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We have calculated the intrinsic magnetic moment of a photon through the intrinsic magnetic moment of a gamma photon created as a result of the electron-positron annihilation with the angular frequency ω. We show that a photon propagating in z direction with an angular frequency ω carries a magnetic moment of
*μ*
_{z} = ±(
*ec*/ω) along the propagation direction. Here, the (+) and (-) signs stand for the right hand and left circular helicity respectively. Because of these two symmetric values of the magnetic moment, we expect a splitting of the photon beam into two symmetric subbeams in a Stern-Gerlach experiment. The splitting is expected to be more prominent for low energy photons. We believe that the present result will be helpful for understanding the recent attempts on the Stern-Gerlach experiment with slow light and the behavior of the dark polaritons and also the atomic spinor polaritons.

In an earlier study [

The first experiment about the magnetic moment of a photon was done in 1896 by Zeeman [

SGE was devised in 1922 to show that electron had a non-zero magnetic moment [

are deflected upward or downward. In the SGE with electron, the electron beam is deflected into two sub-

beams which means that

where

only two possible values which are:

lected upward while the spin-up ones will be deflected downward. Similarly if we have SGE with a positron beam, because of the positive charge, the spin-down positrons will be deflected downward while the spin-up ones will be deflected upward. But when it comes to the gamma photons, which are the composite particles made up of an electron and a positron, the direction of the deflection is determined by the sign of the

We have also found a unique relation between the intrinsic fluxes and the magnetic moments of the particles such as electron, positron and the photon. We show that for all these three particles the unique relation between the intrinsic fluxes and the magnetic moments is given by:

We argue that any photon propagating in z direction with an angular frequency

Because of the two symmetric values of the magnetic moment of a photon, we expect a splitting (deflection) of the beam into two symmetric subbeams in a Stern-Gerlach experiment. We also expect that the splitting will be more prominent for lower energies. We believe that the present work will help to understand the Stern-Gerlach experiment of photons [

Electron-positron (

[

Most recently Saglam and Sahin [

field. We then prove that the intrinsic fluxes of

calculations [

In Appendix II, we introduce the conservation of the quantum flux in collisions: We write the Lagrangian of an electron moving in a uniform magnetic field in z direction then calculate the z-component of the conserved canonical angular momentum J_{c} which has two elements: The conservation of the kinetic angular momentum and the conservation of the magnetic flux. Therefore in the electron-positron annihilation, the conservation of the canonical angular momentum requires both conservation of the spin angular momentum and the conservation of the quantum flux which is originated from the intrinsic fluxes of

The outline of this paper is as follows: In Section 2, we derive the relation between the intrinsic fluxes and the magnetic moments of electron and positron. In Section 3, we calculate the magnetic moment of photon. In Section 4, we give the conclusions.

The definition of the spin magnetic moment vector

Here, g is the Lande-g factor which is equal to 2 for both particles and

Similarly the related z-components of the magnetic moments for spin-up and down positron are:

where

In Appendix I, we calculate the intrinsic fluxes of

respectively. Similarly the intrinsic fluxes for spin-up and down positron are:

If we compare the z-components of the magnetic moments and the spin dependent fluxes for both electron and positron Equations (2a)-(5b) we write the relation between the intrinsic flux and the magnetic moment of both electron and positron.

To calculate the magnetic moment of a photon, our starting point will be the electron-positron annihilation process ending with the creation of two gamma photons with right and left hand circular helicities. After the collision we will have two photons with the same energy

where

a) The electron is in spin-down state and the positron is in spin-down state.

b) The electron is in spin-up state and the positron is in spin-up state.

In Dirac notation, the states (a) and (b) can be defined as:

which build an orthonormalized set. Namely

where

The expectation values of the total z-components of the spin

where

Since the states (a) and (b) are equally probable, the total wave function

which is the quantum entanglement of the states

The expectation value of

where we used Equations (9a), (9b).

The total wave function,

This corresponds to two different gamma photons created with opposite circular helicities [

respectively.

As we calculated in Equations (9a), (9b), the expectation values of the total z-components of the spin

The expectation value of

where we used Equations (14a) and (14b).

If we compare the expectation value of

Now using the Equations (2a)-(3b) we can calculate the expectation values of the z-components of the magnetic moment vectors for the eigenstates

Next, using the Equations (16a)-(16d) we can calculate the expectation value of

If we compare the expectation value of

As we stated above, the final state wave function given in Equation (12) corresponds to two different gamma photons created with opposite circular helicities after the collision. Because the eigenstate

corresponds to the gamma photon with the right hand circular helicity and the second eigenstate

corresponds to the gamma photon with the left hand one.

From the Equations (16c) and (16d), the z-component of the magnetic moment of the eigenstates,

respectively.

Now substituting the relation

which must be true for any photon also. Namely a photon propagating in z direction with an angular frequency _{z} = ± (ec/w) along the propagation direction. Here, the (+) and (−) signs stand for the right hand and left circular helicity respectively.

We have calculated the intrinsic magnetic moment of a photon through the intrinsic magnetic moment of a gamma photon created as a result of the electron-positron annihilation with the angular frequency w. We show that a photon propagating in z direction with an angular frequency _{z} = ± (ec/w) along the propagation direction. Here, the (+) and (−) signs stand for the right hand and left circular helicity respectively. We have also found a unique relation between the intrinsic fluxes and the magnetic moments of the particles such as electron, positron and the photon as well. We show that for all these three particles the unique relation between the intrinsic fluxes and the magnetic moments is given by:

Since for a photon m_{z} is inversely proportional to the angular frequency w, a lower frequency implies a higher magnetic moment. Therefore in the first experiment done by Zeeman [

The authors express their thanks to Professor Mesude Saglam for her encouragement and for her invaluable help.

Solution of Dirac Equation for a free electron moving in a homogeneous magnetic field was given by Saglam et al. [

To calculate the intrinsic quantum flux of a relativistic free electron in a uniform magnetic field within the framework of Dirac theory, we shall follow a similar way that we followed earlier [_{e}. The probability of electric current density for the circular motion along the

Assuming that electron is moving in

Our next objective is to establish a quantized magnetic flux coming from the above current. A current element circulating around z-axis in a circle of radius

where

Substitution of Equations (AI-3) and (AI-6) into Equation (AI-5) gives the total induced quantized magnetic flux:

To calculate the total induced quantized magnetic fluxes for spin-up and spin-down electrons, we substitute the wave functions of

In Equations (AI-8a) and (AI-8b), the spin dependent intrinsic fluxes correspond to m = 0, which are

The Lagrangian of an electron with mass

For an electron moving in the x-y plane in the counter clockwise direction with the angular frequency

The vector potential,

The z-component of the conserved canonical angular momentum J_{c} is given by

where J is the gauge invariant kinetic angular momentum of the electron.

Now using the relation in Equation (6):

We can write the conserved canonical angular momentum J_{c} in terms of the magnetic moment. Substitution of Equation (6) into (AII-3) gives:

Here, Equations (AII-3) and (AII-4) simply state that the conservation of the canonical angular momentum requires also the conservation of the magnetic flux and the magnetic moment as well. In passing, we note that although the above calculations are carried out for an electron, the statement about the conservation of the magnetic flux and the magnetic moment will be valid for a positron as well. The only difference is that, for a positron the (−) signs in Equations (AII-3) and (AII-4) are replaced with the (+) signs.