Документ взят из кэша поисковой машины. Адрес оригинального документа : http://chronos.msu.ru/old/RREPORTS/zaslavsky-aliasing.pdf Дата изменения: Sat Dec 14 13:24:42 2013 Дата индексирования: Fri Feb 28 20:35:14 2014 Кодировка: Поисковые слова: m 2

..

, ( ) , . (aliasing effect) . , . , . , , , , . , , . , «». , . , , ( ), , . , , . , , , ­ ­ , (aliasing effect). , . . , , , . -, , , , -, , . , , , , , , , , , .


. . « , »1. , ­ . . , ( ) . , , . , - , ( ), . , , , , ? , ? , , ­ . , , . , de facto . ( ) , . () , , . . , , , . , , , . S1 , S 2 , S3 , ..., S n . , S S1 S 2 S3 , ..., S n , . , , . , . ( ) , , ­ . , , , , . t0 > 0 ,
1

http://ru.wikipedia.org/wiki/


, , , , , t ( 0 < t0 t , t0 = const ) , (1)

t = t0
(1), (2) . t1 t 2 ,

(2)

t1 ( ) = t2 ( ) = t0 , ( + 1 ) . , . , : . . , , . . . , , . - , . . S1 S2 , S , S1 S2 . [1] , . , , , , . , . . ( ), , ­ , , . , [2], . , , . . , k , , .

k ( t , t ) ,
( t , t + t ) .


k ( t , t ) :

0 p ( t , t
k 0

)

1 ... , p ( t , t ) ...
k 1

(3)

2, 3, ..., ­ . ( t , t )
k k p0 ( t , t ) + p1k ( t , t ) + p2 ( t , t ) + ... = 1 . (4) k ( t , t ) k k k ( t , t ) = 0 p0 ( t , t ) + 1 p1k ( t , t ) + 2 p2 ( t , t ) + ... . (5) t 0 t . , k ( t , t )

µ k ( t , t ) = lim M k ( t , t ) = lim p1k ( t , t t 0 t 0

)

, , (1) , , , µ k ( t , t ) ,

(

t , t + t ) .
(6)

tk = t0 µ k ( t , t ) .
tk t

k µ k ( t , t ) tk = t lim p1 ( t , t ) . = t0 (7) 0 t 0 t t t p1k ( t , t ) ( t ) = lim t 0 t

() . , , t , t . , (7)

tk = t0 ( t ) . t


(8)

. t0 . , (7) t0 , (1), , ,

tk lim p1k ( t , t ) 1 . t t 0

, . , t0 1 , , . lim p1 ( t , t ) < 1 ,
k

t 0

.


, V c . , ,

tk V2 = 1 - 2 1. t c

(9)

(9) . , , (8) . , , ­ . (8) (9), ,

(V) =

1 t

1-
0

V2 c2

(10)

, , , . , . , . , . .

( r , t ) = Ce

j (t -kr

)

= A(r ) e

j ( t

)

,

(11)

= 2 - , r - - , t , k - . E = , p = k . , ,

( r, t1 ) , ( r, t2 ) , ( r, t3 ) , ....



(

t2 - t1 ) , ( t3 - t2 ) , ...

.

-1 t = , - . ( t0 = 0 ), , (11)

.


(

t0 > 0

), , ,

t > 0 (11) , , F . . . ­ , ,



(10), 1 V2 1- 2 . t0 c
V 2 1 - 2 t c
2 0

F= . 2 2

(12)

(13)

(14)

.

2 2 cF = c 1 - 2 t0 , , ,
=

(15)

,

1 t

1-
0

V2 . c2

, cF . , [2], t

t=

, = 0,1, 2, ... .

= A(r ) e


j

2





,

(16)

2



z q :

2



= z + q.
j z

(17)
j q



= A(r ) e
j z

e

.

(18)

z .

= 1 , 1. z - , e , 2 .






2. z - ,

e

j z



= ( -1)





,

, . q

q = ( - z
,

)

= t , z = 0, 2, 4, ... .

(19)

= t , = 2 V2 = - z = - z 1 - 2 , z = 0, 2, 4, ...
t
0

c

(20)

: ( -1) = cos



( )

, sin ( ) 0 .

e

j z

e

j q

,

e

j z

e

j q

= cos ( ) cos ( q ) + j cos ( ) sin ( q ) +

+ sin ( ) sin ( q ) - j sin ( ) cos ( q ) = = cos ( (1 - q ) ) - j sin ( (1 - q , z

))

e


j z

e

j q

=e

- j (1- q

)

(1 - q

)



(1 - q ) = ( - ( - z

))

= ( 2 F - ) t , z = 1, 3, ... .
(21)



V2 z 1 - 2 , z = 1, 3, ... , = 2 F - = 2 F - + t0 c
z = 1

=

2 V2 1- 2 - . t 0 c

(22)

, ,

= 2F =



(). (.1) , « ». F ( . Fold ­ ). , , , , . () .


m ,

= V2 , V 2 . , 1- 2 1- 2 c c cF (22). (22). V = cF
E=


mc

2

mc

2

mc

2 2

=


t
0

c 1- F c

c 1- F c

2

,

(23)

E E ( c ) = c F
2

mm

pl


E ( V

E ( V

)
2

)

E ( c ) = c F

mm
pl

pl

3

E ( c ) = c F

2

mm

5
2

E = mc

E =0 c0F

c1F
z =1

c1F c 2

F

cF 2

z=0

z=2

c

.1. ,



cF mc 2 = 1- t0 . c

(24)

,

mc

2



t0 1

(25)


t pl ,

G = 5, 39056 10-44 s - , , 5 c c = 2 = m pl = 2,17671 10-8 kg - . G c t pl t pl =
(25)

m t0 1 m pl t pl m 1. m pl

(26)

, 1956. .. [3] . (27)

m = m pl . ,

t0 1 . t0 , , t pl

.., , .

=

t0 1 t pl

(28)

, t0 t pl . (28) (24) , ,

cF m = 1- . c m pl

(29)

cF , () , (21), . z = 0,1, 2, 3, ... , z = 2 z

V2 2 z 1 - 2 , z = 0,1, 2, 3, ... = - t0 c z = 2 z ± 1



(30)

V2 = 2 F - + ( 2 z ± 1) 1 - 2 , z = 0,1, 2, 3, ... t0 c



(31)

. cF z , cF - z .


z = 0, 2, 4, ... z = 1, 3, 5, ...

( cF ) = ( cF ) . z = 1, 3, 5, ... z z z = 0, 2, 4, ... ( cF ) = ( cF ) . z z

2 cF z - 2 z 1 - 2 = 2 F ( cF ) - z 2 t0 c cF 1- z c z = 0,1, 2, 3, ...

mc

2



2 cF z + ( 2 z + 1) 1 - 2 , 2 t0 c c 1 - zF c

mc

2



mc 1- z = 1, 2,

2 cF z 2 z 1 - 2 = 2 F ( cF ) - - z 2 c t0 cF z c 3, 4...

2



2 cF z + ( 2 z - 1) 1 - 2 , 2 c t0 c 1 - zF c

mc

2



(10) (12), 2 F ( c

zF

)

=


t
0

c 1 - zF . c

2



2mc

2 2

c 1 - zF c 2mc
2

- (4z + 2

)


t
0

1-

2 cF z = 0, z = 0,1, 2, 3, ... c2

c 1 - zF c
t0 =

2

- 4z


t
0

1-

2 cF z = 0 , z = 1, 2, 3, 4... c2

t pl =


c2m
(28),
pl

cF m z = 1- , z = 0,1, 2, ... , c 2 z + 1 m pl cF m z = 1- , z = 1, 2, 3, ... c 2 z m pl
()

(32)

(33)

( cF ) = ( cF ) = z z

mc

2

m 2 z + 1 mp z = 0,1, 2, ...,



-
l


t
0

2z

m = 2 z + 1 m pl
(34)



=

c

2

mm

pl

( 2 z + 1)

,


( cF ) = ( c z z

F

)

=

mc

2

m
2z m
pl

-


t
0

2z

m
2z m
pl

= 0, z = 1, 2, 3, ...

(35)

. , .. z = 0

Esup = c

2

mm

pl



(36)

. 2 . .
2

E

m pl c





m pl c

2

m2c

2 2

m1c

V c
.2. . , , E

E

pl



. ,

[5], , . [5] , , .., .


(36) . . , ()

.

10

20

. (36) .

Esup 10

20

, ,

0, 00295

(), , . , , , . ( 0 ) . () . . m , , , ,

c1F E = mc 2 .
, . , , , . , . , . , ,

1m c1F = c 1 - 4 mp

,
l

, , , . , , , , , , . , -- ( ), , . , ( ) , 5 10 . , . ,
19


1. 2. 3. 4. 5.

. : . . ­ .: , 1971. ­ 536 . .., .. . ­ .: , 1988. ­ 480 . ., . . . ­ .: , 1982. ­ C. 28 ­ 32. Markov M.A. // Progr. Theor. Phys.: Suppl. Commemoration Issue for 30th Anniversary of the Meson Theory by Dr. H.Yukawa, 1965. Giovanni Amelino-Camelia. Doubly Special Relativity (2002), arXiv:gr-qc/0207049.