# nLab Dirac current

Contents

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

# Contents

## Idea

In the Lagrangian field theory of the Dirac field, the Dirac current is a conserved current whose interpretation is literally the current of the spinor particles that are the quanta of the Dirac field. Therefore the corresponding charge is fermion number. As such the Dirac current (or rather its chiral version) plays a key role for instance in baryogenesis.

The Dirac current is the conserved current which is associated via Noether's theorem I to the infinitesimal symmetry of the Lagrangian given by multiplying the Dirac field by a complex phase.

## Definition

In the usual standard coordinates, the Dirac current is of the form

$\overline{\psi}\gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma \;\in\; \Omega^{p,0} \,.$

For details see at geometry of physics – A first idea of quantum field theory this example.

Created on November 8, 2017 at 13:58:25. See the history of this page for a list of all contributions to it.