Документ взят из кэша поисковой машины. Адрес оригинального документа : http://qfthep.sinp.msu.ru/talks/salinskiy.pdf Дата изменения: Sun Sep 19 18:38:03 2010 Дата индексирования: Mon Oct 1 19:36:46 2012 Кодировка:

New denition of regularity for representation of canonical commutation relations algebras
Sergey G. Salinskiy with M. Mnatsakanova, Yu. Vernov Moscow State University

Septemb er, 2010


Table of content

Intro duction Analytical vectors Main theorem Conclusion


Canonical commutation relations

^ ^^ [p, q ] = -i I L2(-, +) q f (q ) = q f ( q ) , ^ pf (q ) = -i f (q ) ^ q


A denition of regularity

Rellich-Dixmier`s theorem
Op erators

q ^

and

p ^

form regular representation of CCR algebras if:

1. there exist dense domain action of 2. Op erator

D Dq

Dp

invariant under the

q and p ^ ^ 2 + p2 ) (q ^ ^

such that CCR hold on

D

;

is essentially self-adjoint on

D

.

Denition
Any representations of CCR algebras which are unitary equivalent to Schro dinger one are regular.


Weyl form

^^ ^^ eitpeisq = eisteisq eitp ^ U (t) = eitp ^ V (s) = eisq


Analytical vectors
Denition
b e a linear op erator on a Hilb ert space

Let

^ A

H

. A vector

H

is

called entire analytic for

^ A

, if is in the k IN and

domain of

^ Ak

for every



k =0

tk ^k A < + k!
for every

t > 0.


Main theorem

Let's prove that a representation of CCR algebras is regular if there is a dense domain

D

in witch any vector

D

ob eys conditions



k=0

tk k q < , ^ k!



t > 0;
k=0

sk k p < , ^ k!

s > 0

and also any regular representation ob eys this conditions.


Main theorem: part 1

If



is an entire analytic vector for

^ A

then

^ exp(tA)

make sense for

every

t C I

and it is an entire analytic function of

t

.

^^ ^^ ^ p q n - q n p = -i(q n) , ( = d/dq ) ^ ^ ^^ ^ p eisq - eitp p = -i(eitq ) ^ ^ ^ ^ ^ e-itq pn eitq = (p + tI )n


Main theorem: Stone`s and von Neumann`s theorems

^ A U (t) = e
Let

Stone`s theorem
b e a shelf-adjoint op erator on a Hilb ert space

H

. Then

^ itA

, t IR

is a strongly continuous one-parameter family of unitary op erators.

von Neumann`s theorem
If on a Hillb ert space

H

self-adjoint op erators

generators of the unitary groups satisfying the Weyl relations,

^ U (t) = eitp then q and p ^ ^

and

q and p are ^ ^ V (s) = eis

q, ^

satisfy a regular

representation of CCR.


Main theorem: part 2

1 ^^ q = (a + a ), ^ 2
m+n

1 p = (a - a ), ^ ^^ i2 C k k ,

^ ^^ N = a a

=
m

n = (a )n 0 ^

k =0

tk k q ^ k!



k =0

tk k C (2 (m + n + 1)!) k!

k /2

< +


Conclusion

Krein spaces Weyl representation