$S$ Matrix for Gravitational Field. II. Local Measure; General Relations; Elements of Renormalization Theory

A method is suggested for obtaining the local measure in the generating functional for non-linear gauge fields within the framework of covariant $S$-matrix theory. The local measure is obtained explicitly for the case of the gravitational field. The measure obtained is proved to cancel all the divergences of the type ${\delta }^{\mathrm{}\left(4\right)\mathrm{}}\mathrm{}\left(0\right)\mathrm{}$ which arise in the theory due to its nonlinearity. It is proved that the value of the measure that is obtained is also required by the canonical formalism for gauge fields, and thus it secures the unitarity of the $S$ matrix. A new version of the canonical formalism for the gravitational field is given, which leads to the explicit Hamiltonian in terms of independent canonical variables. The quantization procedure in this approach to the canonical formalism is just the usual canonical quantization carried out in the Lorentz-covariant gauge. The new canonical formalism directly gives the value of the local measure in the Feynman integral. It is proved that besides securing the unitarity of the $S$ matrix and eliminating the strongest divergences, the local measure obtained secures the gauge independence of the $S$ matrix. This property results from the fact that the Jacobian of the gauge transformation of the field differentials is not equal to unity, contrary to the statement in previous works. The gauge transformation of the local measure exactly compensates for this Jacobian. The consequences of the gauge invariance of the theory are studied next. The complete set of generalized Ward identities for the Green's functions is obtained in the transverse gauge. The set of quantum equations of motion for the gravitational field is derived, and the problems connected with these equations are discussed. In the framework of the first-order formalism, the quantized Einstein equations are shown to take the form of the Schwinger-Dyson equations. A set of gauge relations for bare vertices is obtained. The analysis of the generalized Ward identities for the Green's functions at the threshold is given. In this connection the gravitational and fictitious-field wave-function renormalization constants $Z_2$ and ${\tilde{Z}}_2$ as well as the fictitious interaction vertex renormalization constant ${\tilde{Z}}_1$ and the infinite number of graviton vertex renormalization constants $Z_1, Z_1^{\mathrm{}\left(2\right)\mathrm{}}, \dots , Z_1^{\mathrm{}\left(n\right)\mathrm{}}, \dots$ are introduced. An infinite set of Ward relations for these renormalization constants is obtained, $Z_1{Z_2}^{-1\mathrm{}}={\tilde{Z}}_1{{\tilde{Z}}_2}^{-1\mathrm{}}$, ${\left(Z_1^{\mathrm{}\left(n\right)\mathrm{}}\right)}^{-1\mathrm{}}={Z_1}^{-n}{Z_2}^{n-1\mathrm{}}$, which leaves only two independent renormalization constants $Z_1$ and $Z_2$ and secures the gauge invariance of the renormalized theory. Further, the new invariance properties of the quantum theory of the gravitational field are investigated, which are connected with peculiarities of the symmetry breaking and with the existence of the dimensional Planck length. The "scale" invariance or "homogeneity" of the theory is proved, which leads by means of Euler's theorem to a new infinite set of relations, obeyed by the Green's functions. Analysis of these relations at threshold gives a new identity for the renormalization constants: $Z_1=Z_2$. Using the fact that the gravitational constant enters the theory only through the dimensionless space-time coordinates, the general off-mass-shell relations for the Green's functions are obtained from the "scale" identities. Possible anomalous singularities of the Green's functions are investigated. The "scale"-invariant regularization is introduced. It is also proved, using the "homogeneity" properties, that the renormalization constants do not depend-on the gauge. As a result, the threshold asymptotic behavior of all Green's functions of the theory is made finite by means of only one renormalization constant—that renormalizing the gravitational constant $\kappa$.