TCP- Invarianz

Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD

Der Artikel TCP- Invarianz basiert auf der Vorlesungsmitschrift von Franz- Josef Schmitt des 3.Kapitels (Abschnitt 1) der Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD.

TCP- Invarianz	Die Maxwell-Gleichungen	Elektrodynamik Schöll
	TCP- Invarianz Maxwell- Gleichungen im Vakuum Induktionsgesetz Energiebilanz Impulsbilanz Eichinvarianz	Elektrostatik Stationäre Ströme und Magnetfeld Die Maxwell-Gleichungen Elektromagnetische Wellen Materie in elektrischen und magnetischen Feldern Relativistische Formulierung der Elektrodynamik

Zeitumkehr T: t -> t´=-t Ladungsumkehr / Konjugation : C : Q  Q´= - Q Paritätsumkehr P : r - > r´= -r ( für den Ortsvektor)

Die Zeitumkehr- Transformation

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

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

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

Denn:

${\begin{aligned}&{\bar {F}}=q{\bar {v}}\times {\bar {B}}\\&{\bar {F}}\in {{T}_{g}},{\bar {v}}\in {{T}_{u}},q\in {{T}_{g}}\Rightarrow {\bar {B}}\in {{T}_{u}}\\&{\bar {B}}=\nabla \times {\bar {A}},\nabla \in {{T}_{g}}\\\end{aligned}}$

Somit folgt jedoch vollständige T- Invarianz der elektromagnetischen Grundgleichungen:

${\begin{aligned}&T:\left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\to \left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\\&T:\left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\to \left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\\&T:\left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\to \left\{-{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\Leftrightarrow \left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\\&T:\left\{\nabla \times {\bar {B}}={{\mu }_{0}}{\bar {j}}\right\}\to \left\{-\nabla \times {\bar {B}}=-{{\mu }_{0}}{\bar {j}}\right\}\\&\\\end{aligned}}$

Kontinuitätsgleichung:

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

Die Gleichungen sind FORMINVARIANT !

Ladungsumkehr ( Konjugation)

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

sind gerade unter C Ungerade unter c sind:

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

• C- Invarianz der Elektro- Magnetostatik:

${\begin{aligned}&C:\left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\to \left\{-{{\nabla }_{r}}\times {\bar {E}}=0\right\}\\&C:\left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\to \left\{-{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=-\rho \right\}\\&C:\left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\to \left\{-{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\\&C:\left\{\nabla \times {\bar {B}}={{\mu }_{0}}{\bar {j}}\right\}\to \left\{-\nabla \times {\bar {B}}=-{{\mu }_{0}}{\bar {j}}\right\}\\\end{aligned}}$

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

Paritätsumkehr: Räumliche Spiegelung/ Inversion

Vertauschung: rechts <-> links

Man unterscheidet:

$P{\bar {r}}=-{\bar {r}}$ -> polarer Vektor und

$P\left({\bar {a}}\times {\bar {b}}\right)=\left(-{\bar {a}}\times -{\bar {b}}\right)=\left({\bar {a}}\times {\bar {b}}\right)$ P- invariant = " axialer Vektor", sogenannter Pseudovektor !!

Seien:

${\bar {a}},{\bar {b}}$ polar, ${\bar {w}},{\bar {\sigma }}$ axial Dann ist

${\begin{aligned}&{\bar {a}}\times {\bar {w}}\quad polar\\&{\bar {a}}\times {\bar {b}},{\bar {w}}\times {\bar {\sigma }}\quad axial\\&{\bar {a}}{\bar {b}}\ skalar:P({\bar {a}}{\bar {b}})={\bar {a}}{\bar {b}}\\&{\bar {w}}{\bar {\sigma }}\ pseudoskalarP({\bar {w}}{\bar {\sigma }})=-{\bar {w}}{\bar {\sigma }}\\\end{aligned}}$

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

Wegen

${\begin{aligned}&{\bar {F}}=q{\bar {v}}\times {\bar {B}}\\&{\bar {F}}\in {{P}_{u}}\\&q\in {{P}_{g}}\\&{\bar {v}}\in {{P}_{u}}\\\end{aligned}}$

ungerade Parität dagegen:

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

Wegen

${\begin{aligned}&{\bar {B}}=\nabla \times {\bar {A}}\\&\nabla \in {{P}_{u}}\\&{\bar {B}}\in {{P}_{g}}\\\end{aligned}}$

P- Invarianz der Elektro- / Magnetostatik:

${\begin{aligned}&P:\left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\to \left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\\&P:\left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\to \left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\\&P:\left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\to \left\{-{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\\&P:\left\{\nabla \times {\bar {B}}={{\mu }_{0}}{\bar {j}}\right\}\to \left\{-\nabla \times {\bar {B}}=-{{\mu }_{0}}{\bar {j}}\right\}\\\end{aligned}}$

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

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