Spinor Representation of Conformal Group and Gravitational Model

TL;DR

This paper explores spinor representations of the conformal group and develops a gravitational model invariant under local transformations using a 15-dimensional vector spacetime derived from the adjoint representation of SO(2,4).

Contribution

It introduces a novel gravitational model based on spinor representations of the conformal group within a 15-dimensional spacetime framework.

Findings

Constructed a spacetime from 15-dimensional vectors in the adjoint of SO(2,4)

Developed a conformally invariant gravitational model

Demonstrated local invariance of the gravitational theory

Abstract

We consider spinor representations of the conformal group. The spacetime is constructed by the 15-dimensional vectors in the adjoint representation of $SO(2,4)$. On the spacetime, we construct a gravitational model that is invariant under local transformation.

Equations

$$({\gamma }_{0}U{)}^{†}=({\gamma }_{0}U{)}^{-1},$$

$$I,{\gamma }^i,i{\gamma }_5,i{\gamma }^{ij},{\gamma }_5{\gamma }^i.$$

$$M_{ij}\equiv \frac{i}{2}{\gamma }_{ij},P_i\equiv \frac{1}{2}({\gamma }_i+{\gamma }_5{\gamma }_i),$$

$$D\equiv \frac{i}{2}{\gamma }_5,K_i\equiv \frac{1}{2}({\gamma }_i-{\gamma }_5{\gamma }_i),$$

$$[D,K_i]=-i{K}_i,[D,P_i]=i{P}_i$$

$$[K_i,P_j]=2i{\eta }_{ij}D-2i{M}_{ij},[K_i,M_{jk}]=i({\eta }_{ij}{K}_k-{\eta }_{ik}{K}_j),$$

$$[P_i,M_{jk}]=i({\eta }_{ij}{P}_k-{\eta }_{ik}{P}_j),$$

$$[M_{ij},M_{kl}]=i({\eta }_{jk}{M}_{il}+{\eta }_{il}{M}_{jk}-{\eta }_{ij}{M}_{jl}-{\eta }_{jl}{M}_{ik}).$$

$$\psi \to {\psi }^{\prime }=U\psi ,$$

$$J_{ij}\equiv M_{ij},J_{i4}\equiv \frac{1}{2}(K_i-P_i),J_{i5}\equiv \frac{1}{2}(K_i+P_i),J_{54}\equiv D.$$

$$[J_{pq},J_{rs}]=i({\eta }_{qr}{J}_{ps}+{\eta }_{ps}{J}_{qr}-{\eta }_{pq}{J}_{qs}-{\eta }_{qs}{J}_{pr}),$$

$$U(x)=e^{-i{\epsilon }_a(x){\Gamma }^a},$$

$$2{\Gamma }^a$$

$$\mathrm{Tr}({\Gamma }^a{\Gamma }^b)=0a\ne b,$$

$$\mathrm{Tr}({\Gamma }^a{\Gamma }^a)=1a=0,8,9,10,12,13,14,$$

$$\mathrm{Tr}({\Gamma }^a{\Gamma }^a)=-1a=1,2,3,4,5,6,7,11.$$

$$X\to X^{\prime }=U^{-1}XU,X\equiv x_a{\Gamma }^a.$$

$$\mathrm{Tr}(XX)$$

$$e^{a\alpha }{e}_a{}^{\beta }=g^{\alpha \beta },e^a{}_{\alpha }{e}^{b\alpha }={\eta }^{ab},$$

$$e_a{}^{\alpha }(x){\Gamma }^a\to U^{-1}(x){\Gamma }^{\alpha }(x)U(x)=e_a{}^{\alpha }(x)U^{-1}(x){\Gamma }^{a}U(x)=e_a^{\prime }{}^{\alpha }(x^{\prime }){\Gamma }^a.$$

$$D_{\alpha }\equiv {\partial }_{\alpha }-ig{A}_{\alpha },A_{\alpha }\equiv A_{a\alpha }{\Gamma }^a,$$

$$A_{\alpha }\to A_{\alpha }^{\prime }=U{A}_{\alpha }{U}^{-1}+\frac{1}{ig}U{\partial }_{\alpha }{U}^{-1}.$$

$$D_{\alpha }\to D_{\alpha }^{\prime }=U{D}_{\alpha }{U}^{-1}.$$

$${\partial }_{\mu }{e}_{i\nu }+g{\omega }_{ij,\mu }{e}^j{}_{\nu }-{\Gamma }^{\lambda }{}_{\nu \mu }{e}_{i\lambda }=0,$$

$$[D_{\mu },{\gamma }_{\nu }]={\Gamma }^{\lambda }{}_{\nu \mu }{\gamma }_{\lambda },$$

$$D_{\mu }\equiv {\partial }_{\mu }+g{\omega }_{\mu },{\omega }_{\mu }\equiv \frac{i}{4}{\omega }_{ij,\mu }{\gamma }^{ij}.$$

$$[D_{\alpha },{\Gamma }_{\beta }]={\Gamma }^{\gamma }{}_{\beta \alpha }{\Gamma }_{\gamma }.$$

$$[[D_{\alpha },D_{\beta }],{\Gamma }_{\gamma }]$$

$$R_{\gamma \delta ,\alpha \beta }^{(15)}=-2\mathrm{Tr}({\Gamma }_{\gamma \delta }[D_{\alpha },D_{\beta }]),{\Gamma }^{\alpha \beta }\equiv \frac{1}{2}[{\Gamma }^{\alpha },{\Gamma }^{\beta }].$$

$$\mathcal{L}=e\left\{\bar{\psi }(2i{\Gamma }_{\alpha }{D}^{\alpha }-m)\psi +\frac{1}{16\pi G^{(15)}}{R}^{(15)}\right\}$$

$$(2i{\Gamma }^a{\partial }_a-m)\psi =0.$$

$$({\partial }_i{\partial }^i-{\partial }^{(5)}{\partial }^{(5)}+{\partial }_{ij}{\partial }^{ij}-{\partial }_i^{(5)}{\partial }^{(5)i}-i{\epsilon }^{ijkl}{\gamma }_5{\partial }_{ij}{\partial }_{kl}$$

$$+2i{\gamma }^{ij}{\partial }^{(5)}{\partial }_{ij}-2i{\epsilon }^{ijkl}{\gamma }_{kl}{\partial }_i{\partial }_j^{(5)}-2i{\epsilon }^{ijkl}{\gamma }_{kl}{\partial }^{(5)}{\partial }_{ij}$$

$$-2{\epsilon }^{ijkl}{\gamma }_5{\gamma }_l{\partial }_i{\partial }_{jk}+2i{\epsilon }^{ijkl}{\gamma }_k{\partial }_{ij}{\partial }_k^{(5)})\psi +m^2\psi =0,$$

Full text

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-1/2 particles and antiparticles,[5] contains a considerable amount of physical 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 $4\times 4$ transform matrix $U(x)$ that satisfies

[TABLE]

has the generators:

[TABLE]

$U(x)$ describes the conformal transformations. In fact, the following[6, 7]

[TABLE]

satisfies the conformal group

[TABLE]

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 $\mu$ and $\nu$, are used to label four-dimensional spacetime vectors. $U(x)$ contains within it local Lorentz transformations. Let a spinor $\psi$ transform as

[TABLE]

then (Spinor Representation of Conformal Group and Gravitational Model) is spinor representations of the conformal group. If we rewrite (Spinor Representation of Conformal Group and Gravitational Model) as

[TABLE]

$J_{pq}$ generates the algebra of $SO(2,4)$:

[TABLE]

where $\eta_{pq}=(1,-1,-1,-1,1)$ and $p,q,r,s=0,1,\cdots 5$.

$U(x)$ can be also written as

[TABLE]

where

[TABLE]

We have projected out the unit matrix $I$. These gamma matrices satisfy

[TABLE]

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 representation, that is, the 15-dimensional coordinates $x_{a}(a=0,1,\cdots 14)$ transform as

[TABLE]

Note that $\gamma_{0}U$ with the generators (2) is the transform matrix of $U(4)$. The space coordinates 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:

[TABLE]

Then, the metric of the tangent space is $\eta_{ab}=(1,-1,-1,-1,-1,-1,-1,-1,1,1,1,-1,1,1,1)$. Let us define a 15-dimensional tetrad

[TABLE]

where indices $a,b,c$ are 15-dimensional vectors in flat space and $\alpha,\beta,\gamma$ are 15-dimensional vectors in curved space. The 15-dimensional tetrad transforms linearly as

[TABLE]

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

[TABLE]

where we introduce gauge fields $A_{\alpha}$, each transforming as

[TABLE]

The covariant derivative transforms as

[TABLE]

In general, the covariant derivative of the tetrad should be zero[9]:

[TABLE]

where $\Gamma^{\lambda}\,_{\nu\mu}$ is the Christoffel symbol and $\omega_{ij,\mu}$ is the spin connection. (18) can be rewritten as

[TABLE]

where

[TABLE]

In direct analogy, we require

[TABLE]

Using this, we can calculate $[[D_{\alpha},D_{\beta}],\Gamma_{\gamma}]$ as

[TABLE]

where $R^{(15)\delta}\,_{\gamma,\alpha\beta}\equiv\partial_{\alpha}\Gamma^{\delta}\,_{\gamma\beta}-\partial_{\beta}\Gamma^{\delta}\,_{\gamma\alpha}+\Gamma^{\delta}\,_{\epsilon\alpha}\Gamma^{\epsilon}\,_{\gamma\beta}-\Gamma^{\delta}\,_{\epsilon\beta}\Gamma^{\epsilon}\,_{\gamma\alpha}$ is the 15-dimensional Riemann curvature tensor. We multiply both sides of (22) by $\Gamma_{\zeta}$ and consider the trace to obtain

[TABLE]

Therefore, the 15-dimensional Riemann curvature tensor is invariant under the transformation of $U(x)$.

Let us now try to form an action with this formalism. We find that the following Lagrangian

[TABLE]

is invariant under the transformation of $U(x)$, where $R^{(15)}$ is the 15-dimensional Ricci curvature and $G^{(15)}$ is the 15-dimensional constant of gravitation. The Dirac’s equation is

[TABLE]

On multiplying (25) by $2\Gamma^{b}\partial_{b}$ from the left, we obtain the high-dimensional Klein–Cordon equation:

[TABLE]

where the summation of $\partial_{ij}$ are limited to $i<j$ and we represent the 15-dimensional coordinates corresponding to (9) as

[TABLE]

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

[TABLE]

Note that the following tetrad also constructs four-dimensional spacetime:

[TABLE]

This would seem to suggest that there is a new internal degree of freedom in our spacetime.