It is well known that at distances shorter than Planck length, no lengthmeasurements are possible. The Volovich hypothesis asserts that atsub-Planckian distances and times, spacetime itself has a non-Archimedeangeometry. We discuss the structure of elementary particles, theirclassification, and their conformal symmetry under this hypothesis.Specifically, we investigate the projective representations of the $p$-adicPoincar\'{e} and Galilean groups, using a new variant of the Mackey machine forprojective unitary representations of semidirect products of locally compactand second countable (lcsc) groups. We construct the conformal spacetime over$p$-adic fields and discuss the imbedding of the $p$-adic Poincar\'{e} groupinto the $p$-adic conformal group. Finally, we show that the massive and socalled eventually masssive particles of the Poincar\'{e} group do not haveconformal symmetry. The whole picture bears a close resemblance to what happensover the field of real numbers, but with some significant variations.
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机译：众所周知，在距离小于普朗克长度的距离处，不可能进行长度测量。沃洛维奇假设断言，在普朗克下的距离和时间上，时空本身具有非Archimedeangeometry。在此假设下，我们讨论了基本粒子的结构，其分类及其共形对称性。具体来说，我们使用Mackey机的新变体对$ p $ -adicPoincar \'{e}和Galilean组的投影表示进行了研究。局部紧实和第二可数（lcsc）组的半直接乘积的统一表示。我们构造$ p $ -adic场上的共形时空，并讨论$ p $ -adic Poincar \'{e}组到$ p $ -adic保形组中的嵌入。最后，我们证明了庞加莱族的块状和所谓的块状粒子不具有保形对称性。整个图片与实数字段上发生的情况非常相似，但有一些显着差异。
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