Единая геометрическая теория классических полей (стр. 1 из 3)

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(dimstein@list.ru)

, .

, 2007 .

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(Ωα⋅µν=Ωα⋅[µν]):

(1) Ω^α_⋅µν=∆^α_µν−∆^α_νµ

# ∆^α_µν – . * ’ . .

∆^α_µν # :

(2)

# K

– , # #

(K_αµν= K_[αµ_]ν), Γ_µ^α_ν – % ( , . 1-3).

$ # #

" $. # & ( ) ’ ( ’ # ) #:

(3)

dds2x2µ µ dxdsα dxdsβ= 0

+∆(αβ)

d 2xµ

(4)

ds2 +Γαµβ dxdsα dxdsβ= 0

(3) #, (4) ’ .

. $ (3) (4) # #, #,

# :

(5) ∆µ(αβ) =Γαµβ

$ (2) ’ # !:

(6) ∆^µ_[αβ] = K^µ_⋅αβ

, # #

#. , # (K_αµν= K_[αµν_] ). . (1) (6) ’

!

(7)

, #,					(Ωαµν=Ω[αµν] ). 1		,
$ #							.
% , #				(7)			.

* ’ .

3. !" " !"# !-" $ % ! && #

, & -

- -% , #, ( ),

’ #, ,

# .

1)

. ( # -

# :

(8) ds² = g_µνdx^µdx^ν

g_µν # ∇_αg_µν= 0,

# ∇_α – # # x^α ( ,

. 4-5).

2) . . 0 ,

, ",

# & . ,

A

, # # (2)

#:

(9) ∆^α_µν=Γ_µ^α_ν+ iA^α_⋅µν

# A_αµν=−A_µαν=−A_ανµ=−A_νµα= A_[αµν_] . . % #

:

(10)

$ # A

# #:

(11) Aαµν=−εαµνσAσ

# A^µ – # , ε_αβµν – 2 3 .

A^µ # # :

(12) Aµ=−

εµαβγAαβγ

( # ’ , # # ’ a^µ:

(13) a^µ= qˆA^µ

# qˆ – ’ #. . ! (13)

’ . % qˆ #

# ! # , , &

( A

~ A^µ ~ 1/qˆ ).

1 " (9) # :

(14) Ω^α_⋅µν= 2∆^α_[µν_] = 2iA^α_⋅µν

$ # "

. * # ,

#

∆^α_µν #

# , # Γ_µ^α_ν ( , . 6).

3) % . 1 - # # ( , . 7): (15) Rα⋅µβν=∂β∆αµν−∂ν∆αµβ+∆ατβ∆τµν−∆ατν∆τµβ

∆αµν

1 - &# " - R

:

(16) Rµν=∂σ∆σµν−∂ν∆σµσ+∆στσ∆τµν−∆στν∆τµσ
. " (9) - # ( , . 8): (17) Rµν= R~µν+ Rˆµν ~ (18) Rµν=∂σΓµσν−∂νΓµσσ+ΓτσσΓµτν−ΓτσνΓµτσ (19) Rˆµν= i∇~σAσ⋅µν− Aτ⋅σµAσ⋅τν	# #
~ 4# Rµν – - ; Rˆµν –	- ,
( ). .	∇~α
# (# Γµαν). (11) , (20) Aτ⋅σµAσ⋅τν=−2(AµAν− gµνAαAα)	!
. (17), (18), (19) (20) - , #: ~ (21) R(µν) = Rµν+ 2(AµAν− gµνAαAα) (22) R[µν] = i∇~σAσ⋅µν
% # (21) (22), -
# ,			.
, - Fµν, # - # : (23) Rµν= R(µν) + iFµν (24) Fµν=∇~σAσ⋅µν

1 F_µν , #

F_µν:

(25) F_µν=

¹ε_µναβF^αβ

2

* (24) (11), & &#, # - (25) :

(26) F_µν=∂_µA_ν−∂_νA_µ

, # " ’ .

. (13) (26) " ’ f_µν

# # #

- :

(27) f_µν=∂_µa_ν−∂_νa_µ= qˆF_µν

. - (21)

# :

(28) R = gµνR(µν) = R~ − 6 AαAα

# R~ = R^~µ_⋅µ – .

1 , # ’ , # #

& ’ . * ’ ’

( ), "

’ – - .

A^µ # -

F_µν & ’ a^µ

" f_µν, & & ’ .

4. ’ $ !"( %’ #$"# #

4 , # -

, ,

:

(29) δ L_G − g d ⁴ x = 0

# L_G – # . 2 , - , # ,

(29). 2 L_G , ( ! ,

- .

* & ’- ( , . 9-10)

- :

(30.1) Rc

(30.2) Rc

R^µνR_αβ

(30.3) Rc

(30.4) Rc(4) ≡δα⋅β⋅γ⋅λ⋅µνστRµνRαβRστRγλ

* " & - #

#, , " & #

. & "& & - (30) # # . * Rc⁽¹⁾ (30.1) # R . (28) (13)

:

(31) Rc(1) = R = R~ −6AαAα= R~ −

qˆ62 aαaα

$ Rc⁽²⁾ (30.2) δ^α_⋅^β_⋅µν &

# - ,

(22)	(24)	’ !
R[µν] = iFµν.	’	!, (25) (27), &# :

(32)

Rc⁽²⁾ f^αβf_αβ qˆ

& (31) (32) #,				- Rc(1)
Rc(2) # &#	#			#
. $	R~ , #			&#
( ! , #	" fαβfαβ,
’			. 1

# & # & & - Rc⁽¹⁾Rc⁽²⁾ , " !

# .

3 L_G

. (§ 2). . ’ #

# L²(R) , # :

(33) L^{2 =}(R − R₀ )² = R² − 2R₀R + R₀²

# R₀ – . 2 L_G L²

&

- :

(34) LG = L2 (Rn →Rc(n) )=Rc(2) −2R0Rc(1) + R02

$ (34) # # & " #

(33). * R₀ , &# L_G ,

# ,

. . " (31) (32) #

#: