Editing TCP- Invarianz

<noinclude>{{Scripthinweis|Elektrodynamik|3|1}}</noinclude>
;Zeitumkehr T: t → t´=-t
;Ladungsumkehr / Konjugation :  C :  Q → Q´= - Q
;Paritätsumkehr P :  r →  r´= -r (für den Ortsvektor)


=== Die Zeitumkehr- Transformation ===

:<math>\begin{align}
& {{T}_{g}}:=\left\{ T-in\operatorname{var}iante\ ObservableA:TA=A \right\} \\
& =\left\{ \bar{r},d\bar{r},a:=\frac{{{d}^{2}}\bar{r}}{d{{t}^{2}}},m,q,\rho :=\begin{matrix}
\lim   \\
\Delta V\to 0  \\
\end{matrix}\frac{\Delta q}{\Delta V},\bar{F}=m\bar{a},\bar{E}=\frac{{\bar{F}}}{q},\Phi ... \right\} \\
\end{align}</math>
Diese Observablen sind "gerade" unter T

Daneben gibt es auch Observablen, die "ungerade" unter T sind:

:<math>{{T}_{u}}:=\left\{ A:TA=-A \right\}=\left\{ \bar{v}:=\frac{d\bar{r}}{dt},\bar{j}=\rho \bar{v},\bar{B},\bar{A} \right\}</math>

Denn:

:<math>\begin{align}
& \bar{F}=q\bar{v}\times \bar{B} \\
& \bar{F}\in {{T}_{g}},\bar{v}\in {{T}_{u}},q\in {{T}_{g}}\Rightarrow \bar{B}\in {{T}_{u}} \\
& \bar{B}=\nabla \times \bar{A},\nabla \in {{T}_{g}} \\
\end{align}</math>

Somit folgt jedoch vollständige T- Invarianz der elektromagnetischen Grundgleichungen:

:<math>\begin{align}
& T:\left\{ {{\nabla }_{r}}\times \bar{E}=0 \right\}\to \left\{ {{\nabla }_{r}}\times \bar{E}=0 \right\} \\
& T:\left\{ {{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \bar{E}=\rho  \right\}\to \left\{ {{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \bar{E}=\rho  \right\} \\
& T:\left\{ {{\nabla }_{r}}\cdot \bar{B}=0 \right\}\to \left\{ -{{\nabla }_{r}}\cdot \bar{B}=0 \right\}\Leftrightarrow \left\{ {{\nabla }_{r}}\cdot \bar{B}=0 \right\} \\
& T:\left\{ \nabla \times \bar{B}={{\mu }_{0}}\bar{j} \right\}\to \left\{ -\nabla \times \bar{B}=-{{\mu }_{0}}\bar{j} \right\} \\
&  \\
\end{align}</math>

{{FB|Kontinuitätsgleichung}}:

:<math>T:\left\{ \frac{\partial }{\partial t}\rho +{{\nabla }_{r}}\cdot \bar{j}=0 \right\}\to \left\{ -\frac{\partial }{\partial t}\rho -{{\nabla }_{r}}\cdot \bar{j}=0 \right\}</math>

Die Gleichungen sind {{FB|forminvariant}}!

==Ladungsumkehr (Konjugation)==

:<math>\begin{align}
& {{C}_{g}}:=\left\{ C-in\operatorname{var}iante\ ObservableA:CA=A \right\} \\
& {{C}_{g}}=\left\{ \bar{F},m,\bar{r},\bar{v},\bar{a} \right\} \\
\end{align}</math>

sind gerade unter C
'''Ungerade unter c sind:'''

:<math>\begin{align}
& {{C}_{u}}:=\left\{ A:CA=-A \right\}=\left\{ \bar{E}=\frac{1}{q}\bar{F},\bar{B},\bar{j},\rho  \right\} \\
& \bar{F}=q\bar{v}\times \bar{B} \\
\end{align}</math>

* C- Invarianz der Elektro- Magnetostatik:

:<math>\begin{align}
& C:\left\{ {{\nabla }_{r}}\times \bar{E}=0 \right\}\to \left\{ -{{\nabla }_{r}}\times \bar{E}=0 \right\} \\
& C:\left\{ {{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \bar{E}=\rho  \right\}\to \left\{ -{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \bar{E}=-\rho  \right\} \\
& C:\left\{ {{\nabla }_{r}}\cdot \bar{B}=0 \right\}\to \left\{ -{{\nabla }_{r}}\cdot \bar{B}=0 \right\} \\
& C:\left\{ \nabla \times \bar{B}={{\mu }_{0}}\bar{j} \right\}\to \left\{ -\nabla \times \bar{B}=-{{\mu }_{0}}\bar{j} \right\} \\
\end{align}</math>

:<math>C:\left\{ \frac{\partial }{\partial t}\rho +{{\nabla }_{r}}\cdot \bar{j}=0 \right\}\to \left\{ -\frac{\partial }{\partial t}\rho -{{\nabla }_{r}}\cdot \bar{j}=0 \right\}</math>

==Paritätsumkehr: Räumliche Spiegelung/ Inversion==

Vertauschung: rechts ↔ links


Man unterscheidet:

:<math>P\bar{r}=-\bar{r}</math>
→ polarer Vektor
und

:<math>P\left( \bar{a}\times \bar{b} \right)=\left( -\bar{a}\times -\bar{b} \right)=\left( \bar{a}\times \bar{b} \right)</math>
P- invariant = " axialer Vektor", sogenannter Pseudovektor!!



Seien:

:<math>\bar{a},\bar{b}</math>
polar,
:<math>\bar{w},\bar{\sigma }</math>
axial
Dann ist

:<math>\begin{align}
& \bar{a}\times \bar{w}\quad polar \\
& \bar{a}\times \bar{b},\bar{w}\times \bar{\sigma }\quad axial \\
& \bar{a}\bar{b}\ skalar:P(\bar{a}\bar{b})=\bar{a}\bar{b} \\
& \bar{w}\bar{\sigma }\ pseudoskalarP(\bar{w}\bar{\sigma })=-\bar{w}\bar{\sigma } \\
\end{align}</math>

:<math>\begin{align}
& {{C}_{g}}:=\left\{ C-in\operatorname{var}iante\ ObservableA:CA=A \right\} \\
& {{C}_{g}}=\left\{ \bar{F},m,\bar{r},\bar{v},\bar{a} \right\} \\
\end{align}</math>

Wegen

:<math>\begin{align}
& \bar{F}=q\bar{v}\times \bar{B} \\
& \bar{F}\in {{P}_{u}} \\
& q\in {{P}_{g}} \\
& \bar{v}\in {{P}_{u}} \\
\end{align}</math>

ungerade Parität dagegen:

:<math>{{P}_{u}}=\left\{  polareVektoren,\bar{r},d\bar{r},\bar{v},\bar{a},\bar{F},\bar{E}=\frac{1}{q}\bar{F},\bar{j}=\rho \bar{v},\bar{A},Pseudoskalare\quad \nabla \cdot \bar{B} \right\}</math>

Wegen

:<math>\begin{align}
& \bar{B}=\nabla \times \bar{A} \\
& \nabla \in {{P}_{u}} \\
& \bar{B}\in {{P}_{g}} \\
\end{align}</math>

P- Invarianz der Elektro- / Magnetostatik:

:<math>\begin{align}
& P:\left\{ {{\nabla }_{r}}\times \bar{E}=0 \right\}\to \left\{ {{\nabla }_{r}}\times \bar{E}=0 \right\} \\
& P:\left\{ {{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \bar{E}=\rho  \right\}\to \left\{ {{\varepsilon }_{0}}{{\nabla }_{r}}\cdot \bar{E}=\rho  \right\} \\
& P:\left\{ {{\nabla }_{r}}\cdot \bar{B}=0 \right\}\to \left\{ -{{\nabla }_{r}}\cdot \bar{B}=0 \right\} \\
& P:\left\{ \nabla \times \bar{B}={{\mu }_{0}}\bar{j} \right\}\to \left\{ -\nabla \times \bar{B}=-{{\mu }_{0}}\bar{j} \right\} \\
\end{align}</math>

:<math>P:\left\{ \frac{\partial }{\partial t}\rho +{{\nabla }_{r}}\cdot \bar{j}=0 \right\}\to \left\{ \frac{\partial }{\partial t}\rho +{{\nabla }_{r}}\cdot \bar{j}=0 \right\}</math>

Nebenbemerkung: Gäbe es magnetische Ladungen, dann wären sie pseudoskalare
Außerdem (Weinberg e.a.) : Schwache Wechselwirkung verletzt die Paritätserhaltung!