Spinor Representation of Conformal Group and Gravitational Model

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by  K. Nishida

Spinor Representation of Conformal Group and
Gravitational Model
Kohzo Nishida∗)
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Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan
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Abstract
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We consider spinor representations of the conformal group. The spacetime is
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v constructed by the 15-dimensional vectors in the adjoint representation of SO(2,4).
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9 On the spacetime, we construct a gravitational model that is invariant under local
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4 transformation.
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∗) E-mail: [email protected]
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Conformal transformations play a widespread role in gravity theories.1),2),3),4) In this
paper, we investigate the conformal transformations of spinors. The spinor, which describes
spin-1/2particlesandantiparticles,5) containsaconsiderableamountofphysical information,
making it a promising target for investigation. In addition, we construct a gravitational
model that is invariant under local transformation.
In general, an arbitrary 4×4 transform matrix U(x) that satisfies
(γ U)† = (γ U)−1, (1)
0 0
has the generators:
I, γi, iγ , iγij, γ γi. (2)
5 5
U(x) describes the conformal transformations. In fact, the following6),7)
i 1
M ≡ γ , P ≡ (γ +γ γ ),
ij ij i i 5 i
2 2
i 1
D ≡ γ , K ≡ (γ −γ γ ), (3)
5 i i 5 i
2 2
satisfies the conformal group
[D,K ] = −iK , [D,P ] = iP
i i i i
[K ,P ] = 2iη D−2iM , [K ,M ] = i(η K −η K ),
i j ij ij i jk ij k ik j
[P ,M ] = i(η P −η P ),
i jk ij k ik j
[M ,M ] = i(η M +η M −η M −η M ). (4)
ij kl jk il il jk ij jl jl ik
Other commutators vanish. Here, we have constructed a flat tangent space at every point on
the four-dimensional manifold, and we indicate the vectors in the four-dimensional tangent
space by subscript Roman letters i, j, k, etc. Greek letters, such as µ and ν, are used to label
four-dimensional spacetime vectors. U(x) contains within it local Lorentz transformations.
Let a spinor ψ transform as
ψ → ψ′ = Uψ, (5)
then (3) is spinor representations of the conformal group. If we rewrite (3) as
1 1
J ≡ M , J ≡ (K −P ), J ≡ (K +P ), J ≡ D. (6)
ij ij i4 i i i5 i i 54
2 2
J generates the algebra of SO(2,4):
pq
[J ,J ] = i(η J +η J −η J −η J ), (7)
pq rs qr ps ps qr pq qs qs pr
where η = (1,−1,−1,−1,1) and p,q,r,s = 0,1,···5.
pq
2
U(x) can be also written as
U(x) = e−iεa(x)Γa, (8)
where
2Γa
≡ (γ0,γ1,γ2,γ3,iγ ,iγ01,iγ02,iγ03,iγ12,iγ13,iγ23,γ γ0,γ γ1,γ γ2,γ γ3). (9)
5 5 5 5 5
We have projected out the unit matrix I. These gamma matrices satisfy
Tr(ΓaΓb) = 0 a 6= b,
Tr(ΓaΓa) = 1 a = 0,8,9,10,12,13,14,
Tr(ΓaΓa) = −1 a = 1,2,3,4,5,6,7,11. (10)
Now, let us suppose that the spacetime transforms as the 15-dimensional vectors in the
adjoint representation of SO(2,4), not the six-dimensional vectors in the fundamental rep-
resentation, that is, the 15-dimensional coordinates x (a = 0,1,···14) transform as
a
X → X′ = U−1XU, X ≡ x Γa. (11)
a
Notethatγ U withthegenerators(2)isthetransformmatrixofU(4). Thespacecoordinates
0
are rotated in the adjoint representation of SU(2). Therefore, it is not unnatural that the
spacetime transforms in the adjoint representation. An invariant can be constructed:
Tr(XX) = x2 −x2 −x2 −x2 −x2 −x2 −x2 −x2
0 1 2 3 4 5 6 7
+x2 +x2 +x2 −x2 +x2 +x2 +x2 . (12)
8 9 10 11 12 13 14
Then, themetricofthetangentspaceisη = (1,−1,−1,−1,−1,−1,−1,−1,1,1,1,−1,1,1,1).
ab
Let us define a 15-dimensional tetrad
eaαe β = gαβ, ea ebα = ηab, (13)
a α
where indices a,b,c are 15-dimensional vectors in flat space and α,β,γ are 15-dimensional
vectors in curved space. The 15-dimensional tetrad transforms linearly as
e α(x)Γa → U−1(x)Γα(x)U(x) = e α(x)U−1(x)ΓaU(x) = e′ α(x′)Γa. (14)
a a a
Next, we propose a gauge model on this spacetime. In our previous paper,8) we have
considered the four- and five-dimensional gauge model. The covariant derivative is given by
D ≡ ∂ −igA , A ≡ A Γa, (15)
α α α α aα
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where we introduce gauge fields A , each transforming as
α
1
A → A′ = UA U−1 + U∂ U−1. (16)
α α α ig α
The covariant derivative transforms as
D → D′ = UD U−1. (17)
α α α
In general, the covariant derivative of the tetrad should be zero:9)
∂ e +gω ej −Γλ e = 0, (18)
µ iν ij,µ ν νµ iλ
where Γλ is the Christoffel symbol and ω is the spin connection. (18) can be rewritten
νµ ij,µ
as
[D ,γ ] = Γλ γ , (19)
µ ν νµ λ
where
i
D ≡ ∂ +gω , ω ≡ ω γij. (20)
µ µ µ µ ij,µ
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In direct analogy, we require
[D ,Γ ] = Γγ Γ . (21)
α β βα γ
Using this, we can calculate [[D ,D ],Γ ] as
α β γ
[[D ,D ],Γ ] = [D ,[D ,Γ ]]−[D ,[D ,Γ ]]
α β γ α β γ β α γ
= {∂ Γδ +Γδ Γǫ −(α ↔ β)}Γ
α γβ ǫα γβ δ
= R(15)δ Γ , (22)
γ,αβ δ
where R(15)δ ≡ ∂ Γδ −∂ Γδ +Γδ Γǫ −Γδ Γǫ is the 15-dimensional Riemann
γ,αβ α γβ β γα ǫα γβ ǫβ γα
curvature tensor. We multiply both sides of (22) by Γ and consider the trace to obtain
ζ
1
R(15) = −2Tr(Γ [D ,D ]), Γαβ ≡ [Γα,Γβ]. (23)
γδ,αβ γδ α β 2
Therefore, the 15-dimensional Riemann curvature tensor is invariant under the transforma-
tion of U(x).
Let us now try to form an action with this formalism. We find that the following La-
grangian
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L = e ψ¯(2iΓ Dα −m)ψ + R(15) (24)
α
(cid:26) 16πG(15) (cid:27)
is invariant under the transformation of U(x), where R(15) is the 15-dimensional Ricci cur-
vature and G(15) is the 15-dimensional constant of gravitation. The Dirac’s equation is
(2iΓa∂ −m)ψ = 0. (25)
a
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On multiplying (25) by 2Γb∂ from the left, we obtain the high-dimensional Klein–Cordon
b
equation:
(∂ ∂i −∂(5)∂(5) +∂ ∂ij −∂(5)∂(5)i −iεijklγ ∂ ∂
i ij i 5 ij kl
+2iγij∂(5)∂ −2iεijklγ ∂ ∂(5) −2iεijklγ ∂(5)∂
ij kl i j kl ij
−2εijklγ γ ∂ ∂ +2iεijklγ ∂ ∂(5))ψ +m2ψ = 0, (26)
5 l i jk k ij k
where the summation of ∂ are limited to i < j and we represent the 15-dimensional coor-
ij
dinates corresponding to (9) as
xa = (x0,x1,x2,x3,x(5),x01,x02,x03,x12,x13,x23,x(5)0,x(5)1,x(5)2,x(5)3). (27)
Our model still cannot explain why the conformal symmetry is broken. However, it
is interesting that the extra dimension and the four-dimensions have a different spatial
structures compared with common high-dimensional models. Through conformal symmetry
breaking, our model would undergo the reduction from 15 down to four dimensions, and the
gauge-fixed tetrad must be
eiµe ν = gµν ei ejµ = ηij, i,j,µ,ν = 0,2,3,4. (28)
i µ
Note that the following tetrad also constructs four-dimensional spacetime:
eiµe ν = gµν ei ejµ = ηij, i,j = 11,12,13,14, µ,ν = 0,2,3,4. (29)
i µ
This would seem to suggest that there is a new internal degree of freedom in our spacetime.
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