Unifying Theory for Casimir Forces: Bulk and Surface Formulations
Abstract
:1. Introduction
- A new, compact and elegant derivation of the Casimir force from the Maxwell stress tensor within both a T-operator approach and a surface operator approach;
- A new surface formula for the Casimir free energy expressed in terms of a surface operator;
- A new path integral-based derivation of a Lagrange and Hamiltonian formulation for the Casimir free energy.
2. Stress-Tensor Approach
2.1. Bulk and Surface Expressions for the Force
2.2. Casimir Free Energy
2.2.1. T-Operator Approach
2.2.2. Surface Operator Approach
3. Equivalence of the Surface-Formula with the Scattering Formula for the Casimir Energy
4. Path Integral Approach
4.1. Lagrange Formulation
4.2. Hamiltonian Formulation
5. Application: Derivation of the Lifshitz Theory
5.1. Lagrange Representation
5.2. Hamiltonian Representation
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Dyadic Green’s Functions and Notations
Appendix B. Derivation of Equation (10)
Appendix B.1. Volumic Representation
Appendix B.2. Surface Representation
Appendix C. Proof of the Force Formula Equation (14)
Appendix D. Partial Wave Expansion and the Scattering Matrix
1. | This restriction is not so severe in practice, since the vast majority of Casimir experiments use test bodies that can be modelled in this way. |
2. | |
3. | Bodies that are only piecewise homogeneous can also be considered by a slight generalization of the homogeneous case. |
4. | For example, one can set , where is any rotationally invariant, smooth non-negative function, supported in a ball of unit radius centered in the origin, such that . |
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Bimonte, G.; Emig, T. Unifying Theory for Casimir Forces: Bulk and Surface Formulations. Universe 2021, 7, 225. https://0-doi-org.brum.beds.ac.uk/10.3390/universe7070225
Bimonte G, Emig T. Unifying Theory for Casimir Forces: Bulk and Surface Formulations. Universe. 2021; 7(7):225. https://0-doi-org.brum.beds.ac.uk/10.3390/universe7070225
Chicago/Turabian StyleBimonte, Giuseppe, and Thorsten Emig. 2021. "Unifying Theory for Casimir Forces: Bulk and Surface Formulations" Universe 7, no. 7: 225. https://0-doi-org.brum.beds.ac.uk/10.3390/universe7070225