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<p>In this paper, we introduce novel reference observables for the sake of establishing a scaling formula in the renormalization group (<i>RG</i>) equation. Firstly, using the transfer matrix method, we calculate the two point observables of the one dimensional (1D) Ising model without an external field under general boundary conditions. The results suggest that the two point observables decay exponentially except at the critical point. Corresponding to the <i>RG</i> procedure underlying the correlation function, we establish a similar procedure for new observables, which is consistent with the findings in physics. Secondly, from a dynamic perspective, we construct a random system via the stochastic quantization method. We calculate the new observables of this random system under the initial distribution satisfying the Dobrushin–Lanford–Ruelle (DLR) equations. Furthermore, we formulate a new renormalization scaling formula with respect to the two point observables. Finally, these results can be extended to any finite point observables, and are independent of the choice of system parameters.</p>

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The Behavior of Observables in Renormalization

  • 期刊论文

  • 发表日期：

    2026年2月14日

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  • Kaiyuan Cui,
  • Fuzhou Gong

摘要

In this paper, we introduce novel reference observables for the sake of establishing a scaling formula in the renormalization group (RG) equation. Firstly, using the transfer matrix method, we calculate the two point observables of the one dimensional (1D) Ising model without an external field under general boundary conditions. The results suggest that the two point observables decay exponentially except at the critical point. Corresponding to the RG procedure underlying the correlation function, we establish a similar procedure for new observables, which is consistent with the findings in physics. Secondly, from a dynamic perspective, we construct a random system via the stochastic quantization method. We calculate the new observables of this random system under the initial distribution satisfying the Dobrushin–Lanford–Ruelle (DLR) equations. Furthermore, we formulate a new renormalization scaling formula with respect to the two point observables. Finally, these results can be extended to any finite point observables, and are independent of the choice of system parameters.