A Gravitational Theory of the Quantum

The synthesis of quantum and gravitational physics is sought through a finite, realistic, locally causal theory where gravity plays a vital role not only during decoherent measurement but also during non-decoherent unitary evolution. Invariant set theory is built on geometric properties of a compact fractal-like subset $I_U$ of cosmological state space on which the universe is assumed to evolve and from which the laws of physics are assumed to derive. Consistent with the primacy of $I_U$, a non-Euclidean (and hence non-classical) state-space metric $g_p$ is defined, related to the $p$-adic metric of number theory where $p$ is a large but finite Pythagorean prime. Uncertain states on $I_U$ are described using complex Hilbert states, but only if their squared amplitudes are rational and corresponding complex phase angles are rational multiples of $2 \pi$. Such Hilbert states are necessarily $g_p$-distant from states with either irrational squared amplitudes or irrational phase angles. The gappy fractal nature of $I_U$ accounts for quantum complementarity and is characterised numerically by a generic number-theoretic incommensurateness between rational angles and rational cosines of angles. The Bell inequality, whose violation would be inconsistent with local realism, is shown to be $g_p$-distant from all forms of the inequality that are violated in any finite-precision experiment. The delayed-choice paradox is resolved through the computational irreducibility of $I_U$. The Schr\"odinger and Dirac equations describe evolution on $I_U$ in the singular limit at $p=\infty$. By contrast, an extension of the Einstein field equations on $I_U$ is proposed which reduces smoothly to general relativity as $p \rightarrow \infty$. Novel proposals for the dark universe and the elimination of classical space-time singularities are given and experimental implications outlined

Average absolute and relative lengths for <i>M. reevesi</i> autosomal SCs.

#<p>SCs were numbered arbitrarily based on their length in each cell.</p

Distribution of MLH1 foci on autosomal SCs from 170 spermatocytes of Chinese muntjac.

<p>The SCs have been divided into five groups based on length, SCs 1–3, SCs 4–6, SCs 7–13, SCs 14–19, and SCs 20–22. The X-axis represents the positions on the SCs from the centromeric end (left) to the distal telomere (right), the marks on this axis are separated by 0.05 of the mean absolute SC length for each group. The absolute length (micrometer) scale is shown on the bottom of the last figure for each group. The y-axis indicates the frequency of MLH1 foci in each 0.05 SC interval. For each group, in order from top to bottom, the histograms show the results for SCs with different number MLH1 foci, and the overall frequencies. Data were not displayed when the MLH1 foci<100. n is the number of MLH1 foci analysed.</p

Spermatocytes at different sub-stages of meiotic prophase I in <i>M. reevesi</i>.

<p>The cells were immunolabeled with antibodies against SYCP3 (red), MLH1 (green) and kinetochores (blue). At leptotene, nuclei displaying as multiple short SYCP3-positive segments and with little association between homologues (A); At zygotene, complete but unpaired SYCP3-positive elements were observed (B); At pachytene, synapsis of homologues was completed with 22 autosomal SCs plus XY pair, MLH1 foci marking recombination sites and 24 centromeric CREST signals were clearly seen, and the appearance of MLH1 recombination sites in sex chromosomes are marked as X and Y (C); At diplotene, SC components disintegrated, recognized by desynapsis of telomeres on some of the bivalents (D). Bar: 10 µm.</p

Mean MLH1 foci per cell, frequency of autosomal bivalents with 0–4 MLH1 foci, percentage of cells with an MLH1 focus in XY pair of testicular samples of <i>M. reevesi</i>.

<p>Mean MLH1 foci per cell, frequency of autosomal bivalents with 0–4 MLH1 foci, percentage of cells with an MLH1 focus in XY pair of testicular samples of <i>M. reevesi</i>.</p

Distribution of MLH1 foci in each autosomal SC of <i>M. reevesi</i>^*.

<p>*Data from 170 cells.</p>#<p>SCs were numbered arbitrarily based on their length in each cell.</p

Relationships between autosomal SC length and meiotic recombination frequency in pachytene of male <i>M. reevesi</i> (n = 170).

<p>(A) Correlation between the frequency of XY pair with an MLH1 focus and total autosomal SC length in a cell. The cells have been divided into four groups based on their total autosomal SC length. (B) A positive correlation between the mean number of MLH1 foci and mean length for individual autosomal SCs (Pearson correlation coefficient = 0.99, <i>P</i><0.0001).</p

Mean distances between two adjacent MLH1 foci in SC groups of different length.

a<p>Significant different , one-way ANOVA, <i>P</i><0.01.</p>b<p>Not included in one-way ANOVA test because <10 observations.</p

The inhibitory effect of SLXL1 antiserum on <i>in vitro</i> zona pellucida binding/penetration of spermatozoa.

<p>(A). Bright (left panel) and fluorescent (right panel) images of eggs and spermatozoa incubated with HTF, control, anti-SLXL1^1–155aa (antiserum) and anti-SLXL1^1–155aa plus SLXL1 (Antiserum+SLXL1). (B) Statistical analysis of binding/penetration of spermatozoa to oocytes. The number of bound spermatozoa on each egg was evaluated. *Denotes significantly different groups compared with control group (p<0.05).</p

Assessment of the effects of SLXL1 antisera on <i>in vitro</i> fertilization, motility and acrosome reaction of spermatozoa.

<p>(A) The inhibitory effect of SLXL1 antisera on <i>in vitro</i> fertilization (IVF) rate. Successful fertilization was indicated by zygote cleavage. Spermatozoa were treated with PBS, preimmune serum, anti-SLXL1^1–155aa, anti-SLXL1^1–155aa+SLXL1, anti-SLXL1^65–155aa and anti-SLX^95–212aa before <i>in vitro</i> fertilization assays were performed. (B). Assessment of effect of anti-SLXL1^1–155aa on the motility of spermatozoa. (C). Assessment of effect of anti-SLXL1^1–155aa on the progressive movement of spermatozoa. (D). Assessment of effect of anti-SLXL1^1–155aa on the acrosome reaction of spermatozoa. **Denotes group that is significantly different (p<0.01) from the control groups.</p