Infrared divergence

Infrared divergence (infrared catastrophe) - a situation of allegedly emitting an infinitely large number of photons with infinitesimal energies in a collision of two charged particles or with a sharp change in the speed of a charged particle. It is a consequence of the divergence of the integral due to the contributions of objects with very low energy (almost equal to zero), or the same thing, due to the physical phenomenon on very large scales.

Infrared divergence is available only in theories with massless particles (such as photons ). These divergences are an effect that a full theory often implies. One way to deal with it is to apply circumcision .

Content

Paradox Description

Process section $d\sigma$ ${\ displaystyle d \ sigma}$ scattering of charged particles with the emission of one additional photon is expressed by the formula: $d\sigma =d\sigma _{0}{\frac {dI}{\omega }}$ ${\ displaystyle d \ sigma = d \ sigma _ {0} {\ frac {dI} {\ omega}}}$ . Here $d\sigma _{0}$ ${\ displaystyle d \ sigma _ {0}}$ - cross section of the scattering process of charged particles with the emission of a certain number of photons, $d I$ ${\ displaystyle dI}$ - total radiation energy , $\omega$ ${\ displaystyle \ omega}$ - radiation frequency. When integrating this formula over frequencies in a certain finite interval from $\omega _{1}$ ${\ displaystyle \ omega _ {1}}$ before $\omega _{2}$ ${\ displaystyle \ omega _ {2}}$ it turns out $d\sigma \sim \alpha \ln {\frac {\omega _{2}}{\omega _{1}}}d\sigma _{u}$ ${\ displaystyle d \ sigma \ sim \ alpha \ ln {\ frac {\ omega _ {2}} {\ omega _ {1}}} d \ sigma _ {u}}$ where $d\sigma _{u}$ ${\ displaystyle d \ sigma _ {u}}$ - scattering cross section of the elastic process. We can approximately assume that $\omega _{2}$ ${\ displaystyle \ omega _ {2}}$ approximately equal to the initial energy of the radiating particle. But the magnitude $\omega _{1}$ ${\ displaystyle \ omega _ {1}}$ can be made arbitrarily close to zero. As a result, the radiation cross section of all possible soft photons tends to infinity. ^[one]

With another method of calculating the average number of photons with a sharp change in the speed of a charged particle: ${\bar {n}}\sim \ln {\frac {L}{\lambda }}$ ${\ displaystyle {\ bar {n}} \ sim \ ln {\ frac {L} {\ lambda}}}$ where $L,\lambda$ ${\ displaystyle L, \ lambda}$ - maximum and minimum integration frequencies. At $\lambda \rightarrow 0$ ${\ displaystyle \ lambda \ rightarrow 0}$ we get that ${\bar {n}}\rightarrow \infty$ ${\ displaystyle {\ bar {n}} \ rightarrow \ infty}$ so that infinitely many photons of zero frequency are always emitted. ^[2]

Paradox explanation

The average number of emitted photons $d{\bar {n}}={\frac {dI}{\omega }}$ ${\ displaystyle d {\ bar {n}} = {\ frac {dI} {\ omega}}}$ where $d I$ ${\ displaystyle dI}$ - classical radiation intensity, $\omega$ ${\ displaystyle \ omega}$ - radiation frequency. Integrating this formula we get: ${\bar {n}}=\int _{\omega _{1}}^{\omega _{2}}{\frac {dI}{\omega }}$ ${\ displaystyle {\ bar {n}} = \ int _ {\ omega _ {1}} ^ {\ omega _ {2}} {\ frac {dI} {\ omega}}}$ . Since soft photons are emitted statistically independently, the probability $\omega (n)$ ${\ displaystyle \ omega (n)}$ radiation ${\bar {n}}$ ${\ displaystyle {\ bar {n}}}$ photons is expressed in terms of their average number by the Poisson formula $\omega (n)={\frac {{\bar {n}}^{n}}{n!}}\exp(-{\bar {n}})$ ${\ displaystyle \ omega (n) = {\ frac {{\ bar {n}} ^ {n}} {n!}} \ exp (- {\ bar {n}})}$ . The cross section of the scattering process with photon emission can be represented as: $d\sigma =d\sigma _{u}\omega (n)$ ${\ displaystyle d \ sigma = d \ sigma _ {u} \ omega (n)}$ . Insofar as $\sum \omega (n)=1$ ${\ displaystyle \ sum \ omega (n) = 1}$ then $d\sigma _{u}$ ${\ displaystyle d \ sigma _ {u}}$ represents the total scattering cross section, accompanied by any soft radiation. The cross section for purely elastic scattering is actually zero. At $\omega _{1}\to 0$ ${\ displaystyle \ omega _ {1} \ to 0}$ average number ${\bar {n}}\to \infty$ ${\ displaystyle {\ bar {n}} \ to \ infty}$ and according to the Poisson formula, the probability of emission of any finite number of photons vanishes. ^[one]

The physical cause of the paradox is the assumption of an infinite radius of the Coulomb field , which leads to the inadequacy of the photon picture for very large wavelengths. To fulfill the condition ${\bar {n}}>1$ ${\ displaystyle {\ bar {n}}> 1}$ wavelengths must be longer $e^{\frac {1}{\alpha }}{\frac {\hbar }{mc}}$ ${\ displaystyle e ^ {\ frac {1} {\ alpha}} {\ frac {\ hbar} {mc}}}$ , which is significantly larger than the radius of the observed part of the universe. Thus, this paradox has a purely theoretical significance ^[2]

Notes

↑ ¹ ² V. B. Berestetskii , E. M. Lifshits , L. P. Pitaevsky Quantum electrodynamics. - M., Fizmatlit, 2001. - c. 482-488
↑ ¹ ² Walter E. Thirring. Principles of quantum electrodynamics. - M., Higher School, 1964. - p. 105-109

Literature

Kaku, Michio. Quantum Field Theory: A Modern Introduction. - New York: Oxford University Press, 1993 .-- ISBN 0-19-507652-4 .
Claude Itzykson, Jean-Bernard Zuber. Quantum Field Theory. - McGraw-Hill , 1980 .-- P. 172/3. - ISBN 0-07-032071-3 .

[Ber-1] ¹ ² V. B. Berestetskii , E. M. Lifshits , L. P. Pitaevsky Quantum electrodynamics. - M., Fizmatlit, 2001. - c. 482-488

[Walter-2] ¹ ² Walter E. Thirring. Principles of quantum electrodynamics. - M., Higher School, 1964. - p. 105-109