???? ????? ??? ???? ? ?????? ??????? ????? ??? ???????? ?? ????? ???????? ? ???? ????? ???? ??????. ?? ????, ??? ????? ? ??? ???? ???????? ?? ????? ?????? (????? ???? ???? ???? ??????? ?????? ???? ?????? ???????). ????? ???? ?? ?? ? ????????? ????? ????? , ??????? ???? ????? ???? ????? ? ??? ????? ??????? ? ???? ?????? ?????? . ????? ??????? ?????? ????? ???? 1918 .

????? ?????? [ ????? ??? ???? | ????? ]

???? ???? ?? ???? ??? ????? ???:

???? ?? ??????? ????? (? ????? ) ?? ? ????? , ???? ???? ????, ??? ?? ???? ???? ????? ??????? ??????? ?? ??????. ^[1]

????? ?????:

?????? ??????'??? ${\displaystyle {\mathcal {L}}(\phi (x_{\mu }),\partial _{\mu }\phi ,x_{\mu })}$, ???? ???????

???????????? ??? ??????????? ?????? ????? ???? ????? ?????? ${\displaystyle \ \alpha }$ ?? ???:

${\displaystyle x\rightarrow y(x,\alpha ),\quad \phi \rightarrow \psi (\phi ,\alpha )}$.

????

${\displaystyle \ x_{\mu }=(x_{0},x_{i})=(t,x_{i})}$

??? ???? ??? ???? ${\displaystyle \ J^{\mu }}$ ?????? ?????? ?????? ${\displaystyle \sum _{\mu }\partial _{\mu }J^{\mu }=0}$.

???? ???? ?? ???

???? g ??? ? ?????? ?- ${\displaystyle \ g_{\nu }^{\mu }=\delta _{\mu ,\nu }}$ ??? ???? ?? ?????? .

???? ?? ???? ???? ?? ????? ????? ${\displaystyle Q=\int d^{3}xJ^{0}}$ ??????

${\displaystyle {\frac {dQ}{dt}}={\frac {dQ}{dx_{0}}}=0}$. ^[2]

????? ${\displaystyle \theta _{\nu }^{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\partial _{\nu }\phi -g_{\nu }^{\mu }{\mathcal {L}}}$ ???? ????? ?????? ?????-?????? ??? ??? ????? ?????????? ??? ???? ???? ?? ????? ?????-?????? ? ?????? ${\displaystyle \ T^{\mu \nu }}$.

???? ??? ? ?????? ???????? [ ????? ??? ???? | ????? ]

??????? ????????, ?? ?????? ??????? ????? ? ?????????? , ???? ???? ??? ?? ????? ????:

??? ${\displaystyle T}$ ?????????? ???????? ????? , ??????? ??????? ???????? G, ???? ????????????:

${\displaystyle T=\exp {\left(-{\frac {i}{\hbar }}\varepsilon G\right)}}$

T ??? ??????? ?? ?????? (?????, ${\displaystyle [T,H]=0}$) ?? ??? ?? G ??? ???? (????) ?? ??????.

????? [ ????? ??? ???? | ????? ]

????? ${\displaystyle [T,H]=0}$ ???? ????? ${\displaystyle T^{\dagger }HT=H}$. ??????? ???? ?? ??? ??? ? ????? ?????? ?? ???????? ?? ${\displaystyle \varepsilon }$, ????? ???? ??????:

${\displaystyle \ \left(1+{\frac {i}{\hbar }}\varepsilon G+O(\varepsilon ^{2})\right)H\left(1-{\frac {i}{\hbar }}\varepsilon G+O(\varepsilon ^{2})\right)}$

????? ????? ?????? ????? ???????:

${\displaystyle \ H+{\frac {i}{\hbar }}\varepsilon [G,H]+O(\varepsilon ^{2})=H}$

????? ?????:

${\displaystyle [G,H]=GH-HG=0}$

???, ???? ???????????? ??? ???? ?? ???????? ????,

${\displaystyle [G,H]=i\hbar {\frac {\partial G}{\partial t}}}$

????,

${\displaystyle \ {\frac {\partial G}{\partial t}}=0}$

?????, G ??? ???? ?? ??????.

??????? [ ????? ??? ???? | ????? ]

????? ???? ??? ???? ???? ?? ???? ?????? ?????:

• ??? ????? ??????? ????? ???????? ????? ? ??? .
• ??? ????? ???? ????? ????? ? ??????? ????? ?????? .
• ??? ????? ???? ??????? ????? ? ??????? ?????? .
• ??? ????? ????? ?????? ????? ? ???????? ???? .

???? ???? ????? ??????, ???? ???? ????? ?? ????? ? ?????? ?? ?????? .

?????? ????? [ ????? ??? ???? | ????? ]

????? ??? ?????? ????? ??????? ???????. ????? ???? ??? ?? ???? ?????? ?????????? (???? ?????? ???? ????? ???? ?????? - ???????). ????? ???, ???? ?????? ???????? ???????? (??????, ???, ??? ??????) ?????? ???? ?????? ???? ??????? ?????????????? - ?????????? ????????????? ?? ???? ???????, ????? ????? ????? ??????? ???? ?????? ???? ??????, ????? ????? ?????? ??? ?????? ???? ????? ?? ???? ??????? ???? ??? ???????? ??????.

??????? ???????? [ ????? ??? ???? | ????? ]

• ????? ?????? ?? ??? ??? (???????)
• ????? ??????? ?? ????? ?????? ?? ??? ???
• ???? ??????? ?? ???? ??? (???????)
• ???? ?? ?????? ??: ????? ?????? ????????? ???? ??????? ??? ???? ???, ???? ??????? ??????????? ?? ???.

????? ?????? [ ????? ??? ???? | ????? ]