Photon time delay due to the

presence of a magnetic field.

Varadero, Cuba 2018

MsC. Lidice Cruz Rodríguez

Dra. Aurora Pérez Martínez

Dra. Elizabeth Rodríguez Querts

Dr. Jorge Rueda

Physics Faculty, Havana University

ICIMAF

ICRANet, Pescara, Italy

Outline Outline

Motivation.

Propagation of photons in a magnetic field:

Dispersion equation in the vacuum

Dispersion equation in the medium

Phase velocity, preliminary results.

Some remarks and future work directions…..

Outline Motivation

It is known how the interaction with the

medium delay the photon propagation.

Higher energy photons arrive before.

How the propagation in the magnetosphere

affects the photon dispersion law and hence

the photon velocity?

Delay in photon propagation is used to

measure pulsars distance!

Polarization tensor, theoretical overview.

In the presence of an external magnetic field B along the 𝑥3 direction the

diagonalization of the polarization tensor leads to the equation:

Having three non vanishing eigenvalues

and three eigenvectors for i=(1,2,3),

corresponding to the three photon

propagation modes

Π𝜇𝜈𝑎𝜈(𝑖)= 𝜅(𝑖)𝑎𝜇

(𝑖)

𝑎𝜇(1)= 𝑘2𝐹𝜇𝜆

2 𝑘𝜆 − 𝑘𝜇 𝑘𝐹2𝑘

𝑎𝜇(2)= 𝐹𝜇𝜆∗ 𝑘𝜆

𝑎𝜇(3)= 𝐹𝜇𝜆∗ 𝑘𝜆

𝑘2 = 𝑘⊥2 + 𝑘∥

2 − 𝜔2

For 𝐤 ⊥ 𝐁 𝐚𝛍𝟏

is a longitudinal and non

physical mode, 𝐚𝛍𝟐,𝟑

are transverse modes.

For 𝐤 ∥ 𝐁, 𝐚𝛍𝟐

is a longitudinal and non

physical mode, 𝐚𝛍𝟏,𝟑

are transverse modes,

Which leads to circular polarized waves

(Faraday effect).

Quantum Faraday EffectDispersion equation

𝑘2 = 𝜅(𝑖)(𝑘2, B)The dispersion equation is then

Eigenvalues of the polarization tensor

From the solution of the dispersion equation the information related to

photon propagation can be obtained.

Quantum Faraday EffectDispersion equation

𝑘2 = 𝜅(𝑖)(𝑘2, B)The dispersion equation is then

Eigenvalues of the polarization tensor

From the solution of the dispersion equation the information related to

photon propagation can be obtained.

Here we will solve the dispersion equation in two particular cases and we

will show preliminary results for the phase velocity.

Quantum Faraday EffectDispersion equation

𝑘2 = 𝜅(𝑖)(𝑘2, B)The dispersion equation is then

Eigenvalues of the polarization tensor

From the solution of the dispersion equation the information related to

photon propagation can be obtained.

Here we will solve the dispersion equation in two particular cases and we

will show preliminary results for the phase velocity.

Photon propagation in the vacuum in the presence of an external magnetic field.

Photon propagation in the medium in the presence of an external magnetic field.

Quantum Faraday EffectPhoton propagation in the vacuum

In general the renormalized eigenvalues of the polarization tensor in the one

loop approximation can be written as

In the range of frequencies with small deviation from the light cone 𝑘2

𝑒𝐵≪ 1

𝜅𝑖(𝑘2, 𝐵) =

2𝛼

𝜋

0

∞

𝑑𝑡

−1

1

𝑑𝜂 𝑒−𝑡𝑏 𝜌𝑖𝑒

𝜁 +𝑘2 𝜂2

2𝑡

𝑓(𝑘2, 𝐵)

𝝌𝒊𝒍 =2𝛼

𝜋

0

∞

𝑑𝑡

−1

1

𝑑𝜂 𝑒−𝑡𝑏 𝑒𝜁0

𝜍

𝑏𝑚2

𝑙 (−1)𝑙

𝑙!𝜌0𝑖 − 𝑙

𝜍

𝑏𝑚2

−1

𝜃𝑖 + 𝛿1𝑙 𝜂2

2𝑡

𝜿𝒊 𝑘2, 𝐵 =

𝑙=0

∞

𝝌𝒊𝒍 (𝒌𝟐)𝒍≈

𝑙=0

𝑛

𝜒𝑖𝑙 (𝑘2)𝑙

Quantum Faraday EffectPhoton propagation in the vacuum

𝜿𝒊 𝑘2, 𝐵 = 𝑘2 ≈

𝑙=0

𝑛

𝜒𝑖𝑙 (𝑘2)𝑙

𝑘2 = −

𝑗2,𝑗3…𝑗𝑛≥𝑛

(−1)𝑗1𝑗1!

(𝑗0+1)! 𝑗2! 𝑗3! … 𝑗𝑛!

𝜒𝑖0𝑗0+1𝜒𝑖2

𝑗2𝜒𝑖3𝑗3 …𝜒𝑖𝑛

𝑗𝑛

(𝜒𝑖1 −1)𝑗1+1

i=2,3

𝒌⊥(𝟐,𝟑)≈ 𝝎𝟐 + 𝒌(𝝎,𝑩)𝟐

𝒗(𝑩,𝝎) =𝝎

𝒌⊥(𝝎,𝑩)

As we can see the phase velocity in general depends on the magnetic field B

and the photon energy.

Quantum Faraday EffectResults

Dispersion equation Phase velocity

ℏ𝜔2

𝑚2

𝑘⊥2/𝑚2

ℏ𝜔 2

𝑚2

𝑣(𝑚/𝑠)

Quantum Faraday EffectPhoton propagation in the medium, diluted gas limit

𝒗(𝑩,𝝎) =𝝎

𝒌⊥(𝝎,𝑩)

Dispersion equation

𝑘2 = 𝜅(𝑖)(𝑘2, 𝜇, B)

𝑘∥(1,3)≈ 𝜔2 + 𝑘(𝜔, 𝐵)2

We have different 𝜅(𝑖) .

Solution for the particular case of 𝐤 ∥ 𝐁.

Diluted gas limit

(classical distribution of particles)

𝑘∥(1,3)≈ − 𝑒𝐵 ±𝑚𝜔 + 2𝑒𝐵 ± 2𝑚𝜔2 − 2𝐴 𝐵, 𝑇, 𝜇 𝑚𝑤

1/2

Phase velocity 𝑣(𝐵,𝜔) =𝜔

𝑘∥(𝜔, 𝐵)

Quantum Faraday EffectResults

𝒗(𝑩,𝝎) =𝝎

𝒌⊥(𝝎,𝑩)

Phase velocity

𝑣(𝑚/𝑠)

ℏ𝜔/𝑚

Photon time delay

ℏ𝜔(𝑒𝑉)

Δ𝑡(𝜇𝑠)

The time delay between radio

frequency photons and micro wave

frequency photons is around 0.7𝜇𝑠

Some remarks

We have solved the dispersion equation in two limit cases, for propagation in

a magnetized vacuum perpendicular to the direction of the constant magnetic

field, and in a medium in the diluted gas limit.

We show the dependence of the phase velocity with the photon energy,

for propagation in the vacuum we found that the higher energy photons are

delay with respect to the lower energy ones.

In the medium (under the limits considered) the higher energy photons

travels faster.

What is coming next…..

ArXiv:gr-qc/9503044v1 25 Mar 1995

Include a more realistic model for the

magnetic field of a rotating neutron star,

dipolar approximation

Combine calculations to describe the

different regions traversed by the

radiation coming from the pulsar.

Estimate delay times and compare with

experimental data which is available.

Varadero, Cuba 2018

Gracias!!!!