Method of virtual quanta

The method of virtual quanta is a method used to calculate radiation produced by interactions of electromagnetic particles, particularly in the case of bremsstrahlung. It can also be applied in the context of gravitational radiation, and more recently to other field theories by Carl Friedrich von Weizsäcker and Evan James Williams in 1934.

Background

In problems of collision between charged particles or systems, the incident particle is often travelling at relativistic speeds when impacting the struck system, producing the field of a moving charge as follows: ^[1]

${\displaystyle E_{1}=-{\frac {q\gamma vt}{(b^{2}+\gamma ^{2}v^{2}t^{2})^{\frac {3}{2}}}}}$

${\displaystyle E_{2}={\frac {q\gamma b}{(b^{2}+\gamma ^{2}v^{2}t^{2})^{\frac {3}{2}}}}}$

${\displaystyle B_{3}={\frac {v}{c}}E_{2}}$

where ${\displaystyle E_{1}}$ indicates the component of the electric field in the direction of travel of the particle, ${\displaystyle E_{2}}$ indicates the E-field in the direction perpendicular to ${\displaystyle E_{1}}$ and in the plane of the collision, ${\displaystyle b}$ is the impact parameter, ${\displaystyle \gamma }$ is the Lorentz factor, ${\displaystyle q}$ the charge and ${\displaystyle v}$ the velocity of the incident particle.

In the ultrarelativistic limit, ${\displaystyle E_{2}}$ and ${\displaystyle B_{3}}$ have the form of a pulse of radiation travelling in the ${\displaystyle {\overrightarrow {e_{1}}}}$ direction. This creates the virtual radiation pulse (virtual quanta) denoted by ${\displaystyle P_{1}}$. Moreover, an additional magnetic field may be added in order to turn ${\displaystyle E_{1}}$ into a radiation pulse travelling along ${\displaystyle {\overrightarrow {e_{2}}}}$, denoted ${\displaystyle P_{2}}$. This virtual magnetic field will turn out to be much smaller than ${\displaystyle B_{3}}$, hence its contribution to the motion of particles is minimal. ^[2]

By taking this point of view, the problem of the collision can be treated as a scattering of radiation. Similar analogies can be made for other processes (e.g. the ionisation of an atom by a fast electron can be treated as photoexcitation).

Bremsstrahlung

In the case of bremsstrahlung, the problem becomes one of the scattering of the virtual quanta in the nuclear Coulomb potential. This is a standard problem and the cross section of the scattering is known as the Thomson cross section:

${\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {1}{2}}({\frac {z^{2}e^{2}}{mc^{2}}})^{2}(1+\cos ^{2}{\theta })}$ ^[3]

The differential radiation cross section per unit frequency is hence:

${\displaystyle {\frac {d\chi }{d\omega d\Omega }}={\frac {1}{2}}({\frac {z^{2}e^{2}}{mc^{2}}})^{2}(1+\cos ^{2}{\theta }){\frac {dI}{d\omega }}}$

where ${\displaystyle {\frac {dI}{d\omega }}}$ is the frequency spectrum of virtual quanta produced by the incident particle over all possible impact parameters. ^[4]

Other applications

Synchrotron radiation

Main article: Synchrotron radiation

In the rest frame of the charged accelerating particle, the emission of synchrotron radiation can be treated as a Thomson scattering problem. This enables the introduction of various corrections into the classical calculation of the power lost by particles while accelerated, such as the quantum correction through the Klein-Nishina formula. ^[5]

Gravitational radiation

Main article: Gravitational waves

When transforming the gravitational field described by the Schwarzschild metric into the rest frame of a passing, relativistic test particle, an effect similar to the relativistic transformation of electric and magnetic fields is observed, and virtual pulses of gravitational radiation are produced. Through this, the cross section of close gravitational encounters and radiative power loss caused by such collisions can be calculated. ^[6]

References

1. Jackson 1999 , p. 559
2. Jackson 1999 , p. 726
3. Jackson 1999 , p. 659
4. Jackson 1999 , p. 730
5. Lieu; Axford; Quenby (1988). "Synchrotron radiation treated by the Weizsacker-Williams method of virtual quanta". Astronomy and Astrophysics. 208: 315–356. Retrieved 25 November 2022.
6. Matzner; Nutku (1973). "On the method of virtual quanta and gravitational radiation". Proceedings of the Royal Society of London. 336: 285–305. Retrieved 25 November 2022.

Bibliography

• Jackson, John David (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN   0-471-30932-X.