The article of record as published may be found at http://dx.doi.org/10.1103/PhysRev.158.1223USDO

Zeleny, W.B.

Calhoun, Institutional Archive of the Naval Postgraduate School

Hamiltonian Form of the Kemmer Equation

The Hamiltonian form of the relativistic wave equation for bosons of spin 0 or 1 was first given by Kemmer.  The problems associated with the redundant components of the wave function were later resolved by Heitler, who eliminated the redundant components by means of projection operators.  We present an alternative treatment which yields essentially the same results as obtained by Heitler, but which retains all components of the wave function.Approved for public release; distribution is unlimited

Calhoun: The NPS Institutional ArchiveFaculty and Researcher Publications Faculty and Researcher Publications Collection1967-06-25Hamiltonian form of the Kemmer equationZeleny, W. B.American Physical SocietyPhysical Review, Second Series, V. 158, no. 5, June 25, 1967, pp.1223-1225http://hdl.handle.net/10945/47487A.. 4~2 LC4 journal of experimental and theoretical physics established by E. L Nichols in 1S93SEcoND SERIEs, VoL. 158, No. 5 25 JUNE 1967Hamiltonian Form of the Kemmer Equation*W. B. ZELENYDepartment of Physics, Nasa/ Postgraduate Schoo/, Monterey, California(Received 28 December 1966)The Hamiltonian form of the relativistic wave equation for bosons of spin 0 or 1 was first given by Kem-mer. The problems associated with the redundant components of the wave function were later resolvedby Heitler, who eliminated the redundant components by means of projection operators. We present analternative treatment which yields essentially the same results as obtained by Heitler, but which retains allcomponents of the wave function.I. INTRODUCTIONA FIRST-ORDER, relativistic wave equation forbosons of spin 0 or 1 was originally derived byKemmer. The Kemmer equation is similar in appear-ance to the Dirac equation, except that the matricesinvolved, the so-called p matrices, obey different com-mutation relations from Dirac's y matrices. Also, theexistence of redundant components in the Kemmerwave function introduces complications in the specifica-tion of observables and their expectation values. Forexample, Kemmer noted an ambiguity in the definitionof the expectation value. In addition, Kemmer's Hamil-tonian for a boson in an electromagnetic field is notHermitian, and, in fact, as we shall see, it is not evenunique.In order to resolve these, and related dif5culties,Heitler proposed a theory in which the redundant com-ponents of the wave function are eliminated. ' InHeitler s theory, a new Hamiltonian, involving only thedynamical components, is obtained. The purpose of thepresent paper is to present an alternative approachwhich is in many respects similar to that of Heitler,except that we retain all components of the wave func-tion. In our approach, we also obtain a new Hamil-tonian, whose dynamical components are, in fact, thesame as Heitler's Hamiltonian.II. THE KEMMER EQUATIONThe Kemmer equation for a free boson of rest massm is'(i8+s+m)P(X) =O,* Supported in part by the Ofhce of Naval Research.' N. Kemmer, Proc. Roy. Soc. (London) A1?3, 91 (1939).' W. Heitler, Proc. Roy. Irish Acad. A49, 1 (1943).' We use the conventions c=k=1 and 8„=8/Bx&,where x'=t,158where the matrices p" satisfypspvpr+papvps —psgvr+ prgvs (2)j'd'x= bz,Vwhere V is the volume of a normalization box, and5+=+1 for positive-energy solutions= —1 for negative-energy solutions,determines the normalizai. ion of f (The fact tha. tJ'j'dt's has the sign of the energy will be shown below. )xI=x, x'= y, x'= s. Also, x„=g„„x",where goo —1, g11=g22 g33= —1, and p&= g&"p, = (E,p). Repeated Greek indices are summedfrom 0—3; Latin indices from 1—3.1223If a 5&(5 representation of the P matrices is used, fdescribes a spinless boson, and if a 10)&10 representationis used, f describes a boson of spin 1. Definingn'= 2(P')' —1and4'o', —the four-vector current density can be taken to bej"=4pV,whereB„j"=0.In particular, since, from Eq. (2),n'P'=PY=P',the charge density isj'= 4'pV-The requirementCopyright 1967 by The American Physical Society.W. B. ZELEN YAs shown by Kemmer, the equationsi Bog'= (i8kgk, P'7 —mP')fand{i~kp'(P')'+~t 1—(P')'7}4'= oare together equivalent to Eq. (1). SettingH'= iBk[jSk,p07 0'—poand making use ofPOPkPO —0(6)which follows from Eq. (2), Eqs. (5) and (6) becomeiaof= EPP,(H'Po+ m) / =0. (10)Equation (10) can be taken as an initial condition andEq. (9) a dynamical equation in Hamiltonian form.Kemmer takes H' as the free-particle Hamiltonian. IfB' is to be Hermitian, we must haveis achieved by means of the projection operatorsandwhich project out the components of subspace I andsubspace II, respectively. In so doing, Heitler obtainsa Hamiltonian which acts only in subspace I, but whichinvolves second derivatives with respect to the spacecoordinates, so that the theory loses its relativisticappearance (though not its Lorentz invariance).We present here an alternative approach, in which allcomponents of the wave function are retained. We ob-tain a new Hamiltonian, which, like Heitler s, involvessecond derivatives, and, in fact, reduces to Heitler's ifwe perform his reduction to subspace I.We begin by noting that because of the form of thecharge density, Eq. (3), we can define the inner productof any two wave functions fi and f& by'pop —po pkt — pk 8 iA 0) = — O'PVod'x (15)To obtain the wave equation for a charged boson inan electromagnetic field, we replace i8„by (i&„eAi,—) ~e then say that an operator Q is "neo-Hermitian"in Eq. (1), obtaining providedDi8„eA„)—P&+mQ(x) =0 (12) 1) 2 1) 2This equation can also be put in Hamiltonian form, ' thetwo equationsi80$= {eA 0+ (i8k eAk) pk, p07—0NP0—«(2~) '~—..(O'P'P" g"'P")}4 —(13)andwheretogether being equivalent to Eq. (12). The operator onthe right-hand side of Eq. (13) could be taken to be theHamiltonian for a charged boson in an electromagneticfield. However, if Eqs. (11) are satisfied, then thisHamiltonian is not Hermitian.GI. OBSERVABLESSuch problems as the determination of observables(as, for example, the above Hamiltonians) and theexistence of the subsidiary conditions (10) and (14)arise in the Kemmer theory because of the excessivenumber of dimensions in the spinor space, there beingmore components in the wave function than necessaryto specify the internal degrees of freedom. Heitler' de-velops a consistent theory by separating the spinorspace into two subspaces, which he calls subspace Iand subspace II. Subspace I involves those componentsof the wave function which determine the dynamicalstate of the boson, and subspace II involves the non-dynamical, or redundant components. The separationfor all gi and P&. From Eq. (15), we see that this isequivalent to the requirementp Q=Qtp . (16)We then require that all observables, and in particularthe Hamiltonian, be neo-Hermitian. 'The eigenvalues of a neo-Hermitian operator are notnecessarily real, but an eigenstate corresponding to anonreal eigenvalue must satisfy jo= —ftPQ—=0, andhence cannot represent a real boson. To see this, supposeQ is neo-Hermitian, and Q =yP. ThenP, P QiPbd'x=b g. P'Pbd'x=a* P, P'Pbd'x.Hence, if a*Nb, then fP,tP'Pbd'x=0.4 For simpli6cation, we drop the reference to the normalizationvolume V.' A similar requirement is imposed by Heitler in his theory.But ptp'Q and ptpg are both real, so if ptpg&0, qmust be real.The orthogonality of eigenstates of a Hermitian opera-tor corresponding to different eigenvalues can also begeneralized to neo-Hermitian operators. Suppose Q isneo-Hermitian, andQ,=af, and QPb= bfbThenHAM ILTONIAN FORM OF KEM MER EQUATION 12250 3 0i BOP = [H'+A (H'P'+ m)] tt,(H'P'+m)P= 0,(17)and from Eqs. (7), (8), and (20), we haveNow the Hamiltonian H is not neo-Hermitian. Note, for all 8 such that (19) is neo-Hermitian. Note 6rsthowever, that in place of Eqs. (9) and (10), we could that from Eq. (2) we havewrite(20)H= H'+m 'iBi,P'P~(H'P'+m) .Equation (17) becomesia,y= HP.(18)If we restrict ourselves to wave functions satisfyingEq. (10), then an eigenstate of H' corresponding toeigenvalue E is also an eigenstate of H correspondingto the same eigenvalue E. Similarly, an eigenstate ofH corresponding to eigenvalue E is also an eigenstateof H' corresponding to the same eigenvalue E. Forsuch a state, we have0= Pt(H'P"+~e)Pd'x= E PtP'Pd'x+m /ted'xHencePtP'Pd'x= —m /ted'x(0,so that E&0, and Jj'd'x= 'J'p"p'pd'x&0 —and hasthe sign of E. As shown by Kemmer, ' the nonzeroeigenvalues of H' are E=+ (p'+nP)"'.The form (18) for the free-particle Hamiltonian isstill not unique. We could add to H the operator8 (H'P'+ m,),where 8 is any dimensionless operator such that (19) isneo-Hermitian. For example, we could take 8=1.Butthis additional term would not change the eigenvaluesof H corresponding to those eigenstates satisfying Eq.(10).' Indeed, using Heitler's viewpoint, the quantity(19) acts entirely in subspace II. To see this, we mustshow that(P')'8(H'P'+~) (P')'= o'Neither would it change the expectation value of the timederivative of any observable, as we shall see below.where the operator A is completely arbitrary. Everysimultaneous solution of Eqs. (9) and (10) is also asimultaneous solution of Eqs. (17) and (10), and viceversa. Hence, Eqs. (17) and (10) are also togetherequivalent to Eq. (1). Since the operator on the right-hand side of Eq. (17) represents the free-particleHamiltonian, we see that this Hamiltonian is not unique.In particular, by appropriately choosing the operator3, we can make the Hamiltonian neo-Hermitian. It iseasily veriGed that a suitable choice isA = m 'ihip-op",yielding the neo-Hermitian HamiltonianP'H'P'= —mP'. (21)IV. EXPECTATION VALUESBecause of the form (3) for jo, and the normalization(4), the expectation value of any observable Q in theSince 8(H'p'+m) must be neo-Hermitian, we haveP'8 =BtP' m'P—'BH'P'+m 'P'H'BtP'. (22)Using Eq. (22), we have(P )'8(H'P'+ ) (P')'=P'(BtP' m'P"B—H'P'+rn 'P"H'8"P')&& (H'Pope) (Po)'. (23)Multiplying out the right-hand side of Eq. (23) andusing Eqs. (20) and (21), we find that all the termscancel. Thus, the quantity (19)acts entirely in Heitler'ssubspace II. It must therefore represent the ambiguityin the Hamiltonian associated with the redundant, ornondynamical, components, and has no physicalmanifestation.A neo-Hermitian Hamiltonian for a boson in anelectromagnetic Geld can be found in the same manneras in the free-particle case. We add to the operator onthe right-hand side of Eq. (13) an operator of the formA((iBi, eA&)P—~(Po)2+m[1 (Po)2j—where, in this case,A=re '(iB,—eA, )p'p'This yields the neo-Hermitian HamiltonianH„=eA p+ (iBi,—eAg)P"P"mp' i e (2m) '—Il „„(pl—"p'p" gl"'p')—+m '(i8; eA,) (i B—i, eA i,)P"P'P". —(24)A term could be added to H, analogous to the term(19) for the free-particle Hamiltonian, only involvingthe operator on the left-hand side of Eq. (14). As inthe case of the free-particle Hamiltonian, such a termwould act only in Heitler's subspace II, and wouldhave no physical consequence. One also readily findsthat the operator on the left-hand side of Eq. (14)itself acts entirely in subspace II, so that Eq. (14) is arestriction on the redundant components only.If we project out of H, the part which acts only inHeitler's subspace I, we obtain(P')'H, (P')'= A.(P')' P'-+m '(iB; eA;) (iBi, —eAi, )P'P'P', —which is the same as Heitler's Hamiltonian. '1226 W. B. ZELEN Ystate f is given by(Q) = —bx PtPoQ|Pd'x,Using Kqs. (7) and (2), the erst term on the right-handside of Eq. (27) can be written(25)which is real because Q is neo-Hermitian. There can beno ambiguity here in the ordering of the factors P' andQ, since the order Qp' would not in general give a realexpectation value. Taking the time derivative of Eq.(25), we haved(Q)l«= —b 0'(H'P'Q P'Q—H)4d'*ibE —lb'p [oH,Q7&d' x=('[H,Q»,bEE—' P"(P')'P'H'Pdsx= —b E 'P ftPrPsPofdsxand the second term can be writtenbs (mE) 'Ps /tH'(Po)sP"PgdsxbxE '—Ps ftP P"Pgd'x.Adding these together gives(x')=p'/E,which is the classical result.so that we may take as the operator representing thetime derivative of Q, V. CONCLUSIONQ=i[H Q7We note that Q is neo-Hermitian, for(P'Q)'= i(~HQ—P'QH)'—= i(Q'O'H— H'P'Q)—i (P'QH— P'HQ—) =P'Q(26)We also note that a term of the form (19) added to thefree-particle Hamiltonian would have no effect on (Q)for states satisfying Eq. (10), because of the fact that(19) is neo-Hermitian. An analogous statement holdsforH, .As an example, a straightforward calculation of thetime derivative of the position operator x' for a freeboson yields=[P' P'7'im 'BsP'PsP' m'—P'P'(H'P'+m) —.(*')=-«t(p')'ped'*+bum 'Ps 4'(P')'P'Ad'x (27)As in the Dirac theory, the quantity x' is not directlymeasurable, but its expectation value is. Suppose wecalculate (x') in a simultaneous eigenstate of the free-particle Hamiltonian and the linear momentum, corre-sponding to eigenvalues E and p, respectively. We have,making use of Eqs. (8) and (10),The problems associated with the redundant com-ponents of the Kemmer wave function can be resolvedwithout resorting to Heitler's reduction to a subspacecontaining only the dynamical components. This resultis achieved by defining a new Hamiltonian, and re-stricting the redundant components of the wave func-tion via Kqs. (10) or (14). Just as in Heitler's theory,one obtains the customary expression (26) for thetime derivative of an observable, and the unambiguousexpression (25) for the expectation value of anobservable. 'ACKNOWLEDGMENTThe author acknowledges the assistance of R. R.-Sousek, whose preliminary calculations led to this work.7 Actually, Heitler omits the sign factor bz from the definitionof the expectation value, but multiplies the eigenvalues of anobservable by 8@ to get the "measurable values. " Thus, for anegative-energy eigenstate of the momentum operator corre-sponding to eigenvalue p, the "measurable value" of the momen-tum would be —p, and the expectation value of the momentum inthis state would also be —p. This system has the advantage thatthe "measurable values, " and the expectation values, of theHamiltonian are positive, even for negative-energy solutions. T'hesame system could be adopted here, but it seems to us somewhatmore artificial than the customary method of including the "norm"of the state in the definition of the expectation value. The problemof the negative-energy solutions is no different here than in theDirac equation, and is similarly resolved by second quantization.(See W. B. Zeleny and A. O. Barut, Phys. Rev. 121,908 (1961).]