Charge conjugation

The charge conjugation or C- parity (for English C harge = charge ) replaces each particle with its antiparticle in quantum mechanical states . It reflects the sign of the charge and leaves the mass , momentum , energy and spin of each particle unchanged.

The electromagnetic and the strong interaction are invariant under charge conjugation (C-invariant for short), i.e. In other words , in the event of scattering or decay, the charge-mirrored states behave like the original states.
In contrast, the weak interaction not C-invariant ( parity violation ): The proportion of the electron , which in weak interactions in an electron neutrino and a - Boson can pass, is in charge conjugation through the part of the positron replaced, which can not take the bosons couples. ${\displaystyle W^{-}}$ ${\displaystyle W}$

Charge conjugation of the Dirac field

In the case of charge conjugation, the Dirac field is transformed into the field that couples with the reverse charge to the electromagnetic potentials . If the Dirac equation (over the double index is to be summed) ${\displaystyle \psi }$ ${\displaystyle \psi _{c}}$ ${\displaystyle e}$ ${\displaystyle A_{0},A_{1},A_{2},A_{3}}$ ${\displaystyle \psi }$ ${\displaystyle n}$

${\displaystyle {\bigl (}\gamma ^{n}\,(\mathrm {i} \,\partial _{n}-eA_{n})-m{\bigr )}\psi =0}$

is satisfied, then the charge conjugate field of the equation ${\displaystyle \psi _{c}}$

${\displaystyle {\bigl (}\gamma ^{n}\,(\mathrm {i} \,\partial _{n}+eA_{n})-m{\bigr )}\psi _{c}=0}$

suffice.

Complex conjugate the first equation yields

${\displaystyle {\bigl (}\gamma ^{n\,*}\,(-\mathrm {i} \,\partial _{n}-eA_{n})-m{\bigr )}\psi ^{*}=0\ .}$

So it satisfies the charge conjugate equation if there is a matrix for which: ${\displaystyle \psi _{c}=B\psi ^{*}}$ ${\displaystyle B}$

${\displaystyle -\gamma ^{n\,*}=B^{-1}\gamma ^{n}B}$

Such a matrix exists for every representation of the Dirac matrices , because all irreducible representations of the Dirac algebra are equivalent to one another, and the Dirac algebra represents just like ${\displaystyle -\gamma ^{n\,*}}$ ${\displaystyle \gamma ^{n}\,.}$

If you write, the charge-conjugate field has the form ${\displaystyle \psi ^{*}=\gamma ^{0\,{\text{T}}}\,{\overline {\psi }}^{\text{T}}}$

${\displaystyle \psi _{c}=C\,{\overline {\psi }}^{\text{T}}}$ with the charge conjugation matrix ${\displaystyle C=B\,\gamma ^{0\,{\text{T}}}\,.}$

Wegen satisfies the charge conjugation matrix ${\displaystyle \gamma ^{n\,\dagger }=\gamma ^{0}\gamma ^{n}\gamma ^{0}}$

${\displaystyle -\gamma ^{n\,{\text{T}}}=C^{-1}\gamma ^{n}C\,.}$

In the Dirac representation of the gamma matrices, the charge conjugation matrix can be written as

${\displaystyle C=\mathrm {i} \,\gamma ^{2}\,\gamma ^{0}={\begin{pmatrix}&-\mathrm {i} \sigma ^{2}\\-\mathrm {i} \sigma ^{2}\end{pmatrix}}}$

be chosen to be real, antisymmetric and unitary , ${\displaystyle -C=C^{-1}=C^{\text{T}}=C^{\dagger }\,.}$

Eigenvalues ​​and eigenstates

For the eigen-states of the C-operator on a particle we have:

${\displaystyle {\mathcal {C}}\,|\psi \rangle =\eta _{C}\,|{\bar {\psi }}\rangle }$.

Since the parity operator is an involution (mathematics) , the following applies

${\displaystyle {\mathcal {C}}^{2}|\psi \rangle =\eta _{C}{\mathcal {C}}|{\bar {\psi }}\rangle =\eta _{C}^{2}|\psi \rangle =|\psi \rangle }$

This only allows eigenvalues , which is the C parity of the particle. ${\displaystyle \eta _{C}=\pm 1}$

However, this means that , and the same quantum charges have, so only neutral systems eigenstates of the C-parity operator may be, d. H. the photon as well as bound particle-antiparticle states like the neutral pion or positronium . ${\displaystyle {\mathcal {C}}\,|\psi \rangle }$ ${\displaystyle {\mathcal {|}}\psi \rangle }$ ${\displaystyle \pi ^{0}}$

literature

• Claude Itzykson , Jean-Bernard Zuber: Quantum Field Theory. McGraw-Hill, New York 1980, ISBN 0-07-032071-3 .

See also

• CP violation
• CPT theorem