Casimir's trick

Casimir's trick , named after the Dutch physicist Hendrik Casimir , is a method for the simple calculation of spin-averaged squared matrix elements in quantum field theories .

General

A term that occurs frequently in quantum field theories is the matrix element of the S matrix , which describes the transition from an initial state to a final state. This matrix element can be graphed using Feynman diagrams and translated into a rigorous mathematical expression. If fermions , i.e. particles with a spin of, are involved, then Dirac spinors , i.e. four-component vectors with additional spin indices , appear in the calculations of the matrix elements . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle 1/2}$

A Lorentz scalar variable is the squared matrix element , which generally contains complex expressions from products of Dirac spinors. However, if the calculation is only interested in a matrix element averaged over all possible spin settings, the matrix element can be converted into a product of traces via Dirac matrices with the help of Casimir's trick , which can be easily carried out on the basis of Dirac algebra. ${\displaystyle \left|{\mathcal {M}}\right|^{2}={\mathcal {M}}^{*}{\mathcal {M}}}$

Details

If the spinors denote incoming [anti] particles in Feynman diagrams and spinors denote outgoing [anti-] particles, then: ${\displaystyle u(k),[{\bar {v}}(k)]}$ ${\displaystyle {\bar {u}}(k)[v(k)]}$

${\displaystyle \sum _{s_{1},s_{2}}\left[{\bar {v}}^{s_{1}}(k)\Gamma u^{s_{2}}(p)\right]\,\left[{\bar {v}}^{s_{1}}(k)\Gamma u^{s_{2}}(p)\right]^{*}={\text{Tr}}\left[\Gamma \left(\gamma ^{\alpha }p_{\alpha }+m_{1}\right){\bar {\Gamma }}\left(\gamma ^{\beta }k_{\beta }-m_{2}\right)\right]}$

${\displaystyle \sum _{s_{1},s_{2}}\left[{\bar {u}}^{s_{1}}(k)\Gamma v^{s_{2}}(p)\right]\,\left[{\bar {u}}^{s_{1}}(k)\Gamma v^{s_{2}}(p)\right]^{*}={\text{Tr}}\left[\Gamma \left(\gamma ^{\alpha }p_{\alpha }-m_{1}\right){\bar {\Gamma }}\left(\gamma ^{\beta }k_{\beta }+m_{2}\right)\right]}$

${\displaystyle \sum _{s_{1},s_{2}}\left[{\bar {v}}^{s_{1}}(k)\Gamma v^{s_{2}}(p)\right]\,\left[{\bar {v}}^{s_{1}}(k)\Gamma v^{s_{2}}(p)\right]^{*}={\text{Tr}}\left[\Gamma \left(\gamma ^{\alpha }p_{\alpha }-m_{1}\right){\bar {\Gamma }}\left(\gamma ^{\beta }k_{\beta }-m_{2}\right)\right]}$

${\displaystyle \sum _{s_{1},s_{2}}\left[{\bar {u}}^{s_{1}}(k)\Gamma u^{s_{2}}(p)\right]\,\left[{\bar {u}}^{s_{1}}(k)\Gamma u^{s_{2}}(p)\right]^{*}={\text{Tr}}\left[\Gamma \left(\gamma ^{\alpha }p_{\alpha }+m_{1}\right){\bar {\Gamma }}\left(\gamma ^{\beta }k_{\beta }+m_{2}\right)\right]}$

An arbitrary matrix denotes the Dirac matrices and an overline denotes the Dirac adjoint . denote the masses of the respective particles / antiparticles. ${\displaystyle \Gamma }$ ${\displaystyle 4\times 4}$ ${\displaystyle \gamma }$ ${\displaystyle {\bar {\Gamma }}=\gamma ^{0}\Gamma ^{\dagger }\gamma ^{0}}$ ${\displaystyle m}$

Math background

The Dirac spinors can be broken down into two independent spinors for particles and for antiparticles. These each fulfill a completeness relation ${\displaystyle u}$ ${\displaystyle v}$

${\displaystyle \sum _{\text{Spins}}u(p){\bar {u}}(p)=\gamma ^{\mu }p_{\mu }+m}$

${\displaystyle \sum _{\text{Spins}}v(k){\bar {v}}(k)=\gamma ^{\mu }k_{\mu }-m}$.

Typical expressions such as, for example , appear in the matrix element . So the squared matrix element is: ${\displaystyle {\mathcal {M}}\propto {\bar {v}}(k)\Gamma u(p)}$

${\displaystyle \left|{\mathcal {M}}\right|^{2}\propto \left[{\bar {v}}(k)\Gamma u(p)\right]\,\left[{\bar {v}}(k)\Gamma u(p)\right]^{*}={\bar {v}}(k)\Gamma u(p)\,{\bar {u}}(p){\bar {\Gamma }}v(k)}$

When adding up over the spin indices, the completeness relation can first be used in the middle pair of Dirac spinors. In the following, it is advisable not to suppress the spin and space-time indices for reasons of comprehensibility, whereby over the spins, over the four Dirac matrices and over the four components of the spinors add up: ${\displaystyle s_{1},s_{2}}$ ${\displaystyle \alpha ,\beta }$ ${\displaystyle \mu ,\nu ,\rho ,\sigma }$

${\displaystyle \sum _{s_{1},s_{2}}\left|{\mathcal {M}}\right|^{2}\propto \sum _{s_{1},s_{2}}{\bar {v}}_{\mu }^{s_{1}}(k)\Gamma ^{\mu \nu }u_{\nu }^{s_{2}}(p){\bar {u}}_{\rho }^{s_{2}}(p){\bar {\Gamma }}^{\rho \sigma }v_{\sigma }^{s_{1}}(k)=\sum _{s_{1}}{\bar {v}}_{\mu }^{s_{1}}(k)\Gamma ^{\mu \nu }\left(\gamma ^{\alpha }p_{\alpha }+m_{2}\right)_{\nu \rho }{\bar {\Gamma }}^{\rho \sigma }v_{\sigma }^{s_{1}}(k)}$.

In component notation it is more obvious that the summation can simply be carried out over, since all objects now commute; it is therefore true ${\displaystyle s_{1}}$

${\displaystyle \sum _{s_{1},s_{2}}\left|{\mathcal {M}}\right|^{2}\propto \Gamma ^{\mu \nu }\left(\gamma ^{\alpha }p_{\alpha }+m_{2}\right)_{\nu \rho }{\bar {\Gamma }}^{\rho \sigma }\left(\gamma ^{\beta }k_{\beta }-m_{1}\right)_{\sigma \mu }={\text{Tr}}\left[\Gamma \left(\gamma ^{\alpha }p_{\alpha }+m_{2}\right){\bar {\Gamma }}\left(\gamma ^{\beta }k_{\beta }-m_{1}\right)\right]}$

The proof runs analogously for the other three cases.

Example: electron-muon scattering

Denote the momentum of the incoming electron and that of the incoming muon , the matrix element of electron-muon scattering in the lowest order in quantum electrodynamics reads : ${\displaystyle p[p']}$ ${\displaystyle k[k']}$ ${\displaystyle e^{-}\mu ^{-}\rightarrow e^{-}\mu ^{-}}$

${\displaystyle {\mathcal {M}}={\bar {u}}(p)(\mathrm {i} e\gamma ^{\mu })u(p'){\frac {\mathrm {i} g_{\mu \nu }}{(p+k)^{2}}}{\bar {u}}(k')(\mathrm {i} e\gamma ^{\nu })u(k)}$

If the spins of the incoming particles are averaged and the spins of the outgoing particles are added up, then Casimir's trick is obtained after twice using it

${\displaystyle {\frac {1}{4}}\sum _{s_{1},s_{2},s'_{1},s'_{2}}\left|{\mathcal {M}}\right|^{2}={\frac {1}{4}}{\frac {e^{4}}{(p+k)^{4}}}{\text{Tr}}\left[\gamma ^{\mu }(\gamma ^{\alpha }p_{\alpha }+m_{e})\gamma ^{\nu }(\gamma ^{\beta }p'_{\beta }+m_{e})\right]\,{\text{Tr}}\left[\gamma _{\mu }(\gamma ^{\alpha }k_{\alpha }+m_{\mu })\gamma _{\nu }(\gamma ^{\beta }k'_{\beta }+m_{\mu })\right]}$

literature

• David Griffiths: Introduction to Elementary Particle Physics . Translated by Thomas Stange. Akademie-Verlag, Berlin 1996, ISBN 3-05-501627-0 .
• Abraham Pais: Inward Bound . Oxford, New York 1986, ISBN 978-0198519713 .