Fortschritte der Physik / Progress of Physics: Volume 34, Number 12 [Reprint 2022 ed.] 9783112613788, 9783112613771

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 34-1986 Number 12

Board of Editors F. Kaschluhn A. Losche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Hohler, Karlsruhe K . Lanius, Berlin F . topuszanski, Wroclaw A. Salarn, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia Z. Zinn-Justin, Saclay CONTENTS: L . L . JENKOVSZKY

Phenomenology of Elastic Hadron Diffraction

791-816

P . BHATTACHARJEE

Do 'Quark-Stars' Exist?

817-827

0 . BERTOLAMI

Brans-Dicke Cosmology with a Scalar Field Dependent Cosmological Term

829-833

E . B . MANOTJKIAN

Charged Particle Emission and Detection Sources and the One-Body and the Many-Body Coulomb Scattering

AKADEMIE-VERLAG • BERLIN ISSN 0015 - 8208

Fortschr. Phys., Berlin 84 (1986) 12, 791 - 853

835-853

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from " 1 " onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written so small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp, ...): black underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c G, k K, o 0, p P, s S, u U, v V, IO W, x X, y Y, z Z). It will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, Mi{, M-j, Wn} Please differentiate between following symbols: a, oc; 0) avoiding the pole at k° = X — ie. For a;0 > x0', we may close the &°-integral contour from below (Im k° < 0) encircling the pole at k° = X — ie. Hence G(x, x') = 0, for x° < x«' (3.6) G(x, x') = i J (x | X) dX {X | x')

,

for

a; 0 > a; 0'.

(3.7)

To describe the scattering process we set up an emission and detection source [16, 17], that is we write:

(3.8)

K{x) = Ki(x)+Kt{x),

where Kx(x), the detection source is switched on after K2(x), the emission source, is switched off. More precisely we choose the sources K^x) and K2(x), idealized in time for simplicity of the analysis, a s : Kx{x)

=

t^x", x)

- T) = K^T, x) d(x° - T)

K2{X) = &2{x, x) d(x? + T) = It^-T, X) b(x* + T)

(3.9)

where T(> 0), a time parameter, that will eventually be led to go to infinity in order that the emission and detection sources be well separated from the interaction region in spacetime. That is for T oo, the sources will be essentially dealing with free particles. For long range interactions, however, it is well known that the particles feel the presence of the potential tail, for T - > oo, "before" and " a f t e r " a collision occurs. This will impose a given time dependent structure on the emission and detection sources which in turn will take into account the propagation characteristics of the particles "before" and " a f t e r " a collision. This propagation characteristic, for x° - > ¿ o o , is very easy to obtain. We suppose that the potential I7(®)=O(-JI7),

LARL-OO

(3.10)

where for a, > 1, the potential is termed as of a short range. The interesting case is the one with oc = 1 corresponding to the Coulomb interaction. For x" ¿ o o , if p is the momentum of a particle, then on dimensional grounds, cf. [4, 7, 8], X rNI^P x° m

(3.11)

Accordingly, if we replace \x\ in U{x) ~ Z70S(|aj|) = cl\x\", for \x] oo, by \x°p\/m, then the propagation characteristic of the particle " a f t e r " or "before" a collision is modified

Fortschr. Phys. 34 (1986) 12

839

by changing the time development operator exp it{—F2/2ra)) to exp (—it(— F2/2m)). exp (—iG(t)), for £ ¿ o o , where G(t) is obtained simply upon integrating Uas(\tp\/m) over t(= x°). That is

m = cjdt>

t -> ± o o .

{^j,

(3.12)

Or | ¿ o o . From (3.13) we'infer that the time evolution operator of a free-particle state is modified to exp —î'c(sgn t)

7ÏI

:lnl ) g2{k'),

(3.25)

where we have defined the double Fourier transform: (3.26)

G(k, k') = / {dx) {dx') e~ikxeik'x' G{x, x'). We write G{k, k') = i{2nf

Ô

-

(3.27)

ô{k - k') + ïGj(k, k').

The first term in (3.27) gives the following contribution to (3.25) :

'J

d3k 2i(Z^m/|fc|)ln((TfcV2m))[g;*(fe) K {k)]. e j i 2 {2n)
oo. All told we may write (3.25) as (Pk

/

( W

d3k' ( W

K

*

( k )

CTik

'

k

'}

K

'[k']'

(3

'29)

where CT{k, k') = ei(zes»/lfel)in((rfc!/2«» AT{k, k')

,

(3.30)

.

(3.31)

die0 rlk0'

/

^ - ci((*'/i«)-t,)T GAk, V) {2n) {2n)

841

Fortschr. Phys. 34 (1986) 12

For T - > oo, k° ^ k / 2 m , k"' ~ fc'2/2m (near the energy shell), the expression for G i ( k , k ' ) in ( 3 . 3 1 ) has been worked out in detail by S C H W I N G E R [ 2 ] , (see also [ 1 0 ] where IYE essentially follow its notation here) and is given by: 2

G j { k , k')

= G°(k) f ( k ,

(¡»(jfc) =

L

(3.32)

k ' ) G ° ( k ' )

where -

^

'

Z

~

6

—

Y

^

2

'

k°

%y

e*ln[(t'-(ftV2m))/«»]

_Ì —

— 2 m

f-

(3.33)

ie

(3 34)

'

Z&^TYb '

-

|fc|

kk'

'

- |fe| |fc'| c o s e .

(3.35)

Due to the presence of the logarithmic phase factor in (3.33), we cannot immediately carry out the P-integration (alsoj the ¿"'-integration) by simply closing the contour from below in evaluating AT(k, k') as given in (3.31). However, in the limit T —> oo, the phase factor exp [iy In (T)] in CT(k, k'), as given in (3.30) cancels out the factor exp X [iy In {k° - fc2/2m)] for k° ~ fc2/2m in (3.33). This is intuitively clear as T ~ 1/ — ( k / 2 m ) ] in these limits. A very lengthy demonstration (see, e.g., [10] — the latter working with the Coulomb scattering by completely different methods) establishes this fact rigorously and yields: 2

lim

C

T

( k , k ' )

=

C ( k ,

k ' )

=

(2Ti? - r f j -

T-t-oo 1

( i

-t- i y ) „..2 2 _ . , 22 e '

ò(k

:

i \ k \ r ( l - i y )

-

k'

v

e ( !

7

—

> c o s

8 ) / 2

The vacuum-to-vacuum transition amplitude may be finally rewritten a s :