Effective scalar field theory of p-adic string.

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Using the effective scalar field theory of the $p$-adic string, we show the equivalence of two previously derived sets of classical Feynman rules: one by the present authors, the other by Brekke et al. We use this equivalence to demonstrate a symmetry of the $p$-adic tree amplitude ${A}^{m}(p)$ under $p\ensuremath{\rightarrow}\frac{1}{p}$. We then present a scalar field theory with a continuous parameter $\ensuremath{\xi}$ that reduces to the two previously discussed examples as special cases.