close

Вход

Забыли?

вход по аккаунту

?

Asymptotic Behaviour of Spectral Functions and Singularities of Wightman Functions and Field Commutators.

код для вставкиСкачать
Annalen der Physik. 7. Folge, Rand 27, Heft 3, 1971, S. 316-318
J. A. Barth, Leipzig
Asymptotic Behaviour of Spectral Functions and
Functions
Singularities of WIGHTMAN
and Field Commutators
By F. KASCIILIWN
and E. ~ V I E C Z O I ~ E K
Abstract
Singulartics of WICHTJIAK
functions and field commutators are studied in localizable
and nonlocalizable cases including the spectral behaviour u(@)= exp (a@), 01 > 0.
Some results are discussed concerning polynomially as well as exponentially
increasing spectral functions and the corresponding singularities of WIGHTMAX
functions and field commutators. For the localizable case [I I the equal-time
behaviour of the field commutator is exhibited, too. Considering nonlocalizable
examples we find singularities of the two-point function in space-like region as
it has been conjectured by e. g. GU~TTIXGER
[2], BAEDAKCI
and SCHROEX
1131.
As an interesting result we mention t h a t in the case of the spectral behaviour
o($) =- exp (q2),
a > 0 the field commutator does not vanish over trhe whole
space-like region. Actually this means t h a t the configuration space WIGIITMAN
functions are not given as boundary values of analytic functions in the standard
manner but require a somewhat modified approach.
function
The starting point is the WIGHTMAN
'
W ( z )-- (0 @(a) @(y)I 0 ) ,
and its spectral representation
2 =x -
y
(1)
W ( z ) = J a4q f?(i@) 8 ( q o )S(q2) o(q2)
which may be rewritten for space-like z in the form
(2)
m
Here
+
+
qp __ -+o)
r, E b O ) = sgn (zo), q -+ 0
defines the continuation into the cut plane and KOdenotes the modified I ~ A K K E L
function.
On the basis of eqs. (l),(2) and (3) we consider polynomially behaved spectral
functions o(q2)== qZn and find
"(z) =
8n 2%!(n
-
(-22
+ 1)!
f iE(zo)q)n+a
(4)
1’. KASCHLUHN
and 13. WIECZOREK
: Asymptotic Behaviour of Spectral Functions
317
By a careful analysis we derived the following expression for the equal-time
behaviour of thc canonical commutator
;o(W(z) - W ( - 2 ) )
=
% ( 2 z ) ~ ( - 1 ) ~ ~ ( -2 nI ) ! !
where thc principal values of higher order poles arc ckfincd according t o GE;T,E’A-UUand STITT.OV
141. For details we refcr t o a previous paper p i .
As an example of nonpolynomial but localizablc type we take W ( z )=
exp (-9 1- is(eo)15) -Isubstraoting the nonintcrestinp first tt,rms. This expression
exists as a gcneralized function on the space of tcst functions analytic in each
neighboiirhood of the real axis. The corrcqondinp spectral function is given by
its order of growth 9 heeing 9 - - AJ/l - 2.V < 112. We note t h a t thc commutator is represented by an infite series 2’ c,, P ) ( z : whose coefficients obey
the critcrium of a localized generalizcd function 161
z2)
~ i n vi c T 3. n ! - 0.
(7)
Lat us now considcr the nonlocalizablr casc o(y2) :
e s p (A:q). To obtain
the WIGHTXASfunction wc start from cq. ( 3 ) ant1 gct by analytic continuation
into the s2 plane
a2)-*5/2
1- terms singular on light cone.
(8)
This expression has a cut in the space-like region which should be absent, if the
original spectral representation eq. (2) would be defined as the boundary value
of an analytic function. Consequently the field cornmutator given by the discontinuity is nonvanishing for
< a2 producing a non-local structure of the
theory. It remains to prove that the result is in accord to the standard relation
W ( z ) = 6n2a( -
z2
1-
ir(zo) q
-
z2
Sdz
@(4W ( z )= SdY &(-d
@(p.)
(9)
where @ ( x ) is a test function and 6 ( y ) its ITOURIEII.transform.
We discuss this question in connection with the next example. The stronger
increasing spectral bchaviour
o(y2) = exp
( a y 2 ) , A:
>0
requires a different approach. Here we apply the method of analytic continuation with respect t o a . For a < 0 we obtain, either from eq. (3) or from a termby-term F ~ C R I E
transform
K
using eq. (4) and a BORIXsummation,
function W-lp.,o(z). Expression (10)
which is closely related to the WTIITTAKER
allows an analytic continuation t o values a > 0 thereby extending the WIGIITMAS function to the case o(q2)= e s p (&a2),a > 0. There is now a cut ovcr the
318
Annalen cler Physik
*
7. Folge
*
Band 27, Heft 3
*
1971
whole space-like region whose discontinuity, however, decreases exponentially.
Correspondingly the field commutator, i. e. the discontinuity of expression (lo),
is nonvanishing in the whole space-like region.
Finally we have to show that expression (10) analytically continued from
cx < 0 to cx > 0 can be identified with t h a t generalized function, which results
from continuation of relation (9). Of course, the class of test functions 6(q)in
momentum space has to be chosen such as to allow the continuation in 01, an
example being the space K of test functions with compact support. Taking
into account the properties of test functions @ ( z ) dual with respect t o the
space K we find for the commutator that continuation both of expression (10)
and relation (9) is possible thereby identifying expression (10) and the generalized
function defined by relation (9) for all values of cx considered. For this purpose
we integrate in z-space around the cut in the zE plane, which in the course of
continuation turns from the positive to the negative real axis.
References
[l] JAFFE,A. M., Phys. Rev. 168 (1967) 1454.
[Z] GUTTINGER,
W., Fortschr. Phys. 14 (1966) 483.
[3] BARDAKCI,
K., u. B. SCHROER,
J. Math. Phys. 7 (1966) 10.
[4] GELFAND,I. M., u. G. F. SHILOV,
Verallgemeinerte Funktionen, Bd. 1, Berlin, Dt. Verl.
Wiss. 1960.
[5] KASCHLUHN,
F., E. WIECZOREK
u. W. ZOELLNERin: Problemi Teorotitscheskoi Fisiki
Moskwa, Nauka 1969.
[6] MEIMAN,
N. N., Zourn. Eksper. i. T. Fisiki 46 (1964) 1502.
B e r l i n , Sektion Physik der Humboldt-Universitat und Institut diir Hochenergiephysik der Deutschen Akademie der Wissenschaften.
Bei der Redaktion eingegangen am 13. April 1971.
Anschr. d. Verf. : Prof. Dr. F. KASCHLUHN,
Sektion Physik d. Univ. Berlin
DDR-108 Berlin, Unter den Linden 6
Dr. E. WIECZOREK
Institut fur Hochenergiephysik der DAW
DDR-1615 Zeuthen. Platanenallee 6
Документ
Категория
Без категории
Просмотров
1
Размер файла
166 Кб
Теги
singularities, behaviour, asymptotic, wightman, field, commutators, function, spectral
1/--страниц
Пожаловаться на содержимое документа