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Carlo Beenakker
Carlo Beenakker
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I think there is a misunderstanding here on what is the expansion parameter.

Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this expansion typically has zero radius of convergence.

Constructive renormalization, instead, does not expand in powers of $\lambda$, but in the number of particles that interact. The interaction strength is not expanded, it is retained to all orders in $\lambda$.

The objective of the theory is to show that this expansion (known as a cluster or Mayer expansion) has a finite radius of convergence. This is typically the case if the interaction decays sufficiently rapidly at large distances.

And yes, convergence of the pressure implies convergence of correlators, see for example https://doi.org/10.1063/1.523040

I think there is a misunderstanding here on what is the expansion parameter.

Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this expansion typically has zero radius of convergence.

Constructive renormalization, instead, does not expand in powers of $\lambda$, but in the number of particles that interact. The interaction strength is not expanded, it is retained to all orders in $\lambda$.

The objective of the theory is to show that this expansion (known as a cluster or Mayer expansion) has a finite radius of convergence. This is typically the case if the interaction decays sufficiently rapidly at large distances.

I think there is a misunderstanding here on what is the expansion parameter.

Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this expansion typically has zero radius of convergence.

Constructive renormalization, instead, does not expand in powers of $\lambda$, but in the number of particles that interact. The interaction strength is not expanded, it is retained to all orders in $\lambda$.

The objective of the theory is to show that this expansion (known as a cluster or Mayer expansion) has a finite radius of convergence. This is typically the case if the interaction decays sufficiently rapidly at large distances.

And yes, convergence of the pressure implies convergence of correlators, see for example https://doi.org/10.1063/1.523040

Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.1k
  • 18
  • 448
  • 651

I think there is a misunderstanding here on what is the expansion parameter.

Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this expansion typically has zero radius of convergence.

Constructive renormalization, instead, does not expand in powers of $\lambda$, but in the number of particles that interact. The interaction strength is not expanded, it is retained to all orders in $\lambda$.

The objective of the theory is to show that this expansion (known as a cluster or Mayer expansion) has a finite radius of convergence. This is typically the case if the interaction decays sufficiently rapidly at large distances.