Chern–Simons States in SO(1,n)Yang–Mills Gauge Theory of Quantum Gravity

Abstract

We discuss a quantization of the Yang–Mills theory with an internal symmetry group $SO(1,n)$ treated as a unified theory of all interactions. In one-loop calculations, we show that Einstein gravity can be considered as an approximation to gauge theory. We discuss the role of the Chern–Simons wave functions in the quantization.

1. Introduction

We discuss a quantization of the Yang–Mills theory with an internal symmetry $SO(1,n)$. When some matter fields are added, then the model can be considered as a candidate for a unified theory of all interactions. The conceptual origin of such an approach to unification came from the paper of Yang [1]. Soon it was followed by Cho [2], Freund [3], MacDowell, and Mansouri [4]. The non-compact internal gauge group also appears in models of supergravity [5,6]. The $SO(1,n)$ model with $n\ge 13$ ($n=13$ in [7], $n=17$ in [8] for string-inspired higher-dimensional theories) could encompass gravity within the Georgi–Glashow [9] or Pati–Salam [10] extensions of the Standard Model (SM) based on $SO\left(10\right)$ internal symmetry group. It is shown in [7] that the Brout–Englert–Higgs symmetry-breaking mechanism will break $SO(1,13)$ symmetry to $SO(1,3)\times SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ in agreement with the SM phenomenology. $SO(1,17)$ is needed (according to [8]) in order to obtain a correct content of Weyl–Majorana spinors in models resulting from higher-dimensional superstring theories. In such a model, the $so(1,n)$ connection is decomposed into the $so(1,3)$ Lorentz connection, the vierbein and the compact gauge fields, which are supposed (after symmetry breaking) to describe weak and strong interactions of SM. The vierbein is used to describe the metric. Then, both the vierbein and the $so(1,3)$ connection are dynamical variables. The decomposition seems ad hoc. There is a scheme where the metric appears as a result of $SO(1,4)$ or $SO(2,3)$ symmetry breaking by scalar fields [11,12] (see also [13]). With the vierbein-connection decomposition, it has been noticed early in the study of the $SO(1,n)$ model [14] that the Yang–Mills Lagrangian can be decomposed into the (non-compact) $SO(1,3)$ Yang–Mills part, the Einstein–Hilbert Lagrangian (in the Palatini form), the cosmological term, and the $SO(n-4)$ (compact) Yang–Mills Lagrangian. Unfortunately, there is an arbitrariness in the coupling constants in this decomposition. Without a coupling to matter fields, the values of these constants (crucial for the phenomenology) cannot be fixed. In addition, the bare constants undergo renormalization (depending on the matter field content) in quantum theory. The project encounters many other problems in the composition of matter fields in the realization of SO(n-4) symmetry breaking for an agreement with the phenomenology of particle physics (for recent reviews, see [8,15]). Concerning the weak interactions, the crucial CP violation is described by a set of left and right Weyl fermions (mirror fermion problem [16]). The problem of the composition of fermions in SM may be related to gravity, as discussed in [17,18,19]. The formulation of classical and quantum gravity in Ashtekar variables [20,21,22,23,24] uses (anti)self-dual connections which are not invariant under spatial reflections. Their coupling to fermions violates parity invariance (on the other hand, determinants of Weyl fermions give parity non-invariant Chern–Simons terms [25,26]). A calculation of the effective action from a coupling of the $SO(1,3)$ connection to fermions can lead to new terms in the model of weak interactions.

The Yang–Mills theory with a non-compact symmetry group [14,19,27,28,29,30] is renormalizable, asymptotically free, and infrared-unstable. The infrared instability can be interpreted as a confinement of the $SO(1,4)$ connection [28], which could justify the classical limit of the Yang–Mills theory to its Einstein form. There are some difficult problems in the Yang–Mills interpretation of gravity. The non-compactness of the internal symmetry group has as a consequence that the energy is not bounded from below. This does not lead to difficulties in the Feynman integral approach to quantum Yang–Mills theory. However, the unitarity of the model is questionable (see a discussion of a possible solution of the problem in [19]). An approach based on the Euclidean formulation does not answer this question. The analytic continuation of $SO(1,n)$ to $SO(n+1)$ allows us to quantize the model on the lattice in the Euclidean framework [31], but the continuation to Minkowski space-time will lead to difficulties. In the canonical quantization, the negative energy could be interpreted as the gravitational energy, but this interpretation needs further investigation. There is still the question of the general coordinate invariance and the equivalence principle in gauge theories of gravity, which require an explanation. In three-dimensional gravity, the gauge invariance and the general coordinate invariance can be unified [32].

In this paper, we concentrate on the gauge field self-interaction. In Section 2, we show that if the Lagrangian is represented as a quadratic form plus the Chern–Simons divergence term; then, a Chern–Simons wave function ${\psi }_{CS}$ does not change in time; i.e., it is an eigenstate. As a consequence, the Schrödinger equation of the wave function $\psi ={\psi }_{CS}\chi$ can be expressed as a diffusion equation for $\chi$ with an (anti)self-dual drift. Then, in the leading order of the ℏ expansion, the solution of the Schrödinger equation is expressed by the solution of the self-duality equation. The higher orders can be obtained from a solution of a stochastic self-duality equation. In Section 3, we explain the method in the Abelian model. The Chern–Simons (CS) state is not invariant under the spatial reflection $\mathbf{x}\to -\mathbf{x}$. Such states may appear from an interaction with Weyl fermions in physical models of weak interactions, but we do not discuss such models (see [17,19]). We calculate correlation functions in CS states. The CS wave function is not normalizable [33]. We show in Abelian gauge theories that a rigorous calculation of correlation functions with a regularized CS wave function has a limit that can be considered as an analytic continuation of formal Gaussian calculations [34,35,36]. Such states are of interest in condensed matter physics in 2 + 1 dimensions (anyons [37], charged vortices with fractional statistics [38]). We expect that they may be relevant in models with gravitational and weak interactions [19]. The Chern–Simons Lagrangian defines a topological field theory related to quantum gravity [39].

Our main concern in this paper is the problem of the undue number of variables in the $SO(1,n)$ Yang–Mills model (Section 4) as compared with the Palatini form of the Einstein–Hilbert action. The $SO(1,n)$ Yang–Mills Lagrangian contains both the vierbeins and the $SO(1,3)$ connection as dynamical variables. It is suggested in [27,28,29] that, owing to the infrared confinement, the connection being confined does not play any dynamical role. In the standard Palatini approach, the functional integration over the Lorentz connection leads to the Einstein–Hilbert action. The procedure is equivalent to the expression of the connection in terms of vierbeins on the basis of classical equations of motion. In Section 5, we perform the functional integral in the ℏ (loop) expansion over the connection in the $SO(1,4)$ Yang–Mills model. As a result, we obtain the effective action, which consists of the classical action and the quantum one-loop correction. We show that if the square of the curvature of the Lorentz $SO(1,3)$ connection is neglected, then until the first order in the expansion in ℏ, we obtain the classical Einstein action (in Palatini form) plus the one-loop quantum correction of order ℏ. The result shows that in the loop expansion, we can consistently control the Palatini (infrared) approximation and the ultraviolet behavior of the model.

In Section 6 we discuss an expansion in ℏ in the $SO(1,n)$ Yang–Mills model by a stochastic perturbation of the self-duality equation. Then, the correlation functions of the Wilson loops can be calculated in regularized CS states in a perturbation expansion.

2. A Total Divergence Term in the Path Integral

The time evolution of the state $\psi \left(A\right)$ dependent on the variable A describing the system is given by the path integral formula (integration is over paths $A_s\left(A\right)$ starting from A, and $\psi$ is the initial condition for the Schrödinger equation)

$${\psi }_t\left(A\right)=\int \mathcal{D}Aexp\left({\displaystyle \frac{i}{\hslash }}{\int }_0^{t}d\mathbf{x}ds\mathcal{L}\left(A_s\right)\right)\psi \left(A_t\left(A\right)\right).$$

(1)

We assume that the Lagrangian can be expressed in the form

$$L=\int d\mathbf{x}\mathcal{L}=\int d\mathbf{x}({\displaystyle \frac{1}{2}}Q\left(A\right)\mathcal{M}Q\left(A\right)+{\partial }_{\mu }{J}^{\mu }),$$

(2)

where $\mathcal{M}$ does not depend on A. Then, the path integral (1) reads (we assume that $\int d\mathbf{x}\nabla \mathbf{J}=0$)

$$\begin{array}{c}{\psi }_t\left(A\right)=exp\left(-{\displaystyle \frac{i}{\hslash }}\int d\mathbf{x}{J}^0\left(A\right)\right)\int \mathcal{D}Aexp\left({\displaystyle \frac{i}{\hslash }}{\int }_0^{t}d\mathbf{x}ds{\displaystyle \frac{1}{2}}Q\left(A\right)\mathcal{M}Q\left(A\right)\right)\hfill \\ \times exp\left({\displaystyle \frac{i}{\hslash }}\int d\mathbf{x}{J}^0\left(A_t\left(A\right)\right)\right)\psi \left(A_t\left(A\right)\right).\hfill \end{array}$$

Let

$$W=i\int d\mathbf{x}{J}^0\left(A\right).$$

We express the initial state $\psi$ in the form

$$\psi =exp(-{\displaystyle \frac{1}{\hslash }}W)\chi \equiv {\psi }_{CS}\chi .$$

(3)

Then

$$\begin{array}{c}{\psi }_t\left(A\right)=exp(-{\displaystyle \frac{W}{\hslash }})\int \mathcal{D}Aexp\left({\displaystyle \frac{i}{\hslash }}{\int }_0^{t}d\mathbf{x}ds{\displaystyle \frac{1}{2}}Q\left(A_s\right)\mathcal{M}Q\left(A_s\right)\right)\chi \left(A_t\left(A\right)\right)\hfill \\ \equiv exp(-{\displaystyle \frac{W}{\hslash }}){\chi }_t\left(A\right).\hfill \end{array}$$

(4)

If the Jacobian $Z^{-1}={\displaystyle \frac{\partial Q}{\partial A}}\simeq const$, then we can change variables $A\to Q\left(A\right)$ so that

$${\chi }_t\left(A\right)=〈\chi \left(A_t\left(Q\right)\right)〉,$$

where Q is a Gaussian variable. We can express this change of variables as

$$Q\left(A_s\right)=\sqrt{i\hslash }{\partial }_s{B}_s,$$

(5)

where $B_s$ (the Brownian motion) has Gaussian distribution with the functional measure

$$\mathcal{D}Bexp\left(-{\displaystyle \frac{1}{2}}\int ds{\left({\partial }_{s}B\right)}^2\right).$$

If we choose $\chi =1$ then $\psi =exp(-{\displaystyle \frac{1}{\hslash }}W)$ and Equation (4) reads

$$\begin{array}{c}{\psi }_t\left(A\right)=\psi \left(A\right)\int \mathcal{D}Aexp\left({\displaystyle \frac{i}{\hslash }}{\int }_0^{t}d\mathbf{x}ds{\displaystyle \frac{1}{2}}Q\left(A\right)\mathcal{M}Q\left(A\right)\right).\hfill \end{array}$$

(6)

Hence, ${\psi }_t\left(A\right)=Z\psi \left(A\right)$ for $\psi =exp(-{\displaystyle \frac{W}{\hslash }})$, where the factor Z does not depend on A. If the Jacobian ${\displaystyle \frac{\partial Q}{\partial A}}\simeq const$ (as will be the case in our models), then the integral on the RHS of Equation (6) is a time-independent constant $det{\mathcal{M}}^{-{\displaystyle \frac{1}{2}}}$. It follows from Equation (6) that $\psi =exp(-{\displaystyle \frac{1}{\hslash }}W)$ does not change over time.

The Formula (3) can be considered either as a similarity transformation of states in quantum mechanics or as a result of a change of the canonical formalism resulting from the presence of an extra time derivative in the Lagrangian [40,41]. In both interpretations, the modified Schrödinger equation for the Lagrangian quadratic in velocity reads

$$i\hslash {\partial }_t\chi ={\displaystyle \frac{1}{2}}\Pi \Pi \chi +({\left({\psi }_{CS}\right)}^{-1}\Pi {\psi }_{CS})\Pi \chi ,$$

(7)

where $\Pi$ is the quantum realization of the canonical momentum.

3. Quantization of an Electromagnetic Field on a Lorentzian Manifold

We consider first the Lagrangian of the electromagnetic field

$$\begin{array}{c}\mathcal{L}=-{\displaystyle \frac{1}{4}}\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }{F}_{\mu \alpha }{F}_{\nu \beta }=-{\displaystyle \frac{1}{8}}\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }(F_{\mu \alpha }\pm {\displaystyle \frac{i}{2}}{\widetilde{ϵ}}_{\mu \alpha \sigma \rho }{F}^{\sigma \rho })(F_{\nu \beta }\pm {\displaystyle \frac{i}{2}}{\widetilde{ϵ}}_{\nu \beta \sigma \rho }{F}^{\sigma \rho })\hfill \\ \mp {\displaystyle \frac{i}{4}}{ϵ}_{\mu \alpha \sigma \rho }{F}^{\mu \alpha }{F}^{\sigma \rho }=-{\displaystyle \frac{1}{8}}\sqrt{-g}{F}_{\mu \alpha }^{\pm }{F}^{\pm \mu \alpha }\mp {\displaystyle \frac{i}{4}}{F}_{\mu \alpha }^{\ast }{F}^{\mu \alpha }\hfill \end{array}$$

(8)

defined on a Lorentzian manifold with the metric $g_{\mu \nu }$ (g denotes the determinant of the metric). In Equation (8),

$${\widetilde{ϵ}}_{\mu \alpha \sigma \rho }=\sqrt{-g}{ϵ}_{\mu \alpha \sigma \rho },$$

(9)

where on the LHS we have the tensor density $\widetilde{ϵ}$ and on the RHS we have the Levi-Civita antisymmetric symbol on the Minkowski space-time defined by ${ϵ}^{0123}=1$, $F_{\mu \nu }^{\ast }={ϵ}_{\mu \nu \alpha \beta }{F}^{\alpha \beta }$.

The last term in the Lagrangian (8) is of the form (2) as

$${\displaystyle \frac{i}{2}}{ϵ}_{\mu \alpha \sigma \rho }{F}^{\mu \alpha }{F}^{\sigma \rho }=i{\partial }_{\mu }\left(A^{\alpha }{ϵ}_{\mu \alpha \sigma \rho }{F}^{\sigma \rho }\right)\equiv i{\partial }_{\mu }{J}^{\mu }.$$

(10)

The self-dual variable $F^{\pm \mu \alpha }$ can be chosen as the Gaussian variable Q. As a consequence of Equation (6), the Chern–Simons state

$${\psi }_{CS}=exp(\pm {\displaystyle \frac{1}{2\hslash }}\int d\mathbf{x}{A}_j{ϵ}^{jkl}{\partial }_k{A}_l)$$

(11)

is an eigenstate. It does not change in time because the Jacobian ${\displaystyle \frac{\partial Q}{\partial A}}\simeq exp\int {\displaystyle \frac{\partial Q}{\partial A\partial A}}=const$. The term ${\partial }_{\mu }{J}^{\mu }$ does not depend on the metric (it is a topological invariant, the Chern–Simons term).

We prove $H{\psi }_{CS}=0$, independently constructing in the canonical formalism the Hamiltonian H for the quantum electromagnetic field on a globally hyperbolic manifold. Then, we can choose coordinates [42] such that the metric takes the form

$$d{s}^2=g_{00}dtdt-g_{jk}d{x}^{j}d{x}^k$$

(12)

(we use Greek letters for space-time indices and Latin letters for spatial indices). Then, the canonical momentum in the $A^0=0$ gauge is

$${\Pi }^j=g^{00}\sqrt{-g}{g}^{jk}{\partial }_0{A}_k.$$

(13)

The Hamiltonian is

$$H={\displaystyle \frac{1}{2}}\int {\displaystyle \frac{g_{00}}{\sqrt{-g}}}{g}_{jk}{\Pi }^j{\Pi }^k+{\displaystyle \frac{1}{4}}\int \sqrt{-g}{g}^{jk}{g}^{ln}{F}_{jl}{F}_{kn}$$

(14)

with the canonical momentum

$${\Pi }^j=-i\hslash {\displaystyle \frac{\delta }{\delta A_j}}.$$

(15)

We prove that on a globally hyperbolic manifold in the metric (12), $H{\psi }_{CS}=0$. This condition requires

$$\begin{array}{c}{\displaystyle \frac{1}{2}}\int g_{00}{(-g)}^{-{\displaystyle \frac{1}{2}}}{g}_{jk}{\Pi }^j{\Pi }^k{\psi }_{CS}=-\int g_{00}{(-g)}^{-{\displaystyle \frac{1}{2}}}{g}_{jk}{ϵ}^{jrl}{ϵ}^{kmn}{F}_{rl}{F}_{mn}{\psi }_{CS}\hfill \\ =-{\displaystyle \frac{1}{4}}\int \sqrt{-g}{g}^{jl}{g}^{rm}{F}_{jr}{F}_{lm}{\psi }_{CS}\hfill \end{array}$$

(16)

The general proof needs some strenuous calculations [41]. It is simple in the case of the orthogonal isotropic metric

$$d{s}^2=a_0^{2}d{t}^2-a^{2}d{\mathbf{x}}^2.$$

(17)

In this metric on the LHS of Equation (16)

$$\begin{array}{c}\int g_{00}{(-g)}^{-{\displaystyle \frac{1}{2}}}{g}_{jk}{\Pi }^j{\Pi }^k{\psi }_{CS}=-\int a_0{a}^{-1}{ϵ}^{jrl}{F}_{rl}{ϵ}^{jmn}{F}_{mn}{\psi }_{CS}\hfill \end{array}$$

(18)

whereas on the RHSk we have

$$\begin{array}{c}{\displaystyle \frac{1}{2}}\int \sqrt{-g}{g}^{jl}{g}^{rm}{F}_{jr}{F}_{lm}{\psi }_{CS}=\int a_0{a}^{-1}{F}_{kl}{F}_{kl}.\hfill \end{array}$$

(19)

Hence, (18) is equal to (19).

Using the transformation (4), we obtain the Schrödinger equation for $\chi$ (a version of Equation (7)):

$${\partial }_t\chi =\int d\mathbf{x}\left({\displaystyle \frac{i\hslash }{2}}{a}_0{a}^{-1}{\displaystyle \frac{{\delta }^2}{\delta \mathbf{A}{\left(\mathbf{x}\right)}^2}}\pm i{a}_0{a}^{-1}{ϵ}_{jkl}{\partial }_k{A}_l{\displaystyle \frac{\delta }{\delta A_j\left(\mathbf{x}\right)}}\right)\chi .$$

(20)

If the term $\hslash {\displaystyle \frac{{\delta }^2}{\delta \mathbf{A}{\left(\mathbf{x}\right)}^2}}$ in the limit $\hslash \to 0$ is neglected, then the solution of Equation (20) is expressed by the solution ${\mathbf{A}}_t\left(\mathbf{A}\right)$ (with the initial condition $\mathbf{A}$) of the self-duality equation

$$\begin{array}{c}{F}_{\mu \nu }^{(\pm )}=\pm {\displaystyle \frac{i}{2}}{\widetilde{ϵ}}_{\mu \nu \alpha \beta }{F}^{\pm \alpha \beta }=\pm {\displaystyle \frac{i}{2}}{ϵ}_{\mu \nu \alpha \beta }\sqrt{-g}{F}^{\pm \alpha \beta }.\hfill \end{array}$$

(21)

Then,

$${\chi }_t\left(\mathbf{A}\right)=\chi \left({\mathbf{A}}_t\left(\mathbf{A}\right)\right)$$

(22)

(written for a time-independent metric).

The transformation to Gaussian variables Q discussed in Equation (5) can be expressed as a stochastic equation corresponding to the ${\psi }_{CS}$ state. In the time-independent isotropic orthogonal metric, it reads (see refs. [43,44,45,46] for a general framework)

$$\begin{array}{c}d{A}_j\left(s\right)=\pm i{a}_0{a}^{-1}{ϵ}_{jkl}{\partial }_k{A}_{l}ds+\sqrt{{\displaystyle \frac{i\hslash a_0}{a}}}d{B}_j\left(s\right),\hfill \end{array}$$

(23)

where the covariance of the Brownian motion is

$$E\left[B_j(t,\mathbf{x})B_k(s,\mathbf{y})\right]=min(t,s)({\delta }_{jk}-{\partial }_j{\partial }_k{▵}^{-1})\delta (\mathbf{x}-\mathbf{y}).$$

(24)

Then, Equation (22) (after averaging over the Brownian motion) determines an exact solution of Equation (20). The solution of the self-duality Equation (21) gives the leading order behavior of $\chi$. The corrections can be calculated by means of an averaging over the Brownian motion.

The Chern–Simons wave function ${\psi }_{CS}$ is not square-integrable. We need $\chi$ in Equation (4) to decay sufficiently fast in order to make $\psi$ square-integrable. It turns out that $\psi$ may preserve some crucial properties of ${\psi }_{CS}$ after a multiplication by $\chi$ (treated as a regularization). As an example, we consider QED on the Minkowski space-time with

$$\chi =exp(-{\displaystyle \frac{\gamma }{2}}(\mathbf{A},\omega \mathbf{A})),$$

(25)

where (,) denotes the $L^2$ scalar product, $\omega =\sqrt{-▵}$ and ${\partial }_j{A}_j=0$. We calculate the correlation functions (the integral over $\mathbf{A}$ is well-defined for $\gamma \hslash >1$) at $t=0$ in the state $\psi ={\psi }_{CS}\chi$. We obtain

$$\begin{array}{c}(\psi ,A_j\left(\mathbf{p}\right)A_k\left({\mathbf{p}}^{\prime }\right)\psi )=\hslash \delta (\mathbf{p}+{\mathbf{p}}^{\prime })({\displaystyle \frac{\hslash \gamma }{{\hslash }^2{\gamma }^2-1}}({\delta }_{jk}-p_j{p}_k{p}^{-2})+{\displaystyle \frac{i}{{\hslash }^2{\gamma }^2-1}}{ϵ}_{jrk}{p}^r{p}^{-2}).\hfill \end{array}$$

(26)

Equation (26) is singular at $\gamma \hslash =1$. It makes sense at $\hslash \gamma =0$. We may treat the RHS of Equation (26) for $\gamma \hslash <1$ as an analytic continuation in the complex $\gamma$ plane of the integral on the LHS of Equation (26). A definition of the Gaussian integral of a function non-integrable in the Riemann sense by means of an analytic continuation appears in the ordinary Feynman oscillatory integrals with the Chern–Simons Lagrangian [34,35,36].

We calculate the loop expectation values in the regularized CS states for the loops $C_1$ and $C_2$

$$\begin{array}{c}(\psi ,{\int }_{C_1}\mathbf{A}d{\mathbf{x}}_1{\int }_{C_2}\mathbf{A}d{\mathbf{x}}_2\psi )=-\hslash 4\pi link(C_1,C_2)+\mathcal{O}(\hslash \sqrt{\hslash }),\hfill \end{array}$$

(27)

where $link(C_1,C_2)$ denotes the Gauss linking number. It follows from Equation (22) that after the time evolution, we obtain the linking number up to the terms of order $\hslash \sqrt{\hslash }$.

4. $\mathit{S}\mathit{O}(\mathbf{1},\mathit{n})$ Yang–Mills Gauge Theory on a Lorentzian Manifold

We consider an $o(1,n)$-valued connection 1-form

$${\Omega }_{\mu }=h_{\mu }^{a\alpha }{J}_{a\alpha }+{\omega }_{\mu }^{ab}{J}_{ab}+{\omega }^{\alpha \beta }{J}_{\alpha \beta }\equiv {\Omega }_{\mu }^{AB}{J}_{AB},$$

(28)

where $J_{AB}$ with $A,B=0,1,\dots ,n$ are the generators of rotations in $AB$ planes of the $O(1,n)$ group, $J_{AB}=-J_{BA}$ , $a,b=0,1,2,3$ , $\alpha =4,\dots ,n$, and $J_{\alpha \beta }$ are the generators of $SO\left(n\right)$ rotations. We have

$$[J_{AB},J_{CD}]={\eta }_{AD}{J}_{BC}+{\eta }_{BC}{J}_{AD}-{\eta }_{AC}{J}_{BD}-{\eta }_{BD}{J}_{AC}$$

where ${\eta }_{AB}$ is the Minkowski metric $(-,+,\dots ,+)$ in n + 1 dimensions.

We may introduce a dimensional parameter $\sigma$ rescaling $h\to \sigma h$. This is equivalent to a rescaling of the $J_{a5}$ generators ($J_{a4}\to \sigma J_{a4}$) in the commutators $J_{AB}$ so that

$$[J_{a4},J_{b4}]={\sigma }^2{J}_{ab},$$

(29)

where $J_{ab}$ satisfy the standard commutation relations of the Lorentz group for the rotations in the $ab$ plane. When $\sigma \to 0$, then $J_{a4}$ will contract to the generators of momenta $P_a$ of the Poincare group. The curvature of the connection $\Omega$ can be defined by the commutator of covariant derivatives $D_{\mu }={\partial }_{\mu }+{\Omega }_{\mu }$

$$[D_{\mu },D_{\nu }]={\mathcal{R}}_{\mu \nu }={\mathcal{R}}_{\mu \nu }^{AB}{J}_{AB},$$

$${\mathcal{R}}_{\mu \nu }^{AB}={\partial }_{\mu }{\Omega }_{\nu }^{AB}-{\partial }_{\nu }{\Omega }_{\mu }^{AB}+f_{CD;MN}^{AB}({\Omega }_{\mu }^{CD}{\Omega }_{\nu }^{MN}-{\Omega }_{\nu }^{CD}{\Omega }_{\mu }^{MN}),$$

(30)

where $f_{CD;MN}^{AB}$ are structural constants of the group $O(1,n)$ (after the rescaling (29)).

For notational simplicity, we restrict our discussion to $n=4$. $n>4$ adds the compact group $SO(n-4)$, which is supposed to describe internal symmetries. We decompose ${\mathcal{R}}_{\mu \nu }^{AB}$ as

$$\begin{array}{c}{\mathcal{R}}_{\mu \nu }^{ab}={\partial }_{\mu }{\omega }_{\nu }^{ab}-{\partial }_{\nu }{\omega }_{\mu }^{ab}+{\omega }_{\mu }^{ac}{\omega }_{\nu }^{db}{\eta }_{cd}-{\omega }_{\nu }^{ac}{\omega }_{\mu }^{db}{\eta }_{cd}+{\sigma }^2(h_{\mu }^a{h}_{\nu }^b-h_{\mu }^b{h}_{\nu }^a).\hfill \end{array}$$

(31)

For the term ${\mathcal{R}}_{\mu \nu }^{a4}{J}_{a4}\equiv R_{\mu \nu }^a{J}_{a4}$, we have

$$\begin{array}{c}{R}_{\mu \nu }^a={\partial }_{\mu }{h}_{\nu }^a-{\partial }_{\nu }{h}_{\mu }^a+{\omega }_{\mu }^{am}{\eta }_{mr}{h}_{\nu }^r-{\omega }_{\nu }^{am}{\eta }_{mr}{h}_{\mu }^r={\left(D_{\mu }h\right)}_{\nu }^a-{\left(D_{\nu }h\right)}_{\mu }^a\equiv {\mathcal{T}}_{\mu \nu }^a,\hfill \end{array}$$

(32)

where ${\left(D_{\mu }h\right)}_{\nu }^a={\partial }_{\mu }{h}_{\nu }^a+{\omega }_{\mu }^{mr}{\eta }_{mr}{h}_{\nu }^r$, ${\eta }_{ab}=(-1,+1,+1,+1)$ is the Minkowski metric and $\mathcal{T}$ is the torsion.

We consider as the non-Abelian generalization of Equation (8) the Lagrangian of the $SO(1,4)$ Yang–Mills field

$$\begin{array}{c}\mathcal{L}=-{\displaystyle \frac{1}{4{\lambda }^2}}\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }{\mathcal{R}}_{\mu \alpha }^{AB}{\mathcal{R}}_{\nu \beta }^{CD}{g}_{AB;CD}\hfill \\ =-{\displaystyle \frac{1}{8{\lambda }^2}}\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }({\mathcal{R}}_{\mu \alpha }^{AB}\pm {\displaystyle \frac{i}{2}}{\widetilde{ϵ}}_{\mu \alpha \sigma \rho }{\mathcal{R}}^{\sigma \rho ;AB})({\mathcal{R}}_{\nu \beta }^{CD}\pm {\displaystyle \frac{i}{2}}{\widetilde{ϵ}}_{\nu \beta \sigma \rho }{\mathcal{R}}^{\sigma \rho ;CD})g_{AB;CD}\hfill \\ \mp {\displaystyle \frac{i}{4{\lambda }^2}}{ϵ}_{\mu \alpha \sigma \rho }{\mathcal{R}}^{\mu \alpha :AB}{\mathcal{R}}^{\sigma \rho ;CD}{g}_{AB;CD}\equiv -{\displaystyle \frac{1}{8{\lambda }^2}}\sqrt{-g}{\mathcal{R}}_{\mu \alpha ;AB}^{\pm }{\mathcal{R}}^{\pm \mu \alpha ;AB}\mp {\displaystyle \frac{i}{4{\lambda }^2}}{\mathcal{R}}_{\mu \alpha }^{\ast AB}{\mathcal{R}}_{AB}^{\mu \alpha }\hfill \end{array}$$

(33)

defined on a Lorentzian manifold with the metric $g_{\mu \nu }$. In Equation (33), ${\widetilde{ϵ}}_{\mu \alpha \sigma \rho }=\sqrt{-g}{ϵ}_{\mu \alpha \sigma \rho }$ as in Equation (9), where on the LHS, we have the tensor density $\widetilde{ϵ}$ , ${\mathcal{R}}_{\mu \nu }^{\ast }={ϵ}_{\mu \nu \alpha \beta }{\mathcal{R}}^{\alpha \beta }$. ${\lambda }^2$ is the dimensionless coupling constant for the Yang–Mills theory. $g_{AB;CD}=Tr\left(J_{AB}{J}_{CD}\right)$ is the Killing form on the algebra $o(1,n)$.

The representation of the Lagrangian (33) leads to the Chern–Simons term

$${\displaystyle \frac{i}{2{\lambda }^2}}{ϵ}_{\mu \alpha \sigma \rho }{\mathcal{R}}^{\mu \alpha }{\mathcal{R}}^{\sigma \rho }=i{\partial }^{\mu }{J}_{\mu }.$$

(34)

Let us note that ${\partial }_{\mu }{J}^{\mu }$ does not depend on the metric (it is a topological invarian, the Chern–Simons term). It has the form

$$J_0={\displaystyle \frac{1}{{\lambda }^2}}{Y}_{CS},$$

where

$$Y_{CS}=-\int d\mathbf{x}Tr({\displaystyle \frac{1}{2}}\Omega d\Omega +{\displaystyle \frac{1}{3}}\Omega \wedge \Omega \wedge \Omega )$$

Specifically, in components in $SO(1,4)$,

$$\begin{array}{c}{Y}_{CS}={ϵ}_{jkl}({\displaystyle \frac{1}{2}}{h}_j^a{\partial }_k{h}_l^b{\eta }_{ab}+{\displaystyle \frac{1}{2}}{\omega }_j^{ab}{h}_k^c{h}_l^d{\eta }_{ac}{\eta }_{bd}\hfill \\ +{\displaystyle \frac{1}{2}}{\omega }_j^{ab}{\partial }_k{\omega }_l^{cd}{\eta }_{ac}{\eta }_{bd}+{\displaystyle \frac{1}{3}}{\eta }_{ra}{\eta }_{pb}{f}_{cd;mn}^{rp}{\omega }_j^{ab}{\omega }_k^{cd}{\omega }_l^{mn}),\hfill \end{array}$$

(35)

where $f_{cd;mn}^{ab}$ are the structure constants of $o(1,3)$. If $h_{\mu }^a$ is interpreted as the tetrad (vierbein), then we can see that the Lagrangian (33) contains the Einstein–Hilbert Lagrangian (in the Palatini formalism) as well as the Yang–Mills part (the square of the $SO(1,3)$ field strength), the cosmological term, and the square of the torsion. In fact, expanding the Lagrangian (33), we get

$$\begin{array}{c}\mathcal{L}=-{\displaystyle \frac{1}{4{\lambda }^2}}\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }{R}_{\mu \alpha }^{ab}{R}_{\nu \beta }^{cd}{\eta }_{ac}{\eta }_{bd}-{\displaystyle \frac{{\sigma }^2}{2{\lambda }^2}}\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }{R}_{\mu \alpha }^{ab}(h_{\nu }^c{h}_{\beta }^d-h_{\nu }^d{h}_{\beta }^c){\eta }_{ac}{\eta }_{bd}\hfill \\ -{\displaystyle \frac{{\sigma }^4}{4{\lambda }^2}}\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }(h_{\nu }^c{h}_{\beta }^d-h_{\nu }^d{h}_{\beta }^c)(h_{\mu }^a{h}_{\alpha }^b-h_{\mu }^b{h}_{\alpha }^a){\eta }_{ac}{\eta }_{bd}\hfill \\ -{\displaystyle \frac{{\sigma }^2}{4{\lambda }^2}}\sqrt{-g}{g}^{kj}{g}^{lm}({\left(D_{k}h\right)}_l^a-{\left(D_{l}h\right)}_k^a)({\left(D_{m}h\right)}_j^b-{\left(D_{j}h\right)}_m^b),\hfill \end{array}$$

(36)

where ${\left(D_{k}h\right)}_l^a={\partial }_k{h}_l^a+{\omega }_k^{ab}{h}_l^b$. Till now, in Equation (33), we did not specify the manifold and the metric $g_{\mu \nu }$ on this manifold. The unification of gravity and gauge theory is based on a choice of the metric, identifying the $J_{a4}$ part of the $so(1,n)$ connection (28) with the vierbein [7,8,14]. It can be seen from Equation (36) that the choice

$$g^{\mu \nu }=h_a^{\mu }{\eta }^{ab}{h}_b^{\nu },$$

(37)

where $h_a^{\mu }$ satisfy

$${\eta }^{ab}{h}_b^{\mu }{h}_{\mu }^c={\eta }^{ac}$$

(38)

leads in Equation (36) to the appearance (as the second term) of the Einstein Lagrangian in the Palatini formalism, if we choose ${\sigma }^2={\displaystyle \frac{{\lambda }^2}{8\pi G}}={\lambda }^2{m}_{PL}^2$, where $m_{PL}$ is the Planck mass. The term four-linear in h in Equation (36) is the cosmological term. The last term is a square of the torsion (32). As a consequence of the identification (37), the $SO(1,4)$ symmetry is broken. The Hamiltonian depends on the Ansatz (37). The Chern–Simons wave function does not depend on the metric. As follows from Section 2 and Section 3, the insertion of the metric (37) does not change the conclusion that ${\psi }_{CS}$ does not evolve in time (it is an eigenfunction of the Hamiltonian). The identification of the connection $h_{\mu }^a$ with the vierbein in the metric (37) has already been discussed by Townsend [14] in classical theory. In quantum theory (as discussed in Section 5), all four terms are separately renormalized. The bare Yang–Mills coupling $\lambda$ tends to zero at small distances (asymptotic freedom), but it can become large at large distances (confinement of the connection; see [27,28,29,30]). When discussing physical consequences of the choice of bare couplings and renormalization, we should consider the $SO(1,n)$ gauge field model as immersed in a unified GUT model. In such a case, the renormalization is changed by the couplings to other fields (with a preservation of the asymptotic freedom). The cosmological constant ${\displaystyle \frac{{\sigma }^4}{{\lambda }^2}}$ acquires a large (positive) contribution depending on the matter field content.

5. Functional Integration over the Lorentz Connection

We perform the functional integral (1) over $\omega$ (assuming that $\psi$ does not depend on $\omega$) in an expansion in ℏ. We obtain as a result the factor $exp\left({\displaystyle \frac{i}{\hslash }}{\mathcal{L}}_{eff}\left(h\right)\right)(1+O\left(\sqrt{\hslash }\right))$ defining the effective action. We treat $h_{\mu }^a$ as an external field and decompose $\omega =A+\sqrt{\hslash }\tilde{\omega }$ into the classical part A and quantum fluctuations $\tilde{\omega }$ as in the background field method. We write

$${\Omega }_{\mu }^{CD}{J}_{CD}=(A_{\mu }^{CD}+h_{\mu }^{CD})J_{CD}+\sqrt{\hslash }{\widehat{\omega }}_{\mu }^{ab}{J}_{ab}\equiv {\tilde{A}}_{\mu }+\sqrt{\hslash }\widehat{\omega },$$

(39)

where $A_{\mu }^{AB}$ has only the $ab$ components and $h_{\mu }^{AB}$ has only $a4$ components. We expand the Lagrangian (36) in $\sqrt{\hslash }\widehat{\omega }$. In order to cancel (in the expansion) the terms proportional to $\sqrt{\hslash }$, we choose $\tilde{A}$ in such a way that

$$\begin{array}{c}{\left({\tilde{\nabla }}^{\mu }\left(\tilde{A}\right){\mathcal{R}}_{\mu \nu }\left(\tilde{A}\right)\right)}^{ab}={\nabla }^{\mu }{\delta }_{ab;cd}{\mathcal{R}}_{\mu \nu }^{cd}+f_{CD;EF}^{ab}{\tilde{A}}^{\mu CD}{\mathcal{R}}_{\mu \nu }^{EF}=0\hfill \end{array}$$

(40)

This is a non-linear equation for $A_{\mu }^{ab}$. We can solve it by iteration. In the lowest order, (linear in $A_{\mu }^{ab}$), only $f_{4c;4e}^{ab}$ terms contribute to the last term in Equation (40). Then, Equation (40) reads (with the term quadratic in A neglected)

$$\begin{array}{c}{\displaystyle \frac{1}{\sqrt{-g}}}{\partial }^{\mu }\sqrt{-g}\left({\partial }_{\mu }{A}_{\nu }^{ab}-{\partial }_{\nu }{A}_{\mu }^{ab}+{\sigma }^2(h_{\mu }^a{h}_{\nu }^b-h_{\nu }^b{h}_{\mu }^a)\right)\hfill \\ +h^{\mu a}({\partial }_{\mu }{h}_{\nu }^b-{\partial }_{\nu }{h}_{\mu }^b+A_{\mu }^{bc}{h}_{\nu }^c-A_{\nu }^{bc}{h}_{\mu }^c)+h^{\mu a}{\Gamma }_{\nu \mu }^{\rho }{h}_{\rho }^b{\eta }_{ab}=0.\hfill \end{array}$$

(41)

When the first term (coming from ${\partial }^{\mu }{R}_{\mu \nu }^{ab}$) is neglected, then we obtain the equation for zero torsion

$$\begin{array}{c}{h}^{\mu a}\left({\partial }_{\mu }{h}_{\nu }^b-{\partial }_{\nu }{h}_{\mu }^b)+A_{\mu }^{bc}{h}_{\nu }^c-A_{\nu }^{bc}{h}_{\mu }^c.+{\Gamma }_{\nu \mu }^{\rho }{h}_{\rho }^b{\eta }_{ab}\right)=0\hfill \end{array}$$

(42)

This equation has the solution [47]

$$A_{abc}={\displaystyle \frac{1}{2}}(G_{bca}+G_{cab}-G_{abc}),$$

(43)

where $A_{abc}=h_a^{\mu }{A}_{\mu bc}$, $G_{cab}=h_{\mu c}{G}_{ab}^{\mu }$ and

$$G_{ab}^{\mu }=h_a^{\nu }{\partial }_{\nu }{h}_b^{\mu }-h_b^{\nu }{\partial }_{\nu }{h}_a^{\mu }.$$

(44)

It follows that by neglecting the derivative term (the first term in (41)) in the functional integration over $\omega$, we obtain in the leading order of the ℏ expansion the Einstein gravity. This could also be seen as a consequence of neglecting the first (Yang–Mills) term in the Yang–Mills action (36) and taking into account only the second (Palatini) term. In an exact $SO(1,4)$ quantum Yang–Mills theory at the one-loop level, we obtain in the functional integral the classical term $exp\left({\displaystyle \frac{i}{\hslash }}L\left(\tilde{A}\right)\right)$ (the tree approximation) and subsequently the one-loop contributions $det{({\tilde{\nabla }}^2{g}_{\mu \nu }+2i{\mathcal{R}}_{\mu \nu })}^{-{\displaystyle \frac{1}{2}}}$ and the Fadeev–Popov determinant, whose form depends on the choice of gauge (see the calculations in [48]). The determinants must be renormalized. It can be seen that at one loop, the $O(1,4)$ gauge theory is asymptotically free [27,28,29,30].

Specifically, we have

$$\begin{array}{c}{\mathcal{R}}_{\mu \nu }^{AB}(\tilde{A}+\sqrt{\hslash }\widehat{\omega })={\mathcal{R}}_{\mu \nu }^{AB}\left(\tilde{A}\right)+\sqrt{\hslash }({\nabla }_{\mu }{\widehat{\omega }}_{\nu }^{AB}-{\nabla }_{\nu }{\widehat{\omega }}_{\mu }^{AB}\hfill \\ +f_{CD;MN}^{AB}({\widehat{\omega }}_{\mu }^{CD}{\tilde{A}}_{\nu }^{MN}-{\widehat{\omega }}_{\nu }^{CD}{\tilde{A}}_{\mu }^{MN}))+O\left(\sqrt{\hslash }\right).\hfill \end{array}$$

(45)

It follows that until the terms of order ℏ,

$$\begin{array}{c}\int \mathcal{L}(\tilde{A}+\sqrt{\hslash }\widehat{\omega })=\int \mathcal{L}\left(\tilde{A}\right)+\sqrt{\hslash }\int \sqrt{-g}{\left({\tilde{\nabla }}^{\mu }{\mathcal{R}}_{\mu \nu }\right)}^{AB}{\widehat{\omega }}^{CD}{g}_{AB;CD}\hfill \\ +{\displaystyle \frac{\hslash }{2}}\int \sqrt{-g}{\widehat{\omega }}^{\mu AB}(-{\tilde{\nabla }}_{\alpha }{\tilde{\nabla }}^{\alpha }{g}_{\mu \nu }{g}_{AB;CD}+R_{\mu \nu }\left(h\right)g_{AB;CD}+{\tilde{\nabla }}_{\mu }^{AB}{\tilde{\nabla }}_{\nu }^{CD}\hfill \\ +2f_{AB;CD}^{EF}{\mathcal{R}}_{\mu \nu EF}\left(\tilde{A}\right)){\widehat{\omega }}^{\nu CD},\hfill \end{array}$$

(46)

where $R_{\mu \nu }\left(h\right)$ is the Ricci tensor of the metric (37) defined by the tetrad h and

$${\left({\tilde{\nabla }}_{\mu }{\widehat{\omega }}_{\nu }\right)}^{AB}={\partial }_{\mu }{\widehat{\omega }}_{\nu }^{AB}+f_{CD;MN}^{AB}{\tilde{A}}_{\mu }^{CD}{\widehat{\omega }}_{\nu }^{MN}+{\Gamma }_{\nu \mu }^{\beta }{\widehat{\omega }}_{\beta }^{AB}$$

The requirement of a vanishing of the $\sqrt{\hslash }$ term leads to Equation (40). There remains

$$\int \mathcal{L}=\int \mathcal{L}\left(\tilde{A}\right)+{\displaystyle \frac{\hslash }{2}}\int \widehat{\omega }\mathcal{M}\widehat{\omega }.$$

The integration over $\widehat{\omega }$ gives the determinant $det{\mathcal{M}}^{-{\displaystyle \frac{1}{2}}}$. It is convenient to choose the $\xi$ gauge leading to the factor

$$exp\left(-{\displaystyle \frac{1}{2\xi }}Tr\int \sqrt{-g}{\left({\tilde{\nabla }}_{\mu }{\widehat{\omega }}^{\mu }\right)}^2\right).$$

(47)

In particular, if $\xi =1$, then the ${\tilde{\nabla }}_{\mu }{\tilde{\nabla }}_{\nu }$ term in Equation (46) is vanishing. The determinant of $\mathcal{M}$ can be represented by means of a Feynman–Kac formula as in [48]. In the gauge (47), the Fadeev–Popov determinant is equal to the determinant of the Laplacian for vector fields $-{\tilde{\nabla }}_{\alpha }{\tilde{\nabla }}^{\alpha }{g}_{\mu \nu }{g}_{AB;CD}+R_{\mu \nu }\left(h\right)g_{AB;CD}$. We can conclude that for the renormalization of the determinant $\mathcal{M}$, we need the terms

$$\begin{array}{c}\int (c_1\sqrt{-g}{g}^{\mu \nu }{g}^{\alpha \beta }{\mathcal{R}}_{\mu \alpha }^{AB}\left(\tilde{A}\right){\mathcal{R}}_{\nu \beta }^{CD}\left(\tilde{A}\right)g_{AB;CD}+c_2\sqrt{-g}{R}^2\left(h\right)\hfill \\ +c_3\sqrt{-g}{R}_{\mu \nu }\left(h\right)R^{\mu \nu }\left(h\right)+c_4\sqrt{-g}),\hfill \end{array}$$

(48)

where $R^{\mu \nu }\left(h\right)$ is the Ricci tensor for the metric (37) and $R\left(h\right)$ is the scalar curvature. In the effective action, there still will be the logarithm of the Faddev–Popov determinant whose renormalization requires the same counterterms as in Equation (48) (with different constants). In Equation (48), besides the square of the $SO(1,4)$ curvature of the connection $\Omega$ appearing in the renormalization of the Yang–Mills theory on a flat manifold (the first term), we have the counterterms required for the renormalization of the determinant of the Laplacian $g_{\mu \nu }{\nabla }^2+R_{\mu \nu }$ for the vector fields on a curved background. Then, the $SO(1,4)$ gauge invariance of the model is broken as a consequence of the identification (37).

The one-loop expansion is easily generalized to $SO(1,n)$. We shall have the coupling of the $S0(1,4)$ gauge field to the $SO(n-4)$ Yang–Mills as in an $SO(1,4)\times O(n-4)$ Yang–Mills theory. If the $o(n-4)$ algebra is denoted as $L_{\alpha \beta }$ then we just make a change in Equation (28) for the background field expansion $\tilde{A}+\sqrt{\hslash }\widehat{\omega }\to \tilde{A}+Y^{\alpha \beta }{L}_{\alpha \beta }+\sqrt{\hslash }\widehat{\omega }$ where $Y_{\mu }$ is the $O(n-4)$ gauge field. It is also possible to achieve a non-trivial extension of internal and space-time symmetries by constructing the Yang–Mills theory with the group $O(p,q)$.

6. Chern–Simons States

The evolution Equation (7) for $SO(1,4)$ theory is a generalization of the one for the Hamiltonian (14). In the gauge ${\Omega }_0=0$, Equation (7) reads

$$\begin{array}{c}i\hslash {\partial }_t\chi =\int {\displaystyle \frac{1}{2}}{g}^{00}{\displaystyle \frac{1}{\sqrt{-g}}}{g}_{jk}({\Pi }^{jAB}{\Pi }^{kCD}{g}_{AB;CD}+2g_{AB;CD}\left({\Pi }^{jAB}{Y}_{CS}\right){\Pi }^{kCD})\chi .\hfill \end{array}$$

(49)

In the quantized model,

$${\Pi }^{jCD}=-i\hslash {\displaystyle \frac{\delta }{\delta {\Omega }_j^{CD}}}$$

(50)

or in components,

$${\Pi }^{ja}=-i\hslash {\displaystyle \frac{\delta }{\delta h_j^a}}$$

(51)

and

$${\Pi }^{jab}=-i\hslash {\displaystyle \frac{\delta }{\delta {\omega }_j^{ab}}}.$$

(52)

In the Hamiltonian evolution Equation (49), we need

$$\begin{array}{c}{\Pi }^{jCD}{Y}_{CS}=i\hslash {ϵ}^{jkl}{\mathcal{R}}_{kl}^{CD}=i\hslash {ϵ}^{jkl}\left({\partial }_k{\Omega }_l^{CD}-{\partial }_l{\Omega }_k^{CD}+f_{GH;EF}^{CD}{\Omega }_k^{GH}{\Omega }_l^{EF}\right)\hfill \end{array}$$

(53)

In components

$${\Pi }^{ja}{Y}_{CS}=i\hslash {ϵ}^{jkl}{R}_{kl}^a=i\hslash {ϵ}^{jkl}\left({\partial }_k{h}_l^a-{\partial }_l{h}_k^a+{\omega }_k^{ab}{h}_l^b-{\omega }_l^{ab}{h}_k^b\right)$$

(54)

and

$$\begin{array}{c}{\Pi }^{jab}{Y}_{CS}=i\hslash {ϵ}^{jkl}{\mathcal{R}}_{kl}^{ab}=i\hslash {ϵ}^{jkl}({\partial }_k{\omega }_l^{ab}-{\partial }_l{\omega }_k^{ab}\hfill \\ +{\omega }_k^{ac}{\omega }_l^{db}{\eta }_{cd}-{\omega }_l^{ac}{\omega }_k^{db}{\eta }_{cd}+{\sigma }^2(h_k^a{h}_l^b-h_k^b{h}_l^a))\hfill \end{array}$$

(55)

It follows already from the change of variables (5) that in the leading order of ℏ expansion of the solution of the Schrödinger equation with the initial condition ${\psi }_{CS}\chi$ is

$${\chi }_t(\Omega )=\chi \left({\Omega }_t(\Omega )\right)+\mathcal{O}\left(\sqrt{\hslash }\right),$$

(56)

where ${\Omega }_t(\Omega )$ is the solution of the self-duality equation (with the initial condition $\Omega$)

$${\mathcal{R}}_{\mu \nu }^{AB}=i{ϵ}_{\mu \nu \alpha \beta }{\mathcal{R}}^{\alpha \beta AB}\sqrt{-g},$$

(57)

i.e.,

$${\mathcal{R}}_{0j}^{AB}=i{ϵ}_{jkl}{g}^{km}{g}^{ln}{\mathcal{R}}_{mn}^{AB}\sqrt{-g}.$$

(58)

The self-duality equation in the gauge ${\Omega }_0=0$ can be expressed as

$$\begin{array}{c}{\partial }_t{\Omega }_j^{AB}=i{ϵ}_{jkl}{g}^{km}{g}^{ln}\sqrt{-g}({\partial }_m{\Omega }_n^{AB}-{\partial }_n{\Omega }_m^{AB}+f_{CD;EF}^{AB}{\Omega }_m^{CD}{\Omega }_n^{EF}).\hfill \end{array}$$

(59)

As the $\Pi \Pi$ part in the Schrödinger Equation (49) is of a higher order in ℏ, we confirm on the basis of the calculations (53)–(55) that the time evolution (56) generated by the classical flow

$$g^{00}{\displaystyle \frac{1}{\sqrt{-g}}}{g}_{jk}{g}_{AB;CD}\left({\Pi }^{kAB}{Y}_{CS}\right){\displaystyle \frac{\delta }{\delta {\Omega }_j^{CD}}}$$

(60)

is the solution of the Schrödinger Equation (49) in the leading order in ℏ. In order to derive an exact solution or calculate higher orders in ℏ, we need a stochastic equation (a generalization of Equation (23)).

As mentioned in the introduction, the Chern–Simons states arise from the total derivative terms in the Lagrangian, which appear in Yang–Mills as well as in other gauge theories of gravity (see [23,24]). The wave functions ${\psi }_{CS}$ are not normalizable. One can regularize these states in order to make them normalizable while still preserving their exceptional properties [40,41]. After the removal of the regularization, the result is the same as a formal (Gaussian) functional integral. In this way, we can perform a perturbative calculation of the Wilson loop variables

$$U_C=TrP(expi{\int }_C{\Omega }_{\mu }d{x}^{\mu }).$$

(61)

We could consider the correlation functions in the ${\psi }_{CS}$ “ground state”

$$\begin{array}{c}<U_{C_1}\dots .U_{C_n}>=\int {\psi }_{CS}^2{U}_{C_1}\dots .U_{C_n}.\hfill \end{array}$$

(62)

In the lowest order of calculations, we obtain

$$\begin{array}{c}<U_{C_1}\dots .U_{C_n}>=1+4\pi \hslash \sum _{kl}link(C_k,C_l)+\dots \hfill \end{array}$$

(63)

where $link(C_k,C_l)$ is the linking number of the curves $C_k$ and $C_l$ and the higher order terms involve expectation values of powers of $\Omega$ and higher orders of ℏ. The result (63) is a consequence of the formula

$$\begin{array}{c}\int \mathcal{D}\omega exp\left(\int {\omega }^j{ϵ}_{jkl}{\partial }^k{\omega }^l\right){\omega }_m\left(\mathbf{x}\right){\omega }_n\left(\mathbf{y}\right)\hfill \\ ={ϵ}_{rmn}(x_r-y_r){|\mathbf{x}-\mathbf{y}|}^{-3}\hfill \end{array}$$

(64)

and a similar formula for the tetrads $h_j^a$.

$Y_{CS}$ (35) is trilinear in $\Omega$ with the propagator ${\partial }^{-1}$. By elementary power counting, a perturbation expansion for a theory with the Lagrangian $Y_{CS}$ is renormalizable in three dimensions. This follows from the formula for the divergence index $\nu$ [45,49] for theories with a propagator $p^{-1}$ as $\nu =d-E{\displaystyle \frac{d-1}{2}}+V(n{\displaystyle \frac{d-1}{2}}-d)$ where d is the dimension, n is the degree of the interaction polynomial, E is the number of external lines, and V is the number of vertices. In our Chern–Simons model, $\nu =d-E$. It follows that we can calculate the correlation functions in a perturbation theory in the Chern–Simons states.

7. Summary and Outlook

We have discussed the special role of Chern–Simons wave functions in Yang–Mills theories. By means of a similarity transformation, they transform the Schrödinger equation into a diffusion-type equation with the (imaginary) diffusion constant proportional to ℏ and an ℏ-independent drift, which is a functional derivative of the Chern–Simons term. As a consequence, a solution of the Schrödinger equation in the leading order of ℏ is expressed by the solution of the self-duality equation. The exact solution of the Schrödinger equation can be expressed by the solution of a stochastic equation, which is a perturbation by noise of the self-duality equation. We have applied this framework to the Yang–Mills theory based on an internal $SO(1,n)$ symmetry group proposed by some authors. We discussed a quantum version of these models in an ℏ expansion. We have derived in detail the one-loop formula for the effective action in $SO(1,n)$ gauge theories, showing that at some limit, the quantum version contains the Einstein gravity interacting with the Yang–Mills connection. We discussed the Chern–Simons wave function and its role in the calculation of correlation functions of Wilson loop variables. The considered wave functions (which are a perturbation of the Chern–Simons wave functions) are of a special form. We believe that states of this form can arise in models describing an interaction of gravity with gauge theories and matter fields. In particular, an effective action arising from weak interactions can lead to Chern–Simons terms. In a prospective Study there remain interesting questions about the consequences of the asymptotic freedom of quantum gauge theories at small distances and the infrared instability (confinement) at large distances for the gravitational part of the $SO(1,n)$ Yang–Mills Lagrangian. The content of matter fields of GUT models could determine the couplings for $SO(n-4)$ symmetry breaking and possibly the CS-type states describing the CP violation in weak interactions. Then, the time evolution of such states (including their scattering amplitudes) could be determined by a solution of the self-duality equations. The gravity itself in the unified model would be described by states invariant under diffeomorphisms (e.g., certain diffeomorphism-invariant square integrable regularizations of the CS wave functions).