Abstract
Based on bulk reconstruction from the finite boundary of the Bruhat-Tits tree, the boundary effective theory is obtained after integrating out fields outside this boundary. According to the $~p$-adic version of Anti-de Sitter/Conformal Field Theory duality, two-point functions of dual theory living on the finite boundary are read out from the effective action. They can be regarded as two-point functions of a deformed conformal field theory over $~p$-adic numbers.
Highlights
It is proposed that physics should be invariant under the change of number fields [1]
We should be able to use either real numbers (R) or p-adic numbers (Qp) [2,3,4] to set up spacetime coordinates and write down the same physical laws. Such number fields should include the set of rational numbers (Q) since all measurement results in physics are rational numbers
One motivation of this paper is to extend the work of Ref. [9], which is to calculate the effective field theory on a finite boundary
Summary
It is proposed that physics should be invariant under the change of number fields [1]. The p-adic version of the anti–de Sitter/ conformal field theory duality (p-adic AdS=CFT) [19,20,21] is proposed in [10,22], which are followed by lots of works, such as [23,24,25,26,27,28,29,30,31,32,33] Among all these references, [9,22] are the most important to this paper. Identifying Tp as the p-adic version of AdS spacetime [22], two-point functions of a deformed CFT over Qp are calculated in Sec. V, where the deformation comes from the “cutoff” of Tp, or in other words, comes from the finite boundary. The measure μ, dx, the p-adic absolute value j·jp, and the edge length L have the dimension of length while p-adic numbers are dimensionless
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