/JA, /^y/-

PROJECTIVE THEORY OF MESON FIELDS AND ELECTROMAGNETICnbsp;PROPERTIES OF ATOMIC NUCLEI

BY

A. PAIS

iht

PROJECTIVE THEORY OF MESON FIELDS AND ELECTROMAGNETIC PROPERTIES OF ATOMIC

NUCLEI

UNIVERSITEITSBIBLIOTHEEK UTRECHT

3598 0097

c'/y

PROJECTIVE THEORY OF MESON FIELDS AND ELECTROMAGNETICnbsp;PROPERTIES OF ATOMIC NUCLEI

PROEFSCHRIFT

TER VERKRIJGING VAN DEN GRAAD VAN DOCTOR IN DE WIS- EN NATUURKUNDEnbsp;AAN DE RIJKSUNIVERSITEIT TE UTRECHT,nbsp;OP GEZAG VAN DEN RECTOR MAGNIFICUSnbsp;DR. H. R. KRUYT, HOOGLEERAAR IN DEnbsp;FACULTEIT DER WIS- EN NATUURKUNDE,nbsp;VOLGENS BESLUIT VAN DEN SENAAT DERnbsp;UNIVERSITEIT TEGEN DE BEDENKINGENnbsp;VAN DE FACULTEIT DER WIS- EN NATUURKUNDE TE VERDEDIGEN OP WOENSDAGnbsp;9 JULI 1941, DES NAMIDDAGS TE 3 UUR

DOOR

ABRAHAM PAIS

GEBOREN TE AMSTERDAM

AMSTERDAM � 1941

N.V. NOORD-HOLLANDSCHE UITGEVERS MAATSCHAPPIJ

AAN VADER EN MOEDER. AAN TINEKE.

(Op verzoek van den promotor, prof. dr. L. ROSENFELD volgt hier, in plaats van het gebruikelijke voorwoord, een korte levensschets.)

19 Mei 1918 w�erd ik te Amsterdam geboren. Daar bezocht ik de derde 5-jarige H.B.S. aan de Mauritskade, (directeur dr. GeRRITS), waar ik veel heb opgestoken.nbsp;In 1935 legde ik het eindexamen af, en liet me in hetzelfde jaar als student aannbsp;de Amsterdamse Gemeente Universiteit inschrijven. Oorspronkelijk koos ik denbsp;fysisch-chemische richting (e), maar veranderde al spoedig van koers, vooralnbsp;onder de invloed van de colleges van prof. MANNOURY. Tenslotte besloot ik, nanbsp;vele gesprekken met enige oudere jaars, de theoretisch-fysische kant op te gaan.

Voor het candidaatsexamen volgde ik de colleges van prof. ClAY, prof. MANNOURY, (zoals gezegd), prof. MlCHELS, prof. PANNEKOEK, prof. HK. DEnbsp;Vries, (ik denk nog steeds met plezier aan het caput over meetkunde van hetnbsp;aantal), prof. WlBAUT en dr. B�CHNER. Op 16 Februari 1938 legde ik hetnbsp;candidaatsexamen a en d af. In dat jaar volgde ik nog enige wiskunde-collegesnbsp;van prof. BROUWER, prof. WeITZENB�CK en dr. FREUDENTHAL.

Het was in deze tijd, dat ik in aanraking gekomen ben met en opgenomen in een kringetje van mensen, die ik het beste zou kunnen karakteriseren door onzenbsp;eigenschap om alles te kunnen lachen, vooral om onszelf. In twe�erlei opzicht isnbsp;dit contact beslissend voor me geweest, (vooral in het tweede).

In het voorjaar van 1938 ging ik naar Utrecht, om bij prof. Uhlenbeck theoretische natuurkunde te studeren. Veel heb ik geleerd van zijn glashelderenbsp;colleges en de colloquia in kleine kring op �220quot;; maar vooral de tijd gedurendenbsp;welke ik met hem aan enige theoretische problemen heb mogen werken, is eennbsp;mooie tijd voor me geweest.

Met grote eerbied herdenk ik hier prof. Ornstein. In de tijd dat ik experimenteel werkte op het Utrechts laboratorium heb ik hem kunnen waarnemen in die hoedanigheid, waarin hij zo groot was; als organisator van een brok fysischnbsp;leven. De bijna dagelijkse gesprekken die ik met hem had in het jaar na hetnbsp;vertrek van prof. UhlENBEQK zijn een kostbare herinnering voor me.

En nog iemand wil ik hier gedenken: Kees VAN LiER.

Voor mijn doctoraal examen volgde ik nog de colleges van prof. BARRAU en prof. WOLFF; dit examen legde ik op 22 April 1940 af. In die cursus heb iknbsp;met dankbaarheid gebruik gemaakt van de gastvrijheid, mij door prof. KRAMERSnbsp;op het theoretisch seminarium te Leiden geboden.

In September 1940 kwam prof. ROSENFELD naar Utrecht en onder zijn leiding heb ik, oorspronkelijk als zijn assistent, mijn studie voortgezet. Ik beschouw hetnbsp;als een voorrecht, dat ik onder zijn toezicht dit proefschrift heb kunnen bewerken.nbsp;Zijn steun, zijn aansporingen en vooral de belangstelling die hij steeds in mijn

persoon stelde hebben me door menig moeilijk moment heengeholpen. Het is vooral van zo grote waarde voor me dat ik in hem iemand gevonden heb, dienbsp;me niet alleen veel op het gebied van de fysica geleerd heeft, (en nog veel zalnbsp;leren, naar ik hoop), maar met wien ik tegelijkertijd ook zoveel contact hebnbsp;kunnen vinden op andere gebieden. De hartelijkheid, die hij en zijn vrouw mijnbsp;steeds betoond hebben, is een grote steun voor me geweest in een moeilijke tijd.

Ik betreur het, dat door de omstandigheden de in deze dissertatie behandelde problemen op sommige punten niet zo behandeld zijn, als dat in mijn voornemennbsp;lag. Ik hoop evenwel later op deze kwesties terug te komen.

Aan hen, die mij het dierbaarst zijn, draag ik dit proefschrift op.

CHAPTER 1.

THE ENERGY MOMENTUM TENSOR IN PROJECTIVE RELATIVITY THEORY.

Summary.

After a survey of the formalism of projective relativity theory (�2) an expression is derived for the S-dimensional energy momentum tensor of annbsp;arbitrary field (� 3). It is proved that, in virtue of the equations which hold fornbsp;the variables describing the field, this tensor is symmetrical and its divergencenbsp;vanishes. This last property expresses the conservation of energy, momentumnbsp;and charge of the system. As an application of the formalism the energynbsp;momentum tensor for the Dirac field is computed in � 4.

� 1. Introduction. Since KaluzaI) in 1921 pointed out that the unification of the gravitational and electromagnetic field mightnbsp;be achieved by introducing a fifth dimension besides space-time ofnbsp;general relativity, several attempts have been made, starting fromnbsp;this idea, to obtain a formalism which satisfies the followingnbsp;requirements:

a. nbsp;nbsp;nbsp;General covariance, (covariant formulation of the �cylinder-condition�).

b. nbsp;nbsp;nbsp;The first set of Maxwell equations follows from thenbsp;postulated structure of 5-dimensional space.

c. nbsp;nbsp;nbsp;The field equations are derivable from a variational principle.

d. nbsp;nbsp;nbsp;The �geodesic� equations of 5-dimensional space representnbsp;the equations of motion of a charged mass point in thenbsp;gravitational and electromagnetic field.

The method of Kaluza which was extended and improved by O. Klein 2) does not fulfill the first condition, as it starts from anbsp;5-dimensianal metrical tensor, the components of which do notnbsp;depend on the fifth coordinate, and which is in fact nothing butnbsp;the ordinary metrical tensor bordered by the electromagnetic vectornbsp;potential, while ^55 is put equal to 1. Einstein and Mayer 3) havenbsp;proposed another method, viz. of adjoining a linear 5-dimensionalnbsp;vector space to every point of the 4-dimensional space-time conti-

nuum. In this case, however, the conditions b and c are not fulfilled.

The projective interpretation which was first introduced by Veblen and Hoffmann 4) considers the 5-dimensional space as anbsp;4-dimensional projective one. The treatment of these authorsnbsp;exhibits the same defect as that of the Kaluza-Klein theory¹),nbsp;but in the projective formalismultimately developed by VAN Dantzig,nbsp;Schouten s) and Pauli 6) the covariance requirement is indeednbsp;satisfied.

In the theory of ScHOUTEN and VAN Dantzig the three-index symbols of projective space are not symmetrical with respect tonbsp;jj. and V. Pauli has emphasized however, that there is no physicalnbsp;argument for not keeping to symmetrical T^,,. Starting from thisnbsp;assumption one then gets a uniquely determined formalism. Thenbsp;metrical tensor g,j,y is assumed to fulfill the (covariant) condition

The last mentioned authors have also discussed the DiRAC theory of the electron in the frame of this formalism and have derived annbsp;expression for the energy momentum tensor of the DiRAC field. Thenbsp;aim of the present paper is to do this for an arbitrary field of whichnbsp;the Lagrange function is given in terms of the field variables.

The problem of the derivation of the energy momentum tensor in general relativity theory has recently been discussed by Belin-FANTE'i') and by ROSENFELD^). These treatments differ methodically and the first mentioned has for our purposes the drawbacknbsp;that the differential conservation laws are only derived in thenbsp;approximation of special relativity. As it seems that no simplenbsp;physical meaning can be attached to a �special projective relativitynbsp;theory� it will be clear that it is more convenient for us to proceed onnbsp;the lines of Rosenfeld�s method which follows closely the ideasnbsp;outlined especially by HlLBERT^).

It should be noted that the situation here differs in two aspects from that in general relativity: first, it is here no more allowed tonbsp;assume that the Lagrangian does not depend explicitly on the coordinates and secondly, the transformation group of projective rela-

1

The projective tensor introduced in equ. (3. 1) loc. cit.¹) has essentially the same properties as the metrical tensor of the KaluZA-KleiN theory.

tivity (hs), in contrast to the group of general relativity (g^), admits only such transformations �¹� X'gt;^ for which X'fquot; is anbsp;homogeneous [unction o[ the first degree of the X^.

� 2. Survey of the formalism of projective relativity theory. We shall first summarize the main features of projective relativitynbsp;theory according to Pauli�s formalism. For the omitted proofs andnbsp;a more detailed treatment of the subject, the reader is referred tonbsp;Pauli�s paper.

a) 5~tensors and 5~tensor densities. The space-time continuum is alternatively described by the inhomogeneous coordinates x� (tonbsp;this description we will refer as �4-space�) or by the homogeneous

coordinates ¹) Xgt;^ (5-space), such that x'^,____x'^ are homogeneous

functions of zeroth degree in X^-.

x�=x�{X^.....X^) = x�(qX^.....(1)

We now define the 5-tensor nbsp;nbsp;nbsp;as a set of quantities which

obey the following transformation law for the group

dX'ih

further, each tensor component shall satisfy the invariant condition that it be a homogeneous function of degree p of X'^, where p isnbsp;given by:

p = n � k.

Therefore, using Euler�s theorem on homogeneous functions and denoting bynbsp;nbsp;nbsp;nbsp;the partial derivative of A\\ \ with respect to X^-.

(3)

Thus the Xgt;^ are components of a contravariant 5-vector field, while the differentials dX^ do not have this property as the secondnbsp;condition is not fulfilled for them. Still one can define

dr = dXK.. dX^

1

We will adhere to the convention that 4-space quantities are marked by Latin and 5-space quantities by Greek indices, so i=1...4, and jU=1...5.nbsp;Summation signs are suppressed according to the usual rule.

as volume-element in 5-space: if we-perform the transformation

X'gt;^ = qX^ nbsp;nbsp;nbsp;(4)

(q any homogeneous function of the X'^ of degree zero), dr transforms according to *)

dr' = dr. nbsp;nbsp;nbsp;(5)

As a consequence of (4) we also have

= , (6)

This enables us to find the degree of 5-tensor density components. For, if t|j;;;j3^is a 5-tensor density and T^l[[['^quot;the correspondingnbsp;5-tensor integral:

=j dr,

we see that the degree of nbsp;nbsp;nbsp;is p � 5, because of (5) and (6)**).

Noting that nbsp;nbsp;nbsp;5 we therefore have:

b) Metric; covariant differentiation. The metric is described by the symmetrical tensor :

Qiiv � gvn- nbsp;nbsp;nbsp;(8)

Further we postulate the following important relation, normalizing the metric in 5-space

g,,rXgt;^ X^=\-, nbsp;nbsp;nbsp;(9)

this relation is invariant for the group hz- The raising and lowering of indices can be performed by introducing the tensor p'��� which isnbsp;connected in the usual way with

Next the three index symbols nbsp;nbsp;nbsp;and

which we assume to be symmetrical in p and v.

rgt;.^ nbsp;nbsp;nbsp;I V quot;1� ggt;-r]fi gfty\ /)

- g'quot;quot;

It is well known that, while T^�;;;^^|p,the derivative of a mixed tensor with respect to A''¹ is not a mixed tensor of rank p�1,nbsp;(except when T is a scalar), om the other hand the covariantnbsp;derivative of

yj 7-�^

does possess this property. Also we remind that for covariant differentiation the product rule holds:

(A:;; 5:;:)

and that the operations of covariant differentiation and raising or lowering of indices are commutable on ac�ount of

Q)-fi\\r - 0, nbsp;nbsp;nbsp;-0'

Finally we introduce the 5-tensor Xfiv-

X,

which satisfies

^^fiv I Q �k \ r �{� Xyf) I ^ - 0.

From (3) and (10a) we infer that

' nbsp;nbsp;nbsp;�f� 2 X/ir ,

so

= yX�'gt; X||^ = iXgt;�-

With the help of (9) one verifies that

X.x = o.

c) Connection between 5~tensors and 4~tensors¹) relations are established by the 40 quantities y^and 7�'^:

7;'^ = x¹iv

1

A 5-scalar is at the same time a 4-scaIar, therefore it is not necessary to add a prefix to �scalar�.

whence, by (1) is defined by

(17)

X = 0. nbsp;nbsp;nbsp;(18)

yj^ transforms like a covariant 5-vector for the group h^, and like a contravariant 4-vector for g^; 7��^, behaves similarly.

With the help of (y��^) we can uniquely connect a contra-variant (covariant) 4-vector with a contravariant (covariant) 5-vector by

(19a)

From (19a) it follows that the metrical 4-tensor, which connects a I and a¹, is related to by

9ik = y^jf�gi.v. nbsp;nbsp;nbsp;(21a)

Furthermore, (17), (18) and (9) give

Y'^J Y'J = df�^�Xr X,

and this relation enables us to connect (starting from (19a)) a contravariant (covariant) 5-vector with a contravariant (covariant)nbsp;4-vector. For, contracting ¹) the first equation (19a) with andnbsp;the second with y'^^ we get

(196)

There is a special type of 5-vectors, namely those for which the corresponding scalar vanishes:nbsp;nbsp;nbsp;nbsp;X^ = 0. Such vectors we denote

by Sft, (or a^¹).

In the same way connections can be established between 5-tensors and 4-tensors of higher rank. For instance

T/fc=yry;'^X, nbsp;nbsp;nbsp;(20a)

� yJ yJ^ 7^' Xv Tap) X y'v ^(o): to.

1

By contraction we understand multiplication followed by summation over the new dummy index (indices).

with

Ti (0) = 7^J X'� Tf,r, r,0) i = Xi^ 7'� . and 7(0) (O) = X^ T,,..

So we see that

if nbsp;nbsp;nbsp;then Tik � Tki,nbsp;nbsp;nbsp;nbsp;Tno)=Tio)i;

if Tf^y = � Tvfi, then Tik � � Tki, Tuo) = � T{o) i, T\o) (o) = 0.

From {20b) and (18) it is seen that we especially have for the metrical 5-tensor:

9i-y = y;J r;*quot; guc Xf^ x. nbsp;nbsp;nbsp;(21 b)

Now we still have to find the relations between the covariant derivative of a 5-tensor in 5-space with the analogous quantity innbsp;4-space. For this purpose we postulate, for any vector a*' (i.e. anbsp;vector for which a'' � 7''i^a^ holds, see above):

= nbsp;nbsp;nbsp;a�! Is.

where a *||; denotes the ordinary covariant derivative in 4-space. Next we define andnbsp;nbsp;nbsp;nbsp;as follows

(22a)

where F is the three-index symbol in 4-space. On account of these definitions the following product-rules hold

(r;* a*)|js � 7;* 7-^� ak\\i 7;*|js at,

(/!* a'�)iis=7\ yj ^'�l\� y'�.ki\s a*-

Furthermore, it can be seen that the postulate then becomes

yf y% y:k\\s �

We now proceed to derive explicit expressions for the covariant derivative of 7^*^ and 7�'^.

We have, by (19b) and (13a)

(y'^k a*)||s =a�'||o X' a\\g ^ ^Xj .

Therefore, using (3), (13), (H) and (16):

Also we have

^quot;.kW s~ y^.k � i -^ey y'':k �

Using a relation similar to (206) which holds for a mixed 5-tcnsor of the second rank, we thus get

y:,ll^ = i {X, JSf- - jsr- X,,.) y\ . nbsp;nbsp;nbsp;(23a)

Likewise we find that (236)

d) The postulate dealing with the electromagnetic field tensor. Introducing

Xik = rj r\ X^y. nbsp;nbsp;nbsp;(24)

we have (comp. (206) and (14)):

X,r = 7-;yJ^Xik. nbsp;nbsp;nbsp;(25)

With the help of (12), (33) and (34) one can prove that¹)

Xik 11 Xii I k Xki I (� = 0. nbsp;nbsp;nbsp;(26)

So if we put Xik proportional to the electromagnetic field Fik, we see that the first set of Maxwell equations is a consequence ofnbsp;the assumed structure of 5-space. The proportionality-factor can benbsp;determined by comparing the field equations of gravitation theorynbsp;(in the absence of matter) combined with those of electrodynamicsnbsp;on the one hand with the corresponding equations that follow fromnbsp;the projective formalism on the other; it turns out to be {2x)'i� cquot;'nbsp;where x is the gravitational constant, thus

_\X2y. �

A-ik �-tik.

From (26) it follows that there exists a 4-vector f,-, such that:

Xik = fk\i � fi I ¹. nbsp;nbsp;nbsp;(26a)

1

Comp, Pauli�), p. 322.

By (27) f i is connected with the electromagnetic vector potential (Pi by

X/ln

tk =-(Pk-

c

We may connect ff^ with a 5-vector �^:

ff. � yiJ'fic; (ff,Xgt;^ = 0),

it can be proved ¹) that

Jfi ~ Xfi ^

where F is an undetermined homogeneous function of the [irst degree of

e) Connection between and the DiRAC matrices. The linearization of the GoRDON-Schr�DINGER equation of the electron rests essentially upon the existence of 4 square matrices y,- whichnbsp;can have no less than 4 rows and columns and which satisfy

7i7k 7k7i � 2dih.

The product of those matrices: 75 = yi 73 7i Is also anticom-mutative with each 7,-, while 75 = 1. These matrix relations can all be comprised in

7y -F Vv7^^ = 2 dfty, (ju,v = 1.....5)

where is the unit matrix. Now just as Tetrode 10) postulated that the matrices 7,- should be generalized to matrix fields describingnbsp;gravitation in 4-space, we can put here for a given metrical fieldnbsp;gf,y, (writing instead of y,^):

o-y �j� ofv 0^ � 2 gpiv.

For given it is in principle possible to find a solution for (comp. SCHR�DINQER 11), which is unique apart from a transformation with any non-singular matrix S:

d/^ = S ^ a^S, (or a'/^ � � 5 ' nbsp;nbsp;nbsp;S).

1

Cf. PAULI, p. 322�323.

10

For reasons that will soon be clear we consider only S-trans-formations of the first type.

Further there exists a matrix A, the �hermitizing matrix� such that

Aa^=:{Aa^) . nbsp;nbsp;nbsp;(33)

As a consequence of (32) A is assumed to transform as follows:

A' = 5 A5. nbsp;nbsp;nbsp;(34)

f) 5~undors. We call

Hi

a 5-undor if, performing (32), it transforms according to

W' = 5-gt; W. nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;(35)

From (34) and (35) we obtain

(1F A)' = ('F A)5.

The quantities AW, W'^Aa^W, W^Aa^ayW etc, are therefore invariant for 5-transformations.

Next we consider a rotation of the Xgt;*\

(36)

is homogeneous of degree zero in Xgt;^), and we will confine ourselves to rotations with Detnbsp;nbsp;nbsp;nbsp; 1. The consideration of

this group, of which the full group of LoRENTZ transformations (including spatial reflections) is an undergroup, is sufficient for allnbsp;physical purposes.

Now it is always possible to find a matrix 2, such that if (with

� glJ.V

o', nbsp;nbsp;nbsp;(37)

then

a'/* = 2�-� a� 2. nbsp;nbsp;nbsp;(38)

11

Such a �^-transformation� only affects the o'*, the quantities and A are scalars with respect to these. Therefore AWnbsp;etc. mentioned above are real scalars, 5-vectors, etc. with respectnbsp;to the transformations (37).

It is possible to connect uniquely a 5-transformation with a ^-transformation by stating that a 5-transformation and the �adjoined� .^-transformation, which shall be performed subsequently, shallnbsp;leave the unchanged. We then have in fact 5 =nbsp;nbsp;nbsp;nbsp;(with

respect to the choice we have made as regards (32)).

� 3. The energy momentum tensor. We now proceed to derive an expression for the energy momentum tensor of an arbitrarynbsp;field. We generally denote the variables describing the field bynbsp;Q(,�). This symbol thus comprises the gravitational field variablesnbsp;(to which we will often refer as Q().)) and the others (Q(a)) whichnbsp;have either tensor or undor character, (we denote them with Q(t)nbsp;and Q(c) respectively). We will, only to fix our thoughts, considernbsp;the Q(a)�s to be covariant tensor components. If in a term the �index�nbsp;(m), (a), (t) or (a) occurs twice, �summation� over all variablesnbsp;Q(c,) or Q{a), etc. is implied.

We will � in contrast to Belinfante 7) and Rosenfeld 8) � not establish the connection with undors by the explicit introductionnbsp;of �5-beinchen� but by the direct consideration of a set of matricesnbsp;varying from point to point and satisfying (31).

In case we only have to do with variables of the type Q(t) (the �tensor case�) we adhere to the customary choice of the Q.{y),nbsp;namely the components of the metrical tensor If there are alsonbsp;undor variables present, (the �general case�), this choice cannot benbsp;maintained; we now take the instead, connected with thenbsp;by (31).

We denote by K the Lagrangian density of the gravitational 5-field in the absence of matter, {K is assumed to depend on Q(y) and their first derivatives only), and by L the Lagrangian density ofnbsp;the arbitrary material field, containing the interactioin of that fieldnbsp;with the g^^-field, (this is the interaction with gravitation and thenbsp;electromagnetic field). L depends on 0(0,), their first derivatives, andnbsp;on Xi^-.

L==L(Qh, nbsp;nbsp;nbsp;(39)

Here is the energy momentum 5-tensor.

In the general case, (42a) is extended, making use of well known properties of variational derivatives and of (31), to

2 \ uCl-i! uO-v

13

In the following we will for this and analogous formulae more simply write

da, �

the symmetrization being understood; we may of course assume the as to be hermitized.

The invariance properties for the group hs of the quantities hitherto introduced enable us to find an explicite expression for Tfivnbsp;in terms of the field-variables. Consider an infinitesimal transformation

X'!^ = Xgt;^-\r nbsp;nbsp;nbsp;(43)

an infinitesimal contravariant 5-vector); we then have by (3):

= nbsp;nbsp;nbsp;(44)

As a consequence of (43), the field variables transform as follows �Qm = (X'i - Q,.) {X^) = c�,, r,

where

r - ^ nbsp;nbsp;nbsp;Qn �1...r

Vi,...,Vp are the tensor indices of Qi'quot;�¹.

Comparing this with (42a) and (42h) one infers that

lt;5Qw

It will appear that it is convenient to introduce the �local variation� (5¹Q(,�) = Q'(,�) (A?) � Q(o,) (A'f�). Thus;

'5Q(-,,) = �¹ Q(t,,) nbsp;nbsp;nbsp;Q(,,,) I fi.

The second part of the right member is the result of the concomitant displacement Q(,�) (A?') �^ Q(.�)

Now j Ldt should be invariant for the group h^, i.e. we have ¹):

^ J'Ldr=J'[d¹ L(Lif�)\p] dr = 0,

1

Cf. H. Weyl, Raum, Zeit, Materie, ed., p. 233; it should be noted that in computing S¹L we need not vary Xf� as 8¹L is the local variation of L.

where and its derivatives must satisfy (44) and all relations that can be obtained from it by differentiation.

On account of (39) the variational derivative of L with respect to Q(c.,) is

�L _ nbsp;nbsp;nbsp;_ f dL A

�Q(amp;j) \�Q(amp;j) I ift

so (47) becomes

If we first consider an infinitesimal linear transformation of X'^, the second derivatives of 1� do no more occur in (49). We thennbsp;obtain from (49) using (44):

(Ar -|- filr [ e) nbsp;nbsp;nbsp;B� ,v Rv^A - 0.

For a general infinitesimal transformation the condition

RrfVi^==o

remains, where the nbsp;nbsp;nbsp;are only restricted ¹) by the derivative of

(44), viz.

= nbsp;nbsp;nbsp;(54)

1

Higher derivatives of (44) need not be taken into account, (although the derivative of (54) still containsnbsp;nbsp;nbsp;nbsp;bon hy counting the number of equations

with which such a tensor relation is aequivalent on the one hand and the number of quantitiesnbsp;nbsp;nbsp;nbsp;f�'|;.|^|g etc., that are involved in such a set of equations on

the other, it is easily seen that the higher derivatives of (43) do not impose any further restriction on and their first and second derivatives.

whence

The wholly undetermined quantity which, on account of (54), has to be introduced into (55), expresses an ambiguity of thenbsp;Lagrange function which is typical for the projective formalism; innbsp;fact we may always replace Q(,�) by a constant times ^^Q(w)|;. (onnbsp;account of (3) if Q(�) is a tensor; for a 5'-undor we have e.g.

is the constant mentioned above; of course one should not perform the summation over (m) in this expression).

Noting that, naturally, ^ cannot be written as the product of a 5-tensor��C(c,),,� with we consequently may, if we introduce thenbsp;convention not to allow products of the typz X^ Q(a)\k to occur in L,nbsp;put �t equal to zero:

^'�quot;'=0: nbsp;nbsp;nbsp;(556)

thus is antisymmetrical in X and /j, from which follows that R!v�\k\fc� 0, We therefore obtain from (53) by differentiation:

I ^ -f- B^_v \ti � 0.

As Av fifv I e 5-vector density we have by (7)

(57)

As only depends on the derivative of Ri.^, it follows that the occurrence of �l would not affect the energy momentum tensor, for

{QIX\, = 0,

on account of (7), being a tensor density.

Now we must bring (56) and (58) in their covariant form. This is easily brought about in the tensor case, for then we have, notingnbsp;that ^L/dQ(.�). C(l),v and RI'^ are tensor densities:

i{'A)

i\v 11 nbsp;nbsp;nbsp;- t\v 11 I\p ^ Xv *

Q(t) 11 ^ � Q(t) I f. c(z),e ,

rpfl

In the general case we may write instead of these last two identities

(60)

17

Q(a):v = 0, nbsp;nbsp;nbsp;(61)

Q(a): - Q(a) I M quot;fquot; ^(a) S

Thus Q(t);^ = Q(t)||^, hut the covariant derivative of a tensor undori3) is not coropletely given by an expression like (59a). Innbsp;this case we can achieve our purpose by using the invariance ofnbsp;� Left for a change in representation corresponding with a .Z-trans-formation in the sense as indicated above ¹). Such an (infinitesimal)nbsp;transformation is given by²):

As a consequence of (62) the variables Q(7) are also affected:

Q(=r) = cf(j);.r �-quot;. nbsp;nbsp;nbsp;(64)

It should be noted that �'A � 0, for nbsp;nbsp;nbsp;is hermitian because

Aa!^ is hermitian and the transformation coefficients s'¹� are real. We now assume that L depends in such a way on Qi,), Q(,)|�,, o'¹nbsp;and �''[e that the (vanishing) variation of /Ldr can be written asnbsp;follows

2

1

See Pauli, p. 350.

2

We have written b' to distinguish these variations from those that follow from (43).

18

where, as already stated, all quantities in the second member of (65) stand for their symmetrized expressions, i.e. one has actuallynbsp;to take half the sum of the quantity written down and its hermitiannbsp;conjugate. From (65) the following identities can be derived

�L.fc dL

a(a) Av ~r nbsp;nbsp;nbsp;Ci(^)Xp � U.

daf-ie nbsp;nbsp;nbsp;�Q^ie

Now with any parallel displacement in 5-space (from P to Q say) corresponds a change of representation, namely a^(P)�^a^(Q),nbsp;with a of the form (62), and it is this connection which causesnbsp;the difference between Q(u)||^ andnbsp;nbsp;nbsp;nbsp;From

(68)

we can readily deduce the quantities corresponding with For, as the a� are linear combinations of a set of constant matricesnbsp;a^, (see Pauli, p. 344), we have

o'*; s = A�?!�g Or, nbsp;nbsp;nbsp;(69)

where is a c-number. One can easily deduce from (68) that the �S�' are connected with by

^ nbsp;nbsp;nbsp;(70)

where

y'::,=-i(A'::,-A:':,). nbsp;nbsp;nbsp;(70a)

Consequently we get for the covariant derivative of and of

af�:

�''ll e = e cfwAv = 0.

(From (72) and (11) it follows that a^\\g -tracting (66) and (67) with we get

19

which is exactly what we need to obtain the covariant form of (60) and (61); for, inserting this we find

Qh�.- nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;(73)

Naturally we must require that in the right hand side of (73) the gravitational quantities do not occur explicitly and indeed this condition is fulfilled except for RY which, by its definition (52), contains dL/0Q(rtp.

In order to eliminate this term from we put

with � EE 0, while d^J, is given by (64). For 5(^) we have thus in the tensor case = 0, while in the general case, accordingnbsp;to (63a), we have for Q(y) = ag ,

�M = � o?(�) nbsp;nbsp;nbsp;�9ee' 9^'^�nbsp;nbsp;nbsp;nbsp;cf(�) %�gt;� �nbsp;nbsp;nbsp;nbsp;(75a)

We now introduce

D^�^'' = i(Rg'^g�^''-Rg^g^^),

and then can write, using the identity (67), and (52), (75) and (75a)

Thus

and we see that in this form Rt'^ does no more contain Q(y) explicitly.

The symmetry of 7�^�� and the conservation laws. Representing the energy momentum tensor in its form (42h) we have from (66):

20

If we form from the variables Q(a) a set of independent variables Q(j) ¹), the field equations are:

dL

consequently

�L

(we have written to distinguish this quantity from the corresponding tensor in (58)). As regards the symmetry of T'^y, on account of {42b), (78) and (75)

The right hand side vanishes however, if we make use of the field equations (79). Therefore T^�� is symmetrical in virtue of thesenbsp;equations. As from (79) it also follows that

n -n

the conservation laws hold under the same restriction:

(81)

Dividing (80) and (81) by \Xg,{g = \ Det ^^,.|). we obtain instead of tensor densities the corresponding tensors. Then (81) is, onnbsp;account of {20b) and (23), aequivalent with

Putting

(82a)

1

We assume to be a homogeneous and linear function of the Q(�) s.

21

where T,* is the energy momentum 4-tensor of the total system minus the tensor referring to the Maxwell field in the absence ofnbsp;matter: Ti^e)ikgt; atnd s* is the charge current density of the systemnbsp;we have, with the help of (27):

� F'* s/t = 0 ,

1^^ � 5*11 A = (1/ g s*)| k � 0.

g = I Det. gtk \.

So the 5 identities (81) are aequivalent with the conservation of momentum, energy and charge!

From (41fe) we now derive two four-dimensional relations by contracting withnbsp;nbsp;nbsp;nbsp;and yj respectively; they are

Ki,=-^Tik.

d-

F'!(o) = T\).

The left members have been computed by Pauli, who finds Kik = Rik -inbsp;nbsp;nbsp;nbsp;F 4 (Pi�Fik-i Oik Fmn Fnlt;

Rik is the contracted RiEMANN-tensor.

Thus (85a) is aequivalent with the equations of Einstein�s gravitation theory, while (856) becomes

i.e. the second set of Maxwell equations.

� 4. Dirac theory in projective form. In case that Q(,) is the 5-undor T we have (see (64), (75) and Pauli 6), p. 351):

S,uy _ nbsp;nbsp;nbsp;_ _ I ^lf,v]nbsp;nbsp;nbsp;nbsp;^[^v] �

22

We now introduce Ag'-

A,=-ka[^v] nbsp;nbsp;nbsp;= -ia^a^ Y^a

Thus, on account of (71), W\\e = ^\e nbsp;nbsp;nbsp;� We are however

still free to replace Ag by a quantity Fg differing from Ag by a multiple of the unit matrix. The choice which is most adapted to ournbsp;purposes is

rg = Ag-lXg,

with ^{1 2})

l=z

We then have similarly

A lXg F.

The expression for these covariant derivatives here chosen differ from those of Pauli by the last term.

Using

�lt;� ai �v � a), �v a� = 2 nbsp;nbsp;nbsp;a,. � lt;5^ �;.),

it is easily seen that (72) can be brought into the form

�'�lie = a^; j O'quot; � af^Ag = 0. nbsp;nbsp;nbsp;(72a)

As regards the covariant derivative of A, we can normalize it in such a way that ³)

A||^ = 0. nbsp;nbsp;nbsp;(91)

We now introduce the DiRAC-equation in 5-space:

This equation can be derived from the following Lagrangian ⁴)

L � Re^{F^ Aat^ F^\^ �?!? A F). \Xg .

1

h is Planck�s constant divided by 2 n.

2

See Pauli 6), p. 359.

3

In (93) and the following formulae we have looked apart from the

4

factor �c^jx (cf. (82)).

23

We can now compute the energy momentum tensor. In doing so we must, according to our assumption as regards the derivation ofnbsp;(65), consider the yl^�s as functions of o'* and This is possible,nbsp;as a solution for Aq satisfying (72a) is

=|(a^.5 ��� � 0'� a^;s).

as can be verified, using (69).

Using further (87) and the analogous relation holding for �.

we find from (76)

he

so

Rt^ = Re � nbsp;nbsp;nbsp;a'' Ov T. \Xg.

(Here we have made use of the fact that Re � A T= 0).

i

For the tensor we then find, using the field equations:

r^v = �Re ! !P A(a^ �^i. ov f'li^)-�? W^Aa^{a^X. a.X^) T\, (94)

where

ag = afquot; Xf,.

Finally we must establish the connection between 5-undors and 4-undors to show the equivalence of (92) with the DiRAC-equationnbsp;in 4-space. To this purpose we introduce 4 matrices a,- by

a,- =

which satisfy (cf. (21a) and (31))

a,- Uk -b a* a, = 2gik, o-i �0 ag at = 0.

From (33) we get

Aak � [A a/t) , (A Oq) = {A ao) .

24

The connection between 5-undors W and 4-undors ip is assumed to be given by

W=ip.Fi-, !F = vt./7-/ nbsp;nbsp;nbsp;(97)

where ip is a homogeneous function of of degree zero, F the homogeneous function of degree one occurring in (29) and Z isnbsp;given by (89a).

We also must find the connection between the Aj^ and the analogous 4-space quantities occurring in the ordinary covariantnbsp;derivative of a¹:

4 .

a¹ 11 / = a¹ I; � ri; a,' -j- a¹ � Ofc yl/ =: 0.

A^ = � i X^I Oq a' � Jg- Xf, Xki

Here

is the gauge-invariant covariant derivative of ip. The term proportional to \Xis so small that its physical consequences (magnetic moment for uncharged particles with spin) are negligible.

From (95) we can derive the 4-dimensional energy momentum tensor and the charge current density, using the prescriptions (20a)nbsp;and (82). This calculation goes in the same way as in Pauli�s casenbsp;and we will give here only the results:

1

See Pauli�), p. 361.

25

Tile = Re~ v A (a* V||^/ ai yj\^k)

(102) and (103) differ from the corresponding expressions found with the usual methods by small terms. As already pointed out, itnbsp;is impossible to decide empirically whether this has to be regardednbsp;as a defect of projective relativity theory in its present form. Finallynbsp;we remark that one can easily verify that s' satisfies (84), notingnbsp;that the divergence of the second term in (103) vanishes on accountnbsp;of � � ai*quot;' and using the field equations.

REFERENCES.

1) nbsp;nbsp;nbsp;Th. KaluzA, S-B. Akad. Wiss. Berlin 1921, 966.

2) nbsp;nbsp;nbsp;O. Klein, Z. Phys. 37, 895, 1926; 46, 188, 1927.

3) nbsp;nbsp;nbsp;A. Einstein and W. Mayer, S-B. Akad. Wiss. Berlin, 1931, 541:nbsp;1932, 130.

4) nbsp;nbsp;nbsp;O. Veblen and B. HOFFMANN, Pbys. Rev. 36,810, 1931.

5) nbsp;nbsp;nbsp;�. A. Schouten, Ann. Inst. H. Poincar�, 5, 49, 1935 (with references tonbsp;previous work by Schouten and van Dantzig).

6) nbsp;nbsp;nbsp;W. Pauli, Ann. Physik, 18, 305, 337, 1933.

7) nbsp;nbsp;nbsp;F. J. Belinfante, Physica, 7, 449, 1940.

8) nbsp;nbsp;nbsp;L. ROSENFELD, M�m. Acad. roy. Belgique, 18, fasc, 6, 1940.

9) nbsp;nbsp;nbsp;D. Hilbert, Gott. Nachr. 1915, 395.

10) nbsp;nbsp;nbsp;H. Tetrode, Z. Phys. 49, 858, 1928.

11) nbsp;nbsp;nbsp;E. SCHR�DINGER, S-B. Akad. Wiss. Berlin, 1932, 105.

12) nbsp;nbsp;nbsp;H. Weyl, Z. Phys. 56, 330, 1929.

13) nbsp;nbsp;nbsp;F. J. Belinfante, Physica, 7, 305, 1940.

CHAPTER 11.

MESON FIELDS IN 5-DIMENSIONAL PROJECTIVE

SPACE.

Summary.

The theory of projective relativity is applied to meson fields; it is shown how to incorporate the MoLLER-ROSENFEI.D theory of nuclear forces in this scheme.nbsp;The two main features of such a treatment are: 1. a reduction of the numbernbsp;of universal constants in the mentioned theory; 2. the automatical introductionnbsp;of the interaction between mesons and the electromagnetic field. After itnbsp;has been shown how to deal with the electron-neutrino field within thisnbsp;formalism, an expression for the energy momentum 5-tensor is derived fromnbsp;which one can obtain the Hamiltonian and the charge current density of thenbsp;system. The commutation rules for meson field variables are also brought in anbsp;more compact form. The Hamiltonian is then transformed by separating off thenbsp;longitudinal electromagnetic field and the static meson field successively and thenbsp;transformation of the current and the density of electric charge is discussed innbsp;detail. Finally, expressions are given for the electric dipole and quadrupolenbsp;moment and for the magnetic dipole moment of a nuclear system.

� 1. Introduction. The idea to describe nuclear forces by a charged field, corresponding with particles (mesons) of integralnbsp;spin and mass intermediate between those of electron and nucleon ¹)nbsp;was first put forward by YuKAWA. The scalar field (meson spinnbsp;zero) originally introduced for this purpose i) does not give thenbsp;right picture of these forces, but the introduction of mesons of othernbsp;type may help to overcome this difficulty. Kemmer 2) has namelynbsp;shown that, assuming the spin of the mesons to be not greater thannbsp;one, there arc four kinds of possible meson fields characterized bynbsp;the transformation properties of the field variables. One may thennbsp;use either a vector field (spin 1) or a pseudoscalar field (spin 0)nbsp;or some suitable combination of them. Further, the best way tonbsp;account for the practical equality of proton-proton and proton-

1

BeliNFANTE has suggested to call the heavy particle of which proton and neutron are different states a �nuclon�. However, if we keep to the custom ofnbsp;using the ending quot;�on� for names of elementary particles, the correct form ofnbsp;the word is, of course, quot;nucleon�.

27

neutron forces seems to be to introduce neutral mesons besides the charged ones in a symmetrical way ¹), as proposed by Kemmer 3).

Adopting this last assumption, MoLLER and ROSENFELD^) (this paper will in the following be quoted as M.R.) have especiallynbsp;advocated a particular combination of a vector and a pseudoscalarnbsp;field which allows to eliminate a term of highly singular characternbsp;(dipole interaction potential) from the expression for the staticnbsp;nuclear interaction. The strength of the coupling between nucleonsnbsp;and these fields is described by four constants: two, characteristic

for the �vector-interaction� scalar interaction�,nbsp;nbsp;nbsp;nbsp;Apart from the condition

|p,^M.R. |2_|^^M.R. |2 nbsp;nbsp;nbsp;jg necessary to eliminate the dipole

potential, they are completely independent. In view of the possibility to obtain stringent tests for this theory, arguments whichnbsp;would enable us to reduce this number of constants would be verynbsp;welcome.

Recently, Moller 6) ²) has pointed out that such a reduction follows from the requirement that the M.R.-theory be invariantnbsp;with respect to a wider group of transformations than the Lorentz-group, namely that of the rotations in a five-dimensional space.nbsp;Moreover, it is then possible to bring the field-equations in a morenbsp;compact form. MoLLER chooses for this space on whose propertiesnbsp;the theory now essentially depends the five-dimensional DE Sitternbsp;space. This, however, seems not to give rise to any significantnbsp;physical consequences. On the other hand, the treatment of thisnbsp;problem from the projective point of view has the advantage that,nbsp;besides the reduction of the constants, the interaction of mesonsnbsp;with the electromagnetic field is automatically introduced. This willnbsp;be studied in the present chapter.

Following M.R., we describe the mesons by three real fields. The quantities referring to each field are distinguished by a bold-printednbsp;index, e.g.

Ft, Fi, Fs,

where Ft and F-i represent the charged and F3 the neutral mesons.

1

Another possibility has been proposed by BethE'').

2

I am much indebted to dr. MoLLER for the communication of his results before publication.

28

Three such quantities are then written as F, F being, (with regard to the index i), a vector in �isotopic spin space�. In the same waynbsp;the nuclear source densities S which determine the real fields innbsp;question arc treated;

S = {Si.S,,S3).

S can be brought into the form

S = r5,

where S is the same for the three fields, and t the isotopic spin vector, (the eigenvalue 1 (�1) of Zg denoting neutron (proton)nbsp;states). Further we need for the following similar symbolical formsnbsp;for electromagnetic field quantities. If qs; is the electromagneticnbsp;vector potential we put

(pi = {0,0,(pi). nbsp;nbsp;nbsp;(1)

As in I we will denote the derivative of a tensor F by F|^ (in 5-space) or F|, (in 4-space), the covariant derivative by F||^ (F||i)¹ The covariant gauge derivative ¹) is indicated by F|[^^ (F|j^,- ) andnbsp;we have:

� 2. The meson field in the absence of other material fields. The mesons are described by a 5-vector 11^, and an antisymmetricalnbsp;5-tensor F^,, defined by ²)

Fur tPuv ^ nbsp;nbsp;nbsp;XJq �

(3)

= Uv|^-Uh.-

1

We prefer this name to quot;covariant gauge-invariant derivativequot; as the latter might suggest the invariance of meson theory with respect to the group

2

of gauge transformations which in fact is not the case.

29

The components of X� in isotopic spin space are (0, 0, X^); 2 is a for the present undetermined constant. For the Lagrangian wenbsp;put¹)

I = - i F�. F^�' - y U^ nbsp;nbsp;nbsp;(4)

/u = nioclh, me,: meson rest mass, so the field equations are

F''�,,, = nbsp;nbsp;nbsp;Q/' � 2F^�' A X,..

L is invariant with respect to rotations in isotopic spin space, and we will now perform a rotation around the 3-axis of that spacenbsp;(phase transformation). If Q(a) is a field variable and q(a) the transformed quantity, such a transformation is given by

Q(clt;),i �� �1' q(a),k ,

(cos X, � sin X, 0' sin X, cos X,nbsp;nbsp;nbsp;nbsp;0 I;

0, 0, nbsp;nbsp;nbsp;1,

x = ^ log F,

where F is an arbitrary homogeneous function of the first degree in X^.

The product of two field variables transforms as follows

Q(�) Q(/5) � q(�) q(/3) �

so (4) becomes

(7)

As a consequence of (6) also (3) and (5) are affected. Considering (3) we have

- @1 (Uk, 1'I nbsp;nbsp;nbsp;Wk, ^ I r) �b (^i [ ^ Uk, V 6^i\v

1

nbsp;nbsp;nbsp;=nbsp;nbsp;nbsp;nbsp;See also the last footnote on p. 22.

30

Now by (6igt;)

P\!ilt;

thus if we identify F with the corresponding function in I, (29):

X| ^ = 2 (X^ nbsp;nbsp;nbsp;/)t),

we get

^/iv Ur I fi nbsp;nbsp;nbsp;U^ I r ^nbsp;nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;�

Consequently, putting 0 C

X � �. nbsp;nbsp;nbsp;, [x the gravitational constant),nbsp;nbsp;nbsp;nbsp;(8)

he |/ 2X

and using {2b) and I (28), formula (3) takes the form

�/ir -- Ur I nbsp;nbsp;nbsp;U^ ![_�'*nbsp;nbsp;nbsp;nbsp;(^)

In the same way (5) must be treated. The result is

P�'l|^r = �u^. nbsp;nbsp;nbsp;(10)

From each equation (9) and (10) two four-dimensional equations can be derived. If we introduce the 4-tensor �,* and the 4-vectornbsp;g,- (cf. I, (20a)):

fik = 7'^j rCfc (fiv = � {ki, nbsp;nbsp;nbsp;(11)

= nbsp;nbsp;nbsp;(12)

and the 4-vector and the scalar u (cf. I (19)) by:

u* = 7'^;^u^, nbsp;nbsp;nbsp;(13)

u =Xi^Ui,, nbsp;nbsp;nbsp;(14)

we have

�^r = 7;J 7-/ �ik nbsp;nbsp;nbsp;/;'� g,-.nbsp;nbsp;nbsp;nbsp;(15a)

Ufc = 7^'�Uk-\-uXf,. nbsp;nbsp;nbsp;(16a)

In order to obtain four-dimensional equations from (9) we first contract this equation with 7^i7'�i^- With the help of I (23) we obtain

iik = lt;Pik vLXik; lt;Pik = vLk\ii � Ui\^_k. nbsp;nbsp;nbsp;(17a)

31

Next we contract with nbsp;nbsp;nbsp;and use (see I (13), (13a))

� � nbsp;nbsp;nbsp;Xf�\\v = ^X;^-

Therefore, (X^f^ = 0):

Xt^ yr Uv\\_f^ = X^^ y��.Ur\\tJi = �j XUfc,

Xf^ y'�_. nbsp;nbsp;nbsp;= ui|_,' � I Xfuk

So

g,-=�u||_,-.

In the same way we proceed with (10) and get

� � nbsp;nbsp;nbsp;u',

g¹'|[A: i -Aquot;'¹ {ik= � /J? U.

By means of the connection (16a) we thus have obtained from (3) and (4) a mixture of a vector and a scalar meson field. Onenbsp;can, however, modify (15a) and (16a) in a covariant way, suchnbsp;that, in the case of special relativity ¹), (3) and (4) give rise to anbsp;set of vector and pseudoscalar field equations.

Consider the quantity

where the brackets denote antisymmetrization with respect to the embraced indices. The tensor in the numerator ²) has only onenbsp;�Kennzahl�: 7], Obviously is a scalar with regard to the group ofnbsp;5'dimensional rotations that have the Det. 1. Furthermore, it is anbsp;pseudoscalar with respect to the full Lorentz-group as follows fromnbsp;well known considerations. Therefore the constant s has the samenbsp;properties, while moreover:

(21a)

1

We have chosen this particular formulation in order to show clearly the connection with special relativity which only is of interest for our presentnbsp;purpose. It is clear that a quite general formulation is possible; to this pointnbsp;we hope to return later.

2

This tensor occurs in a paper by SCHOUTEN, loc. cit.��), p. 60 equ. (16). I thank dr. PODOLANSKI for drawing my attention to this formula.

32

Putting

w = eu, hi = egt,

we may write instead of (15a) and (16a)

= r-f! yjquot; fik e 7;j h,-, nbsp;nbsp;nbsp;(156)

quot;z' � yj^ Ufc � w X/., nbsp;nbsp;nbsp;(166)

which in the case of special relativity denotes a decomposition of a 5-tensor in a 4-tensor and a pseudo 4-vector and of a 5-vector innbsp;a 4-vector and a 4-pseudoscalar. Starting from (156), (166) wenbsp;now obtain from (3) and (4);

�ik = lt;Pik ew Xik, nbsp;nbsp;nbsp;(176)

hf= � W||^,', nbsp;nbsp;nbsp;(186)

ub nbsp;nbsp;nbsp;(196)

h'�\^k Y^Xik��'�= � iu^'w. nbsp;nbsp;nbsp;(206)

The equations (17)�(20) differ from those derived by the usual methods (compare e.g. Bhabha�)) by terms proportional to somenbsp;power of the gravitational constant (on account of I (27)). Thesenbsp;terms define an interaction between vector and scalar (or pseudoscalar) mesons, but only through the intermediary of the electromagnetic field. We need not bother about these somewhat peculiarnbsp;terms, however, as they are too small to have any effect on practicalnbsp;calculations.

Finally we will write down the Lagrangian (7) expressed in 4-dimensional quantities in case we start from (156) and (166);nbsp;using (21a):

I = � i �ik f� y UA u* � I hfc h*

The first term on the right contains the small interactions mentioned above.

� 3. Interaction with nucleons. We introduce a 5-tensor S^r and a 5-vector M^;

�F rAuM !T, nbsp;nbsp;nbsp;(22)

M^=gi nbsp;nbsp;nbsp;T.nbsp;nbsp;nbsp;nbsp;(23)

34

/�q is, according to its definition, a 4-pseudoscaIar; from I, (96) it follows that

(29)

In fact, in the case of special relativity, (gn = ^22 nbsp;nbsp;nbsp;933 =

= �g44 = 1), a realisation of a,-, ag, Pg and A in accordance with I (96) and (29) is given by

both columns refer to the same representation of Qi, Oi. Consequently in either case Aai-tp is a vector, while 1/1 Aagy) is a scalar and xp'^APgxp is a pseudoscalar*). In the following only the fieldnbsp;equations corresponding with (286) will be considered**). Thesenbsp;are

fik = lt;Pik Sifc ew Xik,

Siic = 0 V' ^ A a[ik] ygt;,

hi = �W|[i Sf,

=�fA vP M',

M' = gixp'^ r A at xp, h*|I -f i e A**'f*, = -W M�gt;'.

= gi t A Po xp.

') The iS-transformation corresponding with spatial reflections is in hoth cases: S = ^3.

**} To the general problem of the different kinds of meson fields that are admitted by the projective formalism and by MoLLER�s non-projective theory wenbsp;hope to return later.

35

In order to compare this set of equations with the M.R. formalism we adopt a similar notation as used there for the source densities:

= nbsp;nbsp;nbsp;Sf = P,-Q,

M,- = M, -N, nbsp;nbsp;nbsp;M(�gt; = R.

The expressions (31a)�(34a), (in configuration space of the nucleons), become identical with those of M.R. if we put

Thus in order to describe the forces between nucleons we only need two constants which are related to those in M.R. in the same way asnbsp;the two constants of Moller�s non-projective formalism.

� 4. Interaction with the electron-neutrino field. It is also possible to incorporate in the present scheme the interaction ofnbsp;meson fields with a system of light particles (electrons and neutrino�s) and we will shortly indicate the way of treatment.

Following RozENTAL¹)9) we describe the light particles by

three real fields with the help of an �isotopic spin vector� t, T3 =r = 1 (� 1) referring to the electron (neutrino) state of the particle.

Then, similarly to (22) and (23) we can form a 5-tensor and a

5-vector

2 fi

= !F ria^ W,

W being the electron-neutrino 5-undor. Instead of (26) we now define F^,. by

Ffiv - /\ Uv) 4� S;uv

1

I should like to thank dr. ROZENTAL for the kind communication of his manuscript.

36

and also add a term nbsp;nbsp;nbsp;and a term referring to the free electron-

neutrino field to the Lagrangian (24). As we are free to introduce in the Lagrangian scalars that do not influence the field equationsnbsp;it is further possible to add a term to L of the form

(a and yS are arbitrary constants), describing a direct interaction between the nucleons and the electron-neutrino field (not involvingnbsp;a meson in an intermediate state).

It will be clear that in a theory which does not make use of the condition of covariance in a 5-dimensional space, one can in thenbsp;most general case introduce four new constants instead of the twonbsp;constants a and yS that suffice here. RoZENTAL has shown, however,nbsp;that such a diminution of the number of constants, on the basis ofnbsp;MoLLER�s theory, does not essentially affect the general conclusionsnbsp;regarding the theory of yS-radioactivity and meson disintegration.

� 5. The energy momentum tensor and the charge current density. With the help of the prescription given in I, we willnbsp;derive an expression for the energy momentum tensor, using L innbsp;its form (24) and always writing down tensors instead of tensornbsp;densities.

For (cf. I (77)) we find, using I, (46), (75) and (76)

The terms in L containing explicitly are

I X[^ A llq -f S'*�') -f- ^ X(^ A Uq A ;

on account of X.u X'quot; = nbsp;nbsp;nbsp;however, the second term is

equal to

(Uv U--a. st/si

and in this form it does no more depend explicitly on X''. Therefore

ATquot; = X^* nbsp;nbsp;nbsp;= X (F,., 2 X[v A Uo]). (X'' A

O, O,

We can now immediately obtain the energy momentum 5-tensor by using I (80) and the field equations (26) and (27):

Fve � U'' u,. F'^s nbsp;nbsp;nbsp;U. � L df -

-2 j F^^ (X. A iIa) f.,a (X'' a u^) ! nbsp;nbsp;nbsp;(X[. A (X^AU�)

where nbsp;nbsp;nbsp;is given by I, (94), (the constant rj takes the value

given in (25)). The application of the phase transformation (6) to T^y yields

T^y = � �rs U'' Uv f''� Srg nbsp;nbsp;nbsp;Uv � L lt;5() �

-X ) nbsp;nbsp;nbsp;(X., A UA) f.A (X'' A u^)nbsp;nbsp;nbsp;nbsp;(X[,. Auo]) (X'^Au�^) nbsp;nbsp;nbsp;nbsp;(35)

The energy momentum 4-tensor and the charge current vector Sk can be found at once from Tf^y. In fact we have, denoting thenbsp;energy momentum tensor of the pure Maxwell field by T^e)ikf (seenbsp;also the last footnote on p. 22)

_ nbsp;nbsp;nbsp;.1 V n~'1 rri

ik � Y/a- y.kT,v-\-l(

T^y.

Using (156) and (166) we get

yi! f'.k f-e = (7�; f''��h' x^) (/�'quot; ikm-

= F' iki h' hfc .

The cross terms disappear on account of I, (16) and (18). Similarly

r;igt;^'^y\ s,., = F's./ h'Slt;�gt;.

The terms in (35) containing X� or X'^ explicitly disappear if we contract with y��^. Using (13) and (14) we thus obtain

T\ = -�'�' ffc, f� S/t/ M' Uk-fi'^ u' u^-h' hk

h'sf-L6^ r,�A r�A. (36)

38

T[n)ik is given by I, (102). From (36) we infer that the Hamiltonian is given by ¹)

') L H(e)) dv H(n) L,

H(/, L = - nbsp;nbsp;nbsp;Srt nbsp;nbsp;nbsp;nbsp;U4 -Sf¹ - L,

i nbsp;nbsp;nbsp;.A , he

H(e, = -|(S^ �^).

i-f(n)L has been represented in configuration space, the index (i) referring to the i�th nucleon. (S, � S) is the electromagnetic vector

potential, (lt;�, tf) the electromagnetic field, co is a small term which we will not write down explicitly.

In order to avoid the occurrence of singular terms of the d-func-tion type ²), we add to the Lagrangian the scalar \ S^.,. Squot;quot;. Introducing a vector notation for the dynamical meson variables:

u/t=u,-v,

}xk � r, � lt;h, \^=zw.

we then get

Hif)i = i IF^ -h /r2 (U2 -f V^) i -(F T U M) i (T2-S^)

i(h nbsp;nbsp;nbsp;-h fd W^) - (R W Q 0) i (Q^ - P^). (37)

In the same way the field equations (31) � (34) can be treated.

The charge current density can be calculated by the same methods

1

Generally we denote the Hamiltonian of a system by if it depends on some field variables and their gauge derivativesnbsp;nbsp;nbsp;nbsp;The corresponding

Hamiltonian depending on and nbsp;nbsp;nbsp;denoted by the same letter but

without the symbol L.

2

Cf. M0LLER8). p. 26.

39

and it should be noted that it is the term � nbsp;nbsp;nbsp;in (35)

which here plays an essential part. We obtain

S�� = eyj^ A a^� y (fA Ur)3 f'*.

I nbsp;nbsp;nbsp;nc

f'* is a small term which (apart from a contribution of the �Dirac-type� due to the nucleons, whose divergence vanishes, compare I, (103)) is equal to

[� fhfc�u' w �'* S* (0) M' w]

and it can be shown that nbsp;nbsp;nbsp;0, using the field equations. On

the other hand 5^�(1^,-0 on account of the general theorem proved in I. Therefore, from now on entirely omitting all small terms, wenbsp;may write

nbsp;nbsp;nbsp;(38)

With Sk = S/i and s* == s, �q this leads to the following

expressions (cf. M.R. (61), the nucleon-part of s en p is expressed in configuration space)

S- 5nucl �I� 5ines-^ ^

^(G^^u-FAV rA'?gt;. nbsp;nbsp;nbsp;(41)

1 -� -gt;� nbsp;nbsp;nbsp;-4quot;nbsp;nbsp;nbsp;nbsp;A

P = p��� p�,, = e 2 nbsp;nbsp;nbsp;d (x-rcl'-') ^ (U A F-WA (42)

(� z nbsp;nbsp;nbsp;n c

� 6. Quantization. In the present formalism it is possible to bring the usual commutation rules in a more compact form which isnbsp;especially suitable for practical calculations, and which clearlynbsp;shows the intimate connection between vector and pseudoscalar

40

variables. In fact, the commutation relations of both fields are all contained in

I nbsp;nbsp;nbsp;h cnbsp;nbsp;nbsp;nbsp;A

r/'in nn nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;/nbsp;nbsp;nbsp;nbsp;/\ smnnbsp;nbsp;nbsp;nbsp;.4

[t/i-i, Uv ] =--J- O (x�x ) o nbsp;nbsp;nbsp;,

^/lt;r,4 -p'/iV y^ij^yv �

Here = y'�^ nbsp;nbsp;nbsp;; by ul we understand that the four-dimensional

quantities that can be derived from it, viz. u'k and u' should be taken

at the space time point {x't), and similarly and refer to the

argument (x,(). The isotopic spin indices are for convenience written at the upper side of the symbols. Contracting (43) withnbsp;7^i'i�''k obtain using I, (17) and (21a)

[^M nbsp;nbsp;nbsp;^)gt; K lt;�)]= � y ^ (;�r�jr') [gik�gn ;

this is equivalent with

[LJr{x,t),Fk{x',t)]~^d{x�x')d'^^gik {i,k=l,2,3). nbsp;nbsp;nbsp;(44a)

Contraction with X' gives simply the result that the vector variables commute with the pseudoscalar variables. Finally contraction with X^ X^ gives

In a similar way we can deal with the quantization of the electromagnetic field. If F'¹ is the electromagnetic field (identical with F'¹ in I, (27)) and F^�� the corresponding 5-tensor we put

[F/.4, lt;p'v] = �y ^ (x�x')

= yu.i y�J 7^1,4 y'^,

from which the single set of relations

1

One might think that, on account of the connection of the electromagnetic

41

The commutation rule for the canonical variables of the nucleons is (45)

� 7. Transformation of the Hamiltonian, a) Separation of the longitudinal electromagnetic field. In effecting this separation wenbsp;put:

S = 5|| �h �Sx gt;

�. = �.\\ quot;j-15j_,

S = Sin 4quot; Sex .

where 6|| is the longitudinal and is the transversal electric field

(similarly for B); Sin Is the part of the static potential which is created by the system of nucleons and meson fields, while Sex isnbsp;the contribution of other sources eventually present. We now must

eliminate 6||, S|| and Sin from H. This problem has been extensively dealt with by several authors, so we can confine ourselves here tonbsp;giving the results.

For the first term of H we get

J'H(f,L dv= j Hif) dv� ^Smes B�dv�W(B]_).

The last term denotes a quantity proportional to the square of the perturbation parameter {e^/hcfi^ and therefore generally may benbsp;neglected. At the same time we then must replace the gaugenbsp;derivatives which occur in H^f) by ordinary derivatives, and thus

write F, G, P, 0 instead of F, G, P, 0, where the first group of quantities are the same as in M.R. Similarly we must proceed with

H(n), s and Q.

and the gravitational field, the c-number character of the latter might be affected by (44c). Now from (51) follows (Cf. I (27) and (28)):

and it is easily seen that this does not give rise to any new commutation rule.

42

Further we have

�-gt; nbsp;nbsp;nbsp;-gt;

Snucl

while the last terms of H gives

H{e) dv =J'Hie)� dv G � Q B^^dv, nbsp;nbsp;nbsp;H(e)� = ^{8]_ U^).

Here G is the Coulomb energy of the system, q is the total charge density as given by (42). Infinite electrostatic self energy termsnbsp;have been suppressed. Consequently the total result is (in the

following we write S, H, S, //(e) instead of 8^, Bex. H(e)^ respectively)

H �J H{f) dv -f H{n) -j-J' H{e) dv -\- G �^s Rdv nbsp;nbsp;nbsp;qB dv (46)

with

H,/) = i|F2 G2 /r2 (U^ V^)!-(FT UM)

i ( ^2 rr 2)-(R rr Q 0) 1 (T^-S^) i (Q^-P^), (47)

l T^ nbsp;nbsp;nbsp;1-t(|gt;

--

If we suppose that all wave lengths occurring in the Fourier-development of the electromagnetic field are large compared to the dimensions of the nuclear system, it may be shown that lo)

�� B cfu = e Bo 9 grado 8 (Q gradg) gradg Bq, (49)

sRdv--Sq ^ M 2^0 (Q g^^dg) tig.

where the index 0 indicates that we have to take the value of the quantity at a fixed point of the nuclear system, (its centre of gravity,nbsp;say). Further we have introduced the following quantities referringnbsp;to the nuclear system:

43

its total charge; e = j q dv,

its electric dipole moment: �p = jqx dv,

its magnetic dipole moment: M = 4- � x A s dv,

its electric quadrupole moment: Qik = \ j QXiXtdv.

Instead of (49) and (50) we may also write:

RP {Q grado) S\. nbsp;nbsp;nbsp;(51)

(For processes in which the total energy of the system is conserved the matrix elements of the time derivative on the right of (51)nbsp;vanish).

b) Separation of the static meson field. The equations determining the static fields are:

The equations in the first column define the static field; those in the second one are the same but now written in tensor form, whilenbsp;in the third column the 5-tensor form is indicated. Any quot;tensor�nbsp;labeled with � is the same function of the static variables as thenbsp;corresponding tensor in the former equations is of the ordinary

Terms for which / ^ 3 need not be taken into account, as they are of higher order than the second in and 32¹

1

nbsp;nbsp;nbsp;� [a, [a, . . . [A, B]] . . .] is the number of brackets.

45

We now introduce a set of variables marked with 1 that refer to free mesons. Of course:

1 1

^ 1 1

II = U , F = F , W= W , lt;P=:fP,

while

fj.~^ div F , G = rot Vi , r= � grad W

so

1 -gt; -2 �

5�es = 5�e. ^[S^U-^-2FAN PA^l3. For the second term in the development we have

In calculating this commutator we must use (43); we then get¹)

5fi) = ^ \\i A Uv)3 � (u� A f�'^)3 I �

� ^ � he I � nbsp;nbsp;nbsp;

Jquot;(u� Au.')3 [fs4. nbsp;nbsp;nbsp;(57)

The commutators occurring in the integrals are composed of quantities with the same isotopic index, which has been omitted. The contribution of each term to the 4-vector s�^^^ is then found bynbsp;inner multiplication with y^� and with the help of (15b), (16b) andnbsp;of I (16)�(18). Thus the first term contributes (apart from thenbsp;factor e/hc)

o nbsp;nbsp;nbsp;onbsp;nbsp;nbsp;nbsp;onbsp;nbsp;nbsp;nbsp;o

to Qw: -(FAU)3: tos�):-(FAV)3 = -(FAV)3-;a-2(FAN)3,

1

The first term has been computed making use of:

[u' e, ff¹] = nbsp;nbsp;nbsp;[u' e, fV]nbsp;nbsp;nbsp;nbsp;[a' e,nbsp;nbsp;nbsp;nbsp;.

46

and the second

O

to ew: -(F AUh-{W A

1 nbsp;nbsp;nbsp;onbsp;nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;o

to 7�): (G A U)3 (r A tfOs (S U)3 (P A -^03 �

The third term gives, after multiplication with and keeping in mind that = lt;Jgt; = 0

(�'h a �*')3 [�' �.tile],

so it does not contribute to p(i) while it gives for S(i)

- ^ i � (F'l A F)3 [Ul. V] dv' = i � (� AF)3 [Ui div F].

Now

[Ul. dill F] ^ ^ lt;5 (x-xO = - ^ nbsp;nbsp;nbsp;lt;5 (x-x'),

t ox' nbsp;nbsp;nbsp;I Ox '

therefore the third term gives for S(i) after a partial integration

-(FAV)3 /^-MFAN)3.

Finally the last term of (57) becomes after multiplying with y

� Yc I^ nbsp;nbsp;nbsp; � (w' A 'w)3 [h'4, h'] I

from which we infer that it does not contribute to p(i). The contri-

bution to S(i) can be found in a similar way to the treatment of the preceding term. The result is

O nbsp;nbsp;nbsp;O

(G A U)3 (F A 'F)3 -{s a U)3 - (PA ^)3.

Consequently the complete result of (57) becomes

Pd) ee px = ^ (U A F u A F-V'A��gt;)3,

~(S^U-/*-2F/yN PA f)3. (59)

where 5x = ^(g^� g^ii - fa V- FA V r^w r/\w)^. (60)

nc

Now we must find the third term of (55)

[K, 5-].

The calculation of this commutator goes by the same methods which we have used to find s^y It gives rise to field-independent (f.i.)nbsp;terms as well as to terms quadratic in the meson field components.nbsp;For the calculations in the following chapter we are only interestednbsp;in the former which we here directly give:

f.i. part of 0(2) = Pexch = 7- (U A F)3 ,

nc

f.i. part of s,2) = sexch- ^(S^U-/.-^2FAN PA^)3, nbsp;nbsp;nbsp;(62)

with

5exch = ~(G^U-FAV rA^')3. nbsp;nbsp;nbsp;(63)

Thus, from (58) and (61) we may infer that to the approximation indicated

Qmes - Qmes ~l~ nbsp;nbsp;nbsp;^exch

and similarly from (56), (59) and (62) that

1

-gt;� nbsp;nbsp;nbsp;-y -

3nies - 3nies �1� 3x �1� 5exch

(64) and (65) can immediately be understood if one remembers that any fieldvariable A occurring innbsp;nbsp;nbsp;nbsp;is approximately the

48

sum of a static part A and a new free meson variable A. Inserting this one gets: a) a part of Pn,es depending only on the free meson

O

variables, namely gmesi h) a part depending both on A and A\

c) field-independent part of p^es: Pexch! Similarly for Smes.

We must now apply the same transformation to s� This calculation is quite straightforward. For the developments of thenbsp;next chapter we only need the f.i. part of and it can be seennbsp;that this is simply Thus, summarizing the result of the trans-

formation, we may state that, in order to compute U and Q in our approximation, we may put

Q - Qnucl H� ^exch �

S � 5,,ucl �h �Sexch ?

To calculate 'P we remark that F = �V and

1

exp.

Bringing � and V in a form in which the nuclear variables occur explicitly (cf. M.R. equ. (14)) we get:

with

k - .

�^0

Further we have replaced by 1 which only gives a difference of the second order in the velocities.

Q is found in the same way. Here we make use of

8n ju nbsp;nbsp;nbsp;2

- Mnucl A Mexch �

To compute Mexch we introduce in Sexch the explicit expressions for the static field variables (see also M.R, (37) and (40)) and thennbsp;get

-g2 �U) cp(lt;)/\ (�(�/\

(a(') Vlt;'gt;) 7lt;'')/\ (ot*) A f *gt;) � ��lt;'�) (?)lt;*gt;

2 nbsp;nbsp;nbsp;(o(*) �(*)) (5 (jc�X*'*) A ^ 1 (o*'* V *'gt;) ��'� I (0**)

f2 nbsp;nbsp;nbsp;fX

^(k) � grad*** V*'* � grad^h

Therefore Mexch becomes

The first two terms describe the �exchange� part of the magnetic dipole moment due to the vector field and the other two the contribution of the pseudoscalar field. The separate expressions fornbsp;these two parts are:

4

50

1 nbsp;nbsp;nbsp;_ rik

2 nbsp;nbsp;nbsp;2 fi

' '

These expressions depend on the coordinates of the centre of gravity of the i�th and k�th nucleon (neglecting the differencenbsp;between rrif^ and mp) and it is remarkable that this is no more the

variables, only depends on r,* , and thus is a translation-invariant quantity. The complete expression for the magnetic dipole momentnbsp;of a nuclear system thus is

Finally, we will give here for later purposes the expression for

the time derivative of 5^ which has been computed by M0LLER and Rosenfeld 10):

5^=-^2�(l-tW)alt;''

Z i

^ 2 (rlt;') A lt;''')3 (x(''-x('')) {g\ g\ nbsp;nbsp;nbsp;lt;p {tik).

^nC i^k

In calculating the time derivative of the quadrupole moment we have

51

made use of the same method which was followed to find 'P. We will here merely state the result;

Qik nbsp;nbsp;nbsp;(1�4^�) (�*/�nbsp;nbsp;nbsp;nbsp;�'/')

nbsp;nbsp;nbsp;2quot;nbsp;nbsp;nbsp;nbsp;�Cp (rmn).

^ nC m,n

REFERENCES.

1) nbsp;nbsp;nbsp;H. Yukawa, Proc. phys. math. Soc. Japan, 17, 48, 1935.

2) nbsp;nbsp;nbsp;N. Kemmer, Proc. roy. Soc. A, 166, 154, 1938.

3) nbsp;nbsp;nbsp;N. Kemmer, Proc. Cambridge Phil. Soc. 34, 358, 1938.

4) nbsp;nbsp;nbsp;H. A. Bethe, Phys. Rev. 55, 1261, 1939.

5) nbsp;nbsp;nbsp;C. MoLLER and L. Rosenfeld, D. Danske Vid. Selsk. math.-fys. Medd.,nbsp;17, fasc. 8, 1940.

6) nbsp;nbsp;nbsp;C. MoLLER, D. Danske Vid. Selsk. math.-fys. Medd., 18, fasc. 6, 1941.

7) nbsp;nbsp;nbsp;J. A. Schouten, Ann. Inst. H. Poincar� 5, 49, 1935.

8) nbsp;nbsp;nbsp;H. }. BhabhA, Proc. roy. Soc. A 166, 501, 1938.

9) nbsp;nbsp;nbsp;S. Rozental, in the press.

10) C. MoLLER and L. ROSENFELD, in course of publication.

CHAPTER III.

THE PHOTO-EFFECT OF THE DEUTERON.

Summary.

The photo-disintegration of the deuteron and the capture of neutrons by protons are discussed from the standpoint of the M0LLER-ROSENFELD theorynbsp;of nuclear forces. The general expression for the cross section of the photoelectric effect turns out to be identical in form with the corresponding quantitynbsp;in the old Bethe-PeiERLS theory, while the photo-magnetic cross section containsnbsp;an extra term due to the meson field. As a consequence of the different wavenbsp;functions used for the ground state of the deuteron the cross sections decreasenbsp;more rapidly with increasing photon energy than in the old theory. The absolutenbsp;values for the cross sections are of the same order of magnitude as found empirically, though definite numerical results can as yet not be given, owing to thenbsp;unreliability of the deuteron wave functions used. This circumstance makes anbsp;definite statement with regard to the angular distribution premature. The capturenbsp;cross sections also are of the right order of magnitude and, as in the old theory,nbsp;the 1/u law appears to be a magnetic effect.

� 1. Introduction. The discovery, made by Chadwick and Goldhaberi), that the deuteron can be disintegrated under thenbsp;influence of j'-rays of sufficiently high energy, provides us withnbsp;most valuable information about the interaction of electromagneticnbsp;radiation with nuclear systems. This effect is closely connectednbsp;with the capture process of neutrons by protons, which especiallynbsp;plays a prominent role in experiments with slow neutrons. In thenbsp;earliest treatments that were given of the photo disintegration 2)3)nbsp;as well as of the capture, these effects were considered as photoelectric (PE) processes, (interaction of the electric field of thenbsp;incident wave with the nuclear system). The cross sections thusnbsp;obtained for the PE disintegration were in reasonable agreementnbsp;with experiment, but there turned out to be a difference of severalnbsp;orders of magnitude between theoretical expectations and thenbsp;measured values of the capture cross section. This point wasnbsp;cleared up by the remark of Fermi 4) that, besides the mentionednbsp;processes, one has also to take into account the photomagneticnbsp;(PM) transitions, due to th� interaction of the magnetic field ofnbsp;the incident wave with the magnetic moments of the nuclear

53

particles, (cf. also Breit and Condon 5)): it was shown by him that the slow neutron capture is essentially of magnetic characternbsp;and that the well-known \/v law could be explained on thisnbsp;assumption. Thus, all experimental data known at the time could benbsp;accounted for on a theory based only on the assumption that thenbsp;range of the nuclear forces was small compared to the radius ofnbsp;the electron.

More recent experiments by VON Halban however, seem to indicate a discrepance with theoretical expectations on the angularnbsp;distribution of the disintegration products: while for the PM effect,nbsp;(corresponding to a transition between the 35-state and the i5-statenbsp;of the deuteron), this distribution is isotropic, the contribution of thenbsp;PE effect (a 3S 3p transition) per unit solid angle is proportionalnbsp;to sin 20, 0 being the angle between the incident 7-ray and thenbsp;ejected neutron. Therefore, from the expressions for the differentialnbsp;cross section of both effects, which we will callnbsp;nbsp;nbsp;nbsp;(0) and

c/^magn fjj^d for the ratio of the intensities at 0 = 0, (t?||) and

0 = V2, (^x):

0|| nbsp;nbsp;nbsp;^magn

d -f d (njiy

For ThCquot; 7-rays this ratio was calculated to be 0,29 (assuming the iS-level of the deuteron to be a virtual one, as now seems to benbsp;certain), while the measurements of VON Halban give a value that,nbsp;(considering the experimental uncertainties), lies between 0,01 andnbsp;0,13. This effect, if real, constitutes a difficulty which may benbsp;expected to be cleared up only by a deeper insight in the nature ofnbsp;nuclear forces. It is therefore of interest to see whether our presentnbsp;conceptions of the interaction between nucleons can clarify thisnbsp;point.

We shall here for this purpose adopt the standpoint of the theory of MoLLER and Rosenfeld according to which nuclear forcesnbsp;are described by a mixture of vector and pseudoscalar mesonnbsp;fields ¹). In the next two sections we will treat the PE and PM

1

Recently, a discussion of the PE effect in the frame of the meson theory of nuclear forces was given by Fr�HLICH, HeitLER and KAHN�), assuming thenbsp;interaction to be described by a field of the vector type. However, their �Ansatr�nbsp;is clearly inconsistent with the general electromagnetic properties of nuclearnbsp;systems; we will therefore here not consider their results.

54

effect from this point of view, while in the last section the capture problem has been dealt with. The admixture of a D-state with thenbsp;35-ground state of the deuteron has, on account of its smallness,nbsp;practically no influence on the effects under consideration. We willnbsp;therefore throughout neglect the contribution of this D-state, andnbsp;thus consider the ground state to be purely of the 35 type.

� 2. a) The wave equation of the deuteron. We will first give a survey of the properties of the deuteron wave functions, representing in a slightly different form results obtained by Kemmer 9)nbsp;in a paper on the neutron-proton interaction.

The two nucleons that constitute the deuteron, and all quantities that refer to them, are labeled with the upper indices 1 and 2

respectively; thus, for instance, nbsp;nbsp;nbsp;andnbsp;nbsp;nbsp;nbsp;represent the

spatial coordinates and impulse, (multiplied by the velocity of light c), of the first and second particle. The deuteron is describednbsp;by a 16-component wave functionnbsp;nbsp;nbsp;nbsp;(/r stands for all those sets

of values of the degeneracy parameters that belong to the same energy E). In the frame of reference in which the centre of gravitynbsp;of the deuteron is at rest it satisfies the equation *):

Ho We^ (x) = nbsp;nbsp;nbsp;� grad ^ Mc^ -f (r)^ Te^. (x) = E We^ (x) (2)

with

V= (tO) T�) [g2 gl �(2)] .

According to Kemmer we can classify the non-trivial proper solutions of (2) as follows:

Type la: nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;i � .. triplet state with f = / � 1

rj, ,, corresponds m non-relativistic , nbsp;nbsp;nbsp;�'

Type lb: nbsp;nbsp;nbsp;triplet state with f = i.

lype Ilh: nbsp;nbsp;nbsp;singlet state, (/ = /).

h is Planck�s constant divided by 2 3i.

with

and

The functions Z and z only depend on the relative spatial coordinates; introducing polar variables, (x = c sin amp; cos lt;p,nbsp;y = r sin amp; sin (p. z = r cos #), we obtain the following expressionsnbsp;for them, (to these we will refer as (6));

57

The spherical harmonics are defined as in M.R. equ. (115); we also use the same normalization prescriptions as stated there.

The large radial functions R(j) satisfy

(7) where

for type la, lb: T':=[l�2(�I)-'��quot;']

From (7) it follows that R^, the asymptotic solution for R, is given by

R^=^]/'lcosikr ej); ej = - ^(j l) dj; k = yMEIh (8)

The factor l\/2l7i normalizes R^ in the energy scale; as shown by Bethe and Bacher lo) have for X

The phase constants Sj are essentially fixed by the solution of (7). With the exception of they are negligibly small ifnbsp;(h/ME)'/-2)) ��1, (cf. Bethe and Bacher lo), p. 115).

On account of (8) we may write for the asymptotic solution of the complete large wave function

^E,fi nbsp;nbsp;nbsp;=Bgt;;(E, ju; �amp;, (p) . � cos {krej), (!F large). (10a)

(dgt;), r(|). oW of, Qf, Qf),

Further, the small radial wave functions are related with the corresponding large functions by

Similarly to (10) we can write for the asymptotic expression of the complete small wave function:

(r �^ oo) = J3v . � . i sin (fc r -j- sj), {W small). (lOfc)

b) Interaction with electro-magnetic radiation¹). We now examine the result of an irradiation of the deuteron with a monochromatic polarized y-iay beam, and thus have to insert the operatornbsp;� into the time dependent Schroedinger equation of the deuteron:

I h ^ = {Ho � e-'-f conj) W.

where (see chapter II)

Q = M mU {Qgrad)a-, nbsp;nbsp;nbsp;(12)

S and H are the electric and magnetic amplitude respectively, taken at the centre of gravity of the system. ?gt;, U and Q of course also

1

The developments in this section are analogous to the treatment of the PE effect for the hydrogen atom as given by Bethe'^^).

*) Cf. loc. cit. p. 446.

61

die nbsp;nbsp;nbsp;t

Thus, if we take vt = nbsp;nbsp;nbsp;. t')'} r, so �. r ((-r, we have

dp nbsp;nbsp;nbsp;dE h

/2 = 0

and

__L IR4.2 Mr2\ t

^^ = -nie quot; nbsp;nbsp;nbsp;2- 5: (E, /i;nbsp;nbsp;nbsp;nbsp;lt;p). {E. f^\ Q \Q) ---

With

id

Q{r^oo,�,(p) = \ T^\^ = r3\2B^^.(E.^l\�\Qi)e^�-j\\ nbsp;nbsp;nbsp;(18)

jA.

the differential cross section is, (for a fixed direction of polarization of the y-rays)

d^ = Qvddco, nbsp;nbsp;nbsp;(19)

where q is meant to be the average over the three magnetic substates corresponding with the degeneracy of the ground state.

3. Calculation o[ the cross sections ¹). According to the prescription given in M.R. we must break off our calculations at the first stage which gives non-vanishing contributions to the effect concerned. We thus will not have to go further than to the first ordernbsp;in the velocities, that is to say, we will only have to consider matrixnbsp;elements {F \Q \ 0) which belong to one of the following three types

It should be noted that, even if (F |f3| 0) satisfies this requirement, it still may be negligible in our approximation if Q itself containsnbsp;velocity dependent factors,.

a) The PE effect. We have to consider the transitions due to

=(�?�. Taking the x-axis as direction of propagation of the photon beam, and the z-axis as the direction of its electric vector:

Del = A 9z.

1

I should like to thank dr. MoLLER for the communication of preliminary calculations on the photo-effect which have provided a valuable check of thenbsp;calculations given here.

62

We choose the amplitude A of the fields of the light wave such that it normalizes the radiation to one (polarized) photon per sec.

per cmquot;^

hv^ 2c �

First we consider �quot;i

/^nucl_

-

As the ground state is antisymmetric in the isotopic spins and

^Co(ry'-a'Co=2:

the final state must be antisymmetric with respect to rs�. Taking further into account the behaviour of (22) with regard to rotationsnbsp;and spatial reflections and the fact that the ground state is of thenbsp;type la with 1 = 0, j = 1, we find the following possibilities fornbsp;the states that combine with the ground state, (behind each statenbsp;we have indicated in brackets the spectroscopic symbol that corresponds to the non-relativistic approximation):

la, l=\,j=2{^P2), la, l=\.j=0{^Po),nbsp;ib,i=j=i m).

while the familiar selection rule Am = 0 holds. From (5) it is easily seen that to the first order in the velocities we have for allnbsp;these transitions:

Using (6) the matrix-elements can readily be calculated and we get

63

where we have indicated behind each expression the corresponding magnetic transition; further

I\ =J^ dr .r . R*a{\) Ro,

h=Jdr.rRl{0) Ro.

where Rq is the large radial wave function of the ground state. Now, by (7):

Ra{l)-=Rb {0) = R = R*.

so

I,=h=I=J.

. the second part of which is given by (see equ. (13)):

does, in our approximation, not contribute to the PE effect. This will be shown in the appendix.

With the help of (18) and (19) we can now directly obtain the differential cross section. As the final states all are P-states, thenbsp;phase factors dj may be neglected in this case. We then get, withnbsp;the help of (9) and (2) and neglecting those terms in By thatnbsp;are proportional to (u/c)2, expressions which turn out to be thenbsp;same for the three possible magnetic transitions, so that they directlynbsp;give the average value q. The result is;

cos^ amp; . sin 9 d9 nbsp;nbsp;nbsp;(27)

This is the cross section for a fixed direction of polarization. To obtain the cross- section for an unpolarized photon, we have tonbsp;average over all # corresponding with the same 0, (the anglenbsp;between the direction of the light quantum and that of the neutron).nbsp;This we do by first transforming to other angular variables:

cos �amp; � sin 0 cos ip, sin �amp; d� dlt;^ = sin 0 d@ dw.

64

where yj is related to 0 in the same way as (p to �; we then integrate over all orientations ygt; of the polarization direction (which gives us a �zonal cross section� for an unpolarized photon), andnbsp;multiply the result by d'tp/ln. This gives, on account of the factornbsp;cos2 yj, an averaging factor ]/2 the differential cross section:

The total cross section is

a

jie^ VI r

J

This result is identical in form with that obtained in the Bethe-Peierls theory. Deviations from this simple formula are at most to be expected in the second order with respect to the velocities.

b) The PM effect. To begin with we consider the contribution of the first term in (14), for which we write, noting that the magneticnbsp;vector stands in the y-direction

�Zt = j-A [x A 1(1-r^�*) �'quot;-(l-t'l�) a�!], .

Now

3C^(l-r(V)% = -l , %(l-r(irCo=l gt;Co(l-TyVCo = l. .

so there are allowed transitions to states which are symmetric as well as antisymmetric in the isotopic spins. In the second casenbsp;however (transition to a^D-state) the contributions are vanishinglynbsp;small in the whole energy-region which is of interest; we neglectnbsp;them in the following. If the final state is symmetric in the t's wenbsp;can write

.nucl

the only allowed transitions are to the state of the type lib with

65

j � O (^S), while Am � �1. With the help of (6) we get (for 1nbsp;nbsp;nbsp;nbsp;0 as well as � 1 �gt; 0)

where juq � eh/2Mc is the nuclear magneton. The integral can be simplified by partial integrations. The result is

_ �

{llb,j=0\QZtn\0)=-if^oA^j drRRo : R = Rii(0). (30)

0

The extra magnetic moments of proton and neutron are contained in the second part of (22), namely those terms for whichnbsp;i = k (= 1,2)¹). These terms are �of course� infinite and can onlynbsp;be managed by using a cut-off prescription. Calling their contribution to (19)nbsp;nbsp;nbsp;nbsp;we have

(o�gt; q) (a^o A e) f

Averaging over all directions of q (which we indicate by the overlining of the left member) gives for any componont (g e) of q

(o('gt; q) (ee) = i (o(') e).

Thus if we take for q) (o¹'¹ Ae))i its average over all directions

of p, which seems to be an appropriate way to deal with this quantity, we have to replace it by '/j (�^�gt;nbsp;nbsp;nbsp;nbsp;alt;)�gt;) = � 2t/3nbsp;nbsp;nbsp;nbsp;.

5

1

Other terms of higher order, which also contribute to these extra magnetic moments, have to be discarded according to the prescription given by MoLLER

and ROSENFELD 12).

66

We now cut off by replacing

9l nbsp;nbsp;nbsp;M \ ( A , ,

lim -� --5

e^oAnhc Mm 3 \y.Q J

by a finite quantity which we call a and obtain

For this operator only the i5-state is allowed as final state. The matrix-elements are of the type as indicated in (23). The result is

00

{Wb, j = nbsp;nbsp;nbsp;A\/2 � drRR,.

We notice that

2,�(iih, ;� = 0! nzt 10) =r M � = 01 �ZZ! 0),

the perhaps unexpected factor }/2 on the right arising from the fact that we have to do with the magnetic moment of the deuteron withnbsp;respect to its centre of gravity 12) .

At the moment the only way of dealing with is to fix it with the empirical values for ^ip and (the magnetic moments of protonnbsp;and neutron in units /Iq):

jup � 1 -j- nbsp;nbsp;nbsp;f^N �� � fJ-

This gives

_ ^

{lib, j^O\QZ.t QT:n\0) = -if^oA^{^lp-f^N-i)j drRRo. (32)

0

Finally we have to consider the terms from the second part of (14) with i^k; we call the corresponding operator QZgn whichnbsp;reduces after some simple calculations to

I (oO) A x% (0'2gt; x�) - (�lt;2) A x�)y (aO) x�) j .

1

67

As nbsp;nbsp;nbsp;A�Co = 2z, 'to A'*^�)3 'to = 0. the final states

must be antisymmetric in the isotopic spins. The only states that combine with the ground state turn out to be

Type II b, y = 0, ('5),

Type II b. y = 2, CD).

The matrix-elements are found to be again of the type (23). They are

(36)

. nbsp;nbsp;nbsp;47�nbsp;nbsp;nbsp;nbsp;, v I ^1,

j||2_

c) Numerical estimations. For the numerical evaluation of the cross-sections we have taken for the large radial wave function ofnbsp;the ground state the approximate expression obtained by WiLSONnbsp;under the assumption of a nuclear potential of the type as used here:

Further we take Eq = 2,16 MeV. Assuming

Mm _ 1 M � 10�

we have a = 2,13 and (cf. M.R. (107), (108) (109)):

For the phase of i5 state which occurs as final state in the magnetic transitions we have, according to Bethe and Bacher

sin (5(1 =

where

(/MEo

The plus or minus sign holds according to whether the first excited (iS) state of the deuteron (with the energy E'o) is real or virtual.nbsp;Experiments have decided for the latter possibility. We have takennbsp;E'o = 105.000 eV. Further we put ¹)

(41)

1

See loc. cif. ^�), p. 124 equ. (77b), (77c) and p. 128 equ. (93a).

6(6)=! ^^

We notice that for very large energies both the PE and the PM cross secti9ns decrease likenbsp;nbsp;nbsp;nbsp;, that is more rapidly than in

the �old� theory (oo r~^k). Apart from that we may state in a general way that the results for not too large frequencies are of thenbsp;same order of magnitude as in the old theory. As regards thenbsp;absolute values of the cross sections we have thus reasonable agreement with the measured values, viz. 5.10�28 cm2 (Chadwick andnbsp;GoldHABERI)) or 9.10�28 cm2 (voN HALBAN 6)) for the 3^-rays

1

It should be borne in mind that e is expressed in Heaviside units in accordance with the normalization (21) of A.

70

of ThCquot; (/iv = 2,64 MeV), and 11,6.10�28 for an energy of 6,2 MeV (Allen and Smith i3)).

As regards the angular distribution, it seems that the terms specific for meson theory, (arising from Q'^^gn), do not appreciablynbsp;ameliorate the situation. However, we are unable as yet to decidenbsp;this point, since we have noticed that the wave functions at ournbsp;disposal are much too unreliable to allow definite statementsnbsp;regarding such a sensitive effect (see also the Appendix).

5. Capture of neutrons by protons. The cross-sections for these processes can immediately be inferred from (42) and (43). Wenbsp;have in fact, calling the cross-sections for �electric� and �magnetic�nbsp;capture andnbsp;nbsp;nbsp;nbsp;respectively

^el 9

lt;f.=y

Therefore

We are especially interested in the behaviour of these expressions in the region of thermal neutron energies. In this region we maynbsp;write /S2 fornbsp;nbsp;nbsp;nbsp;(as � 25,7.1022 cm�2).

It is then easily seen that, for thermal neutrons, on k, while ^magn ^ ^�1, can thus be ignored, while, just as in the �old�nbsp;theorynbsp;nbsp;nbsp;nbsp;gives us the well known 1/y law. The agreement with

the experimentally found values: 0,27.10�24 cm2 for a velocity of 2,2.105 cm. sec.�1 (Frisch, v. Halban and KoCH i4j) ^^dnbsp;0,31.10�24 cm2 for a velocity of 2,5.105 cm. sec.�i which wasnbsp;obtained by Amaldi and Fermi is) appears to be satisfactory.

APPENDIX.

To calculate the PE effect we have made use of operator S �gt;. But, as

-�� -V nbsp;nbsp;nbsp;rf

119 = 89 ~ {^9),

cat

71

and as the second term on the right has vanishing matrix elements for the transitions concerned (because of energy conservation), we

may use the operator R 9 just as well. This we will do here; in the centre of gravity system we have

Denoting B.9 hy we have for the first part of this operator: wu A' e

(A' � ic/v A is the amplitude of the vector potential). This gives rise to the following matrix elements:

(la, Z=l,y=2|^liiO) = -eA/a (la, /=1,; = 0|^:, 0) = ^ A'e/^

(I b, j:=\\Ql,\0)=^ A'ely

With

a

/a= jquot;Co(;=l)i?a(l).

0

CO nbsp;nbsp;nbsp;00

/^ = -il/6 � Co(y=:i)/?2(i) Jc2(y = o)/?o.

0 0

00

Iy=jCnj^\)Rb (0).

0

Now (see (25))-/?^ (1) � nbsp;nbsp;nbsp;[Q) = R. Taking into account the

equations (11), we get introducing

CO

!'= I

The second part of (48), (the corresponding operator is called �ei), must be treated in the same way. We obtain expressionsnbsp;entirely similar to (49); we must in (49) only replace

f� r u ^1 9'2 r

where

OO

In =J'dr e-quot;' R Ro-0

Now it is easy to see that

9i g\

2nhc

In fact, R and R^ satisfy the following equations:

R = 0.

Ro = 0.

Subtracting these equations (E Eq = hv\) and multiplying the result by r/2, (50) readily follows.

{F I i3el I 0) - (-F 1 S 'Panel \ 0).

That the matrixelements of (F | S ^exch | 0) indeed vanish in our approximation can also directly be seen. For these are all proportional to

4nhc

and thus are obviously of higher order in the velocities compared to those of �h. Now the matrixelements of �h and Q\\ are of thenbsp;same order, as follows from (52) below, which proves ournbsp;assumption.

Using the right member of (50) to calculate the PE cross section, we find, in the same way as we have derived (42),

.sin^ � . sinOdO^^, (51a) 2 ^

with

On account of (50), (51) and (42) are mathematically equivalent. If one inserts numerical values in these equations, however, onenbsp;meets with a serious discrepancy between the final results sonbsp;obtained which clearly is a consequence of the approximativenbsp;character of the wave functions we have used.

Finally we remark that the first term on the right of (52), arising from Qh is of the same order in the velocities as the second which

arises from Dgi, This completes our proof that the matrix-elements -gt;� -gt;�

of S Pexch vanish in this approximation.

It should be noted that this result is also valid in a pure vector meson theory provided the dipole interaction potential (includingnbsp;cut-off) may be regarded as a perturbation.

74

REFERENCES.

1) nbsp;nbsp;nbsp;J. Chadwick and M. Goldhaber, Nature, 134, 237, 1934.

2) nbsp;nbsp;nbsp;H. A. Bethe and R. Peierls, Proc. Roy, Soc. A 148, 146, 1935.

3) nbsp;nbsp;nbsp;H. Massey and C. Mohr, Proc. Roy, Soc. A 148, 206, 1935.

4) nbsp;nbsp;nbsp;E. Fermi, Phys. Rev. 48, 570, 1935.

5) nbsp;nbsp;nbsp;G. Breit and E. U. Condon, Phys. Rev. 49, 904, 1936.

6) nbsp;nbsp;nbsp;H. V. Halban, Nature 141, 644, 1938.

7) nbsp;nbsp;nbsp;C. MoLLER and L. ROSENFELD, D. Danske Vid. Selsk. math.-fys. Medd., 17,nbsp;fasc, 8, 1940.

8) nbsp;nbsp;nbsp;H. Fr�HLICH, W. Heiteer and B. Kahn, Proc. Roy. Soc. A 174, 85, 1940.

9) nbsp;nbsp;nbsp;N. Kemmer, Helv. Phys. Acta, 10, 47, 1937,

10) nbsp;nbsp;nbsp;H. A, Bethe and R. F, BACHER, Rev. Mod. Phys. 8, 82, 1936.

11) nbsp;nbsp;nbsp;H. A. Bethe, Ann. d. Phys. 4, 443, 1930.

12) nbsp;nbsp;nbsp;C. MoLLER and L. RoSENFELD, in course o} publication.

13) nbsp;nbsp;nbsp;J. A. Allen and N. M. Smith, Phys. Rev. 59, 618, 1941.

14) nbsp;nbsp;nbsp;O. R. Frisch, H, V. Halban and J. Koch, D. Danske Vid. Selsk. math.-fys.nbsp;Medd., 15, fasc. 10, 1937.

15) nbsp;nbsp;nbsp;E. Amaldi and E. Fermi, Phys, Rev. 50, 899, 1936.

CONSPECTUS.

In hujus dissertationis primo capite explicatur theoria jam a compluribus auctoribus elaborata, qua campus gravificus et campusnbsp;electromagneticus in spatio projectivo quintidimensionali complectinbsp;possunt; deinde ducitur expressie generalis tensoris densitatisnbsp;energiae et momenti cujuslibet campi. Eum tensorem legum campisnbsp;impositarum causa symmetricum esse et evanescentem divergentiamnbsp;habere ostenditur, qua secundo loco dicta proprietate energiaenbsp;momenti ac electricitatis conservatie exprimitur. In quarta pata-grapho adhibetur formula generalis ad calculationem tensorisnbsp;energiae et momenti campi Diraciani.

In capite secundo extensione theoriae projectivae ad campos mesicos ostenditur, quomodo in ilia theoria Moller�Rosenfeldiananbsp;de viribus nuclearibus incorporanda sit. Hac tractatione etiamnbsp;minuitur numerus universalium constantium quae camporum mesi.-corum intensitatem determinant, et naturaliter introducitur inter-actio mesonum cum campo electromagnetico. Postquam deindenbsp;ostensum est, quomodo campus electronico-neutrinicus tractandusnbsp;sit, expressione tensoris energiae ac momenti constituta, systematisnbsp;functio Hamiltoniana et electricitatis distributio ducitur. Regulaenbsp;commutationis variabilium campos mesicos describientium in com-pendiosiorem formam traducuntur. Functio Hamiltoniana posteanbsp;transformator separanda longitudinali parte campi electrici et staticanbsp;parte campi mesici. Ejusdem methodi applicatio ad electricitatisnbsp;distributionem transformandam accuratius discutitur et expressionesnbsp;indicantur momentorum dipoli et quadrupoli electrici necnon dipolinbsp;magnetici nuclearis systematis.

In tertio capite photodisintegratio deuteronis et captatio neutronum a protonibus ex theoria Moller-Rosenfeldiana de viribus nuclearibusnbsp;tractantur. Expressionem generalem sectionis efficaciae effectusnbsp;photoelectrici invenitur formaliter identicam esse ei, quae exnbsp;anteriore Bethe-Peierlsiana theoria sequitur, sectionem efficaciamnbsp;photomagneticam autem insuper accessionem ex mesico camponbsp;orientem continere. Sectiones efficaciae energia incidentis photonis

76

crescente celerius minuuntur quam anterioris theoriae ratione. Magnitudines absolutae sectionum efficaciarum ejusdem magnitu-dinis categoriae sunt, atque magnitudines empirice inventae, quam-quam certi numeris expressi effectus nondum dari possunt propternbsp;incertas functiones undarum deuteronis quibus usi sumus. Haec resnbsp;certum judicium de angulari distributione praematurum reddit.nbsp;Sectiones efficaciae captationis eaedem veram magnitudinis cate-goriam habent et, sicut in anteriore theoria, legem 1/u magneticumnbsp;effectum esse apparet.

STELLINGEN

Wil men de mesontheorie zodanig formuleren, dat deze covariant is voor een continue 5-dimensionale groep van transformaties, dannbsp;is van fysisch standpunt de projectieve formulering te verkiezen,

II

In een systeem, bestaande uit een combinatie van een vectori�el en een pseudoscalair mesonveld kan men een rechtstreekse wisselwerking tussen deze velden invoeren, door aan de Lagrange-functienbsp;extra termen toe te voegen. Eist men de in de vorige stelling genoemde invariantie, dan legt dit aan deze termen een groterenbsp;beperking op dan de gebruikelijke eis van invariantie t.o.v. denbsp;Lorentz-groep doet.

III

Metingen van de warmte, die vrijkomt bij een Uraan-splijting zijn niet te interpreteren zonder een gedetailleerde kennis van alle secundaire processen, die op de splijting volgen.

Vgl. M. C. Henderson, Phys. Rev. 58, 774, 1940.

IV

Het is mogelijk, een differentiaalvergelijking op te stellen, die aangeeft hoe een groot-kanoniek ensemble ontstaat.

V

Het vereenvoudigde model van FuRRY voor het kaskade-shower fenomeen kan zodanig verfijnd worden, dat het meer in overeenstemming is met het werkelijke proces. Het aldus verkregen modelnbsp;heeft qualitatief dezelfde eigenschappen als het FURRY-model.

H. Furry, Phys. Rev. 52, 569, 1937.

' '' ^ =::/' •■;■•-nï!. .1: ■• quot;'•■'•, ’-v ■ quot;■

«ojd nbsp;nbsp;nbsp;'nbsp;nbsp;nbsp;nbsp;»

? /■ :,-V '■ • h^l'i'C''-.’

' nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;;ïU:''' vivJiteio; _nKi'!

rgt;:. nbsp;nbsp;nbsp;;;5fJ^^lS :j;-ijT ^ ;rn;;.v

.^V ■.

VI

De berekeningen van Hafstad en Teller tonen aan, dat men uit de gegevens omtrent de bindingsenergie�n en spectra van lichtenbsp;atoomkernen niets met zekerheid kan zeggen over de bouw van dienbsp;kernen.

L. R. Hafstad en E. Teller, Phys. Rev. 54, 681,

1938.

VII

Het is gewenst de werkzame doorsneden voor de foto-desinte-gratie van het deuteron te meten voor zeer hoge foton-energie�n.

VIII

Dirac�s opvattingen betreffende een �mathematical quality in nature� is onaanvaardbaar,

P. A. M. Dirac, Proc. Edinburgh Soc. 59, 122, 1939.

IX

Men kan de definitie van ,,pool van een punt t.o.v. een punten-paar op een rechte� uitbreiden tot ,,poolfiguur van een p-dimen-sionale vlakke ruimte t.o.v. een lineair k-stelsel hyperkwadrieken in een n-dimensionale ruimte�, (p^n). Het aantal dimensies vannbsp;deze laatste figuur isn� \k � p| � 1, (n� 1^1^ � pj), zijn graadnbsp;(^ 1)!nbsp;nbsp;nbsp;nbsp;(P 1)!

(k�p 1)! p

X

Het is zeer gewenst, dat voor candidaten in de natuurkunde hetzij een college over ,,algemene methoden in de fysica � gegeven wordt,nbsp;hetzij in seminaria of colloquia aandacht aan dit onderwerp wordtnbsp;besteed.

XI

Het verdient aanbeveling om in wiskunde-schoolboeken de uitdrukking ,,bewijs uit het ongerijmde� te vervangen door ,,bewijs van de ongerijmdheid van het tegendeel�, (zo men deze methode alnbsp;wenst te handhaven).

Ó3!^,.'!öigt; .jsfjC'fr nbsp;nbsp;nbsp;'' quot;■. ;V..vnbsp;nbsp;nbsp;nbsp;:v. y/'

■sJÏÏrjiU tr.i'v', ’'. : . . '■■riJr.ygt;

■»■■/'■.»(}, sïi .iw , nbsp;nbsp;nbsp;■ gt;' •nbsp;nbsp;nbsp;nbsp;•■■'»gt;' 'jï lt;''’V

I ■ nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;-- -nbsp;nbsp;nbsp;nbsp;■ .nbsp;nbsp;nbsp;nbsp;’»•■nbsp;nbsp;nbsp;nbsp;...