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.
W.
-
{\ displaystyle W ^ {-}}
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)
ψ
{\ displaystyle \ psi}
ψ
c
{\ displaystyle \ psi _ {c}}
e
{\ displaystyle e}
A.
0
,
A.
1
,
A.
2
,
A.
3
{\ displaystyle A_ {0}, A_ {1}, A_ {2}, A_ {3}}
ψ
{\ displaystyle \ psi}
n
{\ displaystyle n}
(
γ
n
(
i
∂
n
-
e
A.
n
)
-
m
)
ψ
=
0
{\ displaystyle {\ bigl (} \ gamma ^ {n} \, (\ mathrm {i} \, \ partial _ {n} -eA_ {n}) - m {\ bigr)} \ psi = 0}
is satisfied, then the charge conjugate field of the equation
ψ
c
{\ displaystyle \ psi _ {c}}
(
γ
n
(
i
∂
n
+
e
A.
n
)
-
m
)
ψ
c
=
0
{\ displaystyle {\ bigl (} \ gamma ^ {n} \, (\ mathrm {i} \, \ partial _ {n} + eA_ {n}) - m {\ bigr)} \ psi _ {c} = 0}
suffice.
Complex conjugate the first equation yields
(
γ
n
∗
(
-
i
∂
n
-
e
A.
n
)
-
m
)
ψ
∗
=
0
.
{\ 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:
ψ
c
=
B.
ψ
∗
{\ displaystyle \ psi _ {c} = B \ psi ^ {*}}
B.
{\ displaystyle B}
-
γ
n
∗
=
B.
-
1
γ
n
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
-
γ
n
∗
{\ displaystyle - \ gamma ^ {n \, *}}
γ
n
.
{\ displaystyle \ gamma ^ {n} \ ,.}
If you write, the charge-conjugate field has the form
ψ
∗
=
γ
0
T
ψ
¯
T
{\ displaystyle \ psi ^ {*} = \ gamma ^ {0 \, {\ text {T}}} \, {\ overline {\ psi}} ^ {\ text {T}}}
ψ
c
=
C.
ψ
¯
T
{\ displaystyle \ psi _ {c} = C \, {\ overline {\ psi}} ^ {\ text {T}}}
with the charge conjugation matrix
C.
=
B.
γ
0
T
.
{\ displaystyle C = B \, \ gamma ^ {0 \, {\ text {T}}} \ ,.}
Wegen satisfies the charge conjugation matrix
γ
n
†
=
γ
0
γ
n
γ
0
{\ displaystyle \ gamma ^ {n \, \ dagger} = \ gamma ^ {0} \ gamma ^ {n} \ gamma ^ {0}}
-
γ
n
T
=
C.
-
1
γ
n
C.
.
{\ 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
C.
=
i
γ
2
γ
0
=
(
-
i
σ
2
-
i
σ
2
)
{\ 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 ,
-
C.
=
C.
-
1
=
C.
T
=
C.
†
.
{\ 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:
C.
|
ψ
⟩
=
η
C.
|
ψ
¯
⟩
{\ displaystyle {\ mathcal {C}} \, | \ psi \ rangle = \ eta _ {C} \, | {\ bar {\ psi}} \ rangle}
.
Since the parity operator is an involution (mathematics) , the following applies
C.
2
|
ψ
⟩
=
η
C.
C.
|
ψ
¯
⟩
=
η
C.
2
|
ψ
⟩
=
|
ψ
⟩
{\ 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.
η
C.
=
±
1
{\ 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 .
C.
|
ψ
⟩
{\ displaystyle {\ mathcal {C}} \, | \ psi \ rangle}
|
ψ
⟩
{\ displaystyle {\ mathcal {|}} \ psi \ rangle}
π
0
{\ displaystyle \ pi ^ {0}}
literature
See also
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