IC/83A1INTERNAL REPORT

(Limited distribution)

.International Atomic Energy Agency

and

Uniteif-itWi.onsTiiucational Scientific and Cultural Organization

Xt^jglHATIONAL CENTRE FOR THEORETICAL PHYSICS

TIME-ENERGY UHCERTAINTY RELATION AMD IRREVERSIBILITY

IN QUANTUM MECHANICS *

It is well known that many difficulties in the probabilistic

interpretation of Quantum Theory are related to the off-diagonal

elements of the density operator. The irreversible "state reduction"

to the diagonal form is the fundamental problem in statistical

mechanics and in measurement theory, l ecau c it cannot be described

in exact terras ty Schrodinger1s or Liouville's equation.

All the attempts to give a general solution to this problem via

approximate Master Equations are, in our opinion, largely unsatisfac-

tory,since specific assumptions and models are used.

In this paper, we propose an irreversible Master equation for state

reduction, which should be generally valid for all systems, since it can

be constructed, just by taking properly into account a finite uncertainty

time T, as required by the time-energy uncertainty relation

R. Bonifacio

International Centre for Theoretical Physics, Trieste, Italy,and

Istituto di Scienae Fisiche dell'Universita di Milano,Milano, Italy.

ABSTRACT

We think that the meaning of time energy uncertainty relation

is rather obscure in terms of a continuous description of time evolution,

whereas it is consistent with a finite difference equation for the density

operator with time steps T ^rs - This equation intrinsically gives

irreversible state reduction to the diagonal form leading to "ergodic"

behaviour even at the classical limit.

T > I*. (1)

This relation comes from ideal Heisenberg's experiments as the

position-momentum relation but its meaning * is not so clear and

unique, since it is not included in the formal theory as a relation

between non-commuting operators.

On the contrary, we think that relation (l) is not satisfactorily

taken into account in the usual description of time evolution.(2)

Following Messiah's book , relation (l) can be discussed in terms of

of the Tamm-Mandelstam inequality.Let A be a generic observable; hence from the general statement of the

uncertainty relation, we have

~ < [A, H] > (2)

MIRAMARE - TRIESTE

May 1983-2-

To "be submitted for publication.

where

and

A A -_\

Hence, if we define the characteristic times

t - AA -(3)

coincides with

we can rewrite equation (2) as fallows

Since the-denominator

one can interpret f A as the time necessary to have some significant

change on the statistical distribution of A , or equivalently, It might be

possible to interpret "EA as the time uncertainty for A to take any

definite value.

In this sense no measurement of A can be performed in a time tft ,

unless t A satisfies the inequality (1). Obviously, here we refer

to the so-called first type measurements; i.e. those' which do

not significantly perturb the state of the system ,

The above argument is easily understood fora free particle represented

by a Gaussian wave packet with position spread A x and group

velocity \l0- 111. .

In this case

furthermore,

[*-"] 1L >rrr\ J

AH = A ( j i ) * v. Af>

- 3 -

hence

t* - 41

iAM (5)

vhere we have used A x A p — •

This example also clearly shovs tHe meaning of T , i.e. the

instant at which the position takes a definite value x cannot be

determined precisely, but with an uncertainty -p _ A Jfc , and

(1) ~ V o

this satisfies the relation (l) v '. —

In our opinion, the usual description of time evolution looks

rather inconsistent with the time-energy relation (l). In fact the

time evolution of the density operator for a system with Hamiltonian K

is given by the von Neumann-Liouville equation

It -to) - e

being the Liouville operator:

= e(6)

and meaning that p-. is the initial density operator which represents

the state of the system as determined at t = t .

The last part of this sentence is, in our opinion, inconsistent with (l).

In fact to determine or prepare p. one must perform some measurement

of some observable and this, in accordance with (1), must have required

-It-

a f ini te measurement (or preparation) time uncertainty T 3^7JJjj -

Only the case with AH infinite allows us to speak about a definite tQ:

according to this simple argument, if one accepts (l) one must think

that the exact meaning of (6) is o'bscure.

So it is for the Liouville-von Neumann equation:

(7)

whose solution is (6) and which reduces to the Schrodinger equation

when

pure state I * P

i i.e. if Po is a pure state. (We remember

that if p. is a pure state 1 * P : 1 and so it will remain at all times

due to the unitary time evolution (6).)

Hence in general we refer to (7) because it includes the Schrodinger

equation for pure states.

In order to introduce a time evolution equation, which takes into

account the relation (1), we propose the following heuristic, but

hopefully, reasonable procedure.

Later we shall propose it as the generalized simplest way describing time

evolution on a temporal grid with steps *"* T o , which is consistent

with (1) and gives "state reduction" reducing to (7) for t -V 0

Equation (6), if H is time independent, can be written as :

where T = t - tD is the evolution time.

As t0 can range from -<*> to t , T can range from 0 to w

Since \0 has an uncertainty t^t,, , the same will affect T.

-5-

Hence let us divide the T axis into interval^ T . Furthermore (for

the meaning of X ), it is meaningful and sufficient to calculate P

only on the time lattice points K"tf with K= 0, 1, ... N.

Accordingly we write the density operator at T = Kt as a

superposition of <? ( )

i.e.:

& (T) = e p with weight function p \t/ ' ] ,

o oWhen we introduce the adimensional continuous time X T , -

so that p [ K T) - ±. P ( K > X ) and ^(Xt)= «"*" c0

This expression can always be introduced to go from a continuous

variable X to a grid H as, for P ( i( , X ) = % ( X - M. ) . these

just give

It is convenient to write : 4 w

a(9*

Now we must impose conditions which guarantee that p (Wtj is a

density operator and that the evolution law obeys translational

invariance.

i) ^ l ^ / X ) must be a positive definite and well behaved normalized

probability function for ?11 K, i.e. : + w

and 9 [V.i\ < a s given by (9), is obviously positive definite,

ii) Translational invariance is obtained if :

tlOJ

with ^lO^ - i- to impoER the initial condition.

iii> We demand that in the continuous limit T-»e> and K -» <x>,

so that T = ICY is finite, we must obtain the Schrodinger evolution:

We also require that P ( K r \ ^ is "centered" on <• A ) K ; K

and has a width <5^ ( X ^ 1 to guarantee that ( (T

A) n =

All these requirements are satisfied by the simplest possible choice

of PU, -\) i.e.:

(12)

. U3)

In fact, via eqs. (9), (10), we have :

Using (91) we get the Poisson distribution

(15)

This is immediately obtained from the integral 1 -function identity •

17-i *-1

e U>0 /

specifying ~\J = 1 + «.<tt and V - < •

These formal operator identities assume a well-defined meaning by

taking matrix elements with energy eigenstates n l*l> =

(16)

Assuming for simplicity that E are non-degenerate, we obtain :

with and where

(17)

given by (15).

Note that C H (X): JK>yl in agreement with (12). We have not shown that

the choice (13) for P[l,X) is the only one which satisfies all the

required conditions, but it appears rather difficult to invent a

simpler one.

In fact, putting U P O ~ VJ lt\ i we can always suppose that

~\J~ = \J [t\ can be expanded as in (16) with ? ; *t , This expansion is

compatible with (9), (10) only if X-~\J*'?"t > which leads to Rc tJ r i-

for the normalization condition v" (o) r 1 • This gives U ; i*\.tLt

which in turn leads to . VM (t) 5 \J - ^l + i«[T;|

Hence just the compatibility of expression (9) with the general expansion

formula (16) leads to (Id) whatever T is.

Note that Re \J = 4. is much stronger than the convergence condition

of expansion (16). Furthermore,the distribution (13), or directly the

Poisson distribution (15), could be simply constructed making well known

simple statistical assumptions. Suppose (N + 1) possible evolution time

intervals with equal probability ^/^T ; hence P\H1/Xj can be simply

constructed as :

- -M (.18)

(i.e. as the probability of not having time evolution in N time intervals).

We stress that the integral identity (16), with gi *ji , for T > 0 .

gives a well-defined generalization of V^r X>\ to the continuous

transformation Vltit] : +o"

V(tjT)-.'J

(19)

-7-

where ff't ,X*\ is the I-probability function :

It t) >0 (£0)

This can be used to extend (91) and (17) in the continuum • In factusing (16) and (20) we have : t

From (19) and (21), for " t - % Q , we have "V"(t,l\-;) £ a n d

pft.t^^e [t) , hence P[i~ ,ty , for "f -=> a , behaves as

in the integral (19)>as required.

For finite t , however, the evolution law (9') is not

unitary, but preserves the so-called semigroup properties for positive

time translation i.e. :

(21')

Another formal way to introduce (21) comes trj considering the Trotter

identity :

122)

where : V U/«> - (i - i-it^

This identity remains true ty formally changing T •* -T i.e.:

(23)

since "Us Ib,-^ s as given by (19).

-9-

Note that ^U,*^1) • a s Riven by (21), can be "obtained" by (?:0

making a "Planck-like mistake", I.e. not taking thp limit 1 ~> 0

Why do we not do the aame with (22)?

If one follows the same procedure in equation (22), however, one

would obtain a non—positive definite, divergent density operator

as T, -J> DO ; even for "X arbitrarily small, but finite, as

can be seen from (21') changing t - ^ - T .

Furthermore,it is easy to verify that ^ 1 ^ / ^ , defined by (21),

obeys the finite difference equation :

(21.)

This is like the Liouville equation (7), in which one substitutes the

time derivative with an incremental ratio over a step interval T, as

one must do in all the calculations.

Note that if H is time dependent one can maintain the integral

expansion (9) or (21) of ? n ,1" ) , whereas equations (19) and (2&)

are no longer deducible from (9) or (21),

Notice that the right-hand side of (24) is estimated at the upper

limit of the discretization interval. We would have instead obtained

the lower limit using the divergent choice :

suggested by (22).

Hence » \t p! 1 appears as a very natural generator of finite step time

transformation which is well behaved for finite "i, , converging for t -^ O-Ut

to the unitary operator £ . From now on we discuss the time

evolution described by (21), where 'X is a characteristic measurement

or preparation time bounded by the inequality :

-t z t, -- JL. .

-10-

We first discuss the solution of equation (24) for the matrix elements

This is given by (17) putting « * t, , i.e. : * \ + ) , I 1+ I 10 ^ ) ? I°J

This can be written in exponential form as :

where the "line width"

a j"*1 n

(25)

and the frequencies rM^^ are given by

Vm<^ = A Met S C J ^ t ± 1TL (Wo, l....(26)Several observations are in order.

Diagonal elements are constant as it must be for an isolated system

for which H is a constant of motion.

Off diagonal elements vanish for tS>OC even for f arbitrarily small,

but finite.

Hence equation (21) or (24) describe irreversible state reduction to

the stationary diagonal form

(27)

In this sense we say that equation (24) is a Master-equation, which

intrinsically contains a phase destroying mechanism ruled by the unique

dephasing time T .

However, unlike usual Master-equations, this is not connected to any

specific system and interaction model of heatbath or collision, but is

simply defined by H and If for all systems.

Furthermore, the rate constants >'/h/wv depend logarithmically on UJ •

This dependence is essential in the Schrb'dinger limit defined by :

, nj j , X I \\ t ^ \

In this limit equation (26) reduces to :

i iX

(28)

giving V "~- Q for ^, -*} Q , as i t must be ; whereas without

the factor log (1 + (j*t ) we would have obtained y^ ^ w> for

Note that the limit (28) cannot be obtained for all (*)_,_. if the

spectrum is unbounded as for a simple harmonic oscillator.

In the opposite case : W ^ m^ ^ ^ V/w ^ £~ ^ % l w \f and VA> *v

approaches the value JT^ + J.TfJ ; i.e. becames independent

f r O m CaJyv, fy^ •

[Up to now we have assumed that E/Vl are non-degenerate, i.e. that H

is the only constant of motion.

If there is degeneracy, say H (iHjm^ = tn l/Vi]'*.'1> , all previous

formulas, containing matrix elements, can be simply generalized by

replacing P^^^ with 9/v, ^ i *' /vn • I n this ease state reduction

to the diagonal form means suppression of non—stationary elements

with/Vv^/Vi , so that f/v,/^ M /m' — ^ P<n m. f M'tn* O/w^ -I

Further implication regarding linewidth and correlation functions will

be discussed elsewhere. Note that the effective frequencies V*,J»H contain

arbitrary frequencies ^"^ - . They come from .the fact that if ?«*« rt

is a solution of equation (?4)7 o IV) e (*J"»)is also a solution

satisfying the same initial condition; i.e. the solution of equation

(24) is determined within periodic functions of period f ,

(29)

-11- -12-

ixrtri?iaaaMHSHf:in 11

These arbitrary "inner" solutions are ruled out on the temporal grid

t = H f , where they take the value 1.

Hence the solution of (24) is univoeally determined only for t s * f i

which implies that the equation (24) predicts the temporal evolution

Hence the time-energy relation (1) is required by (24) provided t > -£—

' ^ A t

since it is required to rule out the inner solutions.

On the contrary,equation (7) predicts a mathematically precise time

evolution of 9 |+) for A t arbitrarily small.

Note that, in tfte Schrodinger limit (28), the frequencies

be obtained by an equation line

where :

Hence the internal degeneracy described by (29) can be obtained in the

Schrodinger limit: t>B>^>'^' tl 1 , adding a "hidden" harmonic-oscillator

like Hamiltonian H1 to H in the Liouville equation. The effect of H'

disappears, as we said, only for t =V{t , but in general it affects the

eigenvalues of H = H + H' of arbitrary quantities

which are inversely proportional to *v

Interpreting this fact as an intrinsic indetermination for the generator

of time translations, one obtains an uncertainty relation :

A W >,

very similar to (1).

In this sense one can say that, if one accepts the equation (24) to

describe time evolution, the limitation t "} — — does not need

to be added, but, in a sense, is intrinsically contained in (24),

We now look for the equation of motion for mean values

where is given by (21) or (24).

From eq. (21) we have :

, jwhere

I* XT

Equivalently from (24) we have :

where we have permuted into trace, i.e.:

(30)

(31)

A),,

(32)

(33)

Hence again equation (33) differs from the usual equation :

d <A>t - _ i . < [ A , H j > t t

since time derivative is substituted with finite difference.

The two equations coincide in the limit t -> 0 . Note that relations (3)

using (33), give :

where

If we now require that inequality (34), (which rigorously comes from (3)

and (33)), must be consistent, at least for some A, with the condition for

-13-

- 15 -

!<&> to be observable, i.e. A ^ A } ^ } A A , one necessarily

obtains the condition X >"CO .In the opposite case : T < to ,

one would obtain from (34) that, for all ir\>t , A <| A >t < A A ,

(which is contrary to the definition.of T as the measurement or

preparation time for p0 to be fixed). Hence if one accepts the equation

(24), where T is the measurement time, it must be, by definition,

A <. A>i«; > A A , at least for the measured quantities; this

implies f .Jta

This again shows that the relation (l) is strictly consistent with the

equation (24), from which (33) has been derived.

This argument is just the exact version of that given in Messiah's

book, without the approximation :

which is necessary if one uses Heisenberg equations for^A^ instead of (33),

Note that the above approximation necessarily fails if A is the position

or momentum of a simple harmonic oscillator at frequency u>a unless OJo"*

Applying (33) to x. and p x , we have :

- / a H(35)

These are finite-difference Ehrenfest equations. They do not reduce to

Hamilton equations, even at the classical limit as it is jisually defined

/

or more generally :

1where

(36]

denotes Poisson brackets of corresponding classical variables.

In fact,even so one obtains finite-difference Hamilton's equations or,

more generally, finite difference Liowille's equation I

which reduce to time derivative equations for "(-=> 0 .

This limit however is largely independent from the classical limj^t (36),

In fact (36) is always exactly verified for free particles or harmonic

oscillators, independently of 0 and A L , so that t Tf *

can be arbitrarily large.

Bow, as shown by Caldirola, finite difference Hamilton equations, generally

give irreversible motion to oscillating systems. This can be generally

understood from (31). Even if ^ " C ^ ^ j i s a n oscillating function the

integral (31) gives an "irreversible" behaviour.

Explicitly calculating ^ X ^ a n d i P«) f o r 3 si™ple harmonic oscillator

at frequency oJ0 , with H = i ^ w p ^p^ + x'j , from (31) or directly from (35),4 -iujot _Lvt -fct

one finds the classical solution in which £

where V and v are given by (26) with

is replaced by

, Hence -—

reversible motion is recovered only in the Schrodinger limit lJJot <i 1 .

For X arbitrarily small, but finite, as t.-* » ., < ^ O a n d i p , > tend

to zero, which coincides with the classical time average.

This result will be generalized in the following. We stress that, if one

calculates i_ ^ •) and < p' >t by (31) or (35), one easily finds that,

for t -9 oo , these quantities approach the time average of the Schrb'dinger

solution. Hence for t.-^ oo ^•*-\ —4 P' t — <''~'* , which is an

equipartition behaviour.

We now show that for systems having discrete energy spectrum, an "ergodic"

limit holds, i.e, :

(37)

where plt^t) is given by (21) and l U ) denotes the tiise average as usually

-15-

-16-

•*~ ' .--. fL-^mm-;jfonr :**• '«A *«•

defined, i.e.:

In fact from (£5) we have :

±

if Tf is finite, in the limit t -» « , we have :

The equation (40) is a consequence of the state reduction (27).

138)

(39)

*" ^ '^M'Mt *

since by definition (38) one easily sees that:

t W»».t U K

Comparing (40) with (41) we have (37),

In the same way using (39) one obtains :

One verifies immediately that (37) can be extended to the degenerate spectrum

case, i.e.: H U ^ ^ E M li,^> ; ^m>;^ni,- = < *Wi | Pl«'*»'?

In this case one obtains :

where a is a degeneration index.

Hence time evolution, as given by (21), for finite*t , leads to the t -»

ergodic limit (37); i.e. ensemble average computed by (21), for fc -*> aO

approaches the time average of the Schrodinger ensemble average.

At the classical limit (36), In which the Schrodinger ensemble average

coincides with classical trajectories, ensemble averages given by (21)

approach classical tine averages.

Finally we show the connection between equation (24) and a "phase

destroying" Master Equation. Changing i into t -tt , equation

can be written in a Lippmann-Schwinger integral form

= elt) - i

Iterating n-times, we get :

EIn the limit (2B) can be approximated by :

a t *i-i

This is a nth-order Master-Equation which, for n = 2, is well known

in the literature to describe "phase destroying" relaxation of

spin-systems or harmonic oscillators.

In conclusion we think that the time-energy relation (l) is consistent

with (2k), in which T IJ. T^TF is the preparation time. This equation

for finite T leads to irreversible state reduction of the density

operator, which in turns implies ergodic behaviour even at the classical

llmit (36).

Acknowledgements :

We are highly indebted to Prof. p. Caldirola also for his continuous

interest, suggestions and encouragement during this work; to Profs.

G.Casati, V.Gorini, L.Lanz and L.A.Lugiato for helpful discussions.

-17--18-

cuiWEHt ::TP PURL.^AIIGM:; AJ.J ii.vi; NAL I.'".PORTS

(1) - This paper originates from Caldirola's pioneering work of finite

difference equations approach to the classical and Quantum Theory ' of

the electron. On these lines also finite differences Heisenberg equations have

been proposed (see A.Jannussis et al.: Nuovo Cimento B 67, 161 (1982)). These

equations however violate commutation relations . Our density operator

approach does not present these difficulties and tries to give a general

meaning for the finite difference approach and an interpretation of the

Caldirola "chronon" fundamental time in terms of time-energy uncertainty

relation.

(la) - P.Caldirola : Hivista Nuovo Cimento 2, n 1 13 (1979).

(lb) - P.Caldirola : Nuovo Cimento A 45, 549 (197B).

(lc) - R.Bonifacio and P.Caldirola : Lett. Nuovo Cimento 33, 197 (1982).

(2) - A.Messiah : MScanique quantique (Dunod-Paris).

(3) - The Caldirola's "chronon", defined as T t - C te where 1 t is

the classical electron radius, can be seen as a limting case

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K. SAEED - I d e n U H c - s U y. --y" v±o ,••.!.-- -1,-L.i . uificier.t condition..

real non-negativeness 31 . n-iu-i-u .i;'t , : n;;j

K. SAEED - The modifies gt-us;. di •,-'.

A.O.E. ATIIMALU - Quark ap. -SJ.••;::structure - II,

M. DURGUT and U.K. PAK - yMryo-:

D.H. SCljJ-lA - Tne role of aax^Lastronomy.

B.F.L. WARD - On the ratit of 1

J. LOKIERSKI - Composite ..TT-S-.VL:.

J, GORECKT - Some remarks 01: r.n

i_za^ion cf polynomial aiatri -fr.

.'. •. i',.':: conjecture of hadror; • .e

.••/or.U:m in QCD O.

':,> in cosmology and gai., Lie

: .''or the D and D meson.

-;::,,-,O3 i t e : ucergra'. i i.y .

i1 '.1\T: ?i.feetive melium

approximation for the r e s i j - . j v i ty ; . . ; cu ia t ions -in l i qu id noble .-a-'-al -

R.P. HAZOUME - A theory for the or ient&t ional order ing in neiaati.l i q u i d s and for the p^msr diap,r;..3 ^:" tt:t :if^nLatic-isotopic t r a n s i . .on.

R, BAQUERO and J . P , CAE3GTVE - The thermodyr:ajnics of supercondu^ti.:

lanthanum.

A.-S .F . OBADA and M.ii. MAllfiAH - Hespo;:Sf and normal modes of a ny^toiii

in tlie e l e c t r i c and magnetic-dipo Le approximation.

M.K. KOLEVA and I .Z . KOSTADINOV - Absence of Hall e f fec t due t ophonon a s s i s t ed hopping in disordered systems.

L. JURCZYSZYH and H. STESLICKA - Surface s t a t e s in the pnv t i i ce of

absorbed atoitis - I I ; approximation of ^.he small radius p o t e n t i a l s .

N.S. AMAGLOBELI, S.M. E.SAKIA. V.R. ./'JiSEVAIilSHTILI, A.M. KrfUJAJKL ,

G.O. KURATASHVILI .and T.P. TOPUBIA - Perhapi -i new unif ied s e a l ! - .

v a r i a b l e for descr ibing the low- and ' : -ii-p.. processes.?

A.S. OKB-EI BAB a, I M.S. EL-G1IASLI - T!.e i r t e g r a l repreat-n^.'-titjil c:'

t he exponentiolJ ; -nvex functions o; i n f i n i t e l y many va r i ab les op.

Hilber t spaces .

LI XINZHOU, WAKCJ Killl'i and ZHAI ." "A:;7.TJ - J-R so:U.;r.:; i ;i BGI;rsuperayninetry i •r:oi-y .

W. MECKI.r'..'IBURG and L. MTZn.-iL.il - . .H-E'-M-O t I T S and dual foriaul:-*'i• :.s

of gauge t h e o r i e s .

I . AfJAWI - In t e rac t ion oi' ;ole!'tric '.r:.1 Hhrj;rnctic ;r.arg:.-.-J - J .

IC/82/197

IC/82/198IHT.REP.*

IC/82/199

IC/82/200

IC/82/201IHT.BEP.*

IC/82/2021MT.REP.*

IC/82/203INT.REP.*

IC/o2/20l4

IC/82/205

IC/82/206INT.REP.*

IC/82/207ERRATA

IC/62/208

IC/82/209

IC/82/210

IC/82/211INT.REP.*

IC/62/212INT.REP.*

IC/82/213INT.REP.*

IC/82/211*

IC/82/215

IC/82/216

IC/82/217INT.REP.*

IC/82/218

IC/82/219INT.REP.*

IC/82/220INT.REP.*

IC/82/221

PREM P. BRIVASTAVA - Gauge and non-gauge curvature tensor copies.

R.P. HAZOUME - Orientational ordering in cholesterics and smeeties.

S. RANDJBAR-DAEMI - On the uniqueness of the SU(2) instanton.

H. TOMAK - On the photo-ionization of impurity centres in semi-conductors .

0. BALCIOGLU - Interaction between ioni2ation and gravity waves in theupper atmosphere.

S.A. BARAN - Perturbation of an exact strong gravity solution.

H.N. BHATTARAI - Join decomposition of certain spaces.

H. CARMELI - Extension of the principle of minimal coupling to particleswith magnetic moments.

J.A. de AZCARRAGA and J. LUKIERSKI - Supersymnetric particles in S = 2superspace: phase space variables and hamilton dynamics.

K.G. AKDENIZ, A. HACINLIYAN and J. KALAYCI - Stability of merons ingravitational models.

N.S. CRAIGIE - Spin physics and inclusive processes at short distances,

S. RANDJBAR-DAEMI, ABDUS SALAM and J, STRATHDEE - Spontaneouscompactification in six-dimensional Einsteln-M&xvell theory,

J. FISCHER, P. JAKES and M. NOVAK - High-energy pp and pp scatteringand the model of geometric scaling.

A.M. HARUN ar RASHID - On the electromagnetic polarizabilities of thenucleon.

CHR.V. CHRISTOV, I.J. PETROV and I.I. DELCHEV - A quaslmoleculartreatment of light-particle emission in incomplete-fusion reactions.

K.G. AKDEHIZ and A. SMA1LAGIC - Merons in a generally covariant modelwith Gursey term.

A.R. PRASANNA - Equations of motion for a radiating charged particlein electromagnetic fields on curved space-time,

A. CHATTERJEE and S.K. GUPTA - Angular distributions in pre-equilibriumreactions,

ABDUS SALAM - Physics with 100-1000 TeV accelerators..

B. FATAH, C. BESHLIU and G.S. GRUIA - The longitudinal phase space(LPS) analysis of the interaction np -+ ppir" at Pn = 3-5 GeV/c.

BO-YU HOU ana GUI-ZHANG TU - A formula relating infinitesimal backlundtransformations to hierarchy generating operators.

BO-YU HOLT - The gyro-electric ratio of supersymmetrical monopoledetermined by Its structure,

V.C.K. KAKANE - Ionospheric scintillation observations.

R.D. BAETA - Steady state creep in qu.irtz.

G.C. GHIRARDI, A. RIMItIL and T. Wi:Bi:ii - Valve preserving quantummeasurements; impossibility ctjeon. W:J -ind lower bounds for the distortion.

i m p u r i t y w i t i i uvl

IC/12/222 BRYAN V. LYNH - Order «Gy corrections to the parity-violatingelectron-quark potential in the Weinberg-Ealam theory; parirviolation in one-electron atoms.

IC/62/223 G.A. CHRI3T0S - Hote on the m.,,,av dependence of <£ qq > from chiralIHT REP * QUAEK

* * perturbation theory.

IC/82/22U A.M. SKULIMOWSKI ~ On the optimal exploitation of subterraneanIHT.KEF.• atmospheres.

IC/82/225 A. KOWALEWSKI - The necessary and sufficient conditions of theIHT,HEP.* optimality for hyperbolic systems with rioa-iifferentiable perforisanc

functional.

IC/82/226 S. GUHAY - Transformation methods Tor constrained optimization,INT.REP.*

IC/82/22T S. TAMGMMEE - Inherent errirs in the methods of approximations:IHT.REP.* a case of two point singular perturbation problem.

IC/82/228 J. STRATHDEE - Symmetry in Kaluza-Klein --henry.INT.REP,•

IC/82/229 S.B, KHADKIKAR and S.K. GUPTA - Magnet.:- moments of light bai-y..in harmonic model.

IC/82/230 M.A, HAMA.SIE, ABDUS SALAM and J. STRAil'UtE - Finitenesa of Ijrok.-nK = 1* super Yang-Mills theory.

IC/82/231 H. MAJLIS, S. SELSEE, D1EP-THE-HUWG and II. PUSZKARSKI - Surfaceparameter characterization of surface vibrations in linear chains,

IC/82/232 A.G.A.G. BABIKER - The distribution of sample egg-cov.nt and it..•INT.REP.* effect on the sensitivity of schistosomiasis tests.

IC/82/233 A. SMAILAGIC - Superconformal curreiri multiplet.INT.REP.*

IC/62/231* E.E. RADESCU - On Van der Waals-like forces in spontaneously brokensupersymmetries,

IC/82/235 G. KAMIEHIARZ - Random transverse Ising model in two dimensions,IHT.REP.*

Ic/82/236 G. DEHARDO and E. SPALLUCCI - Finite temperature scalar pregeometry.