Completeness relation for polarization vectors

In summary, the completeness relation for polarization vectors is a mathematical expression that states any polarization state of light can be represented as a linear combination of two orthonormal polarization states. This relation is significant in the study of polarized light and its interactions with matter. It is derived from the properties of the electric field vector and tells us that light is a transverse electromagnetic wave with two orthogonal components fully describing its polarization state. However, there may be limitations to this relation in certain systems and it does not account for polarization altering phenomena.
  • #1
center o bass
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I'm reading some quantum optics and I stumbled on the following completeness relation for the polarization vectors of the electromagnetic field.

[tex]\sum_{a} (\epsilon_{\vec{k} a } )_i ( \epsilon^*_{ \vec{k} a} )_j = \delta_{ij} -\frac{k_i k_j}{\vec{k}^2}[/tex]


Does anyone know how to derive this relation?
 
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  • #2
Check out page 36 of Cohen-Tannoudji's <Introduction to Quantum Electrodynamics>.
 
  • #3
Thanks! :)
 

Related to Completeness relation for polarization vectors

What is the completeness relation for polarization vectors?

The completeness relation for polarization vectors is a mathematical expression that states that any polarization state of light can be represented as a linear combination of two orthonormal polarization states, typically denoted as horizontal and vertical polarization. This relation is important in the study of polarized light and its interactions with matter.

What is the significance of the completeness relation for polarization vectors?

The completeness relation for polarization vectors is significant because it allows for a complete description of the polarization state of light. This is crucial in many applications, such as in the field of optics, where understanding and controlling polarization is essential.

How is the completeness relation for polarization vectors derived?

The completeness relation for polarization vectors is derived from the properties of the electric field vector that describes the polarization state of light. By considering the properties of the electric field, it can be shown that any polarization state can be represented as a linear combination of two orthonormal polarization states.

What does the completeness relation for polarization vectors tell us about the nature of light?

The completeness relation for polarization vectors tells us that light is a transverse electromagnetic wave, meaning that the electric and magnetic fields are perpendicular to the direction of propagation. It also tells us that the polarization state of light is fully described by two orthogonal components.

Are there any limitations to the completeness relation for polarization vectors?

The completeness relation for polarization vectors is a mathematical concept that has been verified experimentally. However, it assumes that the polarization states are linearly independent, which may not always be the case in certain systems. Additionally, it does not take into account the effects of birefringence or other forms of polarization altering phenomena.

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