# Renormalization for a Scalar Field in an External Scalar Potential

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## Abstract

**:**

## 1. Introduction

#### 1.1. Regularization and Renormalization in External Fields

#### 1.1.1. Analytic Methods

#### 1.1.2. Cutoff Methods

#### 1.1.3. The Pauli–Villars Method

#### 1.2. Power-Law Potentials

## 2. Scalar Field Models

#### 2.1. Lagrangian and Equations of Motion

#### 2.2. Curved-Space Action and Stress-Energy-Momentum Tensor

## 3. Vacuum Expectation Values

#### 3.1. The Square of the Field

#### 3.2. The Stress Tensor

#### 3.3. The Trace Identity

#### 3.4. The Conservation Law

## 4. Renormalization and Interpretation

#### 4.1. Comparison of Pauli–Villars and Point-Splitting “Renormalization”

- $\mathsf{{\rm Y}}=1$: This removes the term $-\mathsf{V}$ from $\langle {\phi}^{2}\rangle $ and recovers [7] (5.11) for $\langle {T}_{\mu \nu}\rangle $ modulo a term proportional to the conserved and covariant tensor ${W}_{\mu \nu}\phantom{\rule{0.166667em}{0ex}}$, so it reproduces the conclusions of [7].
- $\mathsf{{\rm Y}}=\frac{3}{2}$: This completely removes the anomalous ${\mathsf{V}}^{2}{g}_{\mu \nu}$ term from the stress tensor (and again changes the coefficient of the ${W}_{\mu \nu}$ term). In $\langle {\phi}^{2}\rangle $, it changes the $-\mathsf{V}$ to $+\mathsf{V}/2$.

#### 4.2. Renormalization

#### 4.2.1. Model 1

#### 4.2.2. Model 2

#### 4.2.3. The Logarithmic Terms

#### 4.3. Conclusions and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Fulling, S.A.; Settlemyre, T.E.; Milton, K.A.
Renormalization for a Scalar Field in an External Scalar Potential. *Symmetry* **2018**, *10*, 54.
https://doi.org/10.3390/sym10030054

**AMA Style**

Fulling SA, Settlemyre TE, Milton KA.
Renormalization for a Scalar Field in an External Scalar Potential. *Symmetry*. 2018; 10(3):54.
https://doi.org/10.3390/sym10030054

**Chicago/Turabian Style**

Fulling, Stephen A., Thomas E. Settlemyre, and Kimball A. Milton.
2018. "Renormalization for a Scalar Field in an External Scalar Potential" *Symmetry* 10, no. 3: 54.
https://doi.org/10.3390/sym10030054