Spinor-helicity formalism

The spinor-helicity formalism , also Weyl-van-der-Waerden-formalism after Hermann Weyl and Bartel Leendert van der Waerden , is an alternative mathematical formulation of quantum field theories based on the use of spinors and invariants of the special linear group instead of the use based on four-vectors and invariants of the Lorentz group .

Basics

Group theory of the Lorentz group

→ Main article : Representation theory of the Lorentz group

In 3 + 1 space-time -Dimensionen the real Lorentz group is isomorphic to complex specific linear array in two dimensions . This means that each group element of the Lorentz group can be assigned an element of the complex special linear group and each vector in the real four-dimensional space-time on which the Lorentz group operates, a matrix in the space of complex matrices on which the special linear group operated on. This transition occurs through the four Pauli matrices . Let be a four-vector, then: ${\displaystyle SO(1,3,\mathbb {R} )}$ ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle \mathbb {R} ^{1,3}}$ ${\displaystyle 2\times 2}$ ${\displaystyle M_{2}(\mathbb {C} )}$ ${\displaystyle \sigma ^{\mu }}$ ${\displaystyle k_{\mu }}$

${\displaystyle k^{a{\dot {a}}}=\sigma _{\mu }^{a{\dot {a}}}k^{\mu }={\begin{pmatrix}k_{0}+k_{3}&k_{1}-\mathrm {i} k_{2}\\k_{1}+\mathrm {i} k_{2}&k_{0}-k_{3}\end{pmatrix}}}$

The Greek indices denote Lorentz indices that run from 0 to 3, while the Latin indices are called spinor indices and run from 1 to 2. The reverse transformation from to works via ${\displaystyle \mu }$ ${\displaystyle a,{\dot {a}}}$ ${\displaystyle M_{2}(\mathbb {C} )}$ ${\displaystyle \mathbb {R} ^{1,3}}$

${\displaystyle k_{\mu }={\frac {1}{2}}{\bar {\sigma }}_{{\dot {a}}a}^{\mu }k^{a{\dot {a}}}}$

The Lorentz variable is translated via ${\displaystyle k\cdot p}$

${\displaystyle k^{\mu }p_{\mu }={\frac {1}{2}}\varepsilon _{ba}\varepsilon _{{\dot {a}}{\dot {b}}}k^{a{\dot {a}}}p^{b{\dot {b}}}}$

with the totally antisymmetric Levi Civita symbol . In particular, ${\displaystyle \varepsilon }$

${\displaystyle k_{\mu }k^{\mu }=\det(k^{a{\dot {a}}})}$.

The group operation of an element of the Lorentz group with a Lorentz matrix translates as ${\displaystyle k^{\nu }\to k'^{\nu }=\Lambda _{\mu }^{\nu }k^{\mu }}$ ${\displaystyle \Lambda }$

${\displaystyle k^{a{\dot {a}}}\to k'^{a{\dot {a}}}=\zeta _{b}^{a}k^{b{\dot {b}}}{\tilde {\zeta }}_{\dot {b}}^{\dot {a}}}$

with . The matrices are for rotations around an axis with the angle ${\displaystyle \zeta ,{\tilde {\zeta }}=\zeta ^{\dagger }\in SL(2,\mathbb {C} )}$ ${\displaystyle \zeta }$ ${\displaystyle i}$ ${\displaystyle \alpha }$

${\displaystyle \zeta (R_{i}(\theta ))=\exp(-\mathrm {i} {\tfrac {\alpha }{2}}\sigma _{i})}$

and for Lorentz boosts along an axis with rapidity ${\displaystyle i}$ ${\displaystyle \theta }$

${\displaystyle \zeta (B_{i}(\theta ))=\exp(-\mathrm {\tfrac {\theta }{2}} \sigma _{i})}$

where denotes the matrix exponential . ${\displaystyle \exp }$

This can be generalized to the complex Lorentz group that operates on . Then the isomorphism applies and does not necessarily have to be the same and it does not have to apply. ${\displaystyle SO(1,3,\mathbb {C} )}$ ${\displaystyle \mathbb {C} ^{1,3}}$ ${\displaystyle SO(1,3,\mathbb {C} )\cong SL(2,\mathrm {C} )\times SL(2,\mathrm {C} )}$ ${\displaystyle {\tilde {\zeta }}}$ ${\displaystyle \zeta ^{\dagger }}$ ${\displaystyle \kappa ={\tilde {\kappa }}^{\dagger }}$

notation

From it follows that a light-like vector translates into a matrix without full rank . Since is the dimension of two, it follows provided that is. Therefore, the dyadic product can be written: ${\displaystyle k\cdot k=\det(k)}$ ${\displaystyle k^{\mu }}$ ${\displaystyle k}$ ${\displaystyle \operatorname {rg} k=1}$ ${\displaystyle k\neq 0}$ ${\displaystyle k}$

${\displaystyle k^{a{\dot {a}}}=\kappa ^{a}{\tilde {\kappa }}^{\dot {a}}}$

Both and are two-dimensional objects called spinors . The spinor is called holomorphic spinor, the spinor antiholomorphic spinor. An explicit representation of these spinors reads: ${\displaystyle \kappa }$ ${\displaystyle {\tilde {\kappa }}}$ ${\displaystyle \kappa }$ ${\displaystyle {\tilde {\kappa }}}$

${\displaystyle \kappa ^{a}={\frac {1}{\sqrt {k_{0}+k_{3}}}}{\begin{pmatrix}k_{0}+k_{3}\\k_{1}+\mathrm {i} k_{2}\end{pmatrix}}\qquad {\tilde {\kappa }}^{\dot {a}}={\frac {1}{\sqrt {k_{0}+k_{3}}}}{\begin{pmatrix}k_{0}+k_{3}&k_{1}-\mathrm {i} k_{2}\end{pmatrix}}}$

In particular, the spinors can be respectively rescaled by a factor without this changing the matrix . It can be seen that the two spinors are adjoint as long as the vector is real. In the following it is assumed that all vectors that occur are light-like. ${\displaystyle t}$ ${\displaystyle t^{-1}}$ ${\displaystyle k^{a{\dot {a}}}}$ ${\displaystyle k^{\mu }}$

A scalar product of two four-vectors can therefore be written as

${\displaystyle k\cdot p=\kappa ^{a}\varepsilon _{ab}\pi ^{b}{\tilde {\kappa }}^{\dot {a}}\varepsilon _{{\dot {a}}{\dot {b}}}{\tilde {\pi }}^{\dot {b}}\equiv \langle \kappa \pi \rangle [\pi \kappa ]}$

to be written. In this sense, the Levi-Civita symbols take on the role of the metric in the . It applies ${\displaystyle SL(2,\mathbb {C} )}$

${\displaystyle \kappa _{b}=\varepsilon _{ab}\kappa ^{a}}$ and ${\displaystyle {\tilde {\kappa }}_{\dot {b}}=\varepsilon _{{\dot {a}}{\dot {b}}}{\tilde {\kappa }}^{\dot {a}}}$

Analogous to the Bra-Ket notation , the notation for the spinors is:

${\displaystyle \kappa ^{a}=\kappa \rangle \quad \kappa _{a}=\langle \kappa \quad {\tilde {\kappa }}^{\dot {a}}=[\kappa \quad {\tilde {\kappa }}_{\dot {a}}=\kappa ]}$

In particular, is due to the antisymmetry of the Levi-Civita symbol

${\displaystyle \langle \kappa \kappa \rangle =[\kappa \kappa ]=0}$.

A space- or time-like vector can always be by means of

${\displaystyle k=k^{\flat }+{\frac {k^{2}}{2k\cdot p}}p}$

can be decomposed into two light-like vectors. In this example it is called auxiliary vector. ${\displaystyle p}$

Physical implication

Fermions

With the help of the spinor-helicity formalism, the solution of the Dirac equation is trivial. The Dirac equation is:

${\displaystyle (-\mathrm {i} \gamma ^{\mu }\partial _{\mu }-m)\psi =0}$

Where is the momentum of the particle and the Dirac matrices . The approach or leads to ${\displaystyle p_{\mu }}$ ${\displaystyle \gamma }$ ${\displaystyle \psi =u\exp(-\mathrm {i} px)}$ ${\displaystyle \psi =v\exp(\mathrm {i} px)}$

${\displaystyle (\gamma ^{\mu }p_{\mu }-m)u=0}$ and ${\displaystyle (\gamma ^{\mu }p_{\mu }+m)v=0}$

for particles or antiparticles. In the Weyl representation, the following applies

${\displaystyle \gamma ^{\mu }={\begin{pmatrix}0&\sigma ^{\mu }\\{\bar {\sigma }}^{\mu }&0\end{pmatrix}}}$

On the mass shell is also space-like and must therefore be decomposed, so that the Dirac equation in the spinor-helicity formalism ${\displaystyle p^{2}=m^{2}}$

${\displaystyle {\begin{pmatrix}\mp m&\pi \rangle [\pi +{\frac {m^{2}}{\langle \pi \mu \rangle [\mu \pi ]}}\mu \rangle [\mu \\\pi ]\langle \pi +{\frac {m^{2}}{\langle \pi \mu \rangle [\mu \pi ]}}\mu ]\langle \mu &\mp m\end{pmatrix}}{\begin{pmatrix}u\rangle \\u]\end{pmatrix}}=0}$

with an auxiliary "vector" is. It follows ${\displaystyle \mu \rangle [\mu }$

${\displaystyle \psi \rangle \!\!{\big \rangle }\equiv {\begin{pmatrix}\pi \rangle \\\mp {\frac {m}{[\pi \mu ]}}\mu ]\end{pmatrix}}}$ and ${\displaystyle \psi ]\!{\big ]}\equiv {\begin{pmatrix}\pm {\frac {m}{\langle \pi \mu \rangle }}\mu \rangle \\\pi ]\end{pmatrix}}}$

as solutions to the eigenvalue problem. The Dirac spinors are normalized so that . In the massive case, the spinors are particularly dependent on the choice of the auxiliary vector; not in the massless case. ${\displaystyle {\bar {\psi }}\psi =2m}$

Vector bosons

The Maxwell equations

${\displaystyle p_{\mu }\epsilon ^{\mu }=0}$

with the polarization vector has the two solutions ${\displaystyle \epsilon }$

${\displaystyle \epsilon ={\sqrt {2}}{\frac {\pi \rangle [\mu }{[\pi \mu ]}}}$ and ${\displaystyle \epsilon =-{\sqrt {2}}{\frac {\mu \rangle [\pi }{\langle \pi \mu \rangle }}}$

where applies. The normalization was chosen so that the two solutions are orthonormal. ${\displaystyle p^{2}=0}$

The Proca equation for massive vector bosons has the additional solution

${\displaystyle \epsilon ={\frac {1}{m}}\left(\pi \rangle [\pi -{\frac {m^{2}}{\langle \pi \mu \rangle [\mu \pi ]}}\mu \rangle [\mu \right)}$

which corresponds to the longitudinal polarization mode.

Individual evidence

1. ↑ Eduardo Conde and Andrea Marzolla: Lorentz Constraints on Massive Three-Point Amplitudes . arxiv : 1601.08113 .
2. ^ Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 537 (English).
3. ↑ Timothy Cohen, Henriette Elvang and Michael Kiermaier: On-shell constructibility of tree amplitudes in general field theories . arxiv : 1010.0257 .