. U'-.v .. INTERNAL REPORTJ . u . i f * ••••••- -

IC/68/95RNAL R

(Limited distribution)

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

LORENTZ-INVARIANT LOCALIZATION

FOR. ELEMENTARY SYSTEMS.- II.*

, * *A.J. KALNAY

International Centre for Theoretical Physics, Trieste, Italy,

and

Facultad de Ciencias Fisicas y Matema'ticas, Universidad Nacional de Ingenieria, Lima, Peru.

MIRAMARE - TRIESTE

January 1969

* To be submitted for publication.

'"'" Present address: Departamento de Fisica, Escuela de Fisica y Matema'ticas,Universidad Central de Venezuela, Caracas, Venezuela.

LOCALIZATION FOR ELEMENTARY SYSTEMS —II,

ABSTRACT

The ourpose of t h i s paper i s t o f ind t o -what extent fl.ie pos i t i on

•M' non-zero mass-free elementary systems in r e l a t i v i . a t i c cruantrT"

mechanics i s defined (or precluded) "by the requirement t h a t the

desc r ip t ions of a l o c a l i z e d s t a t e as seen from two observers i n

different frames of reference should "be physically consistent.

Neither the existende of a time-component of position operator nor

the compatibility of the space-components of position is assumed.

Hermiticity with respect to the quantum relativistio scalar irroduct

in state vector space is neither assumed nor rejected.

There are three possibilities! (i) Position is represented as

usual by n hermit*an operator: this is the point-type position,

(ii) Non-hermit»an operators are accepted and the k-corap.ono"+, of

position can represent segments of the k-axis with len^h of the

order of l/m; when the operator is not only non-hermit;an, but even

non-normal, thin is the extended-type position, ( i i i ) Non-hermittan

operators are accepted but the segments are reduced to points^ this

is the limiting oase of the extended-type position.

Case (ii) is not studied in this paper. General oonditione are

found, for any spin as regards cases (i) and (iii), and its

consequences obtained for spin, 0 and spin l/2. The results are:

There is no position operator in case (i) but in case "(iii) the

•position operator is unique for spin 0 and unique up to a r-.-r meter

fcr Ppin l/2. ?or both spins, in spite of the Isck of relativistic

hermiticity, the eigenvalues of position are real and the velocity

is the expected one. The eip-enstates of a component of position are

found. The components of position are compatible with, eaoh other for

spin 0 and incompatible for spin l/2. In the former case, the

simultaneous eigenstates of all components are Philips1 class I

localised states.

Related work j a discussed. Part I of the present serif;;; st

for J.C.Gallardo, A.J.Kalnay and S.H.Risemberff, Phys. Rev. MT8,

I W O-1-

I_. INTRODUCTIONA. OEr^JRAl

It could be asked if the search fbr the position operator and its

eif-tenfuncticns (i.e. the localized states) in .rel ativistic quartum

mechanics makes sense, because there is no experimental evidence

that position has a meaning in high-energy physics. But this wcni d

be a dangerous point of view, which could forbid us to measure it

even if it has a physical sense because, in general, an observable is

not measured in an absolute way, but with the air! of the properties

jt has (e.p. its relations with otha1 observables"^, properties which

would not "be suspected without a previous theoretical development.

Perhaps the reason why position has not been measured is because

theoretical physioisti have not been able to obtain the properties

which could allow it to be measured.

On the other hand, the absence of precise experimental information

explains why go many inequivalent solutions war© suggested. The

theoreticians are forced to guess the conditions to be imposed, on the

position operator looking for requirements KMJOSI seem reasonable but

which m-y he not realistic. Indeed, we learnt that several natural

requirements imposed on position to define it resulted in mutual

contradiction (proof: .see the multiplicity of ineauivlent proposals

of position operators and/or localized states made in the last forty

years at a non-decreasing rate).

That is why it is convenient to see if it is not possible

to lessen the requirements imposed on position to define it,

replacing them by an assumption so go&ralth&t it would seem almost

unavoidable if position makes sense. That is why our essential

assumption will be that the description of localization by two

observers in different inertial frames of reference should be physi-

cally consistent (to abbreviate, this property will be later referred to

as Lcrentz-invariance of localization). To use as few restrictions as

possible, w6 even weakened standard quantum mechanical hypothesis.

Such s. starting point could be useful i f in spite of its p-ener-ility,

it is still strong enough to define more or less uniauely t>ifthat

position operator, or to =;how us that it does not e*:i st.The result is/

it is in faot just strong enough to give an answer. (Below, in t! is

introduction, we will p;ive an indication of why this happens, before

froinp: into detail in later Sections).

-2-

Tr if? possible that i'rom a Physical point o" view position

Joes not exist for the fundamental particles, and if we were gure

cf -t.>r'i 3 + .he position operator problem would be solved. But if this

a new difficult question would appear: To understand how by

in? such particles > & world oan be constructed where* , at the

non-relativistic level, position has a meaning. If position

does not exist for the elementary particles, a^ain i t wciO <i be

convenient to be able to deduce i t s • noa-exi stsnce from very

assumptions.

Newton's and Wiener's paper on localized states for elementary s

is a fundamental one in the line of research that tries to deduce

position operator and/or the localized states from general principles.2

Let UP call S the set of states which represent ar elementary system*

(e.n;. an elementary particle) localized at the origin 0 of ^pace-time.

Then Uewtons and Signer's (NW) postulates can "be sunitnarized as

fr-llows:

TvWl : S is a linear set,1 " • O

UV/P : S is invariant under 3-rotations about the cri.^in 0, anr1

under space or tirre reflections.

;T 3 : AM *YG Z __ i~ orthogonal to thfi localised - tntefi obtained'v

"by a displacement in 3-space,

IT-fid : A rra-thernatioal re^ilaritj r condition.

Postulate jr,','2 means that if a system ia*at a point of space-time in

one '.Yfiiie of reference, then i t is alao seen as localized, and at the

same point, when observed from another frame related to the former "by

a rotation or a reflection that leffff.es invar iar-1. the point of 1 ocn'Tt zation .

T1^ this way the observers of both frames would obtain mutually

consistent descriptions of localization.

Theagpostulates define- uniquely the position operator and the

Icoalized states for elementary systems. But i t was immediately

realized that this localization was not Lorentz-invariant because

iT.D.Newtor- and E.P.Wigner, Sev. Mod. Phys. 23 , 400 (1949).9"See Ref. 1, Introduction, for the definition and discussion ofelementary systems.

-3-

if the elementary system was localized at the origin of space-time,

and if we perform a Lorentz transformation that nrese^ves the origin

( i .e . the point of localization), then from the new frame of reference

the system was no laigoc localised i NW'a postulates are incompatible

with Torenttf-invariance of localization.

Philips' and Philip^ and Wigner papers try to avoir! this difficulty

in str ict form in the f irst case, and approximately in the

Let us distinguish between the behaviour under Lorentz transformations

of NVi' position operator and WW localized states. In the standard

forunjl°ti on NYMs position operator ia not the space-part of a 4-veotor.

However, Fieri;ing showed that the formalism should be amend-

ed beoauae hyperplanes where the density of probability of no^ition

is given are not invariant under homogeneous Lorentz transformations.

He found a natural way of doing t .ds , and then he succeeded in

proving that after the reformulation wa3 made, the Kf s 3-position

operator is the spatial part of a 4-veotorf to summarize, Fleming's

reformulation avoids the non-covariant character of STWs position

operator. On the other hand, in the standard formulation as well as in

Fleming's reformulation, a H ' s state J localized at the origin of

spacer-time ( i . e . at a point invariant under homogeneous Lorentz

transformations) is such that i t s Lorentz transform! (e.g. e • ' | )

doee^ot satisfy FW's postulates. This implies that Fleming's

refurnu!Rtjon, in s a t e of i t s interest,does not solve the lack of

Lorentz-i".variance of NW's localization. As regards the present paper

we shall only need to consider such states and such Lorentz

transformations that the support of the density of probability is

left invariant by a Lorentz transformation, o that our conclusions

are valid in the standard formulation as well as in Fleming's.

4G.W.Fleming, Phys. Rev. Ijj2» B 188 (1965).

V o . P h i l i p s , Phys. Kev. 136., B 893 (1964).

"T.O, Philips and E.P, V'igner, He Si t ter Space and. Positive Bnergy.to be published in Group Theory and I t s Application, EJ", Loebl,Fxli + or (Academic Press). They consider the localized states for theunitary representations of 0 ( 2 , l ) .

second. Philip^ postulates' oan be summarized as foil owe:

PI i S is a linear set.

P2 ; So is invariant under all homogeneous Lorentz transformations

(including refleotions) about the origin.7 #

PJ J The localized 3ta1yps are normalizable "by eigendifferentials '.

P4 i SQ i s irreducible.

Philips found atitaa that satisfy thfes*postulates (oalled localiged_

states belonging to olaas I) for the spin zero case, also considered,

with different approaches, by M.I. Shirokov and by Schroder . These

statea do not satisfy NW'3 orthogonality postulate (FW3). This seems> 10

at f i rs t a serdioua difficulty, but Shirokov showed that this ia not

so. Also in Reft. 12 and 15 two possible ways were given to justify

the lack of orthogonality. On the other hand, i t seems difficult to

explain why localization should not be Lorentz-invarinnt.7I t is known that normalisation by eigendifferentiala is a physically

correct requirement for oontinuoua spectrum. See also Reft. 8,9-Q

A.Messiah, Mecanique Q,uantique (Ihinod, Paris, i960).

;J.C.Gallardo. A.J.KSlnay and S.H.Riseniberg, Phys. Rev. 158, I484It is the part I or tne present aeries. ~1C^.I.Shirokov, Ann-;.IhyaiK.10, 60 (1962).

U. Schroder, Ann. Physik 14_, 91 (1964).

"A.J.Kalnay and B,P. Toledo,. N

5A. J.Kalnay, to be published*

12A,J.Kalnay and B.P. Toledo,. Huovo Cimsnto 48,, 997 (1967).

-5-

q[•'or spin zero and spin one-half were found ether solutions of

Philips' postulates which are not of class I. There a.re irore solutions

than physically needed, so that Philips1 axioms should be modified

somewhat toward stronger requirements , One purpose of this paper is

to see if this is possible without destroying Lorentz-invariance of

looalization.

We shall expliciHy assume (Postulate 2, Sec, IT) a quantum

mechanical property that waa not used in Philips' work: position should

be represented (at least in a generalized way) as other v*u'if!bles do

in quantum mechanics, i.e. by an operator whose eigenfunotions stand

for the eigenstates (here the localized states) anct whose eigenvalues

are the oorresponding measurable values. It can be expected that the

combination of this assumption with Lorentz-invariance of localisation

could eliminate many possibilities (and we shall confirm in the next

sections that this is just what happens): for, if the position operator

exists, then it should be constructed with the ingredients of the

theory: these ingredients have a well defined behaviour under a Lorentz

transformation; on the other hand, Lorentz invari ance (jjf localization

restrict*the possible localized states, and the eigenvalue equation

links together the operator and the eigenfunotions (localized states).

This is haw the restrictions appear.

Philips' postulates imply tha-fc^makes sense that a particle be at the

i t d,an asHQoiatpdt

origin of coordinates, and, 1f A posiYion operator exists, this oan

only be done if the component e of position commute1)1 each other. This is

a desirable property but we shall not impose it because we do not want

to introduce avoidable hypotheses % instead we shall show

that Lorentz-lnvarianoe of localization (i.e. relativity") plus the

existence of the position operator (i.e. quantum mechanics) allows us

to deduce that position components must commute for spin zero and that

I t was suggested (see Reft. 5 an(i 15) that instead of P3_,finiteness

of the dimension of S should be postulated, beeause then only class

I localised states would be allowed for spin zero, jin this way

localization would be uniquely defined for spin zero. But this wouldo

preclude localization for spin one-half , and such a difference of

situation does not seem reasonable.

• "T. 0, P h i l i p s , Thesis , Prinoeton Universi ty (1963).

- 6 -

they cannot commute for spin one-half. Localized states where at a

Siven time the system is in a simultaneous eigenstate of the three

spatial components of position will "be called ^-localized states.

States where at a. given tdJMf the system is in an eifrenstate of (at

least) one spatial component of position will be called 1-localized

s_tatesn. NW's and Philips1 localized states are 3-localized states. In

our case we can have l-localized states but we do not know a priori

if 3-localized states are also realizable.

Our postulates 3» 4 and 5 refer to Lorentz-invariance of

localisation. One can try to express this invarianoe in terms of

conditions to be satisfied by the position operator or by the

localised states. Philips and Wigner remarked that the physical meaninr

of covariant ii-vector position operators . is not entirely clear, and

Gurrie, Jordan and Sudarshan showed the ambiguities that appear if

in quantum mechanics one triaa to impose relativistic invariance in terms

of the 3-vector position operator alone (we shall return to this

discussion in See. IV). On the other hand the invariance of localization

under a group of transformations that leave invariant the point of

localization can be expressed, with a sound basis, in term of conditions

on the localised states * , so that we shall use this approacn/^1 tri-fti •

regard to the transformation from one inertial frame to another

uniformly moving one, because -these are the transformaidms for which

the difficulties appear when working with the position operator approach.

As regards 3-rotations, where there is no problem, our conditions

will be imposed on the position operator, because this is easier in

our case(Postulate 3 ) . Behaviour under spaoe translations is expressed

by postulate 5» an& with respect to space reflections we impose, as

usual, that the position operator must not be a pseudovector (postulate 3).

We do not impose any requirement about time inversion. We do not assume

the existence of a time component of position operator.

Remember that time is a c-~number in standard formulations of

relativistic quantum mechanics.

'D. G. Ourrie, T. F. Jordan and E. C. G. Sudarshan, Key..Mod. Phys.

3% 35^ (1963).

- 7 -

Lorentz-invarianoe of localisation (i.e., physical equivalence,

as regards localization, of inertial frames) implies that if a

particle is seen in a region Tv. of spaoe-tirae when observed from an

inertial frame, and if a ohange to another frame is made in such a

way that )v is left invariant, then the particle should appear as

again looalized in ^ when seen from the new frame. This is the only

requirement we shall need to assume as regards invariance of localisation

under changes from a frame to a moving one (Postulate 4), and it is a

self-oonaistenoy requireaent. In Philip*1 paper (3-looalized states)

the region ^L *s a poin* of •paoe-time (see Pg). In our case we can

only assume the existence of 1-looallaed states, so that we shall only be

peTmifredto obtain consequences from Lorenta-invariance of localization

in 2-diraensional planes "^ of the type , e.g., "X • 0 at t » 0", but

this will be a sufficient requirement when combined with the existence

of the position operator*

Usually, hermitian operators are used to represent observables.

However, it is known that normal operators, i.e., operators that12

commute with their heraitian conjugates, can be used for binary12 20 21

observables. It was suggested * * that non-normal (and then non-

hermitian) operators could just be the right tool to deal with a

difficulty peculiar to position, i,e#, that (as is well known) pair

creation precludes localization in a region smaller than a Compton

wave length m™ . In this approach the position is called of the

extended type because the eigenvalues of each component of position

are segments of a length ^ m . The present paper does not refer

to this approach taken in its strict sense, because here we are asking

if position exists in a more traditional point of view» observe that

we gave sense to the proposition Mat t » 0 the system ia at X"* <• 0"

whioh is nonsense for the extended type position.

18

E. C. Kemble, The ffWidamental P r i n c i p l e s of QimntAiir Mechanics wi th

Slen^entar.y A p p l i c a t i o n s (Dover, New York, 1 9 5 ^ ) . See a l s c Sec . J . I of

T?ef. ]2 and Ref. 1Q.-W. E. Br i t in , Ann. of Phys. ^ , 957 (1966).

J. A. OaU.F.rdo, A, J . Kalnay, B, A. Stec and B. P. Toledo, ?Tuoi

Cinertc 48, 100° (1967).? 1 J . A, Oallardo, f\. J . Kfilnay, B. A. Stec and B. P. Toledo, Nuovo

Citnentc 4£, 393. (1967).

- 8 -

Let us now call limiting case of the extended-type position the

case where the operator is s t i l l non-hermittan and even non-normal, but

the above-mentioned segment is no lcmjpr of length U ^ "but reduces

tc a point. As the pure imaginary part of position eiTe12

represent the extension of the segment , the eigenvalues? should be22

real ' " , In Eef, 21 i t was shown "that the operat,or whose eiger, functions

ware Philips' class I localized states was .just a limiting case of the

extended-type position. I t was also shown that a known, decomposition

in pure hermittan and pure antihermitian parts, when combined with

the physical interpretation of non-hermit,tan operators ^iven ir. Tief.

12, implied that the limiting case of the extended-type position bound

together the Lorent?, oovariant "but non-orthogonal Philips' class 1 spir.-piere

localized states with HW's orthogonal but not covariant ]ocnlized

states.

2?""Eigenvalues can be real in spite of the lack of hermiticity because

the scalar product in state vector space for standard, quantum re la t iv is t ie

quantum mechanics is not %%, cwused in non-relativistic quantum mechanics

but l"be Onefjiven in Eq. ( l . l ) . For example, an operator can be non-

hermitlan with respect to the scalar product ( l . l ) but hermi+t,an with

respect to the usual one in non-relativistic quantum mechanics, so that

i t s eigenvalues are real . See also footnote 24.

2 'V. Bargmann a.nd E. P. Wigner, Proc. Natl. Acad. Sci. U.S. . %±, 211

(1948).

2/' . , >

'Other scalar products ware also used (see e.g. Ref, 2S). An operator

that is nnn-hermitian with respect to the scalar product (l.l) could

be hermit L3.n with respect to other scalar products if its form could

be maintained invariant. But this does not help to regain hermiticity

because a change in scalar produot induces a change in the form of the

operator and the operator remains non-hermittan.2'>rr. F. Jordan and N. Mukunda, Phys. Rev. 1 2_, 1842 (19&3).

DS. S. Schweber, An Introduction to Relativist!c Quantum Field Theory,

("LJow, Peterson and Company, Bvanston - Elmsford, 1961),

-9-

Synthesizing, there are three possibil i t ies:k(i) Position operator components X are hermitfcan; this is the usual

hypothesis. The eigenvalues of X are real numbers, i . e. points of12

the k-axis. It, is called the point-type position.

( i i ) Non-hermitfran operators are'dtcepted; the complex eigenvaluesk k k.

a" + i "b of X can represent segments of the k-axia whose centresk k

are located at a and whose length, is b . When this interpretation, isk -^

adopted with b £ Vft , and when i t is required that the operator be

non-normal (and not only non-hermit tan) i t i s called the extended-

type position.

( i i i ) Non-hermitjan operators are accepted but b =0 (rea]op

eigenvalues ) so that the segments degenerate in a point. This is the

limiting oase of the extended-type position.

The set cf postulates we use M(Sec. I I ) exolude case ( i i ) but are

compatible with oases (i) and ( i i i ) . We aooept non-hermitian operators

so as not to prejudge between the usual oase (i) and the case ( i i j ) .

We neither irpose nor reject hermitioity. Our postulates are almost

independent cf the physical interpretation given in Reij. 12, 2j a«H 28 of

a non-hermittan operator as a (limiting case of) extended-type

position, although our consequences will be consistent with them.

Moreover, b = 0 will not be an assumption but a result: we shall allow,

in principle, complex eigenvalues. . 9

The above-mentioned interpretation of^non-hermitian position operator

will only be used as a gueas of the behaviour of oomplex valuesk ka + i b of position under an inhomogeneous Lorentz transformation. Ifthe extended-type position were right, then b' should be invariant

ktonder space translations and a would have the same behaviour under an

inhomogeneous Lorentz transformation as the ordinary real coordinate.

t(tk&*. <f—We shall postulate auch a behaviour cf b and a (Postulates 3 and

5, See. II). On the other hand, we shall not need to know how b changes

under a homogeneous - Lorentz transformation.

To abbreviate, we shall call in. next sections a region of real spaoe-

time any set of points with real coordinates (t, a , a , a') obtained••e have no need to tiae ( .or reject) complex time. Complex space-

time coordinates Were considered by A.Das, J.Math.Phys. 2,45 (1966),J_,52 {l?66), 1, 61 (1P66) and by E.E.H.Shin, J.Math.Phys. 1,174 (1966).

A.J.Kalnay, to be published. See also Ref.l3«

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B, PLAN

T<" Sec. IT. wo s t a t e the p o s t u l a t e s . Before discuss ing sor-e as

of them in crea ter d e t a i l , we give in Sec, I I I a more accurate

of them while obtaining the general mathematical consequences (which

wil l be va l id for any s p i n ) . In Sec, IV we conaider those aspects of

the basic ideas that were not s u f f i c i e n t l y discussed in t h i s In t roduc-

tory Section. Ifh&n th is analysis i s oompletewe obtain in Secj. V and VI

the con sequences for the spin zero and spin one-half oases . In Sec. VII

we discuss the conclusions,

C. CONVENTIONS AND SCALAR PRODUCT TO BE USED

V/e use l i s 0 = t and the same nota t ion and conventions for metric

in Ppace-time, i nd i ce s , vec tors and Dirao matrices as in Par t V of

Ref, 8, The sum convention wi l l only be uned for tensor i n d i c e s . We

work only for1 one-par t io le s t a t e s in p - r ep resen ta t ion , always denoting- • 2 2 1 /2 23

p s +(p ' + m ) ' and. we use the Bargmann-Wigner formalism : t h i sthe sca lar product for apiti s in s t a t e vector space

ty f /t (1.1)and , ives the framework for Seo, III (where formulas valid for any spin

are eon-sidered). A3 is well known, the scalar product (l.l) reduces

to the standard, ones for spin zero and spin one-half, as they are

expressed, e.p;,, in Ref. 26,

Given an operator we 3&y that it is spin-s herinittan if it is

harmit Jan with respect to the scalar product (l.l) for the spin-s

case. As is well known, hermiticit.y of operators representing

physical variables meaPA • hermiticity with respect to the

scalar.product used in state vector apace.AW in/

We c a l l Vr the t o t a l an^mlar momentum tensor and L the o r b i t a l

one. As i s well known (aee e .g . Ref. l )

Lok = - i p o * k , k - 1, 2, 3. (1.2)C1*1 -is)

As in Ref. 8 we d«n©te by C the orbital-variable space and by Cthe spin-variable space, so that the state vector space is C= v. S C ,

-12 -

I I . POSTULATES

We a, ft sums :

"O Position makes sense for a particle,

2^'' Each component of 3~position is represented by an operator K ' w

lues are i ts possible values . An eigenvalue of this

ocn^cnent of position oan "be known with certninty if a,nd cnly i r the

vector that represents the state ( = l-looali^ed state, cf. -iee.I.A)

is an ei^enfunction of X" corresponding to this eigenvalue.

j>_) The set of the X (k * 1, 2, 3) i s a 3-vector operator,

V[_) Localisation is Lorentz-invariant in real apace-time (of.See, I.A)

in the sense that if a particle looks localised in a regionJ^ of real

mmce-time when seen from an inertial frame of reference and if a

homogeneous continuous Lorentz trans formation is made in such a wa;;

that A. is left invariant, then the particle also looks loc-iliaaii in A.

when seen from the new inert ial frame. A region of real spaoe-t'i me

is invariant under a homogeneous Lorentz transformation if the region

is invariant under the corresponding Lorentz transformation of non-

tjuantum special relat ivi ty.

__) The space-translation operators transforms localized states into new

".localized states, "but they only translate the real part of position,

leaving invariant the pure imaginary part.

6) For fixed p, the set of eigenfunctions of a component X' of

position at a given point a + i "b contains a basis of C (notation,

Sec. I.C).

The discussion of postulate 6_ and of the last part of postulate 4_,

as well as the complementary discussion of the remaining assumptions

is given in Sec. IV.

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TJTJ. GETE AL "ATHf ATTCAI CONSSqUEMCES OF THE PQSF.'I./^ES (I-1"" ..'T r-;-T"N

het us ""i^Rt rion^ider the standard case in whri ch tVip posi t ion orera tor

j ^ he1"^ t i. in , i . e . a point - type operator in the monenc] a ture of Refl.

12 and .Sec. I .A, The exis tence of the t rans la t ion operator e j* - R-^r1

tEM any*"TWO. number can be a measurable value of a comione^t

-i t i on . T."'e consider a p a r t i c l e which at the time t = 0 i s in an

ir-ens t.ste ^ of a a?ac»" component of position, We drpw the Varis in h

irection and v;e select its origin at the intersection with the 2-

•Mnens: on?1 nlsne of localization. Postulate _2_(Sec, Tl) implies

J

The index V stands for the remaintwa quantum members that specify the

st'te, if any. Rework in the Heiaenberg picture of time eve1 «+,i on o

that the fact that time equals aero implies that what we c---ll.e X is

*•;. o vH.v.ie at t = 0 of the operator X(t).

T vi t.he ^ore .general case of n on—hermit tan position operator

postulate £ allows us to make aero (with a spatial

translation) only the real part of the measurable val/ie:

Y" •e re , in order slot t o have ao many indices, ire abbreviate

3

u

By perfor^inp; now a new spatial translation a of state Toj we obtain

•3. nev state e ^ ^ , which by postulate ^ ia localized at

a' +1 b ' , ?.o that nostulate 2. implies

31

~0n the other hand, transformation of 3tates by tran.-.lat"in;1:

s ;-Ttially the physical system, or by performing a Lorentz tranaformation

>-v it. !,»/ill be expressed in the "Schrodinger-type picture" of such a

transformation, in which only the state vectors (but not the

o-'era1:ore) are changed. [See e.g. A, S. Wightmann and 3. S. Scbwener,

"!w«j. Rev. 08,, 812 (1955)].

-14-

A? v e . - . ' \ r ^ i^eaj - s p a c e - t i m e ( d e f i n i t i o n s i n Sao. 1 , k) , t h e ivci ,-ri n

• ••i f, •>-,=,. Yoi,,,/ i s l o c a l i s e d i n t h e 2-c.iirr.8nsior.Rl. p lane SC o ^ e e i H a r l fcy

•' •'v- the third spatial coordinate and the time equal to zero. By

-o^tvilate 4_, TV remains invariant under the lorentz tT>ans?crr-.'r+,icvi

r.1' s o g h A - p^ s i n h A

P —* p^ coshX - p° ainh A .

!>• t'.Visa st'-ite vector space the induced transformation is generated by

the component M05 of total angular momentum. Then, postulate ^

ir^'lies tl;nt the transformation e ' *3 (considered, now as q-p

active one) transforms the s ta t s Y&*local ized (in real space-time1* in

t):& '--lane A* into a new state also localized in JV , so +hat rop+ulats

(]n 3i?c. T.A we remark IWwe have no information as reij:9.rSp how b • b

changes, ^o that we- indicate the new b in the generic form h = h ; ) .

I t ip 'Kno-.ra that for spin greater than zero the conditions sa t i s -

fied by the wave functions in the Bargmann-Wigner formalism [e.j.%

Eq. (6,1) for spin. l /? l imply that ths operator repressnttng a

physical yariable can always be written in the form

yv n. yv (3.5a)v/here ^V is a projection operator

.e ST ac'e of the permissible wave functions 7 :

2-1; •} rreli=ivn"+ (:o add to i i an operator &.Q, 3:ich thct -A.

:por OUT purposes i t wi l l he useful to u^e in the 3pin 1/2 fi^.^e th

Lo r e n t s - i n v a r i a n t p r o j e c t i o n o u e r a t o r of Eq. ( 6 , ? ) f

•7 Q

:

32"Severnl projection operators can be used. For our rnirposss i t in

tes t to take advantage of Lorentz invariance *c that we nee theT.orenta-invarjant projection operator, hut then cnve shnnifl he

t/v-'en v;hen working with the hermitian conjugation because this

projection is not orthogonal, J'or other purposes |*see e,,-. ?.ef.

1 ] the f rthOftonal projection operator E is used. Bearing in mind the

••o-.i-vi bi 1 i t ies of usin^ different projection operators, the doubt cci^rj.

nrise th.'-t the use of a special one in Bqs, (3,^) or (^,6) ee operators of },\\e form e.g. E l l 'E , But. t k r e i-i nc yvi'hl (MM,'dven i l (or JL ) an Si' (orJ2 ) oan al-ways be found suolx that

-15-

rir.d i + 5 . erer--)! j sat} on9 fo^ higher spins.

'n. ''?.?) together with postulate _ (and, fcr spii greater tivn

7,ero, postulate 6) allows us to get a f i rs t insight into the form

X . In Appendix A we prove that for an arbitrary spin s,

v:h f> r 3

Bk - Hk(TT ; . . M r , . . . ) , (k - 1,2,31 (

is a j-vector operator that only depends of ri an'l on nny of the

i , , i ; I rrvr*-.rices (heie symbolized by \n )i

^ A *u)*<i)""Imj , . . . , ^ Su, • **,„ , of the Barl s ; 1 23

formalism .Then it follows that , .

(3.'0Firs ex renter! ( c f . Hef. l ) ; We used, here the f i r s t of the well-knownrelaiiomst „ \

[ A , / ] . 0 ,Cop-hinin^ Eqs . ( 3 . 4 ) and ( 3 . 6 ) wi th ( 1 . 2 ) and ( 3 . 8 ) one

1

ie+ us c a l l Vra, g e n e r i c 4 - v e c t o r o p e r a t o r (as p**1 or $ ) and T ^ " a

•ven^rnc Fsetior.d-rank anti5V*wi?t.ric 4 - ^ s ^ s o r o p e r a t o r (.as L or

, ^ ]"), The known commutation r e l a t i o n s they have with t o t a l

,'?'!-i:l r.'-r' rrn^erti;^ TJA imolv-, a s i s trail known,

= V4,

* v 'w iU- /S.-.u r and

The Eos. (3.P) anrl (g.10) will he our basic tools in the remtiihlng

calculationa (Seo. V for spin zero and Sec. VI for spin one-half).

They are valid for any spin.

Before finishing we summarize here some auxili&ryentities to be uned;

= +

-16-

7t will also "be useful to introduce a function Oj (%) which

anp.-rrantly on two varia.bles:

^ it as

so that \> = w l V f l ) t Co~.n-e) (5.12c)

-17-

IV._

( j j Postulate 1_ implies that we deal with

partioles (in general with elementary systems) « not with

;• a r t i c le? ami ixnti par t ic les . Then i t follows that the wave ^ r t i ! - ^

mist be res t r ic ted to the positive mass shell (of, Ref, l ) .

( i O Invariance of localizat ion under a <TOup Qf^°'* tmri ;rf or^'iti f-ns g in

real sp-sce-t.ime can be ex pressed- in a soi;nd basis as a condition on

the s tates (when the theory provides the ,?roup y( ' °r induced

tfan:• !'o''MM.tionr5 f?1 on state-veotor space) in the way shown

for ^-rotations by Newton and Wigner . The procedure can ho i + atefl i.n

the fol'owinr general way: Let K be a region of locr-i.lj nation in innce-

time (real space-ti^e in our' oasa). We call ^.L") C:<*L the subgroup of

(A, thnt leaves invariant the region X, , <5 (%) c : M , the

sub'Tour of in Cursed transformations, and o( JL ) the p^t oi' ^t^ter;

localized in V ; then invariance of local isat ion i s expressed by the

condition that for a l l (fixed) J?- , if t& 5 ( *L) then

^ e ^ (1?) for all ^(I'iZ) . ^Tn wore physical terms: if the region*of localisation is invariant under

the transformation g in real space-time coordinates, and i'' the narticle

(cr, in general, the elementary system) was localised in jv v/her ?;een

f'rori: the old frame, then i t should also look as localized in \_ when

seen from the new frame. Or in more mathematical terrr.s: If %_, is

"invariant under (X X.' , then y ( JV-' should also be invariant under

Eramnl e 1: NVV's 5-1 ocali z at ion , where (i is the (J-0 il.ean ;TOir.,

i s a point in space-tips (e.g. the origin 0 ) , A (o) i s thp set o-1

.p-eneous rotat ions about 0, (j, '(0) i s the set or operators ^ *

) i s the set of NW's s tates n.ocaliBed at 0. W p paper implies that

the 3-localization they defined is invariant under the Galilean .'TOUT.

Example 2; Ph i l ip ' s ^-localization , where uX. i s the Poincare f-roup,

j \ ^ i r, a ^oint in space-time (e .g. the origin 0) , UL (0^ i s the set of

t'ift hrmo-eneous Lorenta transformations about 0, Q 'fo"1 i s the set of

operators t and <5(< ) i s the set of Phi l ips ' st-.+.e^ J

a t 0, P h i l i p s 1 paper implies t h a t the ^ - l o c a l i z a t i o n he Heri>ied ^

i ^vri-r-; r.rt under t h e Po inca re ,»roup,

I?ffir;r-l G 5: f,i;r cane ( l - l oca l i n a t i o n ) , where (Jf. i s the roincr-rB ^ c ; r .

Y\_. ic; a r e a l s p a c e - l i k e 2 - d i m s n 3 i o n a l p l a n e ( e . f . X = t = i "•' ,

i^ +v-e set of the products of Lorantz transformations in the 03 andcU Vu

-18-

a I"? *

12 planes and translations along the axis 1 and 2, MvAy ip the set of

the products of operators of the form e ° , e iL anri11 .22-ia1 1 - ia2 £HQ\

e , and GjCJyis the set of states such that they

ty.. (3.1). The 1-looalization we consider is invariant, under Poincare

•rroup, as it follows from Poatulfites 4, 3 and 5 (for the inva.ria.nce

under e notice that postulate jj_ implies that [X ,M, 9] = 0,

satisfies Eq. (j.l) the same happens with e~lA<. M"?,o that if v satisfies Eq. (j.l) the same happens with e~lA. M " ^ { j ( / ) .

(iii)_ , The 4-vector position operator approach is less general than

the Lorentz-invarianoe approach discussed' in (ii) if the time-component

can be interpreted as the time [there is trouble with the t:.me, as we

will discuss in (v): on the other hand, if X Cfthnot be used as the

time, the physical meaning of 4-position is doubtful ]. Proof; First

part. If X is the time, it follows from the commutation relations

between 4-vectora and angular momentum tensor components that

localization is LorentB-invariant in the sense (ii). Second part. Philips'

class I localized states [Eq. (5.25^] are Lorentz-inva.riant in the

sense (ii) as can be verified by direct inspection. The unique

operator that has these states as eigenfunctions is X = i 3 / J p

(k = 1, 2, 3) and it is not the space-part of a 4-vector, becnuso if it

were, then contradiction would arise with the commutation relations

of '"-vector operators with angular momentum tensor components.

(j_v_) If in the 4-vector position operator approach the components

compute, then Philips' and '"'igner's objection arises.

(v) Thfi physical meaning; of the time-component of a 4—vector position

operator is not clear. It should represent the time. But in standard

formulations of quantum mechanics the time is a c-numher, becru;se it is

the classical time related to the measurement apparatus, (Tbe ] tick of a

time operator is not an imperfection of the theory: see Ref. 33). As X

is an operator, covariance would imply that X is also an operator (again

this follows from the well-knomi commutation relations of 4-vector

operators with angular momentum tensor components). Then X° is a second

time in the theory. But which time? It if! rplated to the physicn] system

we study, but it is not the proper time because X is net Lorentz-invariant,T.?Fe see that tbe 4-vector position operator approach introduces an

C. FfiAler, Communications of the Dublin Insti tute for Advanced Studies,

A.. N°.-5 (1949).A..

-19-

p<-}•-]itiom.il unknown ir& the theory, Viich may or may not have

physical sense.

(vi) Tfe do not mean that the 4-vector position operator approach should

be avoided. A formally eovariant 4-position is. desirnble. But for the

purpcsea of the present paper if'rs assential that the assumptions be as

r.ure as possible. That is why the" approach ( i i ) was preferred.

(vii) Covnrisnt position operator approaches without time component

were proposed in Kefs. 25 and 531 "but i t is not sure that the form

uped for Lorentz-invarianoe is right in the quantum case, as was

shown in Ref, 1? and remarked in Ref. 25.

(viii) From a logical point of view t the form in whioh invariant regions

are obtained could "be objeot»d -to (Postulate A), because i t

I'TftnujfoseF classical meohanics and this paper deals ift quantum theory,

which ia 3U]-:osed to contain classical mecharics as a limit. However

i t was stated-" that quantum mechanics cannot be constructed without

presupposing classical physics. But in any case, i t seems that at least

this part of our Postulates should be considered as stated in a very

prelindn-Ty form.

(ix) Postulate §_ stands for the intuitive meaning of C , i .e .

thnt C has a dimension greater than one for particles with spin, just

because the orbital variables (momentum or position) do not specify

the states completely, and that if only an eigenstate of orbital

variables is prescribed, then arbitrary superposition of spin

states (or, if spin is incompatible with position, eigensta-fees of

a variable related to spin) are allowed. This postulate is obviously

irrelevant for th* snin zr-ro case.

- I. D. Landau and S. M. Lifshitz, '.quantum Mechanics, Ron—Kelativis+-i.c

Theory (Pergamon Press Ltd., 1958).

We have in mind as an analogy,Asimultaneous ei/renstates of the enerpy

of an electron with its third epin component only exist for exceptionalR

cases, "hut then, instead of spin, the helicity can be used .

- 2 0 -

I - SPIff ZERO

A. THE PPSITIOU OPERATOR AND THE 1-LOCALIZED STATES

The only p o s s i b i l i t y for Eq. (3.6b) i s

Rk = Pk g(p°) (C5.D

and the projection operator -A- introduced in Eq., (3.5) is the

identity, so that Eq, (3.9) implies

The unknown e n t i t i e s t o be found a r e A b ^ and ^

Combining Eqs. ( 5 . 2 ) , ( 3 . 8 ) , ( 3 . 1 0 ) , ( 5 . 11 ) and (5 .12 ) , we o b t a i n

° 3 + K2 cosh ( A - 6 ) sinh ( A - $ ) g[ l cosh (A - B )] +

i K cosh (\ - 9 )OJ. (A - 8 ) + i K cosh

- 0, (5.3)

which can be decomposed Into awi and odd .functions with regard to

\-B [remember Eq. (5,12b)]t

[L°5 + i K cosh ( A - 9 )to!(A - 9 )l9 = 0 , (5.4)

K(K cosli ( A - & ) sinh ( A - & ) g[K oosh ( A - 9 )] +

+ i cosh ( A - 9 )oT ( A - 9')]W «-o. (5.5)

Prom Eq. (5.4) we infer

[L0? + i KOO^O)] = 0 (5.6)88

an d

K { eosh (\ -$ ) ufiA -Q ) -00{0)]Wt =0. (5.7)

The outside factor K oan "be eliminated from Eqsj(5'5) and (5.7) beoause

i t cannot be zero [_cf. Eq, (5,11a) J; we mention . t h i s because

otherwise a so lu t ion Y with a factor o(K) could be considered. On

the other hand, a solut ion of Eq. (5.7) of the form ti> = d ( [ . . . ] ) i s

not allowed because W is independent of A , Then one can deduce

from Eq. (5.7) that

00+&( A - 9 ) = % ( 0 ) / ooah (A - £ ). (5.8)

For similar reasons Eq. (5.5) 'implies

JK sinh ( A - 8 ) g[ K cosh ( A - B )] + i % ( A -fc )} v = °- (5.9)As the second term does not depend on.K, g must be of the form

S (a) - N/z (5.10)

where N is a constant-, [it could be thought that there is again a possi-

bility of avoiding this if ^ had a faotor of the type

S(K) b u t < > 0 . ] .

From Eqs. (5.9"1 and (5.IO) we deduce

CO* ( A - 9 ) s i l f tanh ( A - 9 ) , 0-

which replaced together with (5.P) into (3,12c) shews that

b^ = V (0) [oosh (A - 0 ) ] " 1 + i N tanh ( A - £ ) . (5,

As b ' , doeR not depend on p . i t also does not depend on Q f?f. Eq.

( j . l l ) ] , so that dfter differentiat ing with respect to &• we ,^t

- i NT [cosh { A - 9 ) ] ~ 2 . ( r;.l?

After multiplying by [cosh (A - 6 )] the linear independence of the

functions involved (on the variable A - 0 ) implies

. n - 0

i . e . [cf Eos. (5.10), (5.12), (5.6) and

b \ - 0 , i . e . b 5 = b 5o = 0 , foW i ^ ( 5 . 1 6 )

FT or) 5(9, ( j . 6 a ) , (5.1) and (5-15) w© get the posi t ion operator

I t looks lilc& the familiar one of non-relat iviat ic quantum mechor top j i t

has indeed the same form aa the familiar one expressed in p—represent-

a.tion( but, (IS i s well known in the. r e l a t i v i s t i c case th i s

operator i s not hermit*.an, because of the scalar product ( l . l ) .

V ore ever, (/OL i s notoflfiftf1 the momentum representation of the operator

that is multiplicative in configuration space because cf the factor (?) in

Jf ' (5.19)ir nr©*vi cu.31 v

The pos i t ion operator X i s the sameytfionsiaered in Reg, 10, 11and 21. Moreover, i f one writes I t in the form [cf. EQ . (1.21!]

20i t is immediately realized that i t is one of the possible Quantum

versions of PrVce's definition (c) of classical rtptntre of mass t I t

1B not only non-hennitian, but even non-normal, though it^ eigenvalues

-22-

2? 21

r,re r e a l " . It i s the only operator tha t has as ei^ens+.n + es T r i l i p s '

c l a s s - 1 Iocali7;ect s t a t e s f so that i t i s not the space-part of a

. '-vector operator (cf. Sec. IV) , I t i s such tha t= 0 ('>.?

= PVP°. (5(Notice thp.t the velocity Is hermitian).

The l-localized states solutions of Eq. (3.?) (remember h = b5 =are Y

-I p

v.'here f(p ,p ) is arbitrary up to normaliza'bility requirements. The

states ( .23) are not normal!zable to unity (which of course is ne

problem for continuous spectrum), but its eigendifferentin!s should

be normalizable to unity'. Indeed, the p dependence, (v.rbich is +he

only one defined) in the partial eigendifferential'

% a.~ £<n*

(where C i s a normalization constant)docs hot deny i t s normal izabi l i ty ,

B. THE 3-LOCALTZED STATES.

Ecu (5*2l) allows simultaneous e igens ta t e s of a l l components

of X , i . e. 3- loca l ized s t a t e s (cf. Sec. I ) . They are

(f>_, a COH*C. e ( S OZ.)

i . e . P h i l i p s ' c l a s s - I loca l ized s t a t e s . T h e y are non-or thogonal ' , but

t h i s can be j u s t i f i e d ' ' . In Ref. 21 they were ^.tijdied as a

l l ^ i t i n e ; case of the extended-type pos i t i on . '

37•We see that the index V turns out to "be unnecessary in th i s oasej

1 l^tiifantory for a spin zero particle. The point of localization is "a*.

- 23 -

VT. SPTK ONF-HAIF

A. ^TT!<' POSITION OPERATOR AND THE }-LOCAI,TZED STATE'S

A 1-localized s t a t e f (as anv s t a t e ) Trust be such thnf;

-invriri ant projeotion operator [cf, Sq. (.5.5)"' i r t c the

,ioe 0*" thn allowed s t a t e s I F well known:

t )The bracket, i--1 En, ( 3 . 6 ) must be c o n s t r u c t e d with *

• • ' c '

A? the result should be a 3-vector, the only possibility is

A Rl A « A [ tki- % S ' k%„

v / h e r e t h e Z(-\ = S( • ^ ( P ) a r e o r d i n a r y f u n c t i o n s ( i . e . w i t h o u t i r i a t r i c e s

As

Aa redefini ticn of g/-,\ and. S(n\ allows us to impose

e.(±) = °, i = 3, ... , 8, (6.5)without Toning generality. Then, Eqs. (6.3) and (6.5) imply

and

We replace into Bq. (3.9) after using Eqs. (3.8), (3.10)

(X = -i (T* ), (3.11), (3.12) and (3.5c). The result is

- K ct»U ( X - e ) « \ [ k cosl, u - e ) J + i KcesU (X- 9; uj+

-24 -

hich cfin be decomposed as in the spin-zero case:

K A { K c o v K U e ) n U ) j

S i . ( 6 . 8 )

Aan d

toil*

which corresponds t o Eqs. (5.6) and (5.7) of the spin-zero ca^e. Eq.

(r-,9) corresponds to Eq. ( 5 . 5 ) .

"•'e ir,e t i e following,' abbrevia t ions :

ii.( A - e , e ) = c o s h ( A - e ) o j f& ( A - e ) - u / & ( o ) , ( 6 , i ? a )

V(K,X - e , 9 ^ - i c o s h ( A - $ ) g ( 2 ) [ K c o s h ( a . _ e ) ] - i ^ ( 2 ^ ( K ^

' (6 .12b)and

The variable 2; has^the following properties :

^ t ^ K ^ i J I ^ I ^ h e r e pH" = ^ ( p 1 ^ + (p1)1]"*, (6.1?^

14 = I l<Xk+ L A , <XkJ), (6.13c)

-25-

and *L i s - Hil.qrevoord and WoulKuyaen spin-

tensor , which i s the spin associated with Bungs's position

noWe shall omit details of rigour as regards eliminating

the ^ -type solutions because they are similar to the corresponding-

ones of the spin- zero case (Sec. V). On the other hand, for spin zero

the equations were easy to handle because Eq. (5.5) and (5.7)

contained only multiplicative operators. To get a similar simplicity

we first transfer to the right the projection operator-A- in Eq.

(6.11) so that,

(u - ir* 5T) iPokv= 0 e (6.14)and then we multiply thia last equation by (u + i 2" v), In this way

all matrices disappear [cf, Eq, (6.15b)]:

[u2 + (nf2 K2 - 1) v 2 ] ^ - 0, (6.15)

so we can be sure that the bracket itself is a null function, i.e.

[cosh (A - 8 )a/e( A - 8 ) -e?Q(O)f -

= (m~2 K2 - l)[cosh ( A - 0 ) g/2\[K cosh {A -9 ) ] - g ^ ^ K ) ] 2 .

(6.16)

We now wish to find the form of the functions (M and g/?\

compatible with this equation. Let us define

y = cosh ( A -8 ) ,+ f

y (<*)- cosh ( A - 8 ) U/e (A - 9 ) - 00 (0) (6.17and ^

K

I t will be sufficient to r e s t r io t ourselves to X^Q for the ohange

of variable ve made to be one-to-one. Eq. (6.16) looks like

- h(Ky) - h(K). J

and ^Wouthuysen, Huol; Phys. ^0, 1 (1963). W.- Kolsrud, Physica

!v'atheiT.atica Univ. Oslo No. 2? (1965)*" Nuovo Cimento %£, 504 (l 965) and

Uorve^ioa 2_, 141 (1967).

58f'.r, ^o]f3ruf3, Fhysica Uorvegica 2_, 149 (1967).' ' (1968)

Jjrf-.t Burip;e and A. J . Kfilnay, Preprint F. 1A , Facultad de Ciencias Fisicas y

Voteniaticas, Universidad Nacional de Ingenierfa, Lima., Peru.

4°T.1. 3un,5e, Nuovo Cimento 1_, 977 (1955). Cf. Ref. 39.

-26 -

By r l if i 'erenti atir.r; with r e spec t , t o y we f^et

'n'(Ky) = t K"' (trT2K2 -; 1) Y ' (y ) (6.1?)

where the privies stand for d i f f e r e n t i a t i o n with respec t to the whole

mvvunen+. im-.ide the p a r e n t h e s i s . Then we put y = 1 i>•> Eq. (6,19) 'i^i! ^v

the r-e^v.Tt we rep lace K by Ky, By combining th*se equatj f.rfi ve f i>Tr1

Rv ii-i iM'^rentiatinp now with r e spec t to K and ooirbi.nJnr; vn K> +he former

equations we -,ee tha t Y ' t 1 ) ~ ° 3 0 t : a t V'(,y) = i"1 imd

1:(K) = h (0 ) , which together with Eq. (6.18) implies- 'f (y) = r . ^he-i

U,* ( A - 0 ) = <JQ(0) I cosh (A - B ) & (6.?i)

and

r ^ U ) = i C5 / K where G = - i h ( o \ (6.22>

On the o the r hand, a f t e r d r i f t i n g : -/\ . t o the r i g h t E q . ( 6 . 9 ) cr-m be

handled e x a c t l y as it wfcS in t h e sp in zero case wi th S'q. (^ .9 ) (v/ith

5 r e p l a c e d by ?f- i \ ) a n ^ we p;et the same r e s u l t s :

m" = Wo= h; = 0 , b 50 = "b

5 = b = 0,0 o A

"ei'lacim- (6.?l) and (6.24) into (6.10), we get

A [ l ° ? - i G 0 ( 5 ] Y, = 0. (where. h = 0) ,

which, as

^ k = Lck + i , iCVk , (6.26)

o,-n he combined with Eqs. (6.13c) and (3.8) to obtain

y ^ if = (0

I n e i m i l a r f o r m , w e o b t a i n f r o m E q s . ( 6 . 6 ) , ( _ 6 . 2 6 ) , ( 6 . 2 ? N > , ( 6 . 2 ? ) a n d

( 6 . 1 ? 5 c \

Xk - ( i°) L(i + G ) Z 0 k - M 0 k ] A . . (6,28)

We can ver:i fy t h a t Eqs. (6.27) and (6.28) are cons i s ten t with Eq.

( 5 . 1 ) . The quan t i t y G i s a constant [cf. Eq. ( 6 . 2 2 ) ] .

The opera tors K , m p_ and m Pul l a H commute

wavethemselves and (for only on the space of the allowed w

functions) with. -A . We lock for their common ei gen functions \ (r- ^

fcf. Eb. (.6.13b)], ^

-27 -

'f

in the space of the allowed wave functions:

A ****A Se -

(6.2?a)

(6.29b)

(6.29o)

(6.29d)

(The meatiing of the indes1 & will be made olear "baloir. )

As M and m p^ are spin- i hermit'/-an, § and q must be real .

Instead of solving directly Eq. (6.27) or (3.1) it is easier to

look first for the explicit solutiory of Eq, (6.2Q), and use them as a

tool. We do this in the standard representation of Lirac matrices, i.e.

We define an an^le such that

PH e

(6.30)

(6.31)

~ —* —p j 3

and we express p in cy l i nd r i ca l va r i ab les in p-apace, i . e . p , f and p

After a lengthy bat straightforward ca lcu la t ion the general so lu t ion of

Eqs. (6.2Q) can be found:

w-

!"

where

<f) =

-28-

C is a normalization constant and § {<P ) is an arbitrary function

such that

°f ~X • (6 .32c)

The relations

and t ^

•A «*"* ' * (6.32a)

are satisfied,.

Now we use an orthonormal basis § ( f ) in the space of the .functions

I>*,'W- l-V (6.33a)and we restrict ourselves to the subset of the general solutions

V obtained when we replace the general f "by m , We call

% > = 5* ' (6.33b)

From Eq. (6.28) and (6.2Q) we deduce that

i ' r t t- (6.34)

With this equation we are able to find a solution y = i

of Eq. (6.27), i.e. of Eq, (j.l) [remember b5 = b = 0, Eq. (6.24)and

Eq. (6,28)], because

- 0 (6.35)if

5 = 5 U , O = U J + 0) q .. (6.36)

To obtain the general solution of Eq. (6.28), i.e. of Eq. (3.1) we

observe that if the restriction (6,36) is not imposed, the functions

$ span the whole space of the allowed wave functions

(loc-lized or not). [See- Appendix B], This implies that a general

state V localiged at X » 0 can be expressed asCOy

'Oov {—i I *

-29-

Then Eq, (3.1) (where we know that we must put b = b = 0) combined

with ft-i, (fc.J'i) , implies that the more general solution of Bq. (3.1)

where the coefficients c^ are arbitrary up to the restrictions

inmosed by the normalizability requirements.C a M C

Replacing ^ • from Eqs. (6.33b) and (6.32a) the general3

state localized at X » 0 cp.n be expressed as

o«v E * "

-'/»•/.. .1,-1,

(6.38a)

where again f. (?„, ffl ) is arbitrary up to the normal inability

requirecants. (As in the spin zero cage, the function f. remains

indeterminate bec'ause X alone is not a complete set) . In Eqs. (6.38)

and (6.3$), v stands for the set of quantum numbers that specify

the function.!])

Once • Took' iB kno-wn, the other localized states mentioned in

Sec. I l l are also known, i . e . , those transformed through a Lorentz

"aooeleration" along the 3-axie,

transformed with a spatial translation alonp the 3-axis/ (a .spatial

translation along the other axis can be absorbed by a change of the

function f ), or by a product of both operations. In Eq. (6,38b) the \

action of the operator M , = - M over %t can be easily

computed taking into aooount Eqs. (6.37), (6.29) and (6.36).

As in the spin-aero case the states (6.38) are not normalizable '

to unity but a r e normalizable by eigendifferentials, as they

should be .

-30-

Two ei.gen'f'unotions of X corresponding to two different eigenvaluesa and a are not, orthogonal "because if they were,

then ^ Vf J Vp (f) should loe equal to (p r f8ee Eos. (6.38c11 and

( 1 . 0 : of. vieV. l ] , which certainly does not happen. (A similar

discurcfion can "be s:iven fo+ the non*-orthogona.lity of the ei^endif f e r ^ t i al cO

The non-orthogonality implies that the operator X , which i s such that

is not spin-l/2 hermittan,

and even niore, non-normal, i . e .

• [x 3 , ( -x 5 ) + ] = 0 , (6.41)

and t h i s happens for a l l values of the parameter G of Eq, ( 6 .28 ) .22 ' —**

However i t s eigenvalues are r e a l (as for spin ze ro) , so tha t X can a. f

he in te rpre ted as the v>osition operator of a l i m i t i n g case of the

extended-type pos i t ion .

A? X i s a 5-veetor i t fellows from Eq. ( 3 . 1 a ) ' t h a t i f ty i s an

1-local ized s t a t e , then F TMV, (where P = F 0 i s the pa r i t y operator •

i s a lso a l» loca l i zed s t a t e , so tha t s t a t e s can "be constructed such

tha t they are sinrultaneous e igens ta t e s of X and p a r i t y . To obtain

them, we observe from Eq. (6.32b) tjsatP vA* j t (7 ) = t w **" * ( f ) . (6.42)

Then observing Eq. (6.38) we realize that

P t (r 1 = i 1 (P ) ' (6./!3a)i f an^ on! v i f

f (rH, ^ + n = t j f _ t ( p H ^ ) . (6.43b)

To compute the velocity we notice that for every A c - -4 ~>l A. £cf.S^ H.?A)J

because H V = p _/\. .Then, taking into aocount Eq. (6.?B) i t r e su l t s ,

as in the spin zero case, that the velooity i s

(U flrtitr

-31-

As the value of the parameter G remains undetermined it seems

that our set of postulates (Seo.II) should be supplemented by an

additional requirement, "but this will not "be done in this paper.

However, we notice from Eq.(6«28) that the particular caaea

0 m - l/2 and G • 0 have interesting properties:

If G - - 1/2 , X*'« (PQ)"1 M ^ A (6.46)

which has the same form as the unique position operator for spin

zero jcf, Eq.(5«2O)J , and of course it is again one of the possible

quantum versions of Pryce's definition (o) of classical centre of

mass .k 12

On the other hand, i f X i s analysed as a binary va r i ab l e ihrc

f i r s t henni t ian component A _ i s such tha t for a l l allowed s t a t e s f1

and <p ,i f G - 0, ( P , &T¥>) - ( ^ , ^ B i P ) (6.47a)

where

^ B " idK " i p 0 ~ 2 p k + * i m " 1 Y* (6.47b)

i s ju s t Bunge's pos i t ion operator which has the unusual proper ty (for

spin one-half) of having qui te a olose o l a s s i o a l analogue (even in the

i n t e r a c t i n g case) without los ing formal covariance ,

"B, THERE ARE NO 3-LOCALIZED STATES

Let us assume there are 3- local iaed s t a t e s V-* (which a s p a t i a l

t r a n s l a t i o n the or ig in a =• O of coordinates can be thought as the point

of l o c a l i z a t i o n ) . Let us c a l l Y the spaoe of a l l s t a t e s obtained

from 5^Oy with a homogeneous oontinuous Lorenta t ransformation!

- (6.48)

An arbitrary Lorentz transformation leaves invariant the point of

localization, so that postulates 2 and 4 imply

^ V - 0 , (k « 1,2,3) (6.49)

for all iftT. From Eqs. (6.28) and (6.13c) we deduce

**? y - (t + oZ° t / 3

(6.50)

for all ty6 J .

•f is the representation spaoe of a unitary representation of

the homogeneous Lorentz group whose hermitian generators are the M ' .

From Eq,(6.5O) we see that the operators ( + G) V"1 are also

-32-

'" th is unitary representation. ^orV.inp; wit): thin

2" If) of the generators, we^fir p. In this rorrn

vF. (.:h+ i*" a rorresentation that i s f in i te ["remember the rel'i + i ens of

\ e t.v-e (i>.l^"b)l but that i s s t i l l unitary. However, the only f in i te

i •; m«'T:vi or>nl unMary re presentation..of the homogeneous rorsnt?. ^'oup

?; the t r iv i a l one. in which a l l generators are equal to :?ero, FO th'>

r-n'- into account (6.49"b^ w© deduce

t ;;'?•. in impor^flible because M has half-i.nteppral eigenvalues: We

.vert <xt n ccrtradict ion, so there are no J-locnliaerl s t a tes .

- 3 3 -

VII . UT3CP35I0N

Ovv specific results have been stated in Sections III (general

conations), V (spin zero) and VI (spin one-half) and we do not return

to them. We shall only discuss the following general question:

In Sec, T.A we remarked that if the observations of a localised

•3 +ate from different inertial frames of reference should agree, a.nd if a

position operator exists (as i t happens with other quantum mechanical

a) , then i t follows from the present paper that X is not hertrri 1 i an.

feeling is that at least until a better understanding of the so-

or],led "elementary particles" in terms of more fundamental one?, is

obtained, the non-hermitioity of „ the position operator is unavoidable.

Foreowr, i t is possible that X should be of the extended-type TpossiM-

li ty (li)> See, I.A], and not only a "limiting case" [possibility ( i i i ) ] ",

But these ore on],y conjectures,which, in spite of being supported by12 ?0 21

several facts ' ' s t i l l have no sure proof.

On th<=> other hand, what seems an almost ce-rfajn consequence of the present

paper is that . position has no sense , or i t has not a relativistioal ly

invariant meaning, or i t is the only physical variable that enrnot be

Z2~Tfi'^£-nt'e^ hy a-*i operator ,OT the position operator is not hermitAan.

So we can conclude that as regards the localization problem i t seems

alrost, unavoidable to accept at least one strong departure from usual ideas,

Whydid l© write "almost" in the last two previous propositions? Because perhaps

there is s t i l l a narrow 'loophole. T6 get our results i t was essential

to construct with the ingredients of the theory the more general

3-translation invariant 3-vector R of EG. (3.6) (see Sees, V and VI), We

worked in the free-field case, where by analogy with what happens with

other physical variables (momentum, angular momentum, parity, e t c ) i t

cruld be expected that X can be defined even in the absenoe of interactions.

JTowever, interactions are not switched off in the real world. If there ar»

Interactions, additional 3-vectors invariant under 3-translations

and Whidi do not go to zero in the aero interaction limit, can be constructed.

For example, if 5 ,/ i s the Fourier transform of the electromagnetic field

tensor, F (P F T) ^ is an example of such a 3-vector. Perhaps R (and

then X) should be constructed with such vectors. This is the loophole

thv1- perhaps we s t i l l have. But of course, if this bar-pens, position would

again be an exceptional member of the set of physical vn.riab]es.

'"Take into account the discussion of Sec, IA.

-34-

ACKNOWLEDGMENTS

The author is frateful to Professors Abdue Salam and P, Budini for

v.ocH. +,"1 i tv at the International Centre for Theoretical Physics, Trieste.

f-vher? nest of the present paper was w.Tifren), and to the IAEA for -in

A^RO?T nteshi n that made his stay at#,the Centre possible.

I>oc!- oP So. (3., .6).

A.a y} = Ay?Mc(' ®l> (5.5&JL we c&n define

./\H5A = x5 - A i^ 3 A .By r e p e n t e d d i f f e r e n t i a t i o n of Eq, ( 3 , ? ) w i t h r e s p e c t t o t h e p a r a m e t e r s

a , a and a'' anr? by oomtoining; t h e r e s u l t w i t h Eq, (Al) we can o b t a i n

Then, i1' fCp* ) s an ordinary (i, e. without K'-matrices) arbitrary funo+i on

ITof each fiTed stin c opponent of jji , the set of the functions f (p") ,

span a space oontaining C (.both spaces are not equal if we do not

restrict f(p )to ; i.\a.rantee the desired normalization) and for fixed p and

b. postulate 6 (Sec. II) insures that the set of functions U' span £

xe^hen the s e t of the f u n c t i o n s l C f ) T I f ) wi th a r b i t r a r y f and fi

pi ?'i;-'Ce that, c o n t a i n s C ® C so t h a t T3n , (A3) i m p l i e s

[At , r*-r\A-o * ()? 5 n d , a ? / V n o e s m o t c o n t a i n " ^ / " S p [ s e e e . f t . E q . ( 6 . 2 ) f o r s p i n l / 2 l ,

1'rom thin e":jat,i'n it can, he proved that E cannot contain 3/"5 p (eroe^t

in terms that vanish when^, left and right are multiplied by A j so that they

are irrelevant and can "be omittedi

R3 = R3(p , ..., P , . . . ) . (A6)k

5 imji l ies t h a t t h e same happens w i th R , k = 1 , 2 , 3 .

"1 ?

Let nR call (for fixed p and p )

*, = arc sinh '^{v-^ + m2)"1'2] (Bid)

- 35 -

3 -t is real fo^ all real p and such, that

p3 = ( p 2 + m 2 ) 1 / 2

S i n h , . (Bib)"7

Given an a r b i t r a r y square i n t e g r a b l e funct ion f,, \ ( p ) we as soc ia t e with i t

a new function ¥tt (a) defined by

?b*nf alsOj P 1 . i s square i n t e g r a t e in the v a r i a b l e ";, PO tha t

i1, t exists suoh that

which in turn implies

(B4)

T f»t. us new have an arbitrary function f/?\(p) expressed in the

cylindrical coordinates p_, (p and p . From Eq, (B4) we see that the

av Yj 1 i ary fun o t i on

cj:n be expressed as a linear combination of the funotions

(B6)

T>:is j.TT>plies t h a t the given f / p \ ( p ) can be eypre^sed as P "linear

c r""h-i n :-i t \ or of the fun ct i en s

.:'."• thr.-t these - a c t i o n s ?,r.an i_, .

'"n the other h^nd,,'tha aat of Bpiaora w^ defined in Eq. (6.32"b)

i := B. ha^is in the spindr spaoe C [remember tha t the °ner ;;y i s posi t ive

<lei'i"i+e, cf. Sec. IV,

in "iq. (6 .3?b) 3-an the Roe.ce <- ® C . O n the other hand, the functions

^ onn be p r o v e d l i n e a r l y i n d e p e n d e n t . . T ' . ^ i r -,st, i s n b^Ri s

-36-

. 1 —