A causal interpretation of the pauli equation (A)

S U P P L E M E N T O AL V O L U M E I , S E R I E X D E L NUOVO C I M E N T O N. ], 1955

A Causal Interpretation of the Pauli Equation (A).

D. BoH~, R. SCHmLE~ and J. TIolv~No (*)

Faculdade da Filoso]ia, Ci~cias e Letras, Universidade de S~to Paulo - S~to Paulo, Brazil

(ricevuto il 9 0ttobre 1954)

CO~TE~TS: 1. Introduction. 2. Introduction of ((Intrinsic Spin)) in Hydrodynamical Model. 3. Kinematic Description of Rotations. 4. Ca- nonical Relations for the Cayley-Klein Parameter. 6. Relationship Bet- ween Spin and Velocity.

Introduotion.

I n various previous papers (1-8), several new in terpre ta t ions of the non- relat ivist ic q u a n t u m theory wi thout spin have been proposed. Al though these new in terpre ta t ions differ in various details, t hey all have in common t h a t t hey explain the q u a n t u m theory in t e rms of continuous and causally deter-

mined mot ions of various kinds of entities, such as fields and bodies, which are assumed to exist object ively at the microscopic level. Thus far, those

in te rpre ta t ions which have been carried far enough to demons t r a t e in full

(*) Now at Centro Brasileiro de Pesquisas Fisica, Avenida Wenceslau Brs 71, Rio de Janeiro, Brazil.

(1) L. DE BROGLIE: Compt. Rehd., 183, 447 (1926); 184, 273 (1927); 185, 380 (1927). (2) ]~. MADE]LUNG: Zeits. ]. Phys., 40, 332 (1926). (3) D. BOH~: Phys. I~ev., 85, 166, 180 (]952); 89, 458 (1953). (~) T. TAKABAYASI: Prog. Theor. Phys., 8, 143 (1952); 9, 187 (1953). (5) I. FEI~YES: Zeits. ]. Phys., 132, 81 (1952). (6) W. WEIZEL: Zeit. ]. Phys., 134, 264 (1953); 135, 270 (1953). (7) M. SCH~NBERG: NUOVO Cimento, 11, 674 (1954). (s) D. B o I ~ and J. P. VIGIER: Phys. Rev., 96, 208 (1954).

A CAUSAL INTERPRETATION OF THE PAULI EQUATION (A) 49

detail their ability to explain causally all of the features of the Schr6dinger equation without spin have followed one of two general lines of approach.

The first of these general lines, initiated originally by DE BrtoGLI~. (:), and later carried to its logical conclusions by one of the authors of the present paper (3), involves the notion that the Schr6dinger wave function, T(x, t), represents an objectively real but qualitatively new kind of field of force that influences the motion of a body, which latter has a well defined location, ~(t), varying continuously and in a causally determined way with the passage of time. This new field of force produces no important effects at the macroscopic level, but at the atomic level it is, as has been shown in various papers referred to above, able to explain the characteristic new quantum-mechanical properties of matter, which manifest themselves strongly only at this level.

The second general type of causal explanation of the quantum theory is along the lines of the hydrodynamic model i proposed originally by MADE- L~J~G (2), and later extended by TAKA]3AYASI (~) and by SC~6~]3ERG (7). In this model, I~ff[~ represents the density of a fluid, while VS/m represents its local stream velocity (where h u = R exp[iS/h]). In another paper (8), this model has been completed with the aid of the postulate that there is a stable particle-like inhomogeneity in the fluid that moves with the local stream velocity. This inhomogeneity plays a role analogous to that played by the body in the interpretations initiated by I)E Bn.OGHE, while the fluid plays a role

analogous to that played by the T field. In the present paper, we shall find it convenient to work in terms of the

hydrodynamic model, which we shall extend with the aid of various new pos- tulates to the Pauli equation describing an electron with spin. We shall therefore now summarize a few important aspects of the hydrodynamic model without spin, in order to facilitate the description of the nex features that are needed for the treatment of spin. We first write Schr6dinger's equation in the well-known form

(1) ~-~ div\ m/~-- O,

8S (VS) 2 {~ V~R 0 (2) ~ - / + ~ m - + V - - U m R - "

In terms of the Madelung interpretation~ equation (]) then represents the conservation of fluid, while (2) represents the equation which determines the velocity potential, ~q, in terms of the classical potential, V, and the (< quantum potential >> U = - - (~/2m)V2R/R • (h2/4m)((V~)~/2~ 2 - V~/~). As shown by TAKABAYASI (4) and by SC~I6~]3ERG (7), the quantum potential can be inter- preted in terms of a kind of internal stress in the fluid, which depends not,

5 0 D. B O H ~ , R. SCHILLER a n d J . TIOMNO

however , on r itself as is the case with pressures and other in ternal stresses

in an ordinary macroscopic fluid, bu t ra ther , on der ivat ives of Q. In one of the papers referred to previously (s), the addi t ional assumpt ion

was made tha t the fluid is in a s ta te of i r regular f luctuat ion resembling tu rbu len t

mot ion, in which the ac tual densi ty ~, and the ac tual veloci ty v , f luctuate

more or less a t r andom around IT] 2 and V S / m respect ive ly as means. I n these fluctuations, it is permissible to assume tha t v has vor t ex motions, which

however average out to zero. As a resul t of these fluctuations, e lements of

fluid on any one of the mean lines of flow associated with the Madelung fluid

are cont inual ly moving in an i rregular way to other lines of flow, and this

process tends to produce a more or less r andom (< mixing ~> of the fluid. Like

one of the fluid elements, the body-l ike inhomogenei ty, which is carried along

b y the fluid, follows a ve ry i rregular path . I t is then quite easily shown tha t a s ta t is t ical ensemble of such sys tems with an a rb i t r a ry probabi l i ty density,

P ( x ) , of inhomogeneit ics eventual ly decays into one with P = I T I 2. Thus, the

i rregular f luctuations in the motions of the Madelung fluid provide a model explaining in a na tura l way a possible physical origin for the s tat is t ical dis-

t r ibu t ions of the quan t um theory (9).

2. - Introduction of "Intrinsic Spin" in Hydrodynamical Model.

I n the present paper and in a subsequent paper , we shall ex tend the hydrodynamic model to a t r e a t m e n t of the Paul i equat ion for a spinning electron.

Now, the Paul i equat ion is (lo)

(3) i]~ ~t - - 2 m ~7 - - ie c ~-~a + V~P, + 2 m e ((; '$~)Ta ,

where T~ represents a two component spinor with index, a, while A is the

e lectromagnet ic vector potential , V, the to ta l external potent ia l energy (electro-

magne t ic and otherwise), and t~ the magnet ic field.

As in the case of the Madelung model of SehrSdinger 's equation, we mus t

(9) In a previous paper (see third paper of reference (3)) a similar theorem was proved for the model in which the W function is assumed to represent a field of force. In this case, the random fluctuations come from perturbations arising outside the system under investigation.

(lo) We do not include the spin orbit coupling term, ~ . p • e, because a consistent treatment of this term requires a relativistic theory, since it is of the same order of magnitude as the Thomas precession. This term must be dealt with in terms of a causal interpretation of the Dirac equation.

A CAUSAL INTERPRETATION O~ THE PAULI EQUATION (A) 51

define a fluid densi ty and a fluid velocity. Iqow, it is well known tha t the Pauli equat ion admi ts a conserved charge and current given (except for a factor

of c) b y

(4-a) ~ = T ' T ,

h c (4-b) j = ~ ( T * V T - - ~PVT*) - - - c A .

Wri t ing j = or, we m a y then define the veloci ty as

h ( T * V T - - T V T * ) e A (4-c) v - - 2mi T * T - - e "

I t is then evident ly consistent to assume tha t the fluid densi ty is given by ----T*7 t, since the conservat ion equat ion obta ined f rom the Paul i equat ion

can then be in te rpre ted as describing the conservat ion of fluid.

and v do not, however, pe rmi t a complete definition of the physical meaning

of the 7 j funct ion; for since 7 t is a spinor, it contains four independent quant-

ities, as indicated below

(5) T = § id '

where the a, b, c, d are real. On the other hand (4-a), defines only ~ = a % ~ + c 2 § 2 while (4-b) defines only the der ivat ives of the a, b, c, d.

I n order to define all par t s of T more direct ly in t e rms of physical pro-

pert ies of the fluid, we shall therefore have to assume some new propert ies for the fluid. Now, since the Pauli equat ion deals with the electron spin, it

seems na tu ra l to assume tha t our fluid should have a new proper ty , which we m a y call an in t r ins ic angular m o m e n t u m connected with the spin. B y this,

we mean tha t the to ta l angular m o m e n t u m densi ty of the f luid should include, in addit ion to the <( orbi ta l ~> contr ibut ion of m o ~ ( r x v ) , an addi t ional contri- but ion depending on some pa rame te r s connected with the in ternal motions

of each fluid element. To obta in a possible physical p ic ture of where such an

intrinsic angular m o m e n t u m could come from, we m a y suppose, for the sake

of discussion, t ha t the fluid is const i tu ted of molecules, and tha t the spin

mot ion of the const i tuent molecules m a y contr ibute to the to ta l angular mo-

m e n t u m of the system. On the other hand, the same result could be achieved

in a much more general way. For if the fluid has an inhomogeneous structure,

then these inhomogeneit ies will in general have a certain inertia. I f a fluid

e lement is turning, the inhomogeneit ies will tu rn with it and contr ibute to

52 n . BOHM:, R. SCHILLER &lld g. TIOMNO

the to ta l angular momentum. Such inhomogeneities might, for example, be stable or semi-stable pulse-like s tructures formed in the fluid itself, or t hey might be small highly localized vortices or eddies. As far as our purposes in

this paper are concerned, however, the origin of the intrinsic angular mo-

men tum of the fluid is irrelevant. All tha t is re levant is the ~ssumption tha t , for one reason or another, such an intrinsic angular m o m en tu m exists. In order to have a convenient model in terms of which we can work, however, we shall assume in this paper tha t the intrinsic angular momentum is due to the tl lrning of ve ry small qnasi-rigid bodies of which the fluid

is supposed to be const i tu ted (just as ordinary macroscopic fluids are

cons t i tu ted of molecules). We shall see tha t this simplified model is adequate

for giving a causal explanat ion of the Panli equation. On the other hand,

there is no reason inherent in the model why the bodies must always be rigid; or, indeed, even why the inhomogeneit ies must necessarily be regarded as

arising in dist inct bodies out of which the fluid is supposed to be consti tuted. Nevertheless , under the conditions in which the Pauli equation applies, we shall assume t ha t the fluid acts so near ly as if i t were composed of dist inct rigid spinning bodies, tha t we may use this model as a simplifying abst ract ion as one, for example f requent ly simplifies the t rea tment of atoms by replacing them by idealized mass points even though they really ha.re finite sizes.

3. - K i n e m a t i c D e s c r i p t i o n of R o t a t i o n s .

We shall now proceed to develop a kinematic description of the ro ta t ions of a body, which is par t icular ly easy to apply to the Pauli equation, because it works in terms of spinors of the same kind as appear in this equation.

The first problem is to specify the state of ro ta t ion of a body. This can be done in terms of the three Euler angles, 0, ~v and ~v, where 0 represents the angle of the principal axis (1) with a Z axis fixed in space, ~ represents the angle tha t the project ion of th is axis makes on the X-Iz plane, and V represents the angle of ro ta t ion about the principal axis (1) relat ive to the intersections

of the plane of the principal axes (2) and (3), with the X-~ ~ plane. We wish now, however, to connect these angles to a spinor. To do this,

let us imagine tha t we always s ta r t with t h e body in a s tandard orientat ion,

in which its principal axis (1) is directed along the Z axis, which we have fixed in space. W e then make a rotat ion, R(O, ~, y;), which carries the body from

its s tandard orientat ion t'o its actual orientation. This rota t ion can be car r ied out in three steps. First , we make a rotat ion, Rl(~v) through an angle ~v about the Z axis fixed in space (which is at this t ime also the principal axis (1) of the body). We then make a rota t ion R2(O) about the X axis, of an angle, 0. Then we make a ro ta t ion Rs(~v) again about the Z axis fixed in space (but this

A CAUSAL INTERPRETATION OF THE PAULI EQUATION (A) 5 ~

t ime the principal axis of the body is no longer parallel to the Z axis). If the reader will draw a diagram, he will readil ly convince himself tha t these rotat ions give just those described by the Euler angles (11).

Le t us now e~rl T out these operations mathematicul ly . We shall ~ssume tha t the s ta te of the body at rest is to be represented by what we shall call

the s tandard unit spinor (12)fi0 == (10). When we c~rry out the rotat ions des-

cribed above, we shall see tha t a ro ta t ion involving a rb i t ra ry angles~ 0, ~ ~,

can lead to an a rb i t ra ry unit spinor (such tha t f l*fi~-]). Thus, uny unit spinor

can be in terpre ted as a specification on the angles of ro ta t ion of a body. I n terms of spinor notat ion, the first ro ta t ion R~(yJ) is represented by the mat r ix

exp [iaz~fl2] ~-- cos ~/2 ~ i(~z sin ~f12 .

Applying this to the s tandard unit spinor, we get

(6) R l . ( y ~ ) f i o - ~ e x p [ i a z y ~ / 2 ] ( ~ ) = exp [iyJ/2] (10) .

We now apply the rotat ion, R2 ---- exp [iaxOl2] - - cos 02 + iax sin 0/2. We obtain

cos (}/2 (7) R~Rlflo : exp [i~12] i sin 0/2

Applying Ra ~ exp [iazq~/2)], we get

i sin 0t2t/1" ~ [ cos 0/2 cos 0/2] \0] ~-- exp [iy~/2] \ i sin 012] "

tcos 012 e x p [ i ( y J + q J ) 1 2 ] ) : ( f i l ) (8) fl = Rfio = ~i sin 0/2 exp [ i (~ - -~ ) /2 ] ] fi~ "

Writ ing

bl ~- ib2~ (8-a) I~ : b3 ~- ib~] '

(11) For a Clear diagram o3 the Euler angles, see H. GOLDSTEIN: Classical Mechanics (Cambridge Mass., 1950), p. 107.

(13) It will become clear that the spinor that we associate to the standard orientation is arbitrary, and that all that is important is the relationship among spinors corresponding to different orientations. We have chosen the above standard spinor for the sake of simplicity.

54= D. BOHM, R. SCHILLER a n d J . TIOM[NO

we get

(8-b) [ bl = cos 0/2 cos (y, + q~)/2,

] b3 = - - sin 0/2 sin (~v q- ~v)/2,

b~ = cos 0/2 sin (~p -b q~)/2,

b4 = sin 0/2 cos (~f - - ~)/2 �9

The bi ev ident ly sat isfy the relation,

(9) b~ + b ~ q - b ~ - q - b ~ : ] .

The b~ are jus t the well-known Cayley-Klein pa rame te r s of the ro ta t ion group (13) ;

of which ev ident ly only three are independent . B y specifying the b~ we can

solve for the Euler angles, 0, ~, W. Indeed, the specification is two-valued, in the sense t ha t there are two values of the b~ for each set of Euler angles.

Thus, we see t h a t a uni t spinor can be in te rpre ted as defining the or ientat ion

of a body in space. We shall therefore t en ta t ive ly in te rpre t the spinor appear ing in the Paul i equat ion in this way, and then show tha t a consis tent in terpre t - a t ion can so be ob ta ined (14).

4. - Canonical Relat ions for the Cayley-Kle in Parameters .

Now, we shall set up a classical canonical formal ism, in which the Paul i equat ion is used to define the equations of mot ion of the Cayley-Kle in parameters , and therefore the equations governing the rota t ions of the bodies mak ing up the fluid.

Our first step is to write down a classical hami l ton ian f rom which the

Paul i equat ion can be derived. This hami l ton ian is formal ly the same funct ion of the spinor T as t ha t which appears in the ha mi l tonian f rom which the

Paul i equat ion is derived in the usual fo rm of the q u a n t u m theory . This is

~me T o ' ~ P ] dx .

The above hami l ton ian will lead to the correct equat ion for T provided t ha t

we assume the classical Poisson bracke t relat ions

(]1) ~a0 6(X-- X') [T:(X'), T0(*)] = ~

(la) See H. GOLDSTEIN: Chap. 4. Note that we have rotated from body to space axes, while CTOLDSTEIN rotates from space to body axes.

(14) Spinors have already been used to describe the orientation of a rigid body. See, for example, H. B. G. CASBIIE: Rotation Of a Rigid Body in Quantum Mechanics (The Hague, 1931).


For then we obtain just the Pauli equat ion

i h ~ = ih[H, T~] -- 2m --ie T~ + VT~ + 2mine (~'@T)~

where we have used the fact tha t the Poisson bracket of ~ with an integral such as H is defined with the aid of t h e well-known funct ional derivatives.

Thus

(12) [H, F] - - ~T, ~ 7 + ~T, ~T~' + ~'~* ~'~ + ~T* ~T."

In order to simplify the t rea tment , we now split the whole space into volume elements so small tha t T does not change appreciably within them. We may then define T(xm) as the mean value of T in such a region, centering on the point x~. By integrat ing eq. (11) over small regions of x and x ' . centered respect ively at x~, and x~ and dividing by (/IV) 2, we get

~o~ ~(x+,~, x,) [Ta*(X,~), T~(X~)] -- i~ (AV)

We shall now find it convenient to introduce a new spinor,

(13)

We obtain

(14)

We then write

(15)

z = @ T .

[Z~(Xm), Zb(Xn)] ~ . - - i f ~ ( X , m , X n ) a , b .

1 ((al + ia2) t

From the Poisson bracket relations, (14), we then deduce tha t

[al, a2] - [as, a,] - - 1 , (16) ~ [a,,a3] = [a2, a 4 ] = [a2, a s ] = [a2, a a ] = 0 .

Thus al is the momen tum canonically conjugate to a2 and as is canonically

conjugate to a4. The as are evident ly proport ional to the Cayley-Klcin para-

meters , b~ (given by eq. (8-a)). Thus, we have found the canonical relations among the ai, implied by the Pauli equation.

56 D. BOHM, R. SCt t ILLER and J . T IO]gN0

5. - De f in i t ion of Spin Ang u l a r M o m e n t u m .

I n order to m o t i v a t e f u r t he r s teps in the i n t e rp re t a t i on of the Pau l i equat -

ion, let us recal l t h a t in the usual q u a n t u m meehanics~ the to t a l spin is g iven

by (~/2)/T*aTdx. This suggests t h a t (]~/2)(T*aT) is a spin densi ty , and

t h a t S = (~/2)(T*t~T)AV is t he to t a l spin in the smal l e l ement of vo lume AV. Expres s ing T in t e rms of Z t h r o u g h eq. (13), we ge t

(17) S = �89

The to ta l n u m b e r of par t ic les in this region is jus t

(18) A_~ = T * T A V = Z * Z / ] i .

Thus , t he spin per pa r t i c l e is

(19) s - - 2 Z*Z 2 T ' T "

We see t h e n t h a t the m a x i m u m spin per b o d y in a g iven d i rec t ion is h/2. This is in a g r e e m e n t wi th w h a t one gets for t he spin (( obse rvab le ~) in the usual

q u a n t u m t h e o r y (1~).

W i t h the aid of the P .B. re la t ions (11), i t can be shown be means of a

s imple ca lcula t ion t h a t t he componen t s of t he spin vec to r , S, sat isfy t he cycl ical

relat ions, charac te r i s t i c of angular m o m e n t a

(2o) [s~, ,%] = ~ , .

Thus, t he Po i s son-Braeke t re la t ions o b t a i n e d in the de r iva t ion of the Pau l i

equa t ions f r o m a h a m i l t o n i a n are jus t w h a t are needed to lead to t he cor rec t

P .B. r e l a t i ons for t he angu la r m o m e n t a defined in eq. (]7).

(~s) In our theory, the spin vector is a continuous variable, with arbitrary projections on any axis, and with a total magnitude of s2=~2/4 (as can easily be proved by using the Pauli identities). On the other hand, the spin ~( observables )) appearing in the usual quantum theory may have only certain discrete projections on any given axis, while the magnitude of the r observable )) for the total spin is ~h 2. The origin of this difference arises in the circumstance tha t in the model tha t we are proposing here, the spin c( observable )) is, as we shall see later, a kind of statistical or over-all property of the motions of the bodies in the fluid. Thus, there is no reason why it should be identical with the angular momentum of one of the bodies in the fluid, although there will in general of course exist a certain connection between the ~ observable )) and the spins of the bodies, which we shall discuss in more detail in the subsequent paper.


J-(al§ t contains only In the present theory, the basic spinor, Z ~ 2 a3 § ia~l

two canonically independent variables. We shall now see the physical meaning of this reduct ion of the number of basic variables to two. To do this, we find it convenient to go to a new set of canonically independent variables which we take to be two of the Euler angles, ~ and F. In terms of the a~, these are (as can easily be seen from equ. (8-b))

(21) -1 a2 tg 1 a~ F ~ t g -- § ; a l ~s

~0 = tg -1 a2 __ tg_ 1 a4 . a l - a s

By means of a simple calculation, we prove tha t the quan t i ty canonically conjugate to --yJ/2 is ~h, which is proport ional to the to ta l number of particles in the region (see equ. (18)), while the quant i ty canonically conjugate to - - @ 2 is Sz, which is equal to the Z component of the to ta l angular mo-

men tum of these particles. Thus, we have

a ] + a s + . + a ] ) a~ (22-a) ~ ' - - = 2 ' tg-1 § -~ at = 1 ,

~"~1

(22-b) i 1 [ -( ' ~ 1 7 6 tl S z , - - ~ ~ t g -1 - - t g -xgzl --- 1 . a, as/ l

But now we can show by a simple calculation tha t ~ is also equal to the ~ngle ~' made by the project ion of the angular momen tum vector S on the X - Y plane. Thus

i /a l - - ias~( 0 l~(a~ § ia2 t ~[1"(7,111 [as ia4]\--1 0/\as § ia~/= a2a4- alct3.

tg q)'-- ~P*a~!P-- (a l - - iast[O 1]{a~ ia~] a~as § a2a4 a3 ia~]\l O]\a3 ia4/

Now, from eq. (21)

t g ~ = ~ - - ~ a~a4/ alas § a2a4

Thus, we see tha t ~ ' = ~. ~ioreover. the co-lat i tude angle, 0', of the angular 2 a 2 2 2

momentum vector~ is defined by cos 0'-- S~ _ al § 2 - a 3 - a4 But from 2 ~ ~2 �9 S a t § s § s+a~ equ. (8-b)we can calculate the angle 07 made by the principal axis (1) of the

2 s a s ~ a l § a 2 - - 3 body with the Z axis. This is given by cos 0 = a~§ a ss§247 Thus,

0'=~ 0.

,~8 D. BOHM, 1~. SCHILLER a n d J . TIOI~lqO

We conclude then tha t in the Pauli theory, the angular momentum, S, is always pointing along the principal axis (1) of the body. Such a special or ienta t ion of the angular momen tum cannot be mainta ined for the most general kind of torque tha t may act on the body. In the nex t section and in a subsequent paper, we shall see, however, t ha t the Pauli equat ion implies special kind of <( quantum-mechanical torque ~ tha t permits this condit ion to be m~intained as ~ consistent subsidiary condition.

The special relat ion between the direction of S and the principal axis (1)

is what permits us to reduce the number of independent canonical variables in the theory. When this relat ion exists, our theory becomes equivalent to

a theory of the angular momen tum of a point dipole, which has a l ready been t rea ted by several other authors (~6-1s). In our subsequent paper, we shall

see, however, tha t the theory developed here can be generalized to t rea t cases

in which S does not necessarily point along the principal axis.

6. - Relationship Between Spin and Velocity.

Let us now recall tha t equ. (4-c) defines the veloci ty of a part icle in terms of our spinor. Since the spinor is a l ready in te rpre ted in terms of the or ientat ion of a spinning body, eq. (4-c) clearly implies a certain relat ionship between the spin and the velocity. We shall now s tudy this relat ionship and show

tha t i t can be unders tood in terms of a reasonable physieai model. We begin by expressing our spinor (with the aid of eq. (7)) as

/cos 0/2 exp [i(~ + ~)/2]~ ~P • R [i sin 0/2 exp [i(F --~)`/2]] '

where R is real.

Using this value of 5 u in eq. (4-c), we readil ly get

h e A (23) v ---- ~mm (V~f + cos0Vq~)--c "

The above, is however, an expression of the veloci ty in a form tha t has

already long been familiar in classical hydrodynamics . Indeed, in classical hydrodynamics (19), it is shown tha t an a rb i t r a ry veloci ty field can always

(16) G. F. UHLENBECK and S. GOU])S~tTH: Nature, 117, 264 (1926). (17) H. A. KR~,M~RS: Quanteniheorie des Elektrons, (Leipzig, 1938). (is) C. )/I. LATTES, !~/[. SCHENBEI~G and W. SCHUTZER: Anais da Academia Brasi-

leira de Ci~ncias, XIX, no. 3 (September 1947). (1~) See, for example, H. LAMB: Hydrodynamics (Cambridge, 1953), p. 248.

A CAUSAL I N T E R P R E T A T I O N OF THE PAULI EQUATION (A) 5 9

be expressed as

(24) V

v = VS + ~V~--~ A,

where ~ and ~ are scalars, called the Clebsch parameters , and S is also a scalar. The curl of the veloci ty is then

(25) V x v : V ~ x V~ - - _e V x A .

(This is quite a different expression f rom the more common form

v = V S ' + V • in the sense tha t S' and S are different functions, since div (~V~7) is not in general equal to zero).

The Clebseh parameters have recent ly been used by I)]RAC (2o) and and others (21) to t rea t ~ theory of a classical electrified fluid ether. They have also been used by TAKA]~AYASI (4) to t rea t possible vortices in the Ma- delung fluid, and by S c r [ 6 ~ g c ~ (7) to t rea t vor tex motions in a more general context .

In order to establish a basis of comparison with the Pauli theory, we shall first t rea t the ordinary classical hydrodynamics of a charged fluid, capable

of maintaining a pressure gradient, in terms of the Clebsch parameters. One of the advantages of these parameters is tha t they permit the formulation of hydrodynamics in te rms of a variat ional principle. Indeed, if ~ is the density of the fluid, then the hamil tonian is just equal to the to ta l kinetic plus potent ial energy of the fluid (including the potent ia l energy due to compression)

(26)

where A is the vector potential , ~, the scalar potential , e/m the ratio of charge to mass for an element of the fluid in question, and ](~) defines the pressure through P = 8]/8~.

Wow if we adopt essentally the same Poisson Brackets as were derived f rom the Pau]i theory ; i.e.

(27) [e, ~ ] = ~(x - - x ' ) , [(e~), V] - - ,~(x--x'),

(20) p. A. M. DIRAC: P roc. Roy. Soc., A 209, 291 (1951); 212, 330 (1952); 223, 439 (1954).

(21) See, for example, O. BUNEMAN: Proc. Camb. Phil. Soc., 50, 77 (1954).

~ 0 D. BOHI~I, R. SCHILLER and a. TIOMNO

then we obtain the following equations of mot ion

,(2S-a) ~e ~t

3S (28-b) ~t

(~s-c) ~7

,(~s-~) ~7

+ d i v ~ V S + ~ V U - A : ~ - - ~ d i v ( ~ v ) - - - - O ,

---- ~(v-V)~? + (VS -J- ~V~ - - (e/c)A) ~ e OJ 2m + m ~ + ~ = 0 '

d~ 0 + (v.V),~= d~-= '

du + (v .V)u : d-t- = 0 .

Combining (28-b) and (28-d) we get,

Eq. (28-a) is just the equat ion of conservat ion of fluid. Eq. (28-b) (and therefore also (28-e)) is a generalization of the equat ion for the veloci ty po- tentiM, S~ whieli enables us to tMte vor t ie i ty into account ( through ~ and U), Eqs. (28-c) and (28-d) are very interesting, for they say tha t iJ we ]ollow a

moving f luid element, ~ and U are constants of the mot ion (but not at a point fixed in space). Thus, the surfaces, ~(x, y, z, t) = eonst, and ~ =- ~(x, y, z, t) = eonst., define the tubes of flow of the fluid.

To unders tand the motion in more detail, let us focus our a t t en t ion on eq. (28-e). This is ~nMogous to ~ Hami l ton-Jacobi equation, but the equat ion

involves, in addit ion to the electromagnetic potentials, and the pressure potent ia l (P = ~J/3~), a set of equivalent potentials tha t are functions of the Clebseh parameters (~). In fact, the Clebseh parameters contr ibute an ad- dition of 2(Ou/Ot ) to the scalar potent ia l a n d - - ~ V ~ to the vector potential. We m a y thus conclude tha t formal ly they contr ibute to the equat ion of mot ion of a fluid e lement a <~ pseudo-Lorentz force ~>, given by

(29) F ' : e' + v • ,9'

where the <~ pseudo-electric ~> and <~ pseudo-magnetic ~> fields are given

~'=--v f + ~ (#v,;) =~- v~-- ~t vf,

~9'= -- v • (~v~) : -- v~ xV~;.

(32) We are using here an argument due originally to M. SC}IONBE~G: Nuovo Ci- mento, 11, 674 (1954).

h CAUSAL INTERPRETATION OF THE PAULI EQUATION (h) 61

Thus, for the addit ional force, we get

(30) F ' = ~vv ~v ~- --T{V~-- v•215 �9

Writ ing - - v • (V~ • V~;) : (v" V~) V~; - - (v. V~;) V$, we obtain

+(v.V)~-- --(v.V)~:~v~-- v~.

Since from (28r and (28-d) we have d$/dt = 0 and d~l/dt = O, we get /v'_-- 0. Thus the Clebsch parameters formally add to the vector potent ial a set of

quanti t ies tha t produce a pseudo-electromagnetic field for which the pseudo- Loren tz4orce vanishes. Therefore the Clebsch parameters appearing in eq. (28-e) do not al ter the equations of motion of a fluid element (2a). What

they do is to enable us to consider in a canonical formalism an ensemble of trajectories for the fluid elements which have a vor t ic i ty tha t does not come f rom the effects of the electromagnetic vector potential , A. Thus, they make possible a generalization of the Hamil ton-Jacobi theory ; for in the latter , we consider only ensembles in which m y - (e/c)A is derivable from a potent ia l (equal to the action function).

Let us now re tu rn to the Pauli theory. Our first problem will be to compare the classical hamil tonian (26) with the Pauli hamil tonian (10). By means of a simple calculation, we obtain the restilt tha t the Pauli hamil tonian

can be wri t ten as the sum of the following three terms:

where we have

(31-a)

H = H~ + H~. + H~.

H ~ = ~ e ' § - - c A d x ,

f h2 (Ve')~ dx {31-5) H q . = ~ O'

.(31-e) He , .= dx ~ (((V0) 2 + sin 20(V~v) ~) §

4- e ~ (cos 0~2 + cos 0 cos ~g)~ + cos 0 sin ~,9~] ' 2 m e 1

where ~ is the magnetic field.

(23) Note that the fact that dS/dt=d~/dt=O plays an essential role in making this result possible. Later, in connection with spin theory, we are going to have non-zero values for these derivatives; and then, the Clebsch parameters will imply additional terms in the force on a particle.

5 - S u p p l e m e n t o a l N u o v o C i m e n t o .

62 D. BOH:M, R. SCHILLER and a. TIOMNO

In the above equutions, we have, as in eq. (4), defined O ' = [ T [ 2 = ~, where ~' is the densi ty of bodies.

In order to compare the Pauli hamil tonian to the simple hamiltonian, t tc , of classical hydrodynamics, we note tha t ~' is canonically conjugate to --y , /2 , (~' cos 0) to --~o/2. Thus, if we replace ~ by ~', and write

(32) ~ = cos O, ~/h = - - ~v/2 , S/]~ = - - y,/2

then H~ becomes equal to kinetic energy t e rm in eq. (26).

The terms tha t are left over in the Pauli equa t ion are then Hq. and Hs~..

We shall see in eq. (34-b) tha t H leads to the (, quantum-potent ia l ~> te rm of the Madelung fluid. Thus, the two terms, Hr+H~. represent the energy of

a charged Madelung fluid, in which vor tex flow as well as potent ia l flow arc taking place.

As for the th i rd term, H~. , i t is clear first of all t ha t i t contains a par t describing the energy of a spinning dipole in a magnet ic field ~. The magnetic moment of this dipole is et~/2mc.

I f we assume tha t our bodies have a radius of the order of the classical electronic radius, r o = 2 . 8 . J 0 -13 cm, then the magnet ic momen t is #=2erov/C where v is the veloci ty of ro ta t ion of the body at its periphery, and where 2

is a constant of the order of un i ty depending on the charge distr ibution in the body. We then obtain

v h 137

c - 2reCto- 2

Thus, the body nmst, at its periphery, be moving much faster t han light In a non-relativistic theory, such as we are discussing in the present paper, this, of course leads to no difficulties. In a la ter paper, concerned with the

extension of the theory to the Dirae equation, we shall see tha t the spin

motion becomes closely coupled with the mass motion of the fluid, so tha t a

spin angular momen tum of eh/2mc for the body is possible, even when the

speed of its per iphery does not exceed tha t of light, because the fluid motions

are fur ther connected with the body rotation's by a spin-orbit coupling, not present in the non-relat ivist ic theory.

As for the remaining par t of Hs~., this represents an interact ion between the directions of the spins of neighboring bodies, which functions essentially as a spin-dependent addit ion to the (( quan tum potent ia l )>. To obtain a more definite model for this pa r t of the spin energy, we m ay let 6A represent the

to ta l angle between ~wo spin vectors, separated by a distance, 6x. Then, by

going to spherical IJolar coordinates in the space of the spin vectors , we can

A CAUSAL INTERPRETATION OF THE I~AULI EQUATION (A) 63

show tha t

I ~A I'= I VO. 8x I% sin' o t ~7qo. 8x I'

Thus, the energy, H,~. is just an average of ] 8A 12, taken over a small region tha t surrounds the body and weighted with the density, 5. We can therefore interpret H~. physically as the result of an assumed short-range interaction between bodies, which tends to line up the directions of their spins. For example, the energy of two neighboring bodies could be proportional to

(33) U{.t. ---- - - cos (0A) _~_ - - 1 + (0A)' /2.

Le t us now obtain the equations of motion tha t follow from our hamiltonian. To do this, it will be convenient to write H as

+/.,...., where,

H. ---- 2m [(vo) .+ sin. o(vq~) 2] -~ e s.~ m e

[ ] .

The equations of motion then become

(34-a) --Sq--+div q'v = 0

bS (34-b) 8t ~(v .V)~ +

d~ e 8H. (34-c) d t - &] '

dt O~

2 2m< ~' 2 (d)z] + R ' : ~

Eq. (34-a) is then just the conservation equation while eq. (34-b) is the pseudo H~milton-Jacobi equation for a fluid with the usual quantum potential,

v.r 2m e' 4m ( ~ ] ' and a spin-dependent quantum potential, Hs--

-- ~(v. Vb.

Eqs. (34-e) and (34-d), which tell us how the spin directions change with t ime are in precisely the same form as tha t of the equations expressing the

64 D. BOtI1V[~ R. SCHILLER and a. TIO~NO

torque acting on a point dipole, derived previously by Sc~6~]3]~G (18) in a purely classical problem.

We may now, however, obtain another instruct ive form of t h e equations for t h e spin variables by noting tha t H~. can be wri t ten in the form

(35-a) H~,. -- 2m . \8Zj ] + mc

21S I, we also obtain the expression ~qoting tha t ~ - - h

(35-b) . o + = +

We can obtain the equations of motion for s, by writing ~ = S/~, and using the P.B. relations for the angular momentum. We get (with the aid of eq. (35-a))

- x + ( ~ x ~ ) at Q 7 \ / - ~ ~

Thus, the spin vector precesses about the magnetic field, with angular fre- quency, eg2/mc, and there is an additional precession with angular veloci ty

1 _ _ _ 8(~-~iOS) ~ X , . ' which results f rom the torques produced by the neigh-

boring spins. In some ways, one may th ink of this ex t ra torque as due to a kind of (( quantum-mechanical ~> addit ion to the magnetic field. Bu t to make the analogy correct, one must th ink of the magnet ic field in a polarizable medium where each dipole interacts s trongly with its neighbors.

F ina l l y , we shall discuss the motion of an element of fluid. This is best done in terms of the expression for the energy-momentum-st ress tensor. To derive this tensor, we s ta r t with the lagrangian leaving out the vector and

scalar potentials , which do not al ter the problem in any essential way

(37) L = f ~ ' [ aS aT h~ h~ (VQ')~ 8 t + ~ at 8m (V~p ~- ~ V~) 2 - 8m (~,)2

~2 ((v@+ sin~ 0(v~)91 8m

d x .

I t can easily be shown tha t this Lagrangian leads to the correct canonical relat ions (27) between ~ and 7, and between @' ~nd S, and tha t it leads to the Paul i hamil tonian (10).

A CAUSAL INTERPRETATION OF THE PAULI EQUATIO~ (A) ~5

Now the canonical energy-momentum stress tensor is given by (~)

IL tt~ IL ts

iL tt~ ~oo = E , i ( iH i t ) t] -j

where the ]~ represent all the possible field quantities. momentum density, and Too the energy density.

'

(38)

To~ represents the We readily obtain

4m ~x~ ixj ~2 ~,(i0 10 iq~ i~)

+ ~ ix~ ~ ~ ' .~-~ - - + sin 2 0

where ~ and V are given by cq. (32).

The energy per p~rticle is then ~ + ~ , which agrees with what we

obtain with the (( Hamilton-Jaeobi )~ equation (28-e). The momentum per particle is just p ~ VS + ~V~, in agreement with what we have been as- suming. The stress tensor, T~, contains three parts. The first part is just mp'v~vj, which is just the usual term representing the effects of mass motion. The second part cmresponds to the quantum potential and the third part to the effects of the interactions between the spins of neighboring particles. The equation of conservation of momentum can then be written as

i t T , (~'v~) + ~ ~- 0.

tt ~ ~X~ 7

With the aid of the equation for conservation of particles, and the relation

e' [-e ix, t x j = ~ \ o~ ] - i x , ' we obtain

(39) d P _ m +(v .V)v z - - V U - 1 dt ~- ~ ixj (~S.),

(~4) See, for example, G. W]~NTZEL: Quantum Theory o] Fields (New York, 1949).

66 D. BOItM, R. SCIIILLER and J. TIOlgNO

where

S , - ~x/ ~x~ § s in2 0 ,, , . OX i OXj

We see then tha t the fluid e lement moves under the action of the <~ quantum-

~ ~ ' ae ' , which leads to the quantum potential , mechanical >> stress tensor, 4m~' ~x~ ~xj

and an additional quantum-mechanical contr ibut ion to the stress tensor, arising

f rom the interact ion of neighboring spins.

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A causal interpretation of the pauli equation (A)