Differential Geometrical Formulation of Gauge Theory of Gravity

10 Pages·0.1 MB·English

by  Ning Wu

Differential Geometrical Formulation of
Gauge Theory of Gravity
Ning Wu1 ∗, Zhan Xu2, Dahua Zhang1
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2 Institute of High Energy Physics, P.O.Box 918-1, Beijing 100039, P.R.China1

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Tsinghua University, Beijing 100080, P.R.China
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February 1, 2008

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PACS Numbers: 11.15.-q, 04.60.-m, 02.40.-k.
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Keywords: gauge field, quantum gravity, geometry.
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i Abstract
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r Differential geometric formulation of quantum gauge theory of gravity is
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studied in this paper. The quantum gauge theory of gravity which is pro-
posed in the references hep-th/0109145 and hep-th/0112062 is formulated
completely in the framework of traditional quantum field theory. In order to
study the relationship between quantum gauge theory of gravity and tradi-
tional quantumgravity whichis formulated incurved space, it is importantto
find the differential geometric formulation of quantum gauge theory of grav-
ity. We first give out the correspondence between quantum gauge theory of
gravity and differential geometry. Then we give out differential geometric for-
mulation of quantum gauge theory of gravity.
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email address: [email protected]
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1 Introduction
Itisknownthat, inclassical Newton’stheoryofgravity, gravityistreatedasphysical
interactionsbetweentwo massiveobjects, andgravitydoesnotaffectthestructureof
space-time[1]. In Einstein’s general theory of relativity, gravity is treated as geome-
try of space-time[2, 3]. In other words, in general relativity, gravity is not treated as
a physical interactions, it is a part of structure of space-time. Inspired by Einstein’s
general theory of relativity, traditional relativistic theory of gravity and traditional
canonical quantum theory of gravity is formulated in the framework of differential
geometry[4, 5, 6, 7, 8, 9].
Recently, based on completely new notions and completely new methods, Wu
proposesanewquantumgaugetheoryofgravity,whichisbasedongaugeprinciple[10,
12]. The central idea is to use traditional gauge field theory to formulate quantum
theory of gravity. This new quantum gauge theory of gravity is renormalizable[10,
11]. A strict formal proof on the renormalizability of the theory is also given in the
reference [10, 11]. This new quantum gauge theory of gravity is formulated in the
flat physical space-time, which is completely different from that of the traditional
quantum gravity at first appearance. But this difference is not essential, for quan-
tum gauge theory of gravity can also be formulated in curved space. In gauge theory
of gravity, gravity is treated as physical interactions, but in this paper, gravity is
treated as space-time geometry. It means that gravity has physics-geometry duality,
which is the nature of gravitational interactions. In this paper, we first give out the
correspondence between quantum gauge theory of gravity and differential geometry.
Then, we give out the differential geometrical formulation of quantum gauge theory
of gravity.
2 Correspondence Between Quantum Gauge The-
ory of Gravity and Differential Geometry
In gravitational gauge theory, there are two different kinds of space-time: one is the
traditional Minkowski space-time which is the physical space-time, another is gravi-
tationalgaugegroupspace-time whichisnotaphysical space-time. Twospace-times
have different physical meanings. In differential geometry, there are also two differ-
ent space-times. This can be seen from the tetrad field in Cartan form. In cartan
form, there are two different space-times, one is base manifold, another is the tan-
gent space in Cartan tetrad. The correspondence between these two different spaces
in two different theories are: the traditional Minkowski space-time in gravitational
gauge theory corresponds to the tangent space in Cartan tetrad; the gravitational
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gauge group space-time corresponds to the base manifold of differential geometry.
In gravitational gauge theory, the metric in Minkowski space-time is the flat
metric ηµν, which corresponds to the flat metric ηab in Cartan tetrad. The metric
in gravitational gauge group space-time is the curved metric gαβ which corresponds
to the curved metric gµν in differential geometry.
In gravitational gauge theory, Gα is defined by
µ
Gα = δα −gCα, (2.1)
µ µ µ
which corresponds to the Cartan tetrad field e µ. G−1µ corresponds to reference
a. α
frame field ea . That is
.µ
Gα ⇐⇒ e µ, (2.2)
µ a.
G−1µ ⇐⇒ ea . (2.3)
α . µ
Therefore, the following two relations have the same meaning,
gαβ = ηµνGαGβ ⇐⇒ gµν = ηabe µe ν, (2.4)
µ ν a. b.
where gαβ is the metric in gravitational gauge group space-time, and gµν is the
metric of the base manifold in differential geometry. Similarly, the following two
relations correspond to each other,
g = η G−1µG−1ν ⇐⇒ g = η ea eb . (2.5)
αβ µν α β µν ab . µ . ν
Define
△
∂ = e µ∂ , (2.6)
a a. µ
which is the derivative in Cartan tetrad. It corresponds to the gravitational gauge
covariant derivative D in gravitational gauge theory
µ
D ⇐⇒ ∂ . (2.7)
µ a
In gravitational gauge theory, all matter field, such as scalar fields φ(x), Dirac
field ψ(x), vector field A (x), ···, are fields in Minkowski space, which correspond
µ
to fields in Cartan tetrad. In other words, all matter fields in differential geometry
are defined in Cartan tetrad.
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Finally, what is the transformation in differential geometry which corresponds to
the the gravitational gauge transformation? Under most general coordinate trans-
formation, the transformation of Cartan tetrad field is :
∂x′µ
e µ → e′ µ = L be ν, (2.8)
a. a. ∂xν a. b.
where L b is the associated Lorentz transformation. As we have stated before, under
a.
gravitational gauge transformations, there is no associated Lorentz transformation.
In other words, under gravitational gauge transformation,
L b = δb. (2.9)
a. a
Therefore, the gravitational gauge transformation of Cartan tetrad field is
∂x′µ
e µ → e′ µ = e ν, (2.10)
a. a. ∂xν a.
We call this transformation translation transformation. So, gravitational gauge
transformation in gravitational gauge theory corresponds to the translation trans-
formation in differential geometry. According to eq.(2.6), ∂ does not change under
a
translation transformation,
∂ → ∂′ = ∂ . (2.11)
a a a
Eq.(2.10) corresponds to the following transformation in gravitational gauge theory,
Gα → G′α = Λα (Uˆ Gβ). (2.12)
µ µ β ǫ µ
Eq.(2.11) corresponds to the following gravitational gauge transformation in gravi-
tational gauge theory,
D → D′ = Uˆ D Uˆ−1. (2.13)
µ µ ǫ µ ǫ
Finally, as a summary, we list some important correspondences between physical
picture and geometry picture in the following table 1.
3 Differential Geometrical Formulation of Grav-
itational Gauge Theory
In differential geometrical formulation of gravitational gauge theory, all fields are
expressed in Cartan orthogonal tetrad. In differential geometry, gravitational field
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Quantum Gauge Theory of Gravity Differential Geometry
gauge group space-time base manifold
Minkowski space-time tangent space in tetrad
index α index µ
index µ index a
Gα e µ
µ a.
G−1µ ea
α . µ
gαβ = ηµνGαGβ gµν = ηabe µe ν
µ ν a. b.
g = η G−1µG−1ν g = η ea eb
αβ µν α β µν ab . µ . ν
D ∂
µ a
Fα ω¯µ
µν ab
gravitational gauge transformation translation transformation
gravitational gauge covariant translation invariant
Λα ∂x′µ
β ∂xν
Λ β ∂xν
α ∂x′µ
Gα → G′α = Λα (Uˆ Gβ) e µ → e′ µ = ∂x′µe ν
µ µ β ǫ µ a. a. ∂xν a.
G−1µ → G′−1µ = Λ β(Uˆ G−1µ) ea → e′a = ∂xν ea
α α α ǫ β . µ . µ ∂x′µ . ν
gαβ → g′αβ = Λα Λβ (Uˆ gα1β1) gµν → g′µν = ∂x′µ ∂x′νgµ1ν1
α1 β1 ǫ ∂xµ1 ∂xν1
g → g′ = Λ α1Λ β1(Uˆ g ) g → g′ = ∂xµ1 ∂xν1g
αβ αβ α β ǫ α1β1 µν µν ∂x′µ ∂x′ν µ1ν1
D → D′ = Uˆ D Uˆ−1 ∂ → ∂′ = ∂
µ µ ǫ µ ǫ a a a
Fα → F′α = Λα (Uˆ Fβ ) ω¯µ → ω¯′µ = ∂x′µω¯ν
µν µν β ǫ µν ab ab ∂xν ab
Table 1: Correspondence between two pictures of gravity.
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is represented by tetrad field e µ. The field strength of gravitational field is denoted
a.
by ω¯µ ,
ab
1
ω¯µ = − [(∂ e µ)−(∂ e µ)]. (3.1)
ab g a b. b a.
ω¯µ corresponds to the field strength tensor Fα in gravitationalgauge theory. Under
ab µν
translation transformation, ω¯µ transforms as
ab
∂x′µ
ω¯µ → ω¯′µ = ω¯ν . (3.2)
ab ab ∂xν ab
It is known that, in differential geometry, covariant derivative of cartan tetrad
vanishes
D ea = 0. (3.3)
µ . ν
It gives out the following relation
∼a
Γ = ea e νe µΓλ +ea (∂ e ν)e µ, (3.4)
bc . λ b. c. µν . ν µ b. c.
∼a
whereΓλ istheaffineconnexionandΓ istheCartanconnexion. Fromthisrelation,
µν bc
we can obtain
∼a
T = ea e µe νTλ +gea ω¯µ (3.5)
bc . λ b . c . µν . µ bc
∼a
where Tλ and T are torsion tensors
µν bc
∼a ∼a ∼a
T =Γ − Γ , (3.6)
bc bc cb
Tλ = Γλ −Γλ . (3.7)
µν µν νµ
For gravitationalgaugetheory, theaffine connexion is the Christoffel connexion. For
Christoffel connexion, the torsion Tλ vanish. Then
µν
∼a
T = gea ω¯µ. (3.8)
bc . µ bc
It seems that the field strength of gravitational field is related to the torsion of Car-
tan connexion.
Translation transformation of metric tensor g is
µν
∂xµ1 ∂xν1
g → g′ = g . (3.9)
µν µν ∂x′µ ∂x′ν µ1ν1
The metric in Cartan tetrad is denoted as ηab, which is the traditional Minkowski
metric. Under translation transformation, ηab is invariant
ηab → η′ab = ηab. (3.10)
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The lagrangian L for pure gravitational field is selected as
1
L = − ηacηbdg ω¯µ ω¯ν . (3.11)
4 µν ab cd
Using eq.(2.5) and eq.(3.8), we can change the above lagrangian density into the
following form
1 ∼e ∼f
L = − ηacηbdη T T . (3.12)
4g2 ef ab cd
It is easy to prove that this lagrangian is invariant under translation transforma-
tion. The concept of translation invariant in differential geometry corresponds to
the concept of gravitational gauge covariant in gravitational gauge theory.
In order to introduce translation invariant action of the system, we need to
introduce a factor which is denoted as J(C) in gravitational gauge theory. In this
paper, J(C) is selected as
J(C) = −det(g ). (3.13)
µν
q
The action of the system is defined by
S = d4x J(C) L. (3.14)
Z
This action is invariant under translation transformation.
For real scalar field φ, its gravitational interactions are described by
1 1
L = − (∂ φ)(∂ φ)− m2φ2, (3.15)
a b
2 2
where m is the mass of scalar. For complex scalar field, its lagrangian is selected to
be
L = −(∂ φ)∗(∂ φ)−m2φ∗φ. (3.16)
a b
Under translation transformations, φ(x) and ∂ φ(x) transform as
a
φ(x) → φ′(x′) = φ(x), (3.17)
∂ φ(x) → ∂′φ′(x′) = ∂ φ(x). (3.18)
a a a
Using eq.(3.17) and eq.(3.18), we can prove that the lagrangians which are given by
eq.(3.15) and eq.(3.16) are translation invariant.
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For Dirac field, its lagrangian is given by
L = −ψ¯(γa∂ +m)ψ. (3.19)
a
Under translation transformations,
ψ(x) → ψ′(x′) = ψ(x), (3.20)
∂ ψ(x) → ∂′ψ′(x′) = ∂ ψ(x), (3.21)
a a a
γa → γ′a = γa. (3.22)
Under these transformations, the lagrangian given by eq.(3.19) is invariant.
For vector field, the lagrangian is given by
2
1 m
L = − ηacηbdA A − ηabA A , (3.23)
ab cd a b
4 2
where
A = ∂ A −∂ A , (3.24)
ab a b b a
which is the field strength of vector field. This lagrangian is invariant under the
following translation transformations
A (x) → A′(x′) = A (x), (3.25)
a a a
A (x) → A′ (x′) = A (x), (3.26)
ab ab ab
ηab → η′ab = ηab. (3.27)
For U(1) gauge field, its lagrangian is[10, 11, 13]
1
L = −ψ¯[γa(∂ −ieA )+m]ψ − ηacηbdA A , (3.28)
a a ab cd
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where
A = A +gec A ω¯µ , (3.29)
ab ab . µ c ab
A = ∂ A −∂ A , (3.30)
ab a b b a
In eq.(3.28), e is the coupling constant for U(1) gauge interactions. This lagrangian
is invariant under the following translation transformation,
A (x) → A′(x′) = A (x), (3.31)
a a a
A (x) → A′ (x′) = A (x). (3.32)
ab ab ab
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It is also invariant under local U(1) gauge transformation[10, 11, 13].
For SU(N) non-Abel gaueg field, its lagrangian is[10, 11, 13]
1
L = −ψ¯[γa(∂ −ig A )+m]ψ − ηacηbdAi Ai , (3.33)
a c a 4 ab cd
where
Ai = Ai +gec Aiω¯µ , (3.34)
ab ab . µ c ab
Ai = ∂ A −∂ A +g f AjAk. (3.35)
ab a b b a c ijk a b
In above relations, g is the coupling constant for SU(N) non-Able gauge interac-
c
tions. It can be proved the this lagrangian is invariant under SU(N) gauge trans-
formation and translation transformation[10, 11, 12, 13].
4 Summary
In this paper, the geometrical formulation of gauge theory of gravity is studied,
which is performed in the geometrical formulation of gravity[10, 12]. In this picture,
we can see that gravitational field is put into the structure of space-time and there
is no physical gravitational interactions in space-time.
In gravitational gauge theory, all matter field, such as scalar fields φ(x), Dirac
field ψ(x), vector field A (x), ···, are fields in Minkowski space, which correspond
µ
to fields in Cartan tetrad. In other words, all matter fields in differential geometry
are defined in Cartan tetrad.
Ingravitationalgaugetheory,thesymmetrytransformationisgravitationalgauge
transformation, while in differential geometry, the corresponding symmetry trans-
formation is translation transformation. The concept of translation invariant in
differential geometry corresponds to the concept of gravitational gauge covariant in
gravitational gauge theory.
References
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