Utente:Vilnius/Sandbox: differenze tra le versioni

Versione delle 14:13, 31 lug 2023

Versione attuale delle 11:57, 1 ago 2023

(3 versioni intermedie di uno stesso utente non sono mostrate)

Riga 29:

Come sopra, nel caso di più fotoni occorre sommare i corrispondenti termini.

=== Estensione al termine successivo ===

Volendo estendere lo sviluppo della formula al termine successivo [[Andrew Strominger]] e Freddy Cachazo hanno dimostrato che per il gravitone vale la seguente relazione:

<math>{\cal S}' = \sqrt{8\pi G} \frac{\eta p^\mu p^\nu \epsilon_{\mu\nu}}{p \cdot p_G - i \eta \varepsilon}{\cal S}-i\sqrt{8\pi G} \frac{\eta p^\mu ({p_G}_\rho J^{\rho\nu}) \epsilon_{\mu\nu}}{p \cdot p_G - i \eta \varepsilon}{\cal S} + O(p_G^1)</math>,

dove <math>J^{\rho\nu}</math>rappresenta il momento angolare della particella che interagisce con il gravitone.<ref>{{Cita libro|nome=Freddy|cognome=Cachazo|nome2=Andrew|cognome2=Strominger|wkautore2=Andrew Strominger|titolo=Evidence for a New Soft Graviton Theorem|url=https://arxiv.org/pdf/1404.4091|data=aprile 2014|pp=1-3|capitolo=1 .Introduction}}</ref>

[[:en:Francis_E._Low|F.E. Low]] per il fotone

[2] F. E. Low, “Scattering of light of very low frequency by systems of spin 1/2,” Phys. Rev. 96 (1954) 1428–32.

[4] F. E. Low, “Bremsstrahlung of very low-energy quanta in elementary particle collisions,”Phys. Rev. 110 (1958) 974–77.

== Dimostrazione ==

Riga 74 ⟶ 88:

'''propagator.'''

Andrew Strominger - Lectures on the Infrared Structure of Gravity and Gauge Theory, p. 35

Riga 86 ⟶ 98:

Weinberg’s soft graviton theorem<ref name=":0" /> is a universal formula relating any S-matrix element in any quantum theory including gravity to a second S-matrix element which differs only by the addition of a graviton whose four-momentum is taken to zero. Remarkably, the formula is blind to the spin or any other quantum numbers of the asymptotic particles involved in the S-matrix element.

https://dash.harvard.edu/bitstream/handle/1/29374083/1401.7026.pdf;jsessionid=6392FB47A36DFFDF342EC0BC22893C9E?sequence=1

''Consider an amplitude M involving some incoming and some outgoing particles. Now, consider the same amplitude with an additional soft-photon (''<math>\omega_{\text{photon}} \to 0</math>'') coupled to one of the particles. Call this amplitude M'. The two amplitudes are related by''

<math>{\cal M}' = {\cal M} \frac{\eta q p \cdot \epsilon}{p \cdot p_\gamma - i \eta \varepsilon}</math>

''where p is the momentum of the particle that the photon couples to,'' <math>\epsilon</math>'' is the polarization of the photon and'' <math>p_\gamma</math>'' is the momentum of the soft-photon.  ''<math>\eta = 1</math>''for outgoing particles and'' <math>\eta = -1</math>'' for incoming ones. Finally, q is the charge of the particle.''

the proportionality factor relating M and M' is independent of the type of particle that the photon couples to.

== Note ==