|
|
| Line 1: |
Line 1: |
| {{Unreferenced|date=December 2009}}
| | The writer is known by the title of Numbers Lint. His spouse doesn't like it the way he does but what he really likes doing is to do aerobics and he's been doing it for quite a while. Managing people is his occupation. California is our beginning place.<br><br>Also visit my blog over the counter std test - [http://www.neweracinema.com/tube/user/D69E use this link], |
| In [[theoretical physics]], a '''Fierz identity''' is an identity that allows one to rewrite ''[[bilinear]]s{{dn|date=December 2013}} of the product'' of two [[spinor]]s as a [[linear combination]] of ''products of the bilinears'' of the individual spinors. It is named after Swiss physicist [[Markus Fierz]].
| |
| | |
| There is a version of the Fierz identities for [[Dirac spinor]]s and there is another version for [[Weyl spinor]]s. And there are versions for other dimensions besides 3+1 dimensions.
| |
| | |
| Spinor bilinears can be thought of as elements of a [[Clifford Algebra]]. Then the Fierz identity is the concrete realization of the [[Clifford Algebra#Relation to the exterior algebra|relation to the exterior algebra]].
| |
| The identities for a generic scalar written as the contraction of two Dirac bilinears of the same type can be written with coefficients according the following table.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Product
| |
| ! S
| |
| ! V
| |
| ! T
| |
| ! A
| |
| ! P
| |
| |-
| |
| | S × S =
| |
| | 1/4
| |
| | 1/4
| |
| | -1/4
| |
| | -1/4
| |
| | 1/4
| |
| |
| |
| |-
| |
| | V × V =
| |
| | 1
| |
| | -1/2
| |
| | 0
| |
| | -1/2
| |
| | -1
| |
| |-
| |
| | T × T =
| |
| | -3/2
| |
| | 0
| |
| | -1/2
| |
| | 0
| |
| | -3/2
| |
| |-
| |
| | A × A =
| |
| | -1
| |
| | -1/2
| |
| | 0
| |
| | -1/2
| |
| | 1
| |
| |-
| |
| | P × P =
| |
| | 1/4
| |
| | -1/4
| |
| | -1/4
| |
| | 1/4
| |
| | 1/4
| |
| |-
| |
| |}
| |
| | |
| For example the V × V product can be expanded as,
| |
| :<math>
| |
| \left(\chi^\dagger\gamma^0\gamma^\mu\psi\right)\left(\psi^\dagger\gamma^0\gamma_\mu \chi\right)=
| |
| \left(\chi^\dagger\gamma^0\chi\right)\left(\psi^\dagger\gamma^0\psi\right)-
| |
| \frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\psi\right)-
| |
| \frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\gamma_5\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\gamma_5\psi\right)
| |
| -\left(\chi^\dagger\gamma^0\gamma^5\chi\right)\left(\psi^\dagger\gamma^0\gamma_5\psi\right).</math>
| |
| | |
| Simplifications arise when the considered spinors are chiral or [[Majorana spinor]]s as some term in the expansion can be vanishing.
| |
| | |
| ==References==
| |
| A derivation of identities for rewriting any scalar contraction of Dirac bilinears can be found in 29.3.4 of
| |
| {{cite book|author=L. B. Okun|title=Leptons and Quarks|publisher=North-Holland|year=1980|isbn=978-0-444-86924-1}}
| |
| | |
| See also appendix B.1.2 in {{cite book|author=T. Ortin|title=Gravity and Strings|publisher=Cambridge University Press|year=2004|isbn=978-0-521-82475-0}}
| |
| | |
| {{DEFAULTSORT:Fierz Identity}}
| |
| [[Category:Quantum field theory]]
| |
| [[Category:Spinors]]
| |
| [[Category:Mathematical identities]]
| |
| | |
| | |
| {{Phys-stub}}
| |
The writer is known by the title of Numbers Lint. His spouse doesn't like it the way he does but what he really likes doing is to do aerobics and he's been doing it for quite a while. Managing people is his occupation. California is our beginning place.
Also visit my blog over the counter std test - use this link,