Detailed discussions and calculations of quantum Regge calculus of Einstein-Cartan theory

This article presents detailed discussions and calculations of the recent paper “Quantum Regge calculus of Einstein-Cartan theory” in [9]. The Euclidean space-time is discretized by a four-dimensional simplicial complex. We adopt basic tetrad and spin-connection fields to describe the simplicial complex. By introducing diffeomorphism and local Lorentz invariant holonomy fields, we construct a regularized Einstein-Cartan theory for studying the quantum dynamics of the simplicial complex and fermion fields. This regularized Einstein-Cartan action is shown to properly approach to its continuum counterpart in the continuum limit. Based on the local Lorentz invariance, we derive the dynamical equations satisfied by invariant holonomy fields. In the mean-field approximation, we show that the averaged size of 4-simplex, the element of the simplicial complex, is larger than the Planck length. This formulation provides a theoretical framework for analytical calculations and numerical simulations to study the quantum Einstein-Cartan theory.

Figure 1

We sketch a 2-simplex (triangle) $h(x)$ formed by three edges $l_{\mu }(x)=a{e}_{\mu }(x)$, $l_{\rho }(x+a_{\mu })=a{e}_{\rho }(x+a_{\mu })$ and $l_{\nu }(x+a_{\mu })=a{e}_{\nu }(x+a_{\nu })$ [$a=1$] connecting three vertexes $x$, $x+a_{\mu }$ and $x+a_{\nu }$. Assuming three edge spacings $a_{\mu }$, $a_{\nu }$ and $a_{\rho }$ (41) are so small that the geometry of the interior of each 4-simplex and its subsimplex (3- and 2-simplex) is approximately flat, we assign a local Lorentz frame to each 4-simplex. On the local Lorentz manifold ${\xi }^a(x)$ at a space-time point “$x$”, we sketch a closed parallelogram ${\mathcal{C}}_P(x)$ lying in the 2-simplex $h(x)$. Its two edges $e_{\mu }(x)$ and $e_{\nu }^{†}(x)$ are two edges of the 2-simplex $h(x)$, and other two edges (dashed lines) ${\overline{e}}_{\mu }^{†}(x+a_{\nu })$ and ${\overline{e}}_{\nu }(x+a_{\mu })$ are parallel transports of $e_{\mu }^{†}(x)$ and $e_{\nu }^{†}(x)$ along $\nu$ and $\mu$ directions, respectively [see Eqs. (46, 47, 62, 63)]. Each 2-simplex in the simplicial complex has a closed parallelogram lying in it. Group-valued gauge fields $U_{\mu }(x)$ and $U_{\nu }^{†}(x)$ are, respectively, associated to edges $e_{\mu }(x)$ and $e_{\nu }^{†}(x)$ of the 2-simplex $h(x)$, as indicated. The fields $e_{\rho }\equiv e_{\rho }(x+a_{\mu })$ and $U_{\rho }\equiv U_{\rho }(x+a_{\mu })$ are associated to the third edge ($x+a_{\mu }$, $\rho$) of the 2-simplex $h(x)$. The group fields ${\overline{U}}_{\nu }(x+a_{\mu })$ and ${\overline{U}}_{\mu }^{†}(x+a_{\nu })$ indicate the parallel transports of $U_{\nu }^{†}(x)$ and $U_{\mu }(x)$ [see Eqs. (48, 49, 82, 83)] for the zero curvature case. Note that the point ($x+a_{\mu }+a_{\nu }$) is not a vertex of the simplicial complex, points: ($x-a_{\mu }$), ($x-a_{\nu }$), ($x+a_{\mu }+a_{\mu }$), ($x+a_{\mu }-a_{\rho }$), and ($x+a_{\nu }+a_{\rho }$), which are not shown in the sketch, are not vertexes of the simplicial complex as well. Parallel transports ${\overline{e}}_{\nu }(x+a_{\mu })$ and ${\overline{e}}_{\mu }^{†}(x+a_{\nu })$, as well as ${\overline{U}}_{\nu }(x+a_{\mu })$ and ${\overline{U}}_{\mu }^{†}(x+a_{\nu })$ are not associated to any edge of the simplicial complex. Throughout this article, the notations $\overline{e}$ and $\overline{U}$ indicates parallel transports that are not associated to any edge of the simplicial complex.

Figure 2

The smallest holonomy field along a closed triangle path of the 2-simplex $h(x)$: the anti-clocklike orientation $X_h(v,U)$ [left]; the clocklike orientation $X_h^{†}(v,U)$ [right].

Figure 3

We sketch a graphic representation of the dynamical Eq. (165) for the general holonomy field $X_{\mathcal{C}}$ (134). The diagram in the left-hand side of the graphic equation indicates the first term in Eq. (165). The first and second diagrams in the right-hand side of the graphic equation, respectively, indicate the third and second terms in Eq. (165). We indicate the edge $l_{\mu }$, where the local gauge transformation is made. In the right-hand side of graphic equation, the summation over all 2-simplices $h(l)$ associated to this edge $l_{\mu }$ is made.

Figure 4

We sketch a graphic representation of the dynamical Eq. (165) for the smallest holonomy field $X_h(v,U)$ (120). The diagram in the left-hand side of the graphic equation indicates the first term in Eq. (165). The first and second diagrams in the right-hand side of the graphic equation, respectively, indicate the third and second terms in Eq. (165). Note that $A$ and $A^{\prime }$ are the same vertex, so are $B$ and $B^{\prime }$. We indicate the edge $l_{\mu }$, where the local gauge transformation is made. In the right-hand side of the graphic equation, the summation over all 2-simplices $h(l)$ associated to this edge $l_{\mu }$ is made.

Figure 5

We sketch a graphic representation of the dynamical Eq. (165) for the field $X_{\mathcal{L}}$ (151). The diagram in the left-hand side of the graphic equation indicates the first term in Eq. (165). The first and second diagrams in the right-hand side of the graphic equation, respectively, indicate the third and second terms in Eq. (165). Note that $A$ and $A^{\prime }$ are the same vertex, so are $B$ and $B^{\prime }$. We indicate the edge $l_{\mu }$, where the local gauge transformation is made. We also indicate the fermion field $\psi (x_N)$ at staring point $x_N$ and the fermion field $\overline{\psi }(x_1)$ at ending point $x_1$ of the path $\mathcal{L}$. In the right-hand side of the graphic equation, the summation over all 2-simplices $h(l)$ associated to this edge $l_{\mu }$ is made.

Figure 6

In the Planck unit $a=1$, the approximate free energy (179) as a function of the mean-field value $M_h$ (166) is plotted for selected values $\chi =0.03$, 0.3, 3. The minimal values of the approximate free energy ${\mathcal{F}}_{\mathrm{EC}}^{\mathrm{app}}$ locate at the nonvanishing mean-field value $M_h^{*}$. The minimal locations are $M_h^{*}(\chi =0.03)\approx 7.9$, $M_h^{*}(\chi =0.3)\approx 2.1$, $M_h^{*}(\chi =3)\approx 0.8$.