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2.1 Klassicheskii metod Belinfante

2.1.1 Problemy opredeleniya uglovogo momenta v polevoi teorii

Vnachale my korotko izlozhim osnovnye polozheniya original'noi raboty Belinfante [1], vo mnogom dazhe sleduya ego stilyu i oboznacheniyam. Rassmotrim teoriyu nekotoryh dinamicheskih polei predstavlennyh obobshennym simvolom $\psi^A$ s lagranzhianom

$\begin{displaymath} \hat L = \hat L \left(\psi^A, \partial_\alpha \psi^A, \eta_{\mu\nu} \right) \end{displaymath}$

(2.1)

v prostranstve Minkovskogo s metrikoi Minkovskogo $\eta_{\mu\nu}$ . Plotnost' kanonicheskogo tenzora energii-impul'sa v takoi teorii opredelyaetsya obychnym obrazom:

$\begin{displaymath} \hat t_\nu^{\mu} = \sum_A {{\partial \hat L}\over {\partial(\partial_\mu \psi^A)}} \partial_\nu \psi^A -\delta^\mu_\nu \hat L \end{displaymath}$

(2.2)

i sohranyaetsya

$\begin{displaymath} \partial_\mu \hat t_\nu^{\mu} = 0 \end{displaymath}$

(2.3)

na vypolnennyh uravneniyah dvizheniya. Sleduya pravilam obychnoi mehaniki tenzornuyu plotnost' uglovogo (orbital'nogo) momenta sledovolo by opredelit' kak

$\begin{displaymath} \hat {m}_{kl}^{ \alpha} = x_{[k}\hat t_{l]}^{\alpha}. \end{displaymath}$

(2.4)

Zdes' voznikaet problema: opredelennyi takim obrazom uglovoi moment ne sohranyaetsya

$\begin{displaymath} \partial_\alpha \hat { m}_{kl}^{ \alpha} = \hat t_{[kl]} \neq 0 \end{displaymath}$

(2.5)

dazhe s uchetom zakona sohraneniya (2.3). Delo v tom, chto tenzor energii-impul'sa opredelennyi v (2.2) v obshem sluchae ne simmetrichen. Situaciyu v (2.5) mog by spasti sohranyayushiisya simmetrichnyi tenzor energii-impul'sa, kotoryi mozhno bylo by postroit' po drugim pravilam, ili simmetrizovat' uzhe imeyushiisya (2.2).

2.1.2 Simmetrizaciya Belinfante

Vtoruyu iz etih vozmozhnostei ispol'zoval Belinfante [1], kotoryi predlozhil sleduyushee. Pust' pri beskonechno malyh vrasheniyah opredelyaemyh antisimmetrichnym parametrom $\delta \omega^{\mu\nu}$ variacii koordinat i dinamicheskih peremennyh imeyut vid: $\delta x^\nu = x_\mu \delta \omega^{\mu\nu}$ i $\delta \psi^A = \delta \omega^{\mu\nu} {\bf S}_{\mu\nu} \psi^A$ , gde ${\bf S}_{\mu\nu}$ -- operator. Opredelim velichinu

$\begin{displaymath} \hat B_{\lambda\mu\nu} = {\textstyle{\frac{1}{2}}}\left(\hat... ...da} + \hat f_{\mu\lambda\nu} - \hat f_{\nu\lambda\mu}\right), \end{displaymath}$

(2.6)

gde

$\begin{displaymath} \hat f_{\mu\nu}^{ \alpha} \equiv \sum_A {{\partial \hat L}\... ... {\partial(\partial_\alpha \psi^A)}}{\bf S}_{[\mu\nu]} \psi^A. \end{displaymath}$

(2.7)

Dalee velichinu (2.6) ili ee obobsheniya my budem nazyvat' popravkoi Belinfante. Teper' dobavim proizvodnuyu ot (2.6) k kanonicheskomu vyrazheniyu (2.2):

$\begin{displaymath} \hat {t}^{(B)}_{\mu\nu} = \hat t_{\mu\nu} + \partial_\alpha \hat B_{ \mu\nu}^{\alpha}. \end{displaymath}$

(2.8)

Kak i prezhde, v (2.3): $\partial_\mu \hat {t}_\nu^{(B)\mu} = 0$ na uravneniyah dvizheniya, no teper' eshe: $\hat {t}^{(B)}_{\mu\nu} = \hat {t}^{(B)}_{\nu\mu}$ . Blagodarya etim dvum ravenstvam ,,ispravlennyi'' orbital'nyi uglovoi moment:


		$\displaystyle \hat {m}_{kl}^{(B)\alpha} = x_{[k}\hat {t}_{l]}^{(B)\alpha}$

sohranyaetsya $\partial_\alpha \hat {m}_{kl}^{(B)\alpha} = 0$ .

2.1.3 Teorema Neter i metod Belinfante

V original'nom izlozhenii metod Belinfante predstavlen, faktichesti, na urovne uravnenii dvizheniya. Seichas my opishem ego edinym obrazom v ramkah lagranzhevoi teorii. Dlya etogo zapishem lagranzhian (2.1) v obshekovariantnom vide:

$\begin{displaymath} \hat L = \hat L \left(\psi^A, \bar D_\alpha \psi^A, \overline g_{\mu\nu} \right), \end{displaymath}$

(2.9)

gde metrika $\overline g_{\mu\nu}$ mozhet opisyvat' lyuboe fiksirovannoe prostranstvo-vremya, ne obyazatel'no ploskoe. Dlya kovariantnogo lagranzhiana (2.9) zapishem tozhdestvo Neter:

$\begin{displaymath} {\pounds_\xi} \hat L + \partial_\alpha\left(\xi^\alpha \hat L\right) \equiv 0 \end{displaymath}$

(2.10)

s opredelennoi kak v lekcii 1 proizvodnoi Li:


		$\displaystyle \hbox{$\pounds$}_\xi \psi^A = -\xi^\alpha \overline D_\alpha \psi^A + \overline D_\beta \xi^\alpha \left.\psi ^A \right\vert _\alpha^\beta$

i proizvol'nym vektorom $\xi^\alpha$ . Tozhdestvo (2.10) analiziruetsya analogichno tomu kak eto predstavleno v lekcii 1 (bolee detal'no poznakomit'sya s metodom analiza tozhdestv tipa (2.10) mozhno v knige Mickevicha [2]). V rezul'tate (2.10) perehodit v zakon sohraneniya $\partial_\alpha \hat I^\alpha \equiv 0$ dlya toka

$\begin{displaymath} \hat I^\alpha \equiv \hat T^\alpha_\sigma \xi^\sigma - \ha... ... \right\vert^\alpha_\sigma \xi^\sigma +\hat Z_{(\psi)}^\alpha. \end{displaymath}$

(2.11)

Struktura etogo toka vazhna i my ee obsudim podrobno. Pervyi chlen -- eto simmetrichnyi (tak nazyvaemyi metricheskii) tenzor energii-impul'sa polei $\psi^A$ :

$\begin{displaymath} \hat T_{\alpha\beta} \equiv {{\delta \hat L} \over {\delta \... ...equiv 2{{\delta \hat L} \over {\delta \bar g^{\alpha\beta}}}, \end{displaymath}$

(2.12)

vtoroe slagaemoe vyrazheno uzhe izvestnym kanonicheskim tenzorom energii-impul'sa (2.2), tol'ko teper' my zapisyvaem ego v yavno kovariantnom vide:

$\begin{displaymath} \hat t^\alpha_\sigma \equiv {{\partial \hat L} \over {\parti... ...A\right)}} \bar D_\sigma \psi^A - \hat L \delta^\alpha_\sigma. \end{displaymath}$

(2.13)

V tret'em slagaemom v (2.11) glavnuyu rol' igraet spinovyi tenzor:

$\begin{displaymath} \hat \Sigma^{\alpha\beta}_{ \sigma} \equiv - {{\partial \ha... ..._\alpha \psi^A\right)}} \left.\psi^A \right\vert^\beta_\sigma. \end{displaymath}$

(2.14)

My predpolagaem, chto uravneniya dvizheniya


		$\displaystyle {{\delta \hat L} \over {\delta \psi^A}} = 0$

vypolnyayutsya i ne budem bol'she uchityvat' predposlednii chlen v (2.11). Nakonec, strukturu Z-chlena my ne vypisyvaem, no otmechaem, chto on ischezaet na killingovyh vektorah fona.

V silu tozhdestva $\partial_\alpha \hat I^\alpha \equiv 0$ dolzhen sushestvovat' superpotencial, to est' antisimmetrichnaya tenzornaya plotnost'. Deistvitel'no, takoi superpotencial sushestvuet, i zakon sohraneniya $\partial_\alpha \hat I^\alpha \equiv 0$ mozhet byt' zamenen zakonom sohraneniya:

$\begin{displaymath} \hat I^\alpha \equiv \partial_\beta \hat I^{\alpha\beta} \eq... ..._\beta\left( \hat M^{\alpha\beta}_{ \sigma}\xi^\sigma\right), \end{displaymath}$

(2.15)

gde velichina

$\begin{displaymath} \hat M^{\alpha\beta}_{ \sigma} \equiv {{{\partial \hat L} \... ...mu\nu}\right)}} \left.\bar g_{\mu\nu}\right\vert^\beta_\sigma} \end{displaymath}$

(2.16)

antisimmetrichna po verhnim indeksam: $\hat M^{\alpha\beta}_{ \sigma} = - \hat M^{\beta\alpha}_{ \sigma}$ , hotya (2.16) ne pokazyvaet etogo yavno.

Sravnivaya opredeleniya (2.7) i (2.14) my opredelyaem popravku Belinfante tochno takzhe kak v (2.6):

$\begin{displaymath} \hat S^{\alpha\beta\gamma}_{(\psi)} = \hat \Sigma^{\gamma[\a... ...gma^{\alpha[\gamma\beta]} - \hat \Sigma^{\beta[\gamma\alpha]}, \end{displaymath}$

(2.17)

gde $\hat S^{\alpha\beta\gamma}_{(\psi)} = - \hat S^{\beta\alpha\gamma}_{(\psi)}$ . Chtoby uprostit' izlozhenie perepishem zakon sohraneniya (2.15) dlya proizvol'nyh killingovyh vektorov fona $\xi^\alpha = {\bar\xi}^\alpha$ :


		$\displaystyle \hat T^\alpha_\sigma {\bar\xi}^\sigma - \left(\hat t^\alpha_\sigm... ...) = \partial_\beta\left[\hat M^{\alpha\beta}_{ \sigma}{\bar\xi}^\sigma \right]$

i dobavim k obeim chastyam $-\partial_\beta\left(\hat S^{\alpha\beta\gamma}_{(\psi)}\bar\xi_\gamma\right)$ :

$\begin{displaymath} \hat T^\alpha_\sigma {\bar\xi}^\sigma - \left(\hat t^\alpha_... ...gma -\hat S^{\alpha\beta\gamma}_{(\psi)}\bar\xi_\gamma\right]. \end{displaymath}$

(2.18)

V silu opredeleniya (2.17) spinovyi chlen v levoi chasti (2.18) kompensiruetsya dobavkoi Belinfante. Krome togo, okazyvaetsya, chto vyrazheniya v formulah (2.16) i (2.17) sovpadayut v obshem sluchae: $\hat M^{\alpha\beta\gamma} = \hat S^{\alpha\beta\gamma}_{(\psi)}$ ; v silu etogo pravaya chast' (2.18) obrashaetsya v nul', to est' posle korrekcii superpotencial ischezaet. V rezul'tate (2.18) uproshaetsya do ravenstva


		$\displaystyle \hat T^\alpha_\sigma {\bar\xi}^{\sigma} - \left(\hat t^\alpha_\si... ...line D_\beta \hat S^{\alpha\beta}_{(\psi)\sigma}\right) {\bar\xi}^{\sigma} = 0.$

Teper' zapishem simmetrizovannyi s pomosh'yu metoda Belinfante, kak eto bylo sdelano v (2.8), kanonicheskii tenzor energii-impul'sa:

$\begin{displaymath} \hat t^{(B)\alpha}_\sigma = \left(\hat t^\alpha_\sigma + \overline D_\beta \hat S^{\alpha\beta}_{(\psi)\sigma}\right). \end{displaymath}$

(2.19)

Sravnivaya poslednie dva ravenstva nahodim, chto simmetrizovannyi tenzor energii-impul'sa (2.19) raven metricheskomu tenzoru energii-impul'sa (2.12):

$\hat t^{(B)\alpha}_\sigma = \hat T^\alpha_\sigma$ .

Elektordinamika, lagranzhian kotoroi sootvetstvuet forme (2.9),


		$\displaystyle {\hat{\cal L}}^{\dag }=-{1\over 16\pi}\sqrt{-\bar g}\bar g^{\mu\r... ...\mu\nu}F_{\rho\sigma}, F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu.$

yavlyaetsya horoshei illyustraciei izlozhennogo. Kanonicheskii tenzor energii- impul'sa (2.13) i tenzor spina (2.14) preobretayut vid:

$\displaystyle \hat t^{\dag\mu}_{ \nu}$	=	$\displaystyle -{\sqrt{-\bar g}\over 4\pi}\left(F^{\mu\rho}\overline D_\nu A_\rho - {1\over 4}F^{\rho\sigma}F_{\rho\sigma}\delta^\mu_\nu\right),$
$\displaystyle \hat \Sigma^{\dag\mu\rho\sigma}$	=	$\displaystyle -{\sqrt{-\bar g}\over 4\pi}F^{\mu[\rho}A^{\sigma]}.$

Togda popravka Belinfante (2.17) zapisyvaetsya kak


		$\displaystyle \hat S^{\dag\mu\nu\rho}= \hat \Sigma^{\dag\rho[\mu\nu]}+ \hat \Si... ...- \hat \Sigma^{\dag\nu[\rho\mu]}= {\sqrt{-\bar g}\over 4\pi} F^{\mu\nu}A^{\rho}$

i privodit k Belinfante modificirovannomu tenzoru energii-impul'sa (2.19):


		$\displaystyle {\hat T}^{\dag\mu\nu} = {\sqrt{-\bar g}\over 4\pi}\left(F^{\mu\rho}F_{\rho}^{ \nu}+{1\over 4}\bar g^{\mu\nu} F^{\rho\sigma}F_{\rho\sigma}\right).$

S drugoi storony, eto est' izvestnyi simmetrichnyi, kalibrovochno invariantnyi tenzor energii-impul'sa elektoromagnitnogo polya, kotoryi poluchaetsya var'irovaniem ${\hat{\cal L}}^{\dag }$ po zadannoi metrike.

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