Introduction to Casimir Effect (2025 Spring)
Prachi Parashar and K. V. Shajesh
Last updated: Jun 13, 2025. (This page is no more maintained.)

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• General information:
  
  • K. V. Shajesh
  • Meeting time: MWF 2:30-3:30 PM. in Neckers 456.
  • Meeting info: [829 6770 5989, ihbar, link]
  
  
• Resources:
  • Lecture recordings.
  • References:
    • Classical Electrodynamics, by Schwinger, DeRaad Jr., Milton, and Tsai.
    • Prachi's thesis.
    • Casimir-Polder energy for axially symmetric systems.
    • Remarks on the Casimir force for magnetodielectric media.

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COURSE CONTENT:


1. Maxwell equations [20250602v]
   • Maxwell equations in frequency space
   • Electric charge, material with permanent electric polarization, electrically and magnetically polarizable materials.
   • Causal response functions
   • Fourier transform of Heaviside step function
   
   
2. Kramers-Kronig relations [20250606v]
   • Dielectric models
   
   Discussion:
   • Causality is inherently connected to dissipation. However, it is not possible to construct an action for dissipative system. Thus, most field theoretic methods fail at the very outset. Nevertheless, the use of Feynman Green function, in conjunction with Euclidean rotation, seems to capture the relevant responses. Kim Milton has recently found correspondences between starting from Fluctuation dissipation theorem and field theoretic methods.
   • Starting from the actio principle, we seem to be restricted to symmetric and Hermitian susceptibilities. However, this is not satisfactory for discussing most materials, see (Agranovich and Ginzburg, 1966).
   • Polarizaion induced as responses to electric field has been discussed lucidly in (Debye, 1929).
   
   
3. Conservation principles [20250609v]
   • Charge conservation
   • Electromagnetic energy conservation, flux of energy
   • Electromagnetic momentum conservation, flux of momentum
   • Stress tensor
   
   
4. Action for macroscopic electrodynamics
   • conservation of energy for macroscopic case
   
   
5. Schwinger action principle
   • Vacuum to vacuum transition amplitude
   • Source versus field
   
   
6. Correlation function and Green's function
   • Green's dyadic, \( \mathbf{\Gamma}({\bf r},{\bf r}^\prime;\omega) \)
   • Derive \begin{equation} \langle {\bf E}({\bf r},\omega) \varepsilon_0 {\bf E}({\bf r}^\prime,\omega^\prime) \rangle = \frac{\hbar}{i} \Gamma({\bf r},{\bf r}^\prime;\omega) 2\pi \delta (\omega+\omega^\prime). \end{equation}
   • Trace Log formula
   
   
7. Two-body interaction energy
   • Derive \begin{equation} E_{12} = \frac{\hbar c}{2} \int_{-\infty}^\infty \frac{d\zeta}{2\pi c} \,\text{tr} \ln ({\bf 1} - \mathbf{\Gamma}_1 \cdot \mathbf{\chi}_1 \cdot \mathbf{\Gamma}_2 \cdot \mathbf{\chi}_2). \end{equation}
   
   
8. Vector Fourier eigen functions
   • Vector completeness relation
   • Vector orthonormality conditions
   
   
9. Separation of electric and magnetic modes
   • Reduced Green's dyadic
   • Magnetic and electric Green's function
   
   
10. Magnetic Green's function
    • Solution for single interface
    
    
11. Electric Green's function
    • Solution for single interface
    
    
12. Contribution to Casimir energy from electric mode
13. Contribution to Casimir energy from magnetic mode
14. Casimir energy
    • Perfect conductor limit
    
    

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REFERENCES:


1. V. M. Agranovich, and V. L. Ginzburg, Spatial dispersion in crystal optics and the theory of excitons, monographs and texts in physics and astronomy, Vol. XVIII, (1966).
2. P. Debye, Polar Molecules, 1929.

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Last updated on Jun 13, 2025 (and is no more maintained) by K. V. Shajesh (kvshajesh@gmail.com).