found that some ad hoc rules postulated for the covariant derivatives of Pauli sigma matrices and also for the Dirac gamma matrices in General Relativity cover important physical meaning, which is not apparent in the usual matrix presentation of the theory of two components dotted and undotted spinor fields. We discuss also some issues related to the previous one and which appear in a proposed "unified" theory of gravitation and electromagnetism which use two components dotted and undotted spinor fields and also paravector fields, which are particular sections of the even subundle of the Clifford bundle of spacetime.
Conference Paper
We explicitly demonstrate with a help of a computer that Clifford algebra Cℓ(B) of a bilinear form B with a non-trivial antisymmetric part A is isomorphic as an associative algebra to the Clifford algebra Cℓ(Q) of the quadratic form Q induced by the symmetric part of B [in characteristic ≠ 2], However, the multivector structure of Cℓ(B) depends on A and is therefore different than the one of Cℓ(Q). Operation of reversion is still an anti-automorphism of Cℓ(B). It preserves a new kind of gradation in ⋀ V determined by A but it does not preserve the gradation in ⋀ V. The demonstration is given for Clifford algebras in real and complex vector spaces of dimension ≤ 9 with a help of a Maple package ‘Clifford’. The package has been developed by one of the authors to facilitate computations in Clifford algebras of an arbitrary bilinear form B.