Bogolyubov Variables for an Interacting Field System

Abstract

This is the continuation of paper [1], which contained the basic ideas of quantization for nonlinear systems in the neighborhood of nontrivial classical solutions of the equations of motion. Quantization is performed using Bogolyubov variables; this combines an accurate account of conservation laws and perturbation theory and avoids the problem of zero modes. In this paper, the quantization of interacting gravitational and scalar fields is performed using this method.

Expansion of the Gravitational Field Hamiltonian

Let us expand the gravitational field Hamiltonian in a series,

$$H = \frac{1}{{\sqrt \gamma }}\left( {{{\pi }_{{st}}}{{\pi }^{{st}}} - \frac{1}{2}{{\pi }^{2}}} \right) - \sqrt \gamma R{\kern 1pt} .$$

The leading order depends on the classical part alone and is a $c$-number,

$${{H}_{0}} = \frac{1}{{\sqrt F }}\left( {{{F}_{{{{n}_{{st}}}}}}F_{n}^{{st}} - \frac{1}{2}F_{n}^{2}} \right) - \sqrt F R(F),$$

and the next-to-leading order is linear with respect to quantum additions and derivatives with respect to them,

$$\begin{gathered} {{H}_{1}} = \frac{1}{{\sqrt F }}\left( {\frac{1}{2}\left( {{{F}_{{{{n}_{{kl}}}}}}F_{n}^{{kl}} - \frac{1}{2}F_{n}^{2}} \right)} \right.{{F}^{{st}}}{{{\hat {Q}}}_{{st}}} \\ \left. {\frac{{}}{{}} + \,\,\left( {{{{\hat {P}}}^{{st}}}{{F}_{{{{n}_{{st}}}}}} + F_{n}^{{st}}{{S}_{{st}}}} \right) - {{F}_{n}}\left( {{{{\hat {P}}}^{{st}}}{{F}_{{{{n}_{{st}}}}}} + {{{\hat {Q}}}_{{st}}}F_{n}^{{st}}} \right)} \right) \\ - \,\,\sqrt F \left( {\frac{1}{2}{{F}^{{st}}}{{R}_{{st}}}(F){{{\hat {Q}}}_{{st}}} - {{{\hat {Q}}}^{{st}}}{{R}_{{st}}}(F) + {{F}^{{st}}}{{R}_{{st}}}(F,\hat {Q})} \right). \\ \end{gathered} $$

Let us consider the term $a\sqrt F {{F}^{{st}}}{{R}_{{st}}}(F,\hat {Q}).$ We recall that $\Gamma _{{lt}}^{s} = {{\gamma }^{{sp}}}{{\Gamma }_{{ltp}}},$ and the Christoffel numbers are series,

$${{\Gamma }_{{ltp}}} = {{\Gamma }_{{ltp}}}(F) + \frac{1}{g}{{\Gamma }_{{ltp}}}(\hat {Q}) + \frac{1}{{{{g}^{2}}}}{{\Gamma }_{{ltp}}}(A),$$

so

$$\Gamma _{{lt}}^{s} = \Gamma _{{lt}}^{s}(F) + \frac{1}{g}\Gamma _{{lt}}^{s}(\hat {Q}),$$

where

$$\Gamma _{{lt}}^{s}(\hat {Q}) = \left( {{{F}^{{sp}}}{{\Gamma }_{{ltp}}}(\hat {Q}) - {{{\hat {Q}}}^{{sp}}}{{\Gamma }_{{ltp}}}(F)} \right).$$

Taking into account that

$${{\hat {Q}}_{{t{{p}_{{;l}}}}}} = {{\hat {Q}}_{{t{{p}_{{,l}}}}}} - \Gamma _{{lt}}^{m}(F){{\hat {Q}}_{{mp}}} - \Gamma _{{lp}}^{m}(F){{\hat {Q}}_{{mt}}},$$

we note that

$${{\Gamma }_{{ltp}}}(\hat {Q}) = \frac{1}{2}\left( {{{{\hat {Q}}}_{{tp{{;}_{l}}}}} + {{{\hat {Q}}}_{{lp{{;}_{t}}}}} - {{{\hat {Q}}}_{{lt{{;}_{p}}}}}} \right) - 2{{\hat {Q}}_{{mp}}} + \Gamma _{{lt}}^{m}(F),$$

and it can be stated that

$$\Gamma _{{lt}}^{s}(\hat {Q}) = \frac{1}{2}{{F}^{{sp}}}\left( {{{{\hat {Q}}}_{{tp{{;}_{l}}}}} + {{{\hat {Q}}}_{{lp{{;}_{t}}}}} - {{{\hat {Q}}}_{{lt{{;}_{p}}}}}} \right),$$

so the Ricci tensor is also a series,

$${{R}_{{st}}} = {{R}_{{st}}}(F) + \frac{1}{g}{{R}_{{st}}}(F,\hat {Q}) + \frac{1}{{{{g}^{2}}}}{{R}_{{st}}}(F,\hat {Q},A),$$

where

$$\begin{gathered} {{R}_{{st}}}(F,\hat {Q}) = \Gamma _{{st{{;}_{l}}}}^{l}(\hat {Q}) - \Gamma _{{st{{;}_{t}}}}^{l}(\hat {Q}) \\ + \,\,\Gamma _{{st}}^{l}(\hat {Q})\Gamma _{{lm}}^{m}(F) + \Gamma _{{st}}^{l}(F)\Gamma _{{lm}}^{m}(\hat {Q}) \\ - \,\,\Gamma _{{sl}}^{m}(\hat {Q})\Gamma _{{mt}}^{l}(F) - \Gamma _{{sl}}^{m}(F)\Gamma _{{mt}}^{l}(F), \\ \end{gathered} $$

so that

$${{R}_{{st}}}(F,\hat {Q}) = {{\left( {\Gamma _{{st}}^{l}(\hat {Q})} \right)}_{{;l}}} - {{\left( {\Gamma _{{sl}}^{l}(\hat {Q})} \right)}_{{;t}}},$$

and the considered term is equal to

$$\begin{gathered} a\sqrt F {{F}^{{st}}}{{R}_{{st}}}(F,\hat {Q}) = \sqrt F \left( {a{{F}^{{st}}}\Gamma _{{st}}^{l}(\hat {Q})} \right){{,}_{l}} \\ - \,\,\sqrt F \left( {a{{F}^{{st}}}\Gamma _{{sl}}^{l}(\hat {Q})} \right){{,}_{t}} + \sqrt F {{F}^{{st}}}{{a}_{{;t}}}\Gamma _{{sl}}^{l}(\hat {Q}) \\ - \,\,\sqrt F {{F}^{{st}}}{{a}_{{;t}}}\Gamma _{{sl}}^{l}(\hat {Q}); \\ \end{gathered} $$

here,

$${{c}^{s}} = {{a}_{{;t}}}{{F}^{{st}}}.$$

Let us consider the form

$$\sqrt F {{F}^{{st}}}{{a}_{{;t}}}\Gamma _{{sl}}^{l}(\hat {Q}) = \sqrt F {{c}^{s}}\frac{1}{2}{{F}^{{lp}}}\left( {{{{\hat {Q}}}_{{sp{{;}_{l}}}}} + {{{\hat {Q}}}_{{lp{{;}_{s}}}}} - {{{\hat {Q}}}_{{sl{{;}_{p}}}}}} \right).$$

We take into account that the following expression is zero:

$${{F}^{{lp}}}\left( {{{{\hat {Q}}}_{{sp{{;}_{l}}}}} - {{{\hat {Q}}}_{{sl{{;}_{p}}}}}} \right) = 0,$$

since it contains the product ${\text{of}}$ the symmetric and antisymmetric tensors. Similarly,

$$\sqrt F {{F}^{{st}}}{{a}_{{;t}}}\Gamma _{{sl}}^{l}(\hat {Q}) = \frac{1}{2}{{\left( {\sqrt F {{c}^{s}}{{F}^{{lp}}}{{{\hat {Q}}}_{{lp}}}} \right)}_{{;s}}} - \frac{1}{2}\sqrt F {{F}^{{lp}}}{{\hat {Q}}_{{lp}}}c_{{;s}}^{s}$$

and

$$\sqrt F {{F}^{{st}}}{{a}_{{;l}}}\Gamma _{{st}}^{l}(\hat {Q}) = {\text{Div}} + \frac{1}{2}\sqrt F {{F}^{{lp}}}{{\hat {Q}}_{{lp}}}c_{{;s}}^{s} - \sqrt F {{F}^{{sl}}}{{\hat {Q}}_{{lp}}}c_{{;s}}^{p},$$

and finally, the terms take the form

$$a\sqrt F {{F}^{{st}}}{{R}_{{st}}}(F,\hat {Q}) = {\text{Div}} + \sqrt F {{\hat {Q}}_{{st}}}\left( {{{F}^{{sp}}}c_{{;p}}^{t} - {{F}^{{st}}}c_{{;p}}^{p}} \right).$$

Similarly, it can be stated that

$$a\sqrt F {{F}^{{st}}}{{R}_{{st}}}(F,\hat {Q},A) = {\text{Div}} + \sqrt F {{A}_{{st}}}\left( {{{F}^{{sp}}}c_{{;p}}^{t} - {{F}^{{st}}}c_{{;p}}^{p}} \right).$$

Let us consider the following order of these terms:

$$\begin{gathered} a\sqrt F {{F}^{{st}}}{{F}^{{st}}}{{{\hat {Q}}}_{{st}}}{{R}_{{st}}}(F,\hat {Q}) \\ {\text{ = Div}} + \sqrt F {{F}^{{st}}}{{\left( {a\hat {Q}} \right)}_{{;t}}}\Gamma _{{sl}}^{l}(\hat {Q}) \\ - \,\,\sqrt F {{F}^{{st}}}{{\left( {a\hat {Q}} \right)}_{{;l}}}\Gamma _{{st}}^{l}(\hat {Q}). \\ \end{gathered} $$

We introduce the following notation:

$${{r}^{s}} = {{\left( {a\hat {Q}} \right)}_{{;t}}}{{F}^{{st}}},$$

and then direct calculation shows that

$$\begin{gathered} a\sqrt F {{F}^{{st}}}\hat {Q}{{R}_{{st}}}(F,\hat {Q}) = \sqrt F \hat {Q}\left( {{{F}^{{sp}}}r_{{;p}}^{t} - {{F}^{{st}}}r_{{;p}}^{p}} \right) \\ = \sqrt F \left( {{{F}^{{sp}}}c_{{;p}}^{t} - {{F}^{{st}}}c_{{;p}}^{p}} \right){{{\hat {Q}}}_{{st}}}{{{\hat {Q}}}_{{st}}}{{F}^{{st}}} \\ + \,\,a\sqrt F {{F}^{{st}}}\left( {{{F}^{{sp}}}{{F}^{{tr}}} - {{F}^{{st}}}{{F}^{{pr}}}} \right){{{\hat {Q}}}_{{st{{,}_{{pr}}}}}}, \\ \end{gathered} $$

and it can be stated that these terms have the form

$$\begin{gathered} a\sqrt F {{{\hat {Q}}}^{{st}}}{{R}_{{st}}}(F,\hat {Q}) = \sqrt F {{\left( {a{{{\hat {Q}}}^{{st}}}} \right)}_{{;t}}}\Gamma _{{sl}}^{l}(\hat {Q}) \\ - \,\,\sqrt F {{\left( {a{{{\hat {Q}}}^{{st}}}} \right)}_{{;l}}}\Gamma _{{st}}^{l}(\hat {Q}) = \sqrt F \left( {{{F}^{{sp}}}{{a}_{{;tp}}}{{{\hat {Q}}}^{{st}}} - {{F}^{{st}}}{{a}_{{;tp}}}{{{\hat {Q}}}^{{pt}}}} \right) \\ \times \,\,{{{\hat {Q}}}_{{st}}} + a\sqrt F {{F}^{{st}}}\left( {{{F}^{{sp}}}\hat {Q}_{{;tp}}^{{st}} - {{F}^{{st}}}\hat {Q}_{{;tp}}^{{pt}}} \right){{{\hat {Q}}}_{{st}}}. \\ \end{gathered} $$

Therefore,

$$\begin{gathered} {{S}_{1}} = {{{\hat {P}}}^{{st}}}(x{\kern 1pt} '){{F}_{{{{n}_{{st}}}}}}(x{\kern 1pt} ') + F_{n}^{{st}}(x{\kern 1pt} '){{{\hat {Q}}}_{{{{n}_{{st}}}}}}(x{\kern 1pt} ') + a{{H}_{1}}(F,\hat {Q}) \\ = \int\limits_\Sigma {{A}_{{st}}}(x{\kern 1pt} '){{{\hat {P}}}^{{st}}}(x{\kern 1pt} ') + {{B}^{{st}}}(x{\kern 1pt} '){{{\hat {Q}}}_{{st}}}(x{\kern 1pt} '), \\ \end{gathered} $$

$${{A}_{{st}}} = \frac{{2a}}{{\sqrt F }}\left( {{{F}_{{{{n}_{{st}}}}}} - \frac{1}{2}{{F}_{n}}{{F}_{{st}}}} \right),$$

$$\begin{gathered} {{B}^{{st}}} = \frac{a}{{2\sqrt F }}\left( {{{F}_{{{{n}_{{kl}}}}}}F_{n}^{{kl}} - \frac{1}{2}F_{n}^{2}} \right){{F}^{{st}}} \\ - \,\,\frac{{2a}}{{\sqrt F }}\left( {F_{n}^{{st}}F_{n}^{{kl}}{{F}_{{{{n}_{{st}}}}}} - \frac{1}{2}{{F}_{n}}F_{n}^{{st}}} \right) \\ - \,\,a\sqrt F \left( {{{R}^{{st}}} - \frac{1}{2}{{F}^{{st}}}R} \right) - \sqrt F \left( {{{F}^{{sl}}}c_{{;l}}^{t} - {{F}^{{st}}}c_{{;l}}^{l}} \right). \\ \end{gathered} $$