The representation given by Mathews to describe particles of integer spins is modified in such a way that the Hamiltonian does not contain the matrix Co which is a null matrix. An extreme relativistic limit of the modified Hamiltonian is also obtained

Balasubramaniam, K S

Ramanujam, G Alagar

English

IACS Institutional Repository

Indian J. Phys. 61A (1), 63-68 (1993)On relativistic wave equation for particles of integer spinsG AlagarRamanujam and K S BalasubramaniamP G Dcparimcnl of Physics, Nallamulhu Gourulcr Mahalingam College, PoUachi-642 001, India Received 4 M arch J992, accepted J Septem ber 1992Abstract : The represcniauon given by Maihcws lo desenbe particles of integer spins is modified m such a way that the llamilionian docs not contain the matrix which is a null matrix. An extreme relativistic limit of the modified Hamiltonian is also obtainedKeywords : Wave equation, Poincare group generators, Lorentz invanani scalar product, metne operatorPACS Nos. : 03.65.-w, 11.10 Qr1. IntroductionFollowing the work of Weaver et al\1\ regarding the relativistic wave equation for particles of arbitrary spin, Mathews [2] obtained two Hamiltonians of the form,s s//, = E lan/i 2jj.9 + /3 8  ^ scc/i 2ji9/i-  2 =s s//2 = 'Z E  C„ colh 2 n 0 +  ( i  I ,  E C .  coscch 2^  9/ i-o   ^ - 0(1.1)( 1.2)lan/i 6  = pi E.Of ihc above iwo, former is quanlizablc for half-inieger spiris and ihc laiier is quantizable for inter spins. The integer Spin Hamiltonian is rather peculiar [3).The matrix Cq occurring in II2 is a null matrix.This can be seen from the relation,C^ A—fiwhere is a certain projection operator possessing the property, A^A„ = A^S^iv.By virtue of the above relation, we get(1.3)(1.4)(1.5) 1993 lACS64 G Aluf>ar Ramanujam and K S BalasubramaniamC^C„ = = B^Snv, (1.5a)Cf, = {A ^ -A  ^ (/\„  + y\ „) = Suv, (1.5b)= (/\^+/l_^)(/l„ + /!-„) = B ^S^v. (1.5c)For /7 = 0, wc have from cqs. (1.3-1.5)C. = 0The occurrence of Co in iJie RHS of eq. (1.2), makes ihe firsl term in the expansion i.c. E Co CoihO indeterminate since C„ = 0 and ColhO —>In this paper* an attempt has been made to modify the Hamiltonian H2 and other relevant operators in such a way tliat they do not contain the matrix Cq and consequently the above mentioned indeterminate term. This elimination of Co gives us a new representation which we call the modified Mathews representation (MMR). The MMR has one distinct advantage. I ’hc extreme relativistic representation which one obtains from MMR is free from the null matrix C^ ,. Earlier work regarding the extreme relativistic representation [4) contained the matrix Co and its presence was considered to be an unhappy feature of the work.The broad outline of the paper is as follows ;The operator (5) linking the Hamiltonian H2 and the Hamiltonian = j9C is available [5|. (Here C stands for Canonical representation). Taking this operator S and eliminating the matrix C„ from this operator, a new operator R is constructed. By using the operator R, a Hamiltonian //^ is obtained from H, by a similarity transformation. The Hamiltonian H2 thus obtained, remains free from the matrix Co- Relevant expressions for the position, spin and boo.st operators are obtained. Finally, the extreme relativistic limit of the Hamiltonian II2 is obtained,2 . O perator S linking Hi andThe Operator S linking the Hamiltonian /A and //, is of the form.V= Z  [ U + P)B^ + a , U + /J) C ^]M- 0S' = X j  (I + + <73 \  (/ + p) C ^ ]. wherem ^„(E + p) ^ j^(£ + ±Pf^\, = (£ + p f  [ (£  + p f ^  ±(y,p+p<7i = If- = i f  = 1 .(2.1)(2.2)(2.3)(2.4) (2.4a)Eliminating ihc matrix C„ from the above expression, let us define operators R and R ' such that65On relativistic wave equation etc s/? = X j  0  + j9) B/i + (/ + /J) C p] + kiB , (2.5)(2.6)The consistance of eq. (2.5) with cq, (2.6) demands that kik2 = 1. Of the many possibilities, we choose the simplest one, that is, k\= k2 = 1. Accordingly we get,B = Z  [ 0  + /? )« , + (Tt \  (/ + P) C J  + B„sR ' = Z  I u  + P) B„ + CT, /)' \  {f + P) C J  + fl„.^ ~ I*"Lcl us define a wavc-Iunclion y/such thai\f/ = r y ,where the wave function satisfies the Schrodmger ccjuation(2.7)(2.8)(2.9)(2. 10)If Gc represents a poincarc group generator relevant for the representation provided by vt. the corresponding generator G' lor the wave I unction y/' is given by the similarity transformation,G' = RGcR '■ (2.11)Replacing G, by = PR, we obtain the Hamiltonian //'t by the relationU \  = RpER '\ (2.12)By virtue of eq. (2.7) and (2.8) wc gelVn \  =  X  E C ^  c o t / i  2 / J 6 +  P  I  ECf,  c o s c c / i  2^^^ + a^E  B „ . (2 .1 3 )= 1 >1 = 1As desired. The Hamiltonian ll'^ docs not contain the null matrix It may be noted here that in eq. (2.13) the summation runs Irom jj. = 1 and not from /i = 0 as in eq. (1.2)3 . Lorentz invariant scalar product and observable in IMMR In the representation, the Lorentz inviiriant scalar product is of the formiVc-Vc) = (31)Transforming the above equation to Y  representation wc have66whereG Alagar Ramanujam and K S Balasubramaniam (V^ . vO = y/' d \M' = (/?■')* '= il/2m ,)[pjr2 + H'2p] + a,B ,.The expectation value of an operator O in the state is defined as < 0 >  =  jy/'* M'O y / ' d \for O to be real, we should have M'O =(3.2)(3.3)(3.4)The conventional position operator x and the spin operator S do not satisfy thfc above requirement. However, by exploiting the eq. (2.11), suitable well behaving expressions for position and spin observables can be obtained. Replacing Gc by x  and S in the eq. (2.11) we getx = R x B ' .  (3.5)S' = R S R \  (3.6)Since x and S arc observables in ihc y/^  representation x' and S' are observables in the \j/' represention. By replacing R and R ' by their equivalent expressions, the right hand sides of the eq. (3.5) and (3.6) can be evaluated. For spin 1, they are of the Torm,X’ = x+lrrio + 2 E -  (2m„£)'^)/(8m,^)‘'  ^{I + p) B.c^ 'dp + + p)<y-i tip -  B„ CTj T/p- i ( I - p ) C , m p  + i {a-dp^) (m„£/8)''^ +Here,+  < j j{ /  +  p ) C i  O i  -  i p  B d p ^  +  i p B x  p { m ^  + E ^ )  / I p ^ E ^ .S' = S -  Cl i Oj {S X p )  / p  -  B„S + (mJ&E)''^ {I + p) B ^S  + i [ [ 2 p ^  m „ E ) / p ( P m o E ) ' ' ^ ]  (/ -  /J) fl, (S x p )  I p  -  i Q - p )  (S X p )  I p+ ( £ / 2 m {I+ P) Cl i cT3 (5 x p) I p .T = {Xxp)  Ip and A = (73 5.(3.7)(3.8)4 . Boost operatorA representation is said to be completely defined only when we have determined expressions for all the operators of Poincare group in the representation. The Poincare group operators are : Hamiltonian (//), momentum (p), the angular momentum (/) and the boost operator (*).For the modified Mathews rcprcsenuition, the Hamiltonion is given in the equation (2.13). The angular momentum J is given t \ s J ^ ( a x p )  + S,where S in the Spin operator.To complete the description of the representation, the expression for the operator k is to be determined. The boost operator k links the wave functions 0' and \ /  in two different inertial frames. If 0' is the wave function at a space-time point in one frame, the wave function i// in another Lorentz frame is given asy/ = (f-^ichv.k^)(l>\ (4.1)Here, dv is the infinitesimal relative velocity between the two frames. An expression for the Boost operator can be obtained from the similarity transformationk' = R k ,R  \On relativistic wave equation etc 67(4.2)where= i p - ^ ( x U ,  + II, x) + [i (S X p)/(E + m„)IS the boost operator in the \j/c rcprcscnuition. For the Spin 1 case, K' is of the form, k' " tp ~ ~  (x' H ’2 ^ P)!^ + '”□)■5 . Extreme relativistic representationA representation suitable for describing Dirac particles with extreme relativistic energies was first studied by Mendlowitz f6| and later by Cini and louschek f7J. The Cini-Touschekrepresentation given for Spin - ^ was generalised by one of us [4] for arbitrary Spii). For any Spin-S; the extreme relativistic Hamiltonian is of the form-  Yj E C^ ^OoT \ 12(5.1)For integer spin case, the Hamiltonian ///. contains the null matrix and thus exhibits a drawback.In this section, we obtain an cxU’cmc relativistic Hamiltonian II e l^ y projecting the MMR to the extreme relativistic limit. For the integer Spin case, this Hamiltonian H e does not contain the matrix C„ and hence is a better candidate than the one obtained earlierby one of us.To project the Hamiltonian //'z to the extreme relativistic limit, we recall the similarity transformation (2.12)H \ = R ^ E R ' \In the extreme relativistic limit, the ratio m j p  -» 0. Injecting this high momentum approximation into the operators R and R wc have68 G Alagar Ramanujam and K S BalasubramaniamR (m jp  -» 0) -> r  = Sft [{I + P) B^ + O'i {I + P) Cf i )  + Bo (5.3)M - IR ‘ {mjp  -» 0) -» 7^ ' = P^ [(I + P) Bn + a-i (! + P) Cft ) ■¥ fl„(5.4)ft - 15ft = (mj%p)'^ (2plmof \ P,t =  (p/2m„)''^ (mo/2p)". The extreme relativistic Hamiltonian can now be obtained as,H’e = T p E T '  = I  £ [ CA  + a ^ B o E . (5.5)Other operators (C/^) relevant for this extreme relativistic representation can be obtainedfrom the similarity transformation.6 . C onclusion\The representation given by Mathews to describe particles of integer spins has been modified, in such a way that the Hamiltonian docs noi contain the null matrix Cq. By projecting the modified Hamiltonian to the extreme relativistic limit, we have obtained an extreme relativistic representation. Now it remains to be seen whether or not the modified Hamiltonian is quanti/able. Work in this direction is in progress and will be reported in a future publication. »Acknowledgm entThe authors remain extremely grateful to Prof N Namasivayam Principal, N G M College, Pollachi for all his kind encouragcmcni and to ihc referee of this paper for very useful suggestions which vastly improved the presentation of the paper.Rcrcrcnccs(Ij D L Weaver, C L Hammer and R M (Jr) Good 1964 P kys. R ev 13SB 241(2) P M Mathews 1966 Phys. R ev  143 978 985PI P M J Mathews 1967 7 M ath P h ys Sci 1 197[4] G Alagar Ramanujam 1913 Int J  Theor Phys 7 311[5| G Alagar Ramanujam 1975 /n/ J  Theor Phys 12 407[61 Mendlowiiz 1956 Phys R ev  102 527|71 M Cmi and B Touschek 1958 N uovo C im  1 422

On Relativistic Wave Equation for Particles of Integer Spins

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On Relativistic Wave Equation for Particles of Integer Spins

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