# A Physically-Motivated Quantisation of the Electromagnetic Field on Curved Spacetimes

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## Abstract

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**37**:014001, 2016) presented a physically-motivated and explicitly gauge-independent scheme for the quantisation of the electromagnetic field in flat Minkowski space. In this paper we generalise this field quantisation scheme to curved spacetimes. Working within the standard assumptions of quantum field theory and only postulating the physicality of the photon, we derive the Hamiltonian, $\widehat{H}$, and the electric and magnetic field observables, $\widehat{\mathbf{E}}$ and $\widehat{\mathbf{B}}$, respectively, without having to invoke a specific gauge. As an example, we quantise the electromagnetic field in the spacetime of an accelerated Minkowski observer, Rindler space, and demonstrate consistency with other field quantisation schemes by reproducing the Unruh effect.

## 1. Introduction

## 2. Gauge-Independent Quantisation of the Electromagnetic Field

#### 2.1. Classical Electrodynamics

#### 2.2. Gauge Dependence in Electromagnetic Field Quantisation

#### 2.3. Physically-Motivated Gauge-Independent Method

## 3. Gauge-Independent Quantisation of the Electromagnetic Field in Curved Spacetimes

#### 3.1. Classical Electrodynamics in Curved Space

#### 3.2. Particles in Curved Spacetimes

#### 3.3. Covariant and Gauge-Independent Electromagnetic Field Quantisation Scheme

#### 3.3.1. Hilbert Space

#### 3.3.2. Hamiltonian

#### 3.3.3. Electromagnetic Field Observables

#### 3.3.4. Summary of Scheme

## 4. Electromagnetic Field Quantisation in an Accelerated Frame

#### 4.1. Rindler Space

#### 4.2. Electromagnetism in Rindler Space

#### 4.3. Field Quantisation in Rindler Space

#### 4.4. The Unruh Effect

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Further Results of Electromagnetism in Rindler Space

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**Figure 1.**Depiction of a 2-dimensional Minkowski space $\mathbb{M}$. Regions I and III are the future and past light cones of the observer O at the origin, while regions II and IV are the right Rindler wedge (RR) and left Rindler wedge (LR) respectively. The worldline of a uniformly accelerated observer with acceleration $\alpha $ is the displayed line of constant conformal Rindler coordinate $\xi $.

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**MDPI and ACS Style**

Maybee, B.; Hodgson, D.; Beige, A.; Purdy, R. A Physically-Motivated Quantisation of the Electromagnetic Field on Curved Spacetimes. *Entropy* **2019**, *21*, 844.
https://doi.org/10.3390/e21090844

**AMA Style**

Maybee B, Hodgson D, Beige A, Purdy R. A Physically-Motivated Quantisation of the Electromagnetic Field on Curved Spacetimes. *Entropy*. 2019; 21(9):844.
https://doi.org/10.3390/e21090844

**Chicago/Turabian Style**

Maybee, Ben, Daniel Hodgson, Almut Beige, and Robert Purdy. 2019. "A Physically-Motivated Quantisation of the Electromagnetic Field on Curved Spacetimes" *Entropy* 21, no. 9: 844.
https://doi.org/10.3390/e21090844