Discrete vs continuum gravitational diagrams in the soft synchronous gauge
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910016166952960 |
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| author | Khatsymovsky, V. M. |
| author_facet | Khatsymovsky, V. M. |
| contents | Due to the non-renormalizability of gravity, the perturbative expansion has sense, say, for its discrete simplicial (Regge calculus) version. A finite-difference form of gravity action has diffeomorphism symmetry at leading order over metric variations from site to site, and we add a term bilinear in $n^λ(g_{λμ}-g_{λμ}^{(0)})$, $n^λ=[1,-\varepsilon(Δ^{(s)α}Δ^{(s)}_α)^{-1}Δ^{(s)β}]$, to "softly" fix the synchronous gauge $g_{0λ}=g_{0λ}^{(0)}=-δ_{0λ}$ at $\varepsilon\to0$, thus removing singularities at $p_0=0$.
For the symmetric derivative $Δ^{(s)}_λ$, the propagator has a graviton pole at $\sin^2p_0=\sum^3_{α=1}\sin^2p_α$ or, at small $p_α$, at $p_0$ close to 0 or $\pm π$. This pole doubling compared to the continuum does not arise from $\sin^2(p_0/2)=\sum^3_{α=1}\sin^2(p_α/2)$ obtained from the action $\check{S}_{\rm g}$ with the usual derivative $Δ_λ= \exp (ip_λ)-1$ instead of $Δ^{(s)}_λ=i\sin p_λ$ in some terms, including in the k part of some term, and $Δ^{(s)}_λ$ in the 1-k part of that term.
Given the propagator $\check{G}(n,\overline{n})$, we form a principal value type propagator $[\check{G}(n,n)+\check{G}(\overline{n},\overline{n})]/2$ by analytically continuing from real $n=\overline{n}$. Singularities are roughly resolved as $p_0^{-j}\Rightarrow[(p_0+i\varepsilon)^{-j}+(p_0-i\varepsilon)^{-j}]/2$ leading to separate diagram finiteness at $\varepsilon\to0$.
We find that k=1 is needed for this prescription to work properly and match the continuum case. The gauge-fixing term needed for this propagator and its finiteness are considered, the ghost contribution is found to vanish at $\varepsilon\to0$. We use these results in arXiv:2601.03228. Calculations are illustrated by the electromagnetic (Yang-Mills) case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02181 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Discrete vs continuum gravitational diagrams in the soft synchronous gauge Khatsymovsky, V. M. General Relativity and Quantum Cosmology 83C27, 81S40 Due to the non-renormalizability of gravity, the perturbative expansion has sense, say, for its discrete simplicial (Regge calculus) version. A finite-difference form of gravity action has diffeomorphism symmetry at leading order over metric variations from site to site, and we add a term bilinear in $n^λ(g_{λμ}-g_{λμ}^{(0)})$, $n^λ=[1,-\varepsilon(Δ^{(s)α}Δ^{(s)}_α)^{-1}Δ^{(s)β}]$, to "softly" fix the synchronous gauge $g_{0λ}=g_{0λ}^{(0)}=-δ_{0λ}$ at $\varepsilon\to0$, thus removing singularities at $p_0=0$. For the symmetric derivative $Δ^{(s)}_λ$, the propagator has a graviton pole at $\sin^2p_0=\sum^3_{α=1}\sin^2p_α$ or, at small $p_α$, at $p_0$ close to 0 or $\pm π$. This pole doubling compared to the continuum does not arise from $\sin^2(p_0/2)=\sum^3_{α=1}\sin^2(p_α/2)$ obtained from the action $\check{S}_{\rm g}$ with the usual derivative $Δ_λ= \exp (ip_λ)-1$ instead of $Δ^{(s)}_λ=i\sin p_λ$ in some terms, including in the k part of some term, and $Δ^{(s)}_λ$ in the 1-k part of that term. Given the propagator $\check{G}(n,\overline{n})$, we form a principal value type propagator $[\check{G}(n,n)+\check{G}(\overline{n},\overline{n})]/2$ by analytically continuing from real $n=\overline{n}$. Singularities are roughly resolved as $p_0^{-j}\Rightarrow[(p_0+i\varepsilon)^{-j}+(p_0-i\varepsilon)^{-j}]/2$ leading to separate diagram finiteness at $\varepsilon\to0$. We find that k=1 is needed for this prescription to work properly and match the continuum case. The gauge-fixing term needed for this propagator and its finiteness are considered, the ghost contribution is found to vanish at $\varepsilon\to0$. We use these results in arXiv:2601.03228. Calculations are illustrated by the electromagnetic (Yang-Mills) case. |
| title | Discrete vs continuum gravitational diagrams in the soft synchronous gauge |
| topic | General Relativity and Quantum Cosmology 83C27, 81S40 |
| url | https://arxiv.org/abs/2601.02181 |