The First Integral of Einstein’s Equations and Covariant Quantization of General Relativity

The first integral of mathematical motions is energy. Therefore, our first task is to express an expression for the energy of spacetime that follows from Einstein’s original equations:

We use the reference formalism in Einstein’s theory [1] in the spinor representation [2,3], in which the space-time geometry is given by the Hermitian spin vector $e_{A{A}^{\prime }}^{\alpha }\alpha =0,1,2,3,$,$A,A^{\prime }=0,1$. The connection with the metric tensor is given by the relations:

Here is an antisymmetric spin tensor, with the help of which the spin indices are lowered and raised:

The spin coefficients $e_{A{A}^{\prime }}^{\alpha }$ relate the vector and spin representations of tensor space. For example, the Einstein equation in a mixed representation:

In some region $\Omega$ of spacetime, we take an arbitrary Dirac bi-spinor field and introduce the Dirac operator: in which the covariant derivatives of spinors are determined by the self-dual and anti-self-dual complex connections, which are consistent with the metric spin coefficients: where the Christoffel symbols in the absence of torsion are symmetric:

We also introduce in the space of bi-spinors the Hermitian scalar product

It is easy to check that the Dirac operator Equation (8) is Hermitian with respect to this scalar product, given that:

Let’s consider the square of this operator:

We divide the double covariant derivative into symmetric and antisymmetric parts:

(the same for the second component). The symmetric parts are the Beltrami-Laplace operators in the space of spinor fields, which are also Hermitian with respect to Equation (13). The antisymmetric parts for both components are: where are the self-dual and anti-self-dual curvature tensors, which together form the Riemann-Christoffel tensor, and the Ricci tensor:

To complete the construction, we introduce a bi-spinor ${\eta }_{\perp }$ orthogonal to $\eta$,

Obviously,

By collapsing Equation (17) with ${\eta }_{\perp }$, for example, we transform expression Equation (20) as follows:

Here we have added the second term in parentheses "by hand." It is equal to zero due to the symmetry properties (${\overline{F}}_{\alpha \beta C^{\prime }{A}^{\prime }}$ is symmetric with respect to the pair of indices $C^{\prime }{A}^{\prime }$ and antisymmetric with respect to $\alpha \beta$). Expression Equation (28), according to Equation (25), reduces to a certain convolution of the Ricci tensor, and therefore, in accordance with Einstein’s equations Equation (1), is proportional to the energy-momentum tensor of the matter fields:

Putting it all together and introducing the operator where we will write the result of the constructions in the form:

According to Equation (26), this means that the operator W is proportional to the unit operator:

Operator equation Equation (33) is the desired first integral of the Einstein equations, and the constant $\pm M^2$ is its numerical value. The main result is that the derivatives of the spin coefficients of the metric, as well as the matter fields, enter into expression Equation (33) in first-order combinations ($A_{\alpha }^{AB},{\overline{A}}_{\alpha }^{A^{\prime }{B}^{\prime }}$- for the metric), unlike the Einstein equations themselves, which contain second-order derivatives.

The physical meaning of the constant $\pm M^2$ becomes clear from comparing integral Equation (33) with the gravitational constraints H,H_k, arising after $3+1$ splitting of the space-time metric in the formalism of Arnowitt, Deser, and Misner [6,7]. The ten Einstein equations in this formalism include six dynamical equations for the $3D$ metric of a spatial section and four conditions H=H_k on the initial data for these equations. The constraints themselves are integrals of the equations of motion. We will add that they are partial integrals. The result of this paper shows that covariance does not exclude a nonzero value for the first integral of the dynamical equations. Being an eigenvalue of the covariant operator W, the first integral is an invariant functional of the metric tensor and its first derivatives in the considered region $\Omega$ of spacetime. The derivation of identity Equation (33) in this paper reproduces Witten’s constructions [4,5] in his proof of the theorem on the positivity of the gravitational field energy.

The constant $\pm M^2$ is not defined in classical general relativity. As for the energy of an electron in a hydrogen atom, restrictions on the possible values of $\pm M^2$ can be expected in quantum theory. The quadratic dependence of expression Equation (33) on the first derivatives of the dynamic variables and its covariance compel us to seek a formulation of quantum theory that is not associated with the canonical form of action.

As a simple exercise, we will implement this possibility using the example of a particle of mass $m=1$ with one degree of freedom q and energy

In canonical quantum theory we have the commutation relation and its realization in the space of wave functions:

This means that there is no definite classical trajectory of the particle $q\left(t\right)$ anymore. However, we can talk about the distribution of possible trajectories during the quantum transition, similar to the formalism of the Feynman functional integral [8]. Let is the solution of the Schrödinger equation on the interval $t\in [0,T]$, where $\chi (t,q)$ is, in general, a complex function. We divide the time interval into small "pieces" $\epsilon$: $T=N\epsilon$, and compose a functional of the particle trajectory, which in the limit $\epsilon \to 0$ has the form:

Note that the exponential functional Equation (38) is singular as $\epsilon \to 0$. However, if we understand it as the limit of the product Equation (37), and take into account that the variational derivative of the functional is related to the partial derivative of its multiplicative representation by the relation [8] we obtain a meaningful implementation of the velocity operator in the form of a variational derivative:

In order to fully determine the action of the energy operator on the wave functional Equation (38) (potential energy $V\left(q\right)$ is the multiplication operator), we must also consider the square of the velocity operator:

For the first term we have,

Given that $\epsilon \delta \left(0\right)=1$ in the multiplicative representation, the last expression is also well defined. Moreover, for the wave function Equation (36) at each moment of time t, we obtain the usual stationary Schrödinger equation, which determines the particle’s energy spectrum.

Returning to the integral of Einstein’s equations Equation (33), we set the task of preserving covariance at each stage of its construction during its quantization. This is a particularly sensitive issue given the special significance of the time parameter in quantum theory, which is manifested, in particular, in the singular nature of the wave functional Equation (38). In the quantum theory of gravity, we must construct the wave functional $\Psi \left[e\left(x\right),\varphi \left(x\right)\right]$ on the space of world histories of geometry and matter fields in the domain $\Omega$. We expect the functional to be singular, but only in one coordinate dimension. Covariance means that this direction (the direction of evolution) can be chosen arbitrarily. Therefore, we assume that if an arbitrary coordinate system $x^{\alpha }$ is specified in the domain $\Omega$, then it is legitimate to select any of them $x^a\in \left[0,X^a\right]$ as the evolution parameter. The remaining three coordinates $x^{\tilde{a}}$ ($x^{\alpha }=\left(x^a,x^{\tilde{a}}\right)$ ) serve to number the physical degrees of freedom. We will enclose the latter in square brackets, $e_{A{A}^{\prime }}^{\alpha }\left(x\right)=e_{A{A}^{\prime }}^{\alpha }\left(x^a,\left[x^{\tilde{a}}\right]\right)$, which means the $3D$ infinite dimension of this numbering. Along with this, we will also introduce four "quanta" of time ${ϵ}^a$ for the multiplicative representation of the wave functional in each direction of evolution. Now we are ready to realize the partial derivatives ${\partial }_a{e}_{A{A}^{\prime }}^{\alpha }\left(x\right)$ of the spin coefficients in the form of variational derivatives with an appropriate choice of singular exponential representation of the wave functional:

Here, the exponent is a curvilinear integral of the second kind, which is calculated along an arbitrary curve ${\gamma }^{\alpha }\left(\tau \right)$ in the domain $\Omega$. Additional integration is implied for the three coordinates in square brackets. Then the result of the operator in Equation (43) is as follows:

The factor on the right now replaces the partial derivatives ${\partial }_a{e}_{A{A}^{\prime }}^{\alpha }\left(x\right)$, which make up the classical connections $A_{\alpha }^{AB},{\overline{A}}_{\alpha }^{A^{\prime }{B}^{\prime }}$. It must have the appropriate transformation properties to ensure the covariance of the quantum theory. This is achieved by taking the partial derivative ${\chi }_a$ in Equation (45) with respect to the covariant components $e_{\alpha }^{A{A}^{\prime }}$ of the spin coefficients. The functions ${\chi }_a$ themselves are transformed as derivatives with respect to the corresponding coordinates $x^a$. They are such, since covariance requires independence from the choice of coordinates, i.e. independence from the choice of the curve ${\gamma }^{\alpha }\left(\tau \right)$. It follows that where the partial derivative is calculated taking into account the dependence of the fundamental dynamic variables on $x^a$ at a given point. The repeated action of the operator $\widehat{{\partial }_a{e}_{A{A}^{\prime }}^{\alpha }}\left(x\right)$ on $\partial {\chi }_a/\partial e_{\alpha }^{A{A}^{\prime }}\left(x^a,\left[x^{\tilde{a}}\right]\right)$ is determined following the logic of formulas Equations (41) and (42). The partial derivative operators of matter fields are defined similarly.

In this work we do not attach any physical meaning to the wave functional $\Psi$ and the potential U. Only the surface integral has meaning along the boundary of the region under consideration, which determines the boundary value of the wave function:

Formula Equation (48) can be applied to two boundary value problems. The first is the Euclidean quantum theory with one boundary, which defines the no-boundary Hartle-Hawking wave function [9]. The second is the evolution between two boundary spatial sections.

Thus, we have defined the operator $\widehat{W}$, which acts simultaneously in the space of bi-spinors $\eta$ and wave functionals $\Psi$, having the form Equation (44). As a result, we obtain the covariant wave equation which must be satisfied for any $\psi$. It should be considered as a functional-differential equation with respect to the potential $U\left(x^a,e\left(x^a,\left[x^{\tilde{a}}\right]\right),\varphi \left(x^a,\left[x^{\tilde{a}}\right]\right)\right)$. Equation (49) and its solutions are non-singular. Following the work [10], we will call the first integral $\pm M^2$ the proper mass of the universe.