We consider a scalar quantum field $\phi$ with arbitrary polynomial self-interaction in perturbation theory. If the field variable $\phi$ is repaced by a local diffeomorphism $\phi(x) = \rho(x) + a_1 \rho^2(x) +\ldots$, this field $\rho$ obtains infinitely many additional interaction vertices. We show that the $S$-matrix of $\rho$ coincides with the one of $\phi$ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. If tadpole diagrams vanish, the diffeomorphism can be tuned to cancel all contributions of an underlying $\phi^s$-type self interaction at one fixed external offshell momentum, rendering $\rho$ a free theory at this momentum. Finally, we propose one way to extend the diffeomorphism to a non-local transformation involving derivatives without spoiling the combinatoric structure of the local diffeomorphism.

中文翻译：

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变换后的量子场的微扰理论

我们考虑在微扰理论中具有任意多项式自相互作用的标量量子场 $\phi$。如果场变量 $\phi$ 被局部微分同胚 $\phi(x) = \rho(x) + a_1 \rho^2(x) +\ldots$ 所取代，则该场 $\rho$ 获得无限多的附加交互顶点。我们证明 $\rho$ 的 $S$-矩阵与 $\phi$ 之一重合，而不使用路径积分参数。即使基础场具有高于动量二次阶的传播子，该结果也成立。如果蝌蚪图消失，微分同胚可以被调整以消除潜在的 $\phi^s$ 类型自相互作用在一个固定的外部壳外动量下的所有贡献，使 $\rho$ 在这个动量下成为一个自由理论。最后，