P H YS I CA L R EV I E%' VOL UM E 78, N UM 8ER A P R I L 1, 1990

On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

LEsLIE L. FoLDYCase Institute of Technology, Cleveland, Ohio

SIEGERIED A. WQUTHUYsENfUniverse'ty of Rochester, Rochester, New York

(Received November 25, 1949)

By a canonical transformation on the Dirac Hamiltonian for afree particle, a representation of the Dirac theory is obtained inwhich positive and negative energy states are separately repre-sented by two-component wave functions. Playing an importantrole in the new representation are new operators for position andspin of the particle which are physically distinct from theseoperators in the conventional representation. The components ofthe time derivative of the new position operator all commute andhave for eigenvalues all values between —c and c. The new spinoperator is a constant of the motion unlike the spin operator in theconventional representation. By a comparison of the new Hamil-tonian with the non-relativistic Pauli-Hamiltonian for particles ofspin ~, one finds that it is these new operators rather than the

conventional ones which pass over into the position and spinoperators in the Pauli theory in the non-relativistic limit.

The transformation of the new representation is also made inthe case of interaction of the particle with an external electro-magnetic field. In this way the proper non-relativistic Hamiltonian(essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Diracparticle interacting with any type of external field (various mesonfields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast onthe question of why a Dirac electron shows some propertiescharacteristic of a particle of finite extension by an examinationof the relationship between the new and the conventional positionoperators.

INTRODUCTION

~ ~

~FREE Dirac particle of mass rn is described by afour-component wave function 4 satisfying the

Dirac equation,

H+= (Pm+a p)% =i(8+/Bt), (1)

where p is the momentum operator for the particle,e and P are the well known Dirac matrices (assumedhere to be in their usual representation with P diagonal),and units have been used in which h and c are unity.The eigenfunctions of the Hamiltonian operator satisfythe equation:

(8m+~ p)4=~4.

The eigenfunctions are of the form u(p)e ~'*. For eachvalue of the momentum p there are four linearly inde-pendent spinors' u(p) corresponding to the two eigen-values &(m'+ p') & of the energy and the two eigenvalues&1 of the s component of an operator X (dehned laterin Eq. (26)) related to the spin operator for the particle.

Except in certain trivial cases, for a given sign of theenergy at least three of the components of N(p) aredifI'erent from zero. However, two of the four com-ponents go to zero as the momentum goes to zero, whileat least one of the other two components remains finite.In the above representation, for positive-energy eigen-functions, the last two (or lower) components vanishwith vanishing momentum, while for negative-energy

* Supported by the AEC and the ONR.t Now at the Zeeman Laboratory, University of Amsterdam,

Amsterdam, Holland.' See, for example, %.Heitler, The Quantum Theory of Radiation

(Oxford University Press, New York, 1944), Chapter 3. Ke shalluse the term spinor for any four-component column (or row)matrix occurring in the Dirac theory.

eigenfunctions, the first two (or upper) componentsvanish with vanishing momentum.

Of course any spinor 4' may be split into the sum oftwo spinors 4 and X, having only upper and lower com-ponents respectively, in the following manner:

@=I+X, C = (1+P/2)4', X= (1—P/2)%',

but the spinors 4 and X do not represent states ofdefinite energy in the above representation.

In the non-relativistic limit, where the momentum ofthe particle is small compared to m, it is well knownthat a Dirac particle (that is, one with spin —.', ) can bedescribed by a two-component wave function in thePauli theory. The usual method of demonstrating thatthe Dirac theory goes into the Pauli theory in thislimit makes use of the fact noted above that two ofthe four Dirac-function components become small whenthe momentum is small. One then writes out theequations satisded by the four components and solves,approximately, two of the equations for the small corn-

ponents. By substituting these solutions in the remain-

ing two equations, one obtains a pair of equations forthe large components which are essentially the Paulispin equations. '

The above method of demonstrating the equivalenceof the Dirac and Pauli theories encounters difhculties,however, when one wishes to go beyond the lowest orderapproximation. One then 6nds that the "Hamiltonian"associated with the large components is no longer Her-mitian in the presence of external fields because of theappearance of an "imaginary electric moment" for theparticle. Furthermore, all four components must againbe used in calculating expectation values of operators to

' See, f'or example, W. Pauli, Handbuch der I'hysik, 2. AuQ. , Bd.24, Teil 1.

29

L. L. FOLDY AND S. A. KOUTHUYSEN

order sz/c'. Apart from these difEculties, however, thereare some serious questions concerning the properties ofcorresponding operators in the Dirac theory and thePauli theory even in lowest order. Thus, in the Diractheory the operator representing the velocity of theparticle is the operator e whose components have onlythe eigenvalues ~1. On the other hand, in the Paulitheory the operator representing the velocity of theparticle is taken to be p/m whose components haveeigenvalues embracing all real numbers. Furthermore,different components of the velocity operator in theDirac theory do not commute and therefore are notsimultaneously measurable with arbitrary precision,while different components of the velocity operator inthe Pauli theory do commute and can therefore bemeasured simultaneously with arbitrary precision. Onecan well ask how the operators which purportedly repre-sent the same physical variable in the two theoriescan have such completely different properties.

One can conclude from the above discussion that therelation between the Dirac theory and the Pauli theoryis by no means clear from the usual method of de-scending from four- to two-component wave functions,and that further clarjL6cation of the connection betweenthe theories would be desirable. In what follows wepresent an alternative method for passing from four- totwo-component wave functions in the Dirac theorywhich alternative method clarifies many of the questionsleft open by the usual treatment. The method consistsof a transformation to a new representation for theDirac theory, putting the theory in a form closelyanalogous to the Pauli theory and permitting a directcomparison of the two. Specifically, we shall show that:

(1) For a free Dirac particle there exists a representa-tion of the Dirac theory in which, for both relativisticand non-relativistic energies, positive-energy states andnegative-energy states are separately described by two-component wave functions.

(2) There exists in the Dirac theory another positionoperator than the usual one; this operator has theproperty that its time derivative is the operatorp/(m'+P') & for positive-energy states and —y/(m'+ p'-) &

for negative-energy states corresponding to the con-ventional concept of the velocity of the particle. It isthis new position operator (which we call the meanposilion operator) that in the non-relativistic limit isinterpreted as the position operator of the Pauli theory.

(3) While the z component of the spin operator,e= (1/2i) Le)&e], in the Dirac theory is not a constantof the motion, there exists in the Dirac theory anotherspin operator X (which we call the @sean spin opera-tor)whose s component is a constant of the motion. In thenon-relativistic limit, the operator X is the one whichis interpreted as the spin operator in the Pauli theory.

(4) From a study of the transformation to the newrepresentation further insight can be obtained into thequestion of why the Dirac particle has a magneticmoment and why it appears to show a behavior charac-

teristic of a particle with 6nite extension of the orderof its Compton wave-length.

(5) In the presence of interaction, such as with anexternal electromagnetic Geld, one can still make atransformation which leads to a representation involv-ing two-component wave functions. The transforma-tion, which previously could be made exactly, must nowbe made by an in6nite sequence of transformationswhich process leads to a Hamiltonian which is anin6nite series in powers of (1/m), where m is the mass ofthe particle. This series is presumably semi-convergentin the sense that for given external 6elds, a 6nitenumber of terms of the series is a better-and-betterapproximation to the exact Hamiltonian, the larger thevalue of m. For strong interactions which stronglycouple free-particle states of positive and negativeenergy, this representation is of little value; however,for sufticiently weak interactions, a 6nite number ofterms of the series may be employed to obtain relativisticcorrections to any order in (1/m). In this way one canobtain the proper non-relativistic limit for the Hamil-tonian representing a Dirac particle interacting withany type of external Geld.

0"—e.s@

H'= e'zHe 's ie'z(8e 'z/Bt), —(3)

(4)

3 An odd operator in the Dirac theory is a Dirac matrix whichhas only matrix elements connecting upper and lower componentsof the wave function, while an even operator is one having nosuch matrix elements. Of the sixteen linearly independent matricesin the Dirac theory, the matrices 1, p, +=1/2iI e)Cej and peare even, while the matrices e, pe, y'= —ia'n'a' and pg are odd.The product of two even matrices or of two odd matrices is aneven matrix, while the product of an odd matrix and an evenmatrix is an odd matrix. The matrix p commutes with all evenmatrices and anticommutes with all odd matrices in the Diractheory. This last fact allows one to write any matrix as the sumof an odd and an even matrix in a simple way, namely:

~-g(~+ p~pJ+$ {~—p~pI,where the first term on the right is the even part of the matrixand the second term is the odd part of the matrix ~.

THE FREE DIRAC PARTICLE

The essential reason why four components are ingeneral necessary to describe a state of positive ornegative energy in the representation of the Diractheory corresponding to Eq. (1) is that the Hamiltonianin this equation contains odd operators, '

specificallythe components of the operator e. If it is possible toperform a canonical transformation on Eq. (1) whichbrings it into a form which is free of odd operators, thenit will be possible to represent positive- and negative-energy states by wave functions having only two com-ponents in each case, the other pair of componentsbeing identically zero. We shall now show that thereexists a canonical transformation which accomplishesjust this end.

If S is an Hermitian operator, then the transforma-tion,

DIRAC THEORY OF SPI N 1 j2 PARTICLES 31

p=e'sTe 's.) (6)

Let us make a transformation of this type on Eq. (1)with 5 the non-explicitly time-dependent operator:

S= —(i/2m)Pe. pw(p/m),

where w is a function of the operator' (p/m) whose formis to be determined such that B' is free of odd operators.VVith this choice of S one may readily show thatgi' ps'~ —i S pe Sgg

=[ cso(Pw/m)+(Pa p/P) sin(Pw/m)]H=P[m cos(Pw/m)+P sin(Pu/m)]

+a.p/P[P cos(Pw/m) —m sin(Pw/m)]. (g)

We see from this that B' will be free of odd operatorsif we choose

u (P/m) = (m/P) tan-'(P/m),

and with this choice, we obtain

a'= P(m'+ p')»= PE„,

where E„represents the oPerator (m'+P')».Equation (5), with H' given by (10), now has solu-

tions such that the upper components represent positiveenergies and the lower components negative energies,for splitting 4' into its upper and lower components:

O'=C'+X',C'= (1+P/2)e', X'= (1—P/2)4',

reduces Eq. (5) to the two uncoupled equa, tions:

E„C'= i(84'/8»),

E~'=i (BX'/Bt)—. (12)

In order for one to understand completely the natureof the transformation which we have performed, how-ever, it is necessary to investigate in greater detail themanner in which the wave function and certain opera-tors transform. A general wave function can be ex-pressed in the form:

leaves Eq. (1) in the Hamiltonian form:

B'0"= i(8%"/Bt).

(It must be remembered that in a canonical trans-formation of this type a physical variable whoseoperator-representative in the old representation is Thas for its operator-representative in the new repre-sentation the operator:

t 1 Pm+a p'4'p(x) = ~ —1+ u(p') exp(ip' x)dp', (14)

J

t1 Pm+u p'

4' (x) = —1— u(p') exp(ip' x)dp', (15)E,,

+'=ei 4=4 '+4 ' (19)

We may legitimately identify 4+' with 4' and 0 ' withX' since the presence of the factors (1+P)/2 and(1—P)/2 shows that these functions have only upperand lower components, respectively. Thus we see thatindeed the transformation is such as to lead to a repre-sentation in which upper components correspond topositive energies and lower components to negativeenergies.

Furthermore, since

u(p') =(2s.)' ~

@(x') exp( —iy'. x')dx',

where 4'+ represents the positive energy part and 4the negative energy part of +. Now

(1 Pup p'te~s exp{ tan-' —

{&2m p mi

= cos[-', tan —'(P/m)]+ (Pa p/P) sin[-,' tan —'(P/m)]

E„+m» Pe. p E„m»—+2E„p 2E„

P {Pm+ e.p+ PE~ I(16)

[2E„(E+m)7»and hence

(1+P) r 2E„

2 )~ E„+mPm+a p'

1+ u(y') exp(ip' x')dp'= 4', (17)Ey

r1—p) r 2E~'4 '=o*'4

I 2 Jj EgmPm+e p'

)& 1— u(p') exp(ip'. x')dp'=X', (18)E„

0 (x) = )I u(y') exp(iy'x')dp'= ++(x)++ (x), (13)

4A function T(p) of the operator p is to be interpreted {in thecoordinate representation) as dined by its Taylor expansion inpowers of (p-pf»), where po is any constant vector {cnumber),wherever the expansion converges, and by the analytic con-tinuation of this series elsewhere; or, alternatively, by its integral-operator representation:

T{p)4'{x)=2 ~ T{p') exppip'- {x—x') ge{x')dp'dx'.1

we have

wherer~ 2E„.

E(x, x') =(2s)' " E„+m

P(Pm+a y')X i+

(2o)

exp(ip'. (x—x'))dp'. (21)

L. L. FOLDY AND S. A. KOUTHUYSEN

iPa iP(a p)p [aXp]P+ (22)

2E„2E~(E,+m) px etsxe 'sI

But if this is the case, then what is the operator-repre-sentative X in the old representation of the physicalvariable whose operator-representative in the newrepresentation is X'=x? We find

iPa iP(a p)p+[aXp]PX= e—'sxe's =x+ (23)

2E~ 2E~(E~+m) p

In order to interpret this new "position" operator, wecalculate its time derivative:

dX p I pm+a y}=i[H, X—]=—. (24)

dt E~ E~

But since Ap= (Pm+a p)/E„has the value +1 whenapplied to a positive energy wave function and thevalue —1 when applied to a negative energy wavefunction, ' we see that the time derivative of X is justthe "conventional" velocity operator +p/E~ forpositive energy states and y/E„ for negative e—nergystates The o.perator-representative of dX/dt in thenew representation is

dX'/dt=dx/dt=i[H', x]=Py/E, (25)

with the same properties. Ke note also that since theoperator p commutes with S, the operator-representa-tive for the momentum in the new representation isstill p.

By comparing these results with certain well-knowqfacts about the Dirac electron, we may understand thesignificance of our change in representation. In dis-cussions of the Dirac theory one Gnds an analysis of themotion of a free Dirac electron which shows that theelectron performs a very complicated motion, indeed. '

~ E. Schrodinger, Berl. Ber. 419 (1930); 63 (1931).' See, for example, P. A. M. Dirac, The Priecip/es of QuantumMechanics (Oxford University Press, New York, 1935},SecondEdition, Chapter XII. See also in this connection, reference 2, pp.230-231 ~

Since E(x, x ) is not a Dirac delta-function in its spacedependence, the transformation of 0' is an integralrather than a point transformation. In general, 4' at agiven point is constituted from contributions dependingon + over a neighborhood of dimensions of the order ofa Compton wave-length of the particle about the point.Thus a wave function which in the old representationcorresponded to a state in which the particle wasdefinitely located at one point, passes over in the newrepresentation into a wave function which apparentlycorresponds to the particle being spread out over afinite region.

The key to understanding this rather unusual state-ment lies in the fact that in the new representation, theoperator-representative for the position of the particleis no longer the operator x, but the rather complicatedoperator:

but to this is added a second term representing a rapidlyoscillating motion ("Zitterhewegung") which ensuresthat a measurement of the instantaneous value of anyvelocity component shall yield the velocity of light.Our results above show that a corresponding division ofthe position operator for the particle is also possible, thefirst part X (in the old representation) representing asort of mean position of the particle, and the secondpart X—x, oscillating rapidly about zero with an am-plitude of the order of the Compton wave-length of theparticle. While in the old representation the positionoperator x played the dominant role, in the new repre-sentation it is the position operator X', which we shallcall the meun posil-ion operator, ' which plays the domi-nant role. Also, as mill become obvious later when weconsider the interaction of the particle with an externalfield, it is the meun position-operator which is identifiedwith the position operator in the non-relativistic Paulitheory.

The modification in the interpretation of operatorsinvolved in our transformation does not end here,however, but new angular momentum operators alsoappear. In the old representation, the orbital angularmomentum of the particle whose operator-representa-tive is [xXp] and the spin angular momentum of theparticle whose operator-representative is 1/2a are notseparately constants of the motion, ' although theirsum is a constant of the motion. However, as one mayreadily verify, the operators [XXy] and

iP[aXp] [pX[axp]]X=e—E, E„(E,+m)

(26)

whose analogues in the new representation are, respec-tively,

[X'Xp]=e' [XXp]e 's= [xXp], (27)

X'=e'sXe 's=e (2g)

'The operator, which we have designated the mean-positionoperator, has been discovered independently in other connectionsby several authors. M. H. L. Pryce (Proc. Roy. Soc. 150A, 166(1935)) found it useful to introduce this operator (as well as theoperator which we later define as the mean spin angular momen-tum) in connection with the definition of coordinate and intr'insicangular momentum operators in the Born-Infeld theory, and again(Proc. Roy. Soc. 195A, 62 (1948}) in a discussion of relativisticde6nitions of center of mass for systems of particles. In the latterpaper, he also noted that this operator was connected with theusual position operator in the Dirac theory by a canonical trans-formation which is identical with the one performed above in thispaper. T. D. Newton and E. P. signer (Rev. Mod. Phys. 21, 400(1949)) also found this operator in an investigation of localizedstates for elementary systems. The latter authors have also shownthat this is the only position operator, with commuting com-ponents, in the Dirac theory which has locahzed eigenfunctionsin the manifold of positive energy wave functions.

Its velocity can be written as the sum of two parts: thefirst is essentially the conventional velocity operator

p pm+a p)

E„ E„

DIRAC THEOR Y OF SP IN 1 j2 PARTI CLES 33

Tmxz I. Table of operator-representatives of dynamical variables in old and new representations.

Dynamical variable

Position

Momentum

Hamiltonian

Velocity

Orbital angular momentum

Spin angular momentum

Mean position

Mean velocity

Mean orbital angular momentum

Mean spin angular momentum

Operator-representative in oldrepresentation

p—= (h/i) yH=Pm+e p

fxXpj

e=(1f2i)feXaj

iPe iP{e p)p+feXpjP="+2E, 2E,(E,+m) p

p Pm+a p

fXXp]itefeXpj fpX fo'XpjjE„E„(E,+m)

Pm+a p

Operator-representative in newrepresentation

iPe+iP(e p)p —fXpjP2' 2E~{E~+m)p

p=pe'= p(m&+ p }~—=pE,

+Pp (e p)p+E„E(E„+m)e'=(~e-e-p)/E.

[x'Xp]ip[eXp] pX[eXp]~+ E, E„(E,+m)

PpX'=—

fX'Xpj= fxXpj

are separately constants of the motion. %e shall denotethe physical variables of which these operators are theoperator-representatives as the meae-Orbital angularmomentum and mean spin angular momentum, respec-tively, of the particle. It is again these variables whichare conventionally identified with the orbital angularmomentum and the spin angular momentum of theparticle in the non-relativistic Pauli theory. For con-venience, we have listed in Table I the operator-repre-sentatives in the old representation and in the newrepresentation of the physical variables of principalinterest in the Dirac theory.

DIRAC PARTICLE IN AN EXTERNALELECTROMAGNETIC FIELD

%e consider now the case where a Dirac particleinteracts with an external field such as the electro-magnetic field. Before proceeding to the general case,however, it is instructive to consider an elementaryspecial case in order to clarify some points regardingthe classifica, tion of states as either of positive or nega-tive energy. Let us consider first a Dirac particle movingin a weak static electric field derivable from a scalarpotential p. The Hamit. tonian is then

H=Pm+e p —ey.

For the case of a free particle, states were classified as ofpositive or negative energy according as they corre-sponded to values of +1 or —1, respectively, for theoperator (Pm+a p)/E~ which was a constant of themotion. In the present case, however, this operator doesnot commute with the Hamiltonian and is thereforenot a constant of the motion. If we regard the electric

field as a perturbation, then one can say that the electricfield induces transitions of the particle between thepositive- and negative-energy states of a free particle.This is one way of viewing the physical situation.

On the other hand, one knows that for suflicientlyweak fields the Hamiltonian above possesses a com-plete set of eigenfunctions with energy eigenvalueswhich may be classified according to whether they arepositive or negative. There exists for these weak fieldsa clear-cut distinction between these two sets of sta-tionary states' since they are separated by a relativelylarge energy gap of order 2m. Furthermore, the wavefunctions corresponding to positive energies show abehavior of the particle appropriate to a particle ofpositive mass, ' in that the particle tends to be localizedin regions of low potential energy; while the negative-energy solutions show a behavior of the particle appro-priate to a particle of negative mass, in that the particletends to be localized in regions of high potential energy.

Either of the two descriptions of the behavior of theparticle in a weak field given above is of course correct,although the distinction between what are called posi-tive- and negative-energy states is diferent in the twodescriptions. However, the question of terminology forpositive- and negative-energy states being left to ourown choice, we are free to choose our definitions in sucha way as to give the more graphic (and perhaps more

It is assumed that the constant in the energy is chosen so thatzero energy occurs approximately midway in the energy gapbetween the two sets of states.' It is perhaps better to classify the states as states of positiveor negative mass rather than energy, since the addition of aconstant to the energy (by adding a constant to q, for example)may upset energy classification but not the mass classification.

L. L. FOLDY AND S. A. WOUTHUYSEN

intuitively satisfying) description of the actual physicalevents which are being described. In this spirit we feelthat the second description is to be preferred since ithas a perfectly reasonable classical limit. It mould bedificult indeed to picture classically the motion of aparticle in a weak field in terms of transitions betweenfree-particle motions with positive and negative mass.

Consider now what happens when the particle inter-acts with strong rather than weak fields. Under suchcircumstances, the division of states into those ofpositive and negative mass is no longer clear-cut, sincethe energy, separation of the two sets of states isreduced to a relatively small amount. Furthermore, thewave functions describing these states no longer appro-priately describe the motion of a particle of fixed signof mass according to our customary notions. In fact, ifwe try to interpret the wave function in these terms,we encounter certain well-known paradoxes —the Kleinparadox, for example. While the energy of any sta-tionary state will still have a definite sign, the state-ment that the particle is in a state of positive energywill no longer carry with it the validity of any intuitiveconceptions as to the behavior of a classical particlewith positive energy, and there will be little quali-tative difference between certain states of positiveenergy and certain states of negative energy. Hence, inthe presence of strong fields, the usefulness of a de-scription in terms of positive and negative-energy stateswill be lost.

These same ideas may be carried over to the case ofmore general interactions. If there are any advantagesto be accrued by the employment of two-componentwave functions to describe states of positive and nega-tive energy for a free particle, we may expect theseadvantages to be present still in the case of weak inter-actions, but they can scarcely survive in the case ofstrong interactions. To define weak interactions in amore quantitative fashion, we prescribe that the inter-action terms have no time Fourier components com-parable with or greater than m, so that no transitionsbetween free-particle states with energy differences ofthis latter order of magnitude are possible; and that theinteraction terms have no space Fourier componentscomparable with or greater than m, so that no transi-tions between free-particle states with momentumdifFerences of this order of magnitude are possible.Under these conditions, if the initial state is one inwhich no high momentum states occur, then the mo-mentum of the particle will remain small compared tom under the inQuence of the interactions. In these cir-cumstances we have essentially a non-relativisticproblem, and hence it will be in the domain of non-retatisistic praMems that a representation by means oftwo-component wave functions may be expected to beof value.

In the presence of interaction it no longer appearspossible to make a single, simple, canonical trans-formation to a representation in which the Hamiltonian

is free of odd operators, but instead one can always makea sequence of transformations, each of which eliminatesodd operators from the Hamiltonian to one higher orderin the expansion parameter" 1/m. In this way oneobtains in the new representation a Hamiltonian freeof odd operators which is an infinite power series inpowers of 1/m. While it can hardly be expected thatthis series is convergent, the series is presumably anasymptotic or semi-convergent series in the sense thatthe sum of a finite number of terms of the series is abetter-and-better approximation to the true Hamil-tonian, the larger the value of m, provided the magni-tude of the interaction remains fixed. The usefulnessof the series is then contingent on the interactions beingsufBciently weak compared to m in just the sensedescribed above for non-relativistic problems.

Turning our attention now to the explicit calculationin the case where a Dirac particle is subject to inter-actions, we note first that the Hamiltonian operator canalways be put in the form'

p= pm+8+ 8, (29)

where b is an even operator and 8 is an odd operator,both of which may be explicitly time-dependent. Weassume (and this is ordinarily the case) that h and 8a,re of no lower order in (1/m) than (1/m)'. Considernow the canonical transformation generated by theHermitian operator

S= —(i/2m)P e. (3o)

We note that if the operator S may be regarded assmall, then we can make an expansion in powers of(1/m) of the Hamiltonian in the new representation:

(ge—is)Ep= e'slee 's ie's~—

at &

BS 1 BS=H+—+i S, H+——

Bt 2 Bt

i' 18S+—S, $, 8+——+. . (31)

2J 3 at

Carrying this out explicitly for the Hamiltonian aboveand retaining terms only to order (1/m)', we obtain

P 1 86H'= pm+8+ e' — 8, [n, h]+

2m Sm2 N

j. 88 p 1——P +—[6, 8)— n'y ", (32)2m Bt 2' 3el2

"The expansion parameter may equally well be regarded as 1/cwhere c is the velocity of light. The actual dimensionless expansionparameters are the operators (A/mc)~ and (h/mc')(8/Bt). Fromthis, one sees that the successive terms of the series will decreaserapidly in magnitude if the interaction potentials do not varyappreciably in space over a Compton wave-length of the particleand in time over the period required for light to travel a Comptonwave-length. This is equivalent to the restrictions imposed aboveon the space and time Fourier components of the jnteraction,

DIRAC THEOR Y OF SP IN 1/2 PARTICLES

where we have made use of the fact that P commuteswith all even operators and anticommutes with all oddoperators in the Dirac theory. Remembering that theproduct of two even or two odd operators is an evenoperator and that the product of an odd and an evenoperator is an odd operator, we see that we have re-moved by this transformation all odd operators of order(1/m)' from the Hamiltonian although odd operators oforder (1/m) and higher remain and are given in thesecond line.

From this we see that by a sequence of furthercanonical transformations, the generator of the trans-formation at each step being chosen to be

S= —(i/2m) p (odd terms in Hamiltonian oflowest order in 1/m), (33)

we may successively remove odd terms from the Hamil-tonian to any desired order in (1/m). If we continuethis process inde6nitely, we obtain, as mentioned earlier,a Hamiltonian which is an infinite power series in (1/m)completely free of odd operators.

I.et us now apply this method to the case where theDirac particle interacts with an external electromag-netic 6eld. In this case the Hamiltonian is given by

H=pm —eq+a (p —eA), (34)

where y and A are the scalar and vector potentials ofthe electromagnetic field evaluated at the position ofthe particIe. In accordance with the procedure outlinedabove, we then 6rst make the canonical transformationgenerated by

Si——— Pc (p —eA),2m

obtaining then for the new Hamiltonian,

a,= pm e&+ —(p e—A)' — p~—H2m 2m

e e iea EX(p—eA)+ divE — Pe E

4m' Sm2 2m

1[(y—eA)' —ee H]e (p—eH)+

jm2

to terms of order (1/m)'. Following with the two canon-ical transformations generated by

i f'e

S2= — p — pa E2m 2m

i e BESs= —p — pe'

2m 4m' Bt

1

1L(p —eA)' —ee.H]e. (p—eA),

3m2

we eliminate odd operators from the Hamiltonian oforder (1/m) and (1/m)' respectively and obtain theHamiltonian

eH3= pm eq—+ (p—eA—)' —pe H

2m 2m

e eo EX (p—eA)+ divE+ ., (35)

4m' 8m'

+8m'

BC'divE C =i, (36)

Bt

which will be recognized as essentially the Pauli equa-tions for a non-relativistic particle of spin —, interactingwith the electromagnetic 6eld. The presence of theterms corresponding to the interaction of the anomalousmagnetic moment of the particle with the magneticGeld and the spin-orbit interaction is evident. The termproportional to divK is a well-known correction" to thePauli theory arising from the Dirac theory, and it isresponsible for a relativistic shift of the 5 levels in thehydrogen atom (not to be confused with the Lamb-Retherford line shift).

The reason for the explicit appearance of these addi-tional interaction terms (as well as further terms ofhigher order in 1/m) in the new representation can nowbe understood in the light of our physical interpretationof the transformation to the new representation. In theold representation the Dirac particle interacted withthe electromagnetic 6eld only at its position. However,a particle which in the old representation was locatedat a point is in the new representation spread out overa region of dimensions of the order of a Compton wave-length in the space of its mean-position variable, X'=x.But in the new representation the interaction betweenthe particle and the electromagnetic 6eld is expressedin terms of the values of the electromagnetic-Geld quan-tities at its mean position Hence, one-must e.xpect aseries of correction terms of the nature of a multipole

"C. G. Darwin, Proc. Roy. Soc. 118A, 654 (1928).

free of odd operators to order (1/m)'. In the above8=curlA and E= —Vqr —BA/Bi are the magnetic andelectric 6eld strengths evaluated at what is now essen-tially the mean posi-tion of the particle. By furthercanonical transformations we could remove oddoperators of still higher order, but we shall limit ourdiscussion of the Hamiltonian in the new representationto terms of order (1/m)'.

Since the Hamiltonian (36) is free of odd operators,its eigenfunctions are two-component functions cor-responding again to positive and negative energies. Forpositive energies, the Schrodinger equation is

1 e em —ey+—(p—eA)' — o"8— o EX(y—eA)

2m 2m 4m2

J. LE I TE LOPES

expansion of the Geld, since the particle is actuallyspread out over a finite region in the space of its meae-positioe variable F.rom this point of view the explicitappearance of the magnetic-moment interaction andthe accompanying spin-orbit interaction in the Hamil-tonian is to be expected. In fact, the term in divE,which has previously been regarded as of rather mys-terious origin, can now also easily be understood sinceit comes from the fact that the electric charge in thenew representation is also spread out over a Gniteregion. In a static potential, the particle then movesaccording to a suitable average of the potential overthis region. But in lowest order such an averagingprocess is known to lead to a term proportional to theI.aplacian of the potential and this is just the characterof the term in divE.

(In employing a finite number of terms in a Hamil-tonian such as (35) it must be remembered that wavefunctions, transition matrix elements, and expectationvalues of operators computed from the Hamiltonian areonly correct to terms of the order in (1/m) to whichterms in the Hamiltonian are retained. Thus in the caseof (36) it would not be consistent to retain only terms oforder (1/m) in the Hamiltonian and then to employterms of this order in second-order perturbation theorywhere they generate terms of order (1/m)'. )

The method employed above for reducing the Hamil-tonian to non-relativistic two-component form for thecase of interaction with an external electromagneticGeld can be employed generally for interaction of aDirac particle with any type of external Geld, such asvarious types of meson Gelds. With meson Gelds oneobtains, in diGerent cases, various types of spin inter-action with the meson field and also various types ofspin-orbit coupling terms such as have been employedin discussions of spin-orbit coupling in nuclei. The dis-cussion of the reduction to non-relativistic form in thecase of the many-particle theory is left for a later com-munication.

ACKNOWLEDGMENTS

This investigation arose out of a conversation betweenthe authors and Dr. A. S. Wightman to whom we areindebted for subsequent valuable discussions on thissubject. One of the authors (L.L.F.) wishes to expresshis gratitude to the University of Rochester for thehospitality extended to him in the period in which thiswork was initiated. This work was supported at CaseInstitute of Technology by the AEC and at the Uni-

versity of Rochester by the joint program of the ONRand the AEC.

P H YSI CAL R EVI EW VOLUM E 78, NUMBER 1 APRI L 1, 1950

The Nucleon Magnetic Moment in Meson Pair Theories

J. LEITE LOPES*Institute for Advanced Study, Princeton, Asm Jersey~*

(Received November 30, 1949)

The contribution to the nucleon magnetic moment from an interaction of the nucleon with a spinor orscalar meson pair Geld is calculated. In both cases it is found to be logarithmically divergent.

'HE covariant formulation of the pseudoscalarmeson theory' together with the concepts of

mass and charge renormalization' were applied by Case'to the computation of the anomalous magnetic momentof nucleons. He showed that finite results are thenobtained. This is essentially due to the fact that one isnow able to isolate and incorporate' the divergencesinto the mass and charge of the nucleon field thusleaving a convergent expression for the magnetic mo-

*John Simon Guggenheim Memorial Fellow, on leave ofabsence from the University of Brazil, Rio de Janeiro.

*~ Present address: Faculdade Nacional de FilosoGa, Rio deJaneiro, Brazil.' S.Kanesawa and S.Tomonaga, Prog. Theor. Phys. 3, 1 (1948);Y. Myiarnoto, Prog. Theor. Phys. 3, 124 (1948).

~ See F. J. Dyson, Phys. Rev. 75, 486, 1736 (1949).3K. M. Case, Phys. Rev. 76, 1 (1949); compare also J. M.

tuttinger, Helv. Phys. Acta 2j., 483 (1948); M. Slotnick and%. Heitler, Phys. Rev. 75, 1645 (1949); S. D. Drell, Phys. Rev.76, 427 (1949),

ment. This separation of the reactive terms representsan improvement over the previous treatment of thisand related problems. 4 On the other hand, divergencedifhculties are still encountered in the magnetic momentcalculation based on the vector meson theory withtensor coupling.

It is the aim of this note to report on the applicationof the renormalization program to the computation ofnucleon magnetic moments by assuming a meson pairinteraction (containing no derivatives of the fields)between the heavy particles. Two cases have beenconsidered for the meson field, namely, the scalar (orpseudoscalar, which here amounts to the same) and the

4 Finite results for the nucleon magnetic moment were previ-ously obtained by J. M. Jauch (Phys. Rev. 63, 334 (1943)) inthe conventional theory by using the lambda-limiting process;the nucleon Geld was treated non-relativistically.

~ K. M. Case, Phys. Rev. 75, 1440 (1949).