S p in -1/2 W ave E quations in
R elativistic Q uantum M echanics
D onald Stephen S ta u d te
Ju n e, 1993
A thesis su b m itted for th e degree of
S ta te m e n t
T h e (1972) idea to c o n stru ct a sp in -1 /2 analogue of th e spin-0 Feshbach- V illars equation belongs to Dr. B. A. R obson an d eq u atio n l . V I I . l was his (1972) a tte m p t at th e co n stru ctio n of such an equation. He (1991) suggested to m e th a t I a tte m p t to o b ta in th e energy sp ectru m of th e hydrogen ato m using eq u atio n l . V I I . l . T he (1990) idea to stu d y th e rep resen tatio n th eo ry of th e L orentz group belongs to Dr. J . M. C. F. G ovaerts.
However, all of th e m a te ria l given in ch ap ters 2, 3, 4, 5 an d 6 is solely my own original work, except w here explicitly acknowledged otherw ise.
I developed this m aterial in th e period from Ju ly 1992 to J u n e 1993.
A c k n o w le d g e m e n ts
I would like to th a n k my supervisor Dr. B. A. R obson for in tro d u cin g me to his problem of developing a sp in -1 /2 wave eq u atio n w ith indefinite m etric in rela tivistic q u a n tu m m echanics, and for his patience an d in itial guidance to enable m e to u n d e rs ta n d his studies of th e subject. I th a n k him also in his role as H ead of D e p artm en t for m aking it possible for m e to u n d ertak e a PhD a t the A u stralian N ational U niversity an d for arran g in g th e financial su p p o rt.
I would also like to th a n k Professor C. Jarlskog an d R ickard L indkvist for a r ranging a m ost fru itfu l an d rew arding six m o n th visit to Stockholm s U niversitet, d u rin g th e p erio d M arch-S eptem ber 1992. It was du rin g th is p erio d th a t I derived th e first m a jo r resu lts of th is thesis.
I also wish to th a n k Dr. J . M. C. F. G ovaerts for his invaluable insights, in sp ira tio n and co n sta n t encouragem ent th ro u g h o u t th e course of th is thesis.
I wish to th a n k Dr. A. M. P en d rill for arran g in g a research position for m e in G öteborg, beginning A ugust 1993. T his has allowed me to co n cen trate on my thesis over th e last few m o n th s, w ith o u t having to find fu tu re em ploym ent as well.
In my developm ent of th e m aterial given in ch ap ters 2, 3, 4, 5 an d 6, I have b e n efited from invaluable discussions w ith an d com m ents from th e following people: Professors K. I. G olden, C. Jarlskog, A. Klein, B. L au ren t, I. Lindgren, B. H. J. M cK ellar, P. M innhagen, J . Sucher, J. A. W heeler;
D rs F. B astianelli, J. M. C. F. G ovaerts, C. J . Griffin, H. H ansson, H. H. Lee, U. L indgren, E. L in d ro th , H. N ordstrom , A. M. P endrill, R. G. S tau d te, B. Sund- borg;
an d P. A. Blackm ore, J. P. C ostella, J . A. Daicic, D. J . D aniel, K. Halliwell, S. Lane, K. Laughlin an d S. A. M oten.
I wish to th a n k D rs M. A ndrew s, J. M. C. F. G ovaerts an d B. A. R obson for th e ir assistance du rin g th e proofreading stage of this thesis.
I wish to th a n k th e A C T A cadem y of S p o rt, P erry Blackm ore, Steve Craig, Rob Jessop, R ickard L indkvist an d all my o th e r friends who have c o n trib u te d to these rew arding years of my life.
A bstract
T his thesis is concerned w ith th e developm ent of a new sp in -1/2 wave equa tio n in relativ istic q u a n tu m m echanics. T his eq u atio n is th e sp in -1/2 analogue of th e spin-0 Feshbach-V illars equation.
T h e thesis begins in ch ap ter 1 w ith a review of th e su b ject of relativistic wave equations and its place in th e developm ent of m odern physics. C h ap ter 2 sees th e derivation of th e new equation (hereafter referred to as th e F V 1 /2 eq u atio n ). T his derivation is p resen ted at th ree levels of u n d erstan d in g , an d is com bined w ith an analysis of th e eq u atio n using elem ents of th e underlying m ath em atics of th e theories of special relativ ity an d q u a n tu m m echanics. T he analysis justifies th e m ethods of derivation of th e equation, an d its su itab ility as a wave eq u atio n in relativistic q u a n tu m m echanics. It is shown th a t the F V 1 /2 eq u atio n is different to th e D irac eq u atio n and o th e r equations in th e lite ra tu re .
O ne of th e striking features of th e F V 1 /2 eq uation is th a t it contains a less restrictiv e dynam ics th a n th e conventional D irac equation. C h ap ter 3 exam ines th e solution space and proves some resu lts using th e discrete sym m etries C, P an d T , which enable a subset of th e solution space to be used for physical applications. It is possible to use a subset of solutions which contain th e usual D irac negative energy states, or it is possible to use a subset w hich contains only positive energy states, b u t includes states of opposite signs of charge. T he second su b set corresponds to th e physical p ro p erties of particles an d an tip articles and suggests th a t th e F V 1 /2 eq u atio n could be useful in atom ic physics.
o b tain ed in this c h ap te r (again) suggest th a t th e F V 1 /2 eq u atio n could be useful for calculations in atom ic physics.
C h ap ter 5 discusses im p o rta n t fu rth e r work which should be u n d ertak en if th e F V 1 /2 eq u atio n is to becom e useful to physics. T he developm ent in th e previous chapters co n cen trated on th e F V 1 /2 eq u atio n in relativ istic q u an tu m m echanics. It should be also considered as a q u an tised field equation. T h e two electron problem a n d th e calculation of physical q u an tities are crucial tests of th e F V 1 /2 eq u atio n ’s applicability in atom ic physics. T h e m a th e m atica l analysis in c h ap te r 2 can b e extended. Suggestions for gaining a deeper m ath em atical u n d e rstan d in g of th e less restrictive dynam ics of th e F V 1 /2 eq uation are given.
T a b le o f C o n te n ts
Title page
Statement of originality of research
Acknowledgements
Abstract
Table of Contents
Preface
Notation
C h a p te r 1. I n tr o d u c tio n
1.1. T h e h istorical developm ent of relativistic wave e q u a tio n s ... 1
1 .II. A b rief overview of R Q M ... 4
1 .III. O p erato rs and ex p ectatio n values in R Q M ... 7
l.IV . Free p article solutions in R Q M ...8
l.V . T h e no n -relativ istic l i m i t ... 10
l.V I. H istorical developm ent, c o n tin u e d ... 11
l.V II. In itial m otivations for a sp in -1/2 analogue of th e FVO e q u a tio n ... 13
1. V III. O u tlin e of this th e s is ... 17
C h a p te r 2. D e r iv a tio n s a n d a n a ly sis o f th e F V 1 / 2 e q u a tio n 2.1. In tro d u c tio n ... 21
2.II. A first derivation of th e F V 1 /2 e q u a tio n ... 25
2.III. T h e K G 1 /2 an d D irac equations in spinor n o ta tio n ... 27
2.IV. C om parison of th e solutions of th e KG 1/2 and D irac e q u a tio n s... 31
2. V. A second derivation of th e F V 1 /2 e q u a tio n ... 32
2.VI.. T h e indefinite m etric form alism in R Q M ... 36
2.VII. A th ird derivation of th e F V 1 /2 e q u a tio n ... 49
2.V III. V arious linearisation procedures applied to th e second o rd er e q u a tio n s ... 53
2.IX. O p erato rs an d ex p ectatio n values for th e F V 1 /2 e q u a ti o n ... 59
65
2. XL The eight solutions...
C h a p te r 3. T h e u se o f th e F V 1 / 2 e q u a tio n in p h y sic a l p r o b le m s
3.1. Introduction...69
3.11. The number of solutions...70
3.111. The non-relativistic limit and C
for the FVO and Dirac equations... 71
3.IV. The non-relativistic limit and C for the FV1/2 equation...78
3. V. A physical example illustrating the C interpretation... 83
3.VI. CPT and the eight solutions... 83
3. VII. Conclusions... 90
C h a p te r 4 . T h e h y d r o g e n a to m
4.1. Introduction...91
4.11. The hydrogen atom solution... 93
4.111. Discussion of the hydrogen atom solution... 102
4.IV. Comparison of the hydrogen atom solution
using the FV1/2 and Dirac equations... 107
4. V. Comparison of the hydrogen atom solution
using the FV1/2 and KG 1/2 equations... 113
4. VI. Further discussion of the hydrogen atom
solution using the FV1/2 equation... 115
4. VII. Conclusions...116
C h a p te r 5. F u r th e r w ork
5.1. Introduction... 118
5.11. Second quantisation...120
5.111. Atomic physics...120
5.IV. Extensions of chapter 2 ... 121
5. V. Extensions of chapter 3 ... 123
5. VI. Extensions of chapter 4 ...124
5.VII. Some other questions...125
C h a p te r 6 . C o n c lu s io n s
P r e fa c e
T his thesis is concerned w ith th e co n stru ctio n of a new sp in -1/2 relativistic wave equation. T h ere already exists a well established s p in -1/2 relativistic wave eq u atio n , th e D irac equation, which was derived in 1928 an d has been used widely an d successfully since th a t tim e. W hy th e n should one search for an o th er spin- 1 /2 relativ istic wave equation? A nd even if one is found, w h at should be done w ith it in th e context of a PhD thesis?
T h ere have been a tte m p ts in th e lite ra tu re to co n stru ct o th e r sp in -1/2 rel ativ istic wave equations, b u t by in large th e D irac eq u atio n has rem ained th e com m only accepted an d used equation. It is shown in th is thesis th a t th ere is yet an o th er sp in -1 /2 relativistic wave equation, being th e sp in -1/2 analogue of th e spin-0 Feshbach-V illars equation. To avoid th is eq u atio n also falling by th e wayside, w hat should be done?
N atu re knows if th e new equation has any m erit, if it provides a b e tte r s ta r t ing po in t th a n th e D irac eq u atio n for the solution of certain physical problem s, if it provides insights into N atu re itself. However, few physicists will consider using th e new equation if all one can say is ‘here is a new eq u atio n , it m ight work, b u t d o n ’t ask me how ’, w hen m any physicists have already toiled for years to learn how to apply th e D irac equation to physical problem s. In stead , in order to m ake a new eq u atio n useful to physicists, one m u st be able to say ‘here it is, it does work, this is how it works, an d this is what one should do to use i t ’. One m u st also com pare it in detail w ith th e D irac eq uation (and m eth o d s based upo n th e D irac equation) so th a t physicists can readily u n d e rsta n d th e differences and readily choose which equation they wish to use in th eir studies of aspects of N ature.
Preface 2 By establishing firm m ath em atical results one has a sta rtin g po in t on which to launch fu rth er investigations into answ ering th e rem aining ‘if ’s, ‘b u t ’s, ‘d o n ’t know ’s an d ‘m ay b e’s of using th e F V 1 /2 equation. I have tried to answ er as m any fu n d am en tal questions as possible in th e scope available for one PhD thesis, so th a t th e vast n u m b er (and th ere will be a vast n u m b er w hen a tte m p tin g to provide an a ltern ativ e to som ething stu d ied intensively for 65 years) which rem ain unansw ered can be approached m ore easily in fu tu re work.
T h e preceding p a ra g rap h s provide some ex p lan atio n of m y app ro ach to this thesis, b u t th e question ‘why should one search for a n o th er sp in -1/2 relativistic wave e q u atio n ’ is still unansw ered. T he initial m otivations of D irac w hen he derived his equation in 1928 were som ew hat different to a m odern a priori ap proach to co n stru ctin g relativistic wave equations. M odern physics now has on th e experim ental side th e phenom enology of elem entary p articles an d condensed m a tte r physics, an d on th e theoretical side group theory, m any-body physics and Q F T . A fter D irac’s eq u atio n th e search for o th er relativ istic wave equations was no t concluded, b u t continued in th e lite ra tu re , w ith b o th em erging experim en ta l resu lts an d theo retical techniques being in co rp o rated into th e studies. D irac him self rem arked, ‘. .. a tru e advance will be m ade only w hen some fu n d am en tal a lte ra tio n is m ade, ju s t ab o u t as fu n d am en tal as passing from th e KGO equation to th e D irac e q u a tio n .’, [Dirac, (1975)]. B. A. R obson becam e in terested in in try in g to co n stru ct sp in -1/2 indefinite m etric relativistic wave equations an d gave a sem inar on this in 1972 [Robson, (1972)].
N evertheless, th e D irac equation has rem ained th e d o m in an t sp in -1/2 rela tiv istic wave eq u atio n to date.
My involvem ent in th e su b ject of sp in -1/2 relativ istic wave equations began w hen I becam e R o b so n ’s sudent in 1989 and this thesis is a record of progress m ade since th a t tim e on co n stru ctin g a sp in -1/2 analogue of th e FVO equation an d then m aking it useful to physics.
Preface 3 Initially, as a stu d en t finishing u n d e rg ra d u ate th eo retical physics a t th e U ni versity of M elbourne, I was in trig u ed an d im pressed by th e m arvels an d b eau ty of th eo retical physics, th e elegant m ath em atics, conciseness an d power of Q F T , general relativ ity an d p article physics. T h ere seem ed so m uch to learn and a longing to spend m ore tim e celebrating th is b eau tifu l p a rt of N atu re. My first im pression of R o b so n ’s research was th a t it was a way to fu rth e r stu d y these aspects of m odern physics a t th e A u stralian N ational University.
N o ta tio n
4~ Vectors
g ^ u — d ia g ( l, —1, —1, —1)
V» = (Vo, V u V 2, V 3) = (Vo, Vt ) o r (V0 , V ) , VM = 9 llv V v
Pu = ( E / c , p) c a n o n ic a l 4 -m o m e n tu m
7Tß = (tt/c, 7t) m e c h a n ic a l 4 -m o m e n tu m
= (A o , A ) e le c tro m a g n e tic 4 -p o te n tia l
TT/x = P/i - e A ^ /c
d „ = ( A ) , ö ) = ( ( l / c ) £ , - V )
— >■ i h d n , 7 r M —> i h D ß q u a n tis a tio n
pn = TikM kß is a t tim e s used fo r convenience in e x p o n e n tia ls
D ß = dß + ( i e / h c ) A fl m in im a l c o u p lin g
a.b = a^b*1 = a^bo — a.b w h e re = (ao,a) a n d bß = (6o,b) sca la r p ro d u c t
E ß i / = d ß A i / d i / A ß
fs T D p r o d u c t o f g a m m a m a tric e s in th e s ta n d a rd re p re s e n ta tio n
r e la tio n s h ip betw een th e s ta n d a rd a n d th e W e y l re p re s e n ta tio n o f th e g a m m a E = - V A 0 - ( l / c ) { d A / d t ) , B = V x A
C onstants
h , c are used e x p lic itly e xcep t w here s ta te d , k = m c / h
G a m m a ( D i r a c ) m.atrices
7n = ( 7 o ,7 ) , (70 = 7 ° )
N otation 2
<x, T{ conventional P au li m atrices
V = I ' D ,
Spinor quantities
Spinor q u an tities are defined exactly as those in ch ap ter 3 of [Lifshitz, B erestetskii an d P itaevskii, (1971)].
[Aa/3] = A0I2 + o.A A af3 is th e spinor equivalent of th e electrom agnetic 4-
p o te n tia l
In n er product notation
T h e n o ta tio n is sim ilar to th a t used in [Pease, (1965)]. H H ilbert space
* com plex conjugate, T tran sp o se t = *T
^ adjoint of a m a trix , # = *T for th e u n ita ry inner p ro d u c t |w ) a b stra c t vector in some inner p ro d u ct space
w ( f ) =
( C |w )
For infinite dim ensional inner p ro d u c t spaces th e rep resen tativ e of |w ) is o b tain ed by specifying w (f)V f.N am es o f theories
QM q u a n tu m m echanics
RQ M relativ istic q u a n tu m m echanics N RQM non -relativ istic q u an tu m m echanics
N otation 3 N am es o f equations
th e KGO eq u atio n th e spin-0 K lein-G ordon equation
th e K G 1 /2 eq u atio n th e sp in -1 /2 K lein-G ordon (F e y n m a n /G ell-M ann) equa tio n [Feynm an an d G ell-M ann, (1958)]
th e FVO eq u atio n th e spin-0 Feshbach-V illars eq u atio n (equation (2.15) [Fes- hbach an d V illars, (1958)] )
th e F V 1 /2 eq u atio n th e sp in -1/2 Feshbach-V illars type eq u atio n derived in c h ap te r 2 of this thesis
W avefunctions
In ch ap ters 1, 4, an d 5; and sections 2.VI-2.IX m ost w avefunctions are labelled by th e sym bols ^ ( x ) , 4/(x), </>(x), A(x), £(x), 7j(x),/ ( r ) , etc. w ith su b scrip ts to in d icate to which eq u atio n they belong a n d /o r to fu rth e r define them . In sections 2.II-2.V , 2.X, 2.XI and ch ap ter 3, w avefunctions are conveniently w ritten as if in stead of i/>(x). In general th e Schrödinger rep re se n ta tio n is assum ed, th e w avefunctions are tim e-dependent.
In a few cases a w avefunction is w ritten as th e tran sp o se of a row vector so th a t it can be w ritten as p a rt of norm al tex t. For exam ple i f = (a , ß ) T represents
T h e entries a and ß can som etim es represent colum n vectors w ith two or four com ponents them selves, and th e n o ta tio n i f = (c t , ß ) T is used to m ean i f = ( aT , ß T ) T, so th a t i f is indeed a colum n vector. To explicitly w rite (a T , ß T ) T
would be cum bersom e, an d hence th e sh o rth a n d ( a , ß ) T is used. 4/d(x) th e D irac w avefunction
N otation 4
'I 'f v o(x) th e FVO w avefunction ^ f v i/ 2(x ) th e F V 1 /2 w avefunction
4,p v i / 2(x ) th e F V 1 /2 w avefunction, using specifically ^ p V1/ 2 = ♦<8)^0 if± th e w avefunctions of th e equations (i JJ) ± K)if± = 0
£a , €1 1lß ’i V w avefunctions of spinor equations
^ w avefunctions of spinor equations w ritten in Feshbach-V illars type no ta tio n
In th e th ird ch ap ter th e qu an tities ( 4 > o , X o ) T , {4>o ^ X o ) T ’> a n d (</>o,X o ) T are used. T h e prim es indicate th a t, in general, <^o 7^ ^0 / ßo a n d Xo 7^ Xo 7^
Xo-Hydrogen atom definitions
T h e atom ic u n its used are those of [Bethe and Salpeter, (1957)], page 3.
yj™ spherical spinors defined according to equation (16.80) of [M erzbacher, (1970)], page 928 of [Messiah, (1985)]
0 ( / , / ' 0) an g u lar wavefunctions / ( r ) , g(r), h ( r) rad ial w avefunctions
F ( a , ß , x ) = \ F \ ( a , ß , x ) confluent hypergeom etric function
Conserved current densities and H am iltonians
T hese are labelled by and H respectively, w ith su b scrip ts to in d icate which eq u atio n they belong to. In general, H , ra th e r th a n f7 (x ), is used to specify th e H am ilto n ian o p e ra to r in the position rep resen tatio n
O ther useful quantities
if = i f ^ °
= ' y 0 T i f *
N otation 5 C, P, T th e discrete tran sfo rm atio n s; charge conjugation, sp atial inversion, tim e reversal
x, r 3-dim ensional sp atial coordinate
P L G , p ro p er Lorentz group T he group of tran sfo rm atio n s com prising ro tatio n s in 3-dim ensional space plus Lorentz boosts
C H A P T E R 1
Introduction
I. T h e h is to r ic a l d e v e lo p m e n t o f r e la t iv is tic w a v e e q u a tio n s
In th e early p a rt of th e tw en tieth cen tu ry th e fou n d atio n s of th e theories of q u a n tu m m echanics and special relativ ity were established*. T hese theories re quired a su b sta n tia l revision an d extension of th e ideas an d concepts of n in eteen th cen tu ry classical physics. Q u an tu m m echanics (QM ) set o u t to explain th e newly discovered atom ic p henom ena an d in doing so necessitated a re-exam ination of such concepts as m easurem ent of a physical system an d th e notion of p a r ti cles itself. T he m a th e m atica l form alism of QM was based u p o n th e use of a H ilbert space. A new fu n d am en tal co n stan t (P lan ck ’s co n stan t, h) appeared, being a m easure of th e m inim al effect a m easurem ent can have upo n a system . T h e form al stru c tu re of QM first ap p eared in 1926 w hen H eisenberg published his ‘m a trix m echanics’ while Schrödinger produced his ‘wave m echanics’, which were soon shown to be equivalent. T he relativ ity theories h a d already been de veloped by 1915 an d required an extensive revision of th e notion of space and tim e, an d in th e general theory, gravity. T h e physical predictions of th e special th eo ry of relativity, published by E instein in 1905, differed from classical physics m ost strongly as th e speed of objects approached a new lim iting value, c.
Hence by 1926 physicists h ad two young and pow erful theories based upon different concepts w ith th eir in itial applications to different frontiers of observable phenom ena. It did no t take long for physicists to a tte m p t to synthesize these
1. Introduction 2 two theories**. How should th e two theories be com bined? Should one s ta rt w ith relativ ity and som ehow im pose concepts an d corrections from QM ? Should QM be th e fou n d atio n an d th e essential aspects of relativ ity be in co rp o rated ? Should a com pletely new th eo ry be in itiated ? Should th e general th eo ry of relativ ity be considered, or is th e special theory sufficient initially?
T h e idea of a ‘re la tiv istic ’ QM soon followed, as a way to in co rp o rate th e special th eo ry of relativ ity in to QM. It was designed to provide an analogous form alism to th a t of QM. To co n stru ct such a th eo ry it ap p eared sufficient (based on th e theo retical knowledge and experim ental evidence at th e tim e) to replace th e non-rela.tivistic energy-m om ent urn relatio n by th e correct relativ istic relation a n d th en proceed analogously to th e developm ent of QM. It was hoped th a t relativ istic wave equations, as relativistic generalisations of Schrödinger’s wave eq u atio n , could be used for QM problem s of a single p article m inim ally coupled to a classical ex tern al electrom agnetic field.
T h e form alism of (non-relativistic) QM, hereafter referred to as N RQM , relates physical q u an tities to m ath em atical objects in a (generalisation of * *) H ilbert space H. T h e sta te of a physical system is rep resen ted by an a b stra c t vector in an d is denoted as |ip). T he p ro b ab ility in te rp re ta tio n is th a t a norm alised s ta te |ip) has inner p ro d u ct
** In fact Schrödinger h ad already considered a wave eq u atio n based upo n the relativ istic energy-m om entum relatio n E 2 = (p c )2 + (m e2)2 before he published his fam ous eq u atio n based upo n E = p 2/(2 m ).
* refer to sections 2.VI an d 2.IX for a m ore precise ex p lan atio n of this
T h e w avefunction of a system in th e position rep resen tatio n is given by
?/>(x). It
satisfies Schrödinger’s equation (in position space),
1. Introduction 3 Schrödinger’s eq u atio n ** is o b tain ed by using n = n2/(2 m ) (w here 7rM = (7r/c , 7r)
is th e m echanical 4-m om entum and p ß = ( E / c , p) = 7rM + e A ^ / c is th e canonical 4-m om entum ) w ith th e replacem ent —> iTidll. D ynam ical variables are rep re sented by H erm itian o p erato rs 0 in th e H ilbert space. T h e eigenvalues of 0 give th e possible m easurem ents of th e dynam ical variable an d th e ex p ectatio n values of 0 represent th e average of m any such m easurem ents on identically p rep ared system s. A u n ita ry tra n sfo rm atio n U can be perform ed on th e sta te \ip) which preserves th e p ro b ab ility in te rp re ta tio n . If \ip') = U\ij>), th en
w w ) = w u ' u m
= <«/#) •
W h at h ap p en s if th e relativistic energy m om en tu m relatio n n ^ n 11 = (m e)2 is in stead used as a s ta rtin g point?
T h e first published relativistic wave eq u atio n was th e K lein-G ordon equa tion, hereafter referred to as th e KGO equation, which is
( D ^ D r + K2) *kgo(x) = 0 . (1 ./.4 )
R a th e r th a n providing an identical form alism to th a t of N R Q M , th e use of th e KGO eq u atio n im m ediately required a revision of th e p ro b ab ility in te rp re ta tio n . T h e zeroth com ponent of th e conserved 4-vector which rep resen ted th e p ro b a bility density in N RQM was no longer positive definite. How could a s ta te have negative pro b ab ility ? (At th e tim e an tip articles h ad no t yet been recognised in experim ents, so th ere was no reason to consider j ° as a charge density) T h e in definiteness of j ° was th e first sign of a new degree of freedom b ro u g h t in by th e use of relativ istic wave equations. T h e use of a second o rd er eq u atio n bro u g h t a n o th er co m p licatio n -th e equation was no longer in H am ilto n ian form. Is th ere a way to linearise th is eq u atio n to recover th e H am iltonian form ?
1. Introduction 4
D irac published his now fam ous eq u atio n in 1928 which was a linear equation re la ted to th e free p article KGO equation. T h e D irac eq u atio n is a 4-com ponent eq u atio n in H am ilto n ian form , w ith positive definite j ° . It also explains th e spin of th e electron, which a t th e tim e was th e d o m in an t p article stu d ie d using QM. B ut why should th e new equation have four com ponents in stead of two? Tw o could be expected to describe th e spin, b u t th ere were four, which provided a new puzzle. C onnected w ith th is was th e problem th a t th e eigenvalues and ex p ectatio n values of th e H am iltonian for th e free p article states were no longer positive definite. N egative free particle energies? T his was a n o th er m an ifestatio n of th e (at th e tim e new an d m ysterious) degree of freedom ap p earin g in relativistic wave equations.
II. A b r ie f o v e r v ie w o f R Q M
Before th e discussion of th e historical developm ent of relativ istic wave equa tions is continued, it is useful to consider a little m ore closely ju s t w h at is th e RQ M of th e D irac an d KGO equations. T his overview of RQ M is in ten d ed only to give an o utline of th e p a rts of the su b ject relevant to th e h isto rical discussion.
A ‘rela tiv istic ’ QM of th e KGO eq u atio n analogous to N RQM can n o t be given directly, because th e KGO equation is not in H am ilto n ian form . T h ere is, however, a n o th er equation, known as th e Feshbach-V illars eq u atio n [Feshbach an d V illars, (1958)] (hereafter referred to as th e FVO eq u atio n ), which is sim ply th e KGO eq u atio n re w ritte n in H am iltonian form *. T his is achieved by rew riting th e second ord er eq uation as two first order in th e tim e derivative equations. T he FVO eq u atio n is an equation, which to g eth er w ith th e D irac equation, provides a convenient com parison of RQM w ith th e form alism of N RQM . T h e D irac equa tio n is for sp in -1 /2 p articles, while th e FVO eq uation is for spin-0 particles. It tu rn s ou t th a t th e FVO equation is th e only sim ple way of rew riting th e KGO eq u atio n in H am iltonian form , th e five-com ponent eq u atio n , which is also a
1. Introduction 5 earisatio n of th e KGO equation, is not in H am iltonian form (see th e footnote on page 66 of [Lifshitz, B erestetskii and P itaevskii, (1971)]).
T h e D irac eq u atio n is
( « Y ^ - k)\Pd(x) = (* Ip - /c)’I 'd(x) = 0 , (1.77.1) w here is a 4-com ponent w avefunction. For th e purposes of this b rief discus sion th e sta n d a rd rep resen tatio n of th e gam m a m atrices will be used;
7° = I 20 0
—12 7 — — (T (1.77.2)
T h e FVO eq uation is
l
2ih(d/dt)'f?Fvo(x)
=77
fvo^ fvo(x), w ith*
fv° = - £
d2 ( -
i- i ) + m c 2 ( i _ ° i ) + ^
•
d - " - 3)
T h e FVO eq u atio n is derived as follows [Feshbach an d V illars, (1958)] , using w hat will be referred to in th is thesis as th e Feshbach-V illars lin earisatio n procedure. C onsider th e KGO eq uation (1.7.4), given by
(D ’‘D„ + «2) ^kgo(x) = 0 . ( 1 .//.4 ) T his can be w ritten as
(D
l
-D
2 + k2)1'kgo(x) = 0 . (1.77.5)T h e Feshbach-V illars linearisation procedure involves w riting th is second order in th e tim e derivative equation as two first ord er in th e tim e derivative equations. Define 'Ffvo(x) = (<£(x),x(x ))T , with
Y x ) = ^=(1 + z‘A)/Y^KGo(x),
x (x ) = ^=(1 - ?770/ « ) ^
kgo(x ) • (1.77.6)
1. Introduction 6
- i« Do(<£(x) -
x (x )) +
( - D 2+ k2)(</>(x) + x (x )) = 0 , (1.77.7)while eq uation (1.77.6) can be w ritten
? Tin
W x ) - x (x)) = ^ W x ) + x(x)) .
(1 .//.8 )
K
R earran g in g these equations gives:
A)(<MX) - x (x ))
= —
D
2+
k2) ( 0 (x) 4- x (x )) , (1.77.9)7)o(0 (x ) + x ( x )) = zac(<^(x) - x ( x )) . (1.77.10) T aking sum s an d differences of these equations gives
=
+ x (x )) + ^mc2 + eAo)^(x )’
(1.77.11)
ih§tX^ =
+
+ (_mc2 + eAo)x(x )-
(1.77.12)
W ritte n as a 2-com ponent equation for th e w avefunction (^ (x ), x ( x ) ) T , equations (1.77.11) an d (1.77.12) to g eth er form th e FVO eq u atio n w ith H am ilto n ian given by eq u atio n (1.77.3).
It was m entioned above th a t th e p ro b ab ility in te rp re ta tio n of N RQM is given by eq u atio n (1.7.1). T his resu lt is derived from a conserved cu rren t density as follows: If j-/i is a 4-cu rren t density, th en it is conserved if d^j^1 = 0. C onsider th e Schrödinger eq u atio n ih (d /d t)r p (^ ) = 7 f^ (x ) an d co n stru ct th e following relatio n ,
1. Introduction 7
T his can be w ritten in th e form d^j*1 = 0, w ith
J° = 0*(x)V»(x) , (1.77.14)
j = - ^ ( V ( x ) V ^ ( x ) - ( V ^ ( x ) ) ^ ( x ) Nj - — A ^ * ( x ) 0 ( x ) . (1.77.15)
Z m \ J m e
W h at h ap p en s in RQ M ? For th e D irac equation, eq u atio n (1.77.13) can be again used (w ith * replaced by *T = t ) , which also leads to a conserved cu rren t density w ith
3° = ^ d(x) ^d(x), j = ^ { ) ( x ) a ^ D(x) , (1.77.16) which is conventionally w ritten as j 1* = 4 'd(x)7 /z,I 'd(x). Hence j ° is again positive definite an d th e original p ro b ab ility in te rp re ta tio n of N RQM is regained. However, for th e FVO equation, th e conserved cu rren t density becom es
3° = 4,fvo(x)t3 ^fvo(x) = |0 ( x ) |2 - |x ( x ) |2 , (1.77.17)
ifi (
j = ^FVo(x )r 3(r 3 + zt2) V ^fvo(x) - (V ^ F V o (x ))t r 3( r 3 + ir2)\I'fvo(x)
_ m e A ^ F V0(x )r 3(r 3 + it2) ^fvo(x). (1.77.18)
j ° is no longer positive definite, and th is corresponds to th e use of an indefinite in n er p ro d u c t (this resu lt will be discussed in m ore d etail in th e next ch ap ter).
I I I . O p e r a t o r s a n d e x p e c t a t i o n v a lu e s in R Q M
1. Introduction 8 in n er p ro d u c t given above by equation (1.7.1). For th e D irac eq u atio n , the form alism is based u p o n an inner p ro d u c t space w ith inner p ro d u c t
( * d| * D > =
J
# D!,(0 * D a (C K ,
£
(1.777.1) from which th e resu lts for o p erato rs an d ex p ectatio n values follow analogously to N RQM . For th e FVO equation, however, th e inner p ro d u c t is indefinite (im p ro p er),< * F V o | * F VO> =
J
* J . v o ( f ) r
3
# F V O ( C K
• ( 1 .I I I .2 ) Hence th e m a th e m atica l form alism m ust be generalised. For exam ple, th e defin itio n of th e ex p ectatio n value of an o p e ra to r Q, becomes:(Ü) = j
4 v o (C )T 3 fi(C )'W C )d f .
(1 .///.3 )
T his definition is discussed in section 2.IX. T h e inner p ro d u c ts of th e D irac and FVO equations are extensions of th e inner p ro d u c t of N R Q M , which ‘c o n ta in ’ th e relativ istic (an d also in th e D irac case th e spin) degree(s) of freedom .
T h e FVO eq u atio n is ju s t the KGO eq u atio n rew ritte n as a 2-com ponent equation. A conserved cu rren t density can also be derived for th e KGO equation, which is found to be identical to th e conserved cu rren t density for th e FVO eq uation, ju s t w ritten using different q uantities.
IV . F ree p a r tic le s o lu tio n s in R Q M
W hile th e D irac and FVO equations provide inner p ro d u c t form alism s, these form alism s include th e new degree of freedom m entioned earlier th a t is not p resen t in NRQM . T h e free p article solutions of th e D irac an d FVO equations p resen t a convenient first glance a t this new degree of freedom .
1. Introduction 9
0S (x) = _ L e - i<E^- p'x)/ ', = ~ e ~ i k l , ( l ./ V .l )
V N V N
w here N is a n o rm alisatio n factor, k. x = k^x^, p^ = HkM, an d E > 0. T h e state ■0s(x) has energy ( H) = -\-E. T he (non-relativistic) P au li eq u atio n for sp in -1 /2 p articles has two free p article solutions:
0 P « =
(1./V .2)w here ( a ,/? ) T is a norm alised 2-com ponent (n on-relativistic) spinor. A gain th e energy is (H) = + E . T h e KGO equation has two free p article solutions:
^kgo(x) = . (1./V .3)
w here N' is a n o rm alisatio n factor so th a t j °KG0 tran sfo rm s as th e zero th com pon en t of a 4-vector and E > 0. N ote th a t th ere are now two signs possible of i E t / h in th e exponential, representing different solutions. j %G0 is opposite in sign for th e two solutions. T h e FVO equation has two free p article solutions:
*
F
V
o
(
x
)
=
^
(
i
)
e
"
“
'
x
’
{1JVA)
=
•
(1 J™ )
T hese have (H) = + E in b o th cases, an d j°Fy Q is again opposite in sign for the two cases. T h e D irac eq u atio n has four free particle solutions:^ d(x)
^ d(x)
1
y/N
777\
\
)
— ik.x e
e+ik.x 7
5
(1./V .6)
1. Introduction 10 w here now (q,/3)t an d (a ' , ß ' ) T are norm alised 2-com ponent spinors in th e rest fram e. T h e free p article solutions of th e D irac eq uation have positive definite j ° , however th e first two have (H ) = -f-E, while th e second two have ( H ) = —E .
T he new degree of freedom ap p ears for th e FVO eq u atio n as th e need to generalise th e concept of p ro b ab ility density, to a density which is positive definite for one type of solution, an d negative definite for th e o th e r ty p e of solution. A charge density fits th is criterion, for some charge (such as th e electric charge) w hich can take eith er sign. However, for th e D irac eq u atio n , th e degree of freedom ap p ears as th e need to generalise th e concept of kinetic (-f rest m ass) energy to include th e possibility of b o th positive and negative kinetic (+ rest m ass) energies.
C are will have to be tak en when developing th e ‘re la tiv istic ’ QM form alism using th e FVO an d D irac equations due to th e presence of th is new degree of freedom . T h e use of th e indefinite inner p ro d u ct for th e FVO eq u atio n will lead to a need to carefully s ta te th e in te rp re ta tio n of eigenvalues an d ex p ectatio n values in RQM . W hile th e inner p ro d u ct of th e D irac eq u atio n is positive defi n ite, negative eigenvalues and ex p ectatio n values occur which are not present in N R Q M , th is also leads to th e need to carefully re-exam ine th e in te rp re ta tio n of these quan tities.
V . The non-relativistic lim it
In th e lim it th a t c —> oo, or th a t th e rest energy m e 2 is m uch g re a ter th a n all o th er energies in a given problem , it is n a tu ra l to expect th e relativ istic wave equations to reduce to th eir non-relativistic co u n terp arts. T his m eans th a t th e FVO eq u atio n should reduce to th e Schrödinger equation, an d th e D irac equation to th e Pauli equation. In fact they do.
1. Introduction 11 su b ject.
V I . H is to r ic a l d e v e lo p m e n t , c o n tin u e d
In 1928, after D irac published his equation, th ere were two m ain relativistic wave equations in th e lite ra tu re , th e D irac an d KGO equations. T h e first of a series of m a jo r developm ents of this situ a tio n was an ex p lan atio n of th e negative energy states of th e D irac equation. D irac proposed th a t all of th e negative energy states are norm ally filled an d hence th e P auli principle could be used to explain why positive energy electrons did not m ake tra n sitio n s in to negative energy electrons. T h e exp lan atio n also included th e possibility of exciting a negative energy electron ou t of th e negative energy sea, leaving behing a hole, which should behave as a positively charged electron. D irac in itally th o u g h t th a t these holes m u st be the p ro to n s, as no o th er positively charged particles h ad been observed, b u t P auli poin ted ou t th a t th e hole and th e electron m u st have th e sam e m ass.
T he ex p erim en tal observations of firstly th e p o sitro n in 1933 an d la te r th e Lam b shift [Lamb an d R etherford, (1947)] were regarded respectively as first th e confirm ation of th e D irac equation and th en th e need for som ething m ore. T he D irac eq u atio n was retain ed , b u t a th eo ry w ith new concepts in tro d u ced , leading to th e co n stru ctio n of th e m anifestly covariant QED. In th e two decades betw een th e derivation of th e D irac equation as a ‘re la tiv istic ’ QM eq u atio n and its use as a qu an tised field equation, th e application of group th eo ry in theoretical physics h a d blossom ed, w ith im p o rta n t co n trib u tio n s by nam es such as W igner and Weyl. T his led to an ex am in atio n of th e D irac eq uation from a n o th er view point as well.
W hile these developm ents in general led to an acceptance of the D irac equa tion, o th e r relativ istic wave equations were published, for exam ple th e works of
1. Introduction 12 wave equation to describe a single p article m oving in an ex tern al field, b u t th e various equations described particles w ith different in te rn a l degrees of freedom , in p a rtic u la r spin and charge. T h e co n stru ctio n of Q ED used th e original equa tions w ith new concepts in a tte m p tin g a m ore com plete description of aspects of th e physical world.
QED b ro u g h t an elegant an d powerful form alism which provided highly ac c u ra te predictions which agreed well w ith experim ents. However m a th e m atica l tricks w ith no a p p aren t physical ju stificatio n h ad to be in tro d u ced to ren o r m alise th e infinities ap p earin g in p e rtu rb a tio n theory. R en o rm alisatio n seemed physically acceptable, infinite ren o rm alisatio n seem ed odd. N evertheless, QED quickly surpassed RQ M as th e dom inant theory, alth o u g h RQ M was no t a b a n doned com pletely as it still provided th e b est sta rtin g po in t for calculations in atom ic physics.
T h ere were various responses to th e infinities question. T h e th eo ry could be retain ed as is, an d some fu tu re experim ental ju stificatio n sought, especially w ith regards to th e (yet to be explored) sh o rt distance lim it. P erh ap s in stead a new sta rtin g eq u atio n could be found an d th e sam e QED form alism could be applied to th e new equation. A ttem p ts a t finding such an eq u atio n included m ultim ass field theories [Pais and U hlenbeck, (1950)], indefinite m etrics in QED [Nagy, (1966)], a unified field theory of elem entary p articles [Heisenberg, (1966)], an d D irac him self said th a t ‘. . . a tru e advance will be m ade only w hen some fu n d am en tal a lte ratio n is m ade, ju s t ab o u t as fu n d am en tal as passing from the KGO eq u atio n to th e D irac e q u atio n .’, [Dirac, (1975)]. R obson [Robson, (1972)] gave a sem inar w here he presented a can d id ate for a sp in -1 /2 relativ istic wave eq u atio n w ith an indefinite m etric.
1. Introduction 13 th eo ry unifying th e electrom agnetic and weak in teractio n s in to one gauge field th eo ry was developed, followed by Q CD, th e corresponding th eo ry for stro n g in teractio n s. These form w h at is know n to d ay as th e (m inim al) sta n d a rd m odel. W ith th e ad ju stm en t of roughly 20 p aram eters to fit th e ex p erim en tal d a ta , the sta n d a rd m odel explains th e observable p henom ena of p article physics to a high degree of accuracy. E x p lan atio n s for th e values of some of these p a ram eters and also for th e inclusion of gravity into th is scheme have been sought producing theories such as strin g theory, various extensions of th e s ta n d a rd m odel, super- sym m etry, supergravity, G U Ts, T O E s etc. V arious m eth o d s of han d lin g the infinities have been developed, including lattice gauge theories.
Hence m odern theoretical physics has seen m uch conceptual an d m a th e m a t ical developm ent since th e tim e w hen D irac first p ro d u ced his equation. O th er areas of th eo retical physics such as atom ic physics an d condensed m a tte r physics have also seen m a jo r advances and use highly developed versions of QM, Q F T , a n d RQM .
Given these extensive developm ents by m any g reat physicists over th e p ast 65 years, one m u st indeed ask, why should one even consider retu rn in g to the in itial form alism of RQM to search for a new equation???
V I I . I n itia l m o t iv a tio n s for a s p in - 1 /2 a n a lo g u e o f t h e FVO e q u a tio n
C onsider th e question [Robson, (1972) an d (1989)]; ‘why d o esn ’t th e D irac eq u atio n give th e correct free particle predictions in R Q M ?’ It is easily shown th a t th e FVO an d KGO equations produce th e correct free p article predictions in RQM*. R obson also considered th e question; ‘given th a t o u r theo retical and ex p erim en tal knowledge is so m uch g reater th a n it was in 1928, is it possible to
1. Introduction 14 co n stru ct a new relativ istic wave equation a priori, w ith different requirem ents th a n D irac u sed ? ’ D espite th e cu rren t day acceptance of Q ED , th e use of the D irac eq u atio n produces infinities in p e rtu rb a tio n th eo ry an d th ere still has been no definite physical ex p lan atio n to ju stify th e infinite ren o rm alisatio n required. T h e sh o rt distan ce regim e aw aits experim ental exploration, an d this provides a m a jo r hope for a resolution of this problem .
T he question of using an altern ativ e sp in -1/2 eq u atio n to resolve th e infini ties question has been addressed (see th e references in th e previous section), b u t so far w ith o u t success. R obson suggested th a t th e use of a sp in -1/2 relativ is tic wave eq u atio n w ith indefinite m etric in RQM m ight: 1) provide th e correct free p article predictions in RQ M , explaining th e relativ istic degree of freedom in term s of th e charge degree of freedom of electrons seen in n a tu re ; 2) provide a RQ M description consistent w ith th e possibility of p a ir p ro d u ctio n (a s ta te of an electron can create a p air resulting in two electrons and one p o sitro n which overall should have th e original no rm alisatio n + 1 , an d hence v irtu al p airs can be created an d a n n ih ilated w ith o u t altering th e value of (V’lV’); 3) since indefinite m etric equations ten d to be of higher order th a n th e D irac equation, th e p ro p a g ato r should have m ore powers of th e m om entum in th e d en o m in ato r, p erh ap s m aking th e Feynm an integrals ‘m ore convergent’. R obson suggested two possible can d id ates for such an equation, including a sp in -1/2 analogue of th e (indefinite m etric) FVO equation. He co n stru cted his sp in -1 /2 analogue in direct analogy to th e derivation of th e Pauli equation from th e Schrödinger equation. T h e Pauli eq u atio n is o b tain ed from th e Schrödinger equation by th e replacem ent of 7T2
by (cr.7r)2, th e replacem ent of n by l27r, and th e replacem ent th e w avefunction V>s(x) by 'ippfa) = i p ( a , ß)T , w here (o ,/? )T is a two com ponent (non-relativistic) spinor an d x)> is sim ilar to ^>s(x). Hence R obson obtained;
1. Introduction 15
^l/2(x)
Hl/2^1/l(X-)
> ( l . V I I . l )h 1/2 =
(ih<T.D) 2 , , „ . 2 ( 2 m ) +
(ihor. D) 2
( 2 m )
(tfe<T.D) 2
( 2 m )
(ih(T. D) 2
+
( 2 m ) — m e
eAol2 0
0
e A0l 2 1 ’
(1.V 7J.2)
( 1 .V //.3 )
Here (ft, ß ) is again a 2-com ponent (non-relativistic) spinor.
R obson asked m e (S eptem ber 1991) to a tte m p t to o b ta in th e relativistic energy sp ectru m for th e hydrogen ato m using eq u atio n ( l . V I I . l ) . I was able to derive th e exact solution an d show th a t th e sp ectru m o b tain ed was identical to th a t o b tain ed using th e KGO equation. T his result suggests th a t th e m eth o d used to derive eq u atio n ( l . V I I . l ) does no t include th e spin in a relativistically correct m an n er, as th e result o b tain ed is for spin-0 ra th e r th a n sp in -1 /2 particles. U pon learning of th is resu lt, R obson th en suggested th e ad d itio n of ad-hoc spin- o rb it term s to th e H am iltonian # 1/2, in an a tte m p t to o b ta in th e sam e energy sp ectru m for th e hydrogen ato m as th a t given by th e D irac equation. T h e idea was to find term s which m ight indicate how to o b tain a correct sp in -1/2 analogue of th e FVO equation. T h e conventional spin-orbit term s did no t give th e desired resu lt, however an ad-hoc te rm of th e form
H! = C
(< r.L )/r2 (< r.L )/r21. Introduction 16 p o ten tials. It will be shown in section 4.II th a t th e correct F V 1 /2 equation, when expanded in powers of (Z a)2 for sm all Z a , produces a series of term s w ith th e first term given by eq u atio n ( 1.V I I A ) . T h e stu d y of th e hydrogen ato m m ade it clear to b o th m yself an d R obson th a t th e m eth o d of co n stru ctio n of equation ( l . V I I . l ) needed a careful re-exam ination, som ething which h a d in fact already been suggested back in J a n u a ry 1990 [Govaerts, (1990)].
A lthough * th e replacem ent 7r2 by (<r.7r)2 successfully produces th e (non- relativistic) P au li equation, m uch care m ust be tak en w hen using such ad-hoc replacem ents. P a rtic u la r care m ust be given to m inim al couplings for sp in -1/2 particles in th e relativ istic case [Barone, (1973)], [Costella, (1993)], [Lifshitz, B erestetskii and P itaevskii, (1971)]. T h ere is a far b e tte r m eth o d to co n stru ctin g sp in -1/2 relativ istic wave equations th a n by such replacem ents, even th o u g h th ere is a relativ istic analogue of th e replacem ent 7T2 —>• ( < r . 7 r ) 2 , which is —» 7t 2 =
( l ll'Rn ) ( l v'Kv)- T his is to derive sp in -1/2 relativistic wave equations a priori using th e correct m a th e m atica l q u an tities, which are spinor irreducible rep resen tatio n s of th e p ro p er Lorentz group, an d th eir p ro d u c t rep resen tatio n s. Such an equation will a priori be a m anifestly covariant sp in -1/2 eq uation an d it is shown in section 2.I ll th a t th e first ord er equation co n stru cted using this m eth o d is th e D irac equation, while th e second order equation is th e KG 1/2 equation.
T he FVO eq u atio n is a specific linearisation of th e KGO equation, correct for spin-0 particles. It is not m anifestly covariant, so its sp in -1 /2 analogue (if it exists) will also not be m anifestly covariant, and hence a priori will not be an eq u atio n co n stru cted directly using spinor irreps. However, th e FVO equation is ju s t th e KGO eq uation re w ritten in Feshbach-V illars ty p e n o ta tio n , so it is n a tu ra l to search for th e F V 1 /2 equation as being ju s t th e KG 1/2 eq u atio n re w ritte n in Feshbach-V illars type n o tatio n . Since th e KG 1/2 eq u atio n is co n stru cted directly from th e correct m ath em atical quan tities, a priori th e eq u atio n is a prom ising can d id ate to correctly describe sp in -1/2 particles an d th e problem s of ad-hoc
1. Introduction 17 replacem ents such as th e m eth o d to derive eq uation ( l . V 1 1 .1) will no t be present. Hence if a sp in -1/2 analogue of th e FVO equation is c o n stru cted by rew riting th e KG 1/2 eq u atio n in Feshbach-V illars type n o ta tio n , it will also be a correct eq u atio n for sp in -1 /2 p articles, providing th a t th e Feshbach-V illars lin earisatio n p ro ced u re is valid for sp in -1/2 p articles, an d produces a valid equation.
V III. O utline o f th e th esis
T his thesis is not an a tte m p t to ju stify or fulfill R o b so n ’s suggestion given in th e previous section th a t th ere m ight exist an indefinite m etric s p in -1/2 rel ativ istic wave eq u atio n th a t satisfies th e th ree p ro p erties given in th e previous section.
Given th e considerations regarding equation ( l . V 1 1 .1) in th e previous sec tion, before any th o u g h t is given to satisfying th e th ree p ro p erties, th e following m u st be established:
a) firstly, it is im p o rta n t to find a correct relativistic wave equation. O therw ise very little indeed has been achieved.
T h en , given th a t th ere are already a num b er of relativ istic wave equations, b) it is im p o rta n t to com pare this equation w ith existing equations in th e lite r a tu re to show it is indeed som ething new.
T h e existing relativ istic wave equations have been stu d ied extensively over m any years from several different angles. Hence,
c) one m u st estab lish th e p ro p erties of th e new equation. Does it give th e correct physical p redictions? C an a consistent m ath em atical ju stificatio n of th e equation be given? Does it produce different answ ers to th e o th e r equations?
P erh ap s th en , th e th ree p ro p erties can be reconsidered. However, th e following should also be tak en into consideration:
1. Introduction 18 M any physicists would say th a t th e use of th e D irac eq u atio n w ith ren o rm al isation in Q ED overrides th e need to satisfy th e th ree pro p erties. M oreover, th ere is a great deal of w orking knowledge for th e ap p licatio n of th e D irac equation in m od ern physics. Even if it was found th a t th ere exists an eq u atio n th a t satisfies a), b), c) an d th e th ree pro p erties, would it m ake any c o n trib u tio n to physics? W ould it provide a b e tte r description of n a tu re ? O ne m u st develop a theory of th e eq u atio n to an ex ten t th a t physicists find it an a ttra c tiv e a ltern ativ e to the conventional th eo ry for use in th eir work.
This thesis is concerned with the m otivations a), b), c), and d).
1. Introduction 19 I hope th a t th is thesis lays an im p o rta n t fou n d atio n for fu tu re work in these directions.
T he preface (to th is thesis) is designed to place th is thesis into th e context of physics research an d its relevance to th e world today.
T his in tro d u cto ry ch ap ter plays several roles. It provides a historical back gro u n d to th e su b ject of relativistic wave equations an d its cu rren t sta tu s in physics. It discusses briefly th e existing relativ istic wave equations an d why the D irac equation is widely accepted today. It th en asks some questions which provided th e m a jo r p a rt of th e in itial m otivations for th e stu d y of equation ( l . V I I . l ) . It describes why th e need developed to find an a ltern ativ e equation an d w hat requirem ents th is altern ativ e should satisfy. It th en gives an outline of th e resu lts p resen ted in ch ap ters 2, 3, 4, 5 and 6.
C h ap ter 2 launches directly into th e derivations an d analysis of the F V 1 /2 equation. T h ree derivations of th e F V 1 /2 equation are presen ted , w ith successive derivations a t a deeper level of u n d erstan d in g . T h e F V 1 /2 eq u atio n is presented as a can d id ate for a sp in -1 /2 relativistic wave eq u atio n in RQM . T he theory of RQ M is a com bination of th e special th eo ry of relativ ity an d of QM. T he m a th e m atica l form ulation of these two underlying theories is based upo n the rep resen tatio n th eo ry of th e Lorentz group and the th eo ry of in n er p ro d u c t spaces respectively. T h e new equation is considered in some d etail from th e view point of these m a th e m atica l theories, b o th to gain an u n d e rstan d in g of its p ro p erties an d to a tte m p t to ju stify it as a valid RQM equation. T h e F V 1 /2 eq u atio n is com pared in d etail w ith th e D irac and o th er RQ M equations. T h e RQ M of th e F V 1 /2 eq uation is th en developed.
C h ap ter 3 focusses on th e uses of th e F V 1 /2 eq u atio n an d also th e physical in te rp re ta tio n of its eight solutions. It provides some significant resu lts which are especially useful for applying th e equation to atom ic physics.
1. Introduction 20
C h ap ter 5 suggests im p o rta n t fu rth e r work which m u st be done to develop th e eq u atio n so it can be applied widely in physics.
C H A P T E R 2
D e r iv a tio n s an d a n a ly sis o f t h e F V 1 / 2 e q u a tio n
I. I n tr o d u c tio n
T his ch ap ter contains th e derivations an d an analysis of th e F V 1 /2 equation. T h e derivations are presented a t th ree levels.
I first derived th e F V 1 /2 equation in Stockholm du rin g Ju ly 1992 from eq u atio n (9.11) (which is th e KG 1/2 equation) in F ey n m an ’s book, ‘Q u an tu m E lectro d y n am ics’ [Feynm an, (1961)]. At th a t tim e my u n d e rstan d in g was lim ited to th e knowledge th a t equation ( l . V I I . l ) was not a relativ istic sp in -1/2 wave equation, th e reason being th a t th e m eth o d of derivation from th e FVO equation did not ‘include th e sp in ’ in a correct relativistic m an n er. E q u atio n ( l . V I I . l ) also did no t produce th e correct hydrogen ato m sp ectru m . At this stage it was not clear w hether th ere in fact was a correct relativ istic wave eq u atio n th a t was a sp in -1 /2 analogue of th e FVO equation, and even if th ere tu rn e d o u t to be such an equation, was it only th e D irac equation? C ould it be used for physical problem s? C ould a ‘relativ istic’ QM form alism be o b tain ed using it? Hence I was searching for a correct F V 1 /2 equation, b u t h ad n o t yet derived any of th e m a terial in this or subsequent chapters.
2. D erivations and analysis o f the F V 1 /2 equation 22 derivation tu rn s ou t to be m ath em atically correct, it was p o in ted out to me* th a t th e resulting eq u atio n was not p a rity invariant. I th en realised th a t one should derive Feshbach-V illars type equations from b o th decoupled p a rts of th e KG 1/2 equation, an d th en consider b o th of th em to g eth er as one eq u atio n to give a p a rity invariant equation. At this tim e Professor B. L au ren t suggested th a t I should m ath em atically analyse th e equation before p u b licatio n in th e lite ra tu re an d these com m ents provided th e in itial m otivations to begin th e analysis which ten m onths la te r has becom e this chapter.
Hence th is ch ap ter begins in section 2.II w ith my original derivation of th e F V 1 /2 eq u atio n from th e KG 1/2 equation in th e Weyl rep resen tatio n of th e g am m a m atrices. T his derivation provides a sim ple in tro d u ctio n to th e F V 1 /2 equation. Some in itial com m ents on th e resulting eq u atio n an d how it com pares to th e FVO eq u atio n are m ade.
One im m ediately asks: is this equation valid? is it different to th e D irac equation? These questions guided me in my in itial analysis of th e equation. T h e FVO eq u atio n has been established in th e lite ra tu re [Feshbach and V illars, (1958)], [Greiner, (1990)] an d it is sim ply th e (well accepted) KGO equation re w ritte n using Feshbach-V illars type n o tatio n . T h e F V 1 /2 equation is th e KG 1/2 eq u atio n rew ritten using Feshbach-V illars ty p e n o tatio n . To establish th e validity of th e F V 1 /2 equation, it is necessary to first show th e correctness of th e K G 1 /2 equation. T his is done in section 2.I ll by co n stru ctin g th e K G l/2 an d D irac equations a priori using spinor irreps of th e PL G and th e ir p ro d u ct rep resen tatio n s. T h e KG 1/2 (and hence F V 1 /2 ) eq u atio n is th en shown in section 2.IV to be essentially different to th e D irac eq u atio n by considering th e solution sets of th e KG 1/2 and D irac equations.
W ith these resu lts it is now possible to give a second derivation of th e F V 1 /2 eq u atio n **. It is now u n d ersto o d w here F ey n m an ’s eq u atio n comes from an d why it is a correct relativ istic sp in -1 /2 equation. M oreover, section 2.Ill, th e section
* by Drs H. H ansson an d U. Lindgren
2. D erivations and analysis o f the F V 1 /2 equation 23 on th e spinor n o ta tio n , shows th a t it is th e correct eq u atio n to derive th e F V 1 /2 eq u atio n from . Section 2.IV, th e section which shows th a t F ey n m an ’s equation is different to th e D irac equation, is essential because it m eans th a t th e F V 1 /2 eq u atio n is som ething new. If it was only th e D irac equation, one w ould have to seriously ask th e question, ‘can th ere be an y th in g gained from fu rth e r stu d y of th e e q u atio n ? ’ T h e second derivation presented in section 2.V has two essential features no t present in th e first derivation. Firstly, th e F V 1 /2 eq u atio n should be w ritten in a form as close to th e FVO eq u atio n as possible. T h e definition of th e F V 1 /2 w avefunction using K ronecker p ro d u cts proves convenient for this p u r pose. T h e second extension of th e original derivation is th a t th e F V 1 /2 equation is derived in an a rb itra ry gam m a m a trix rep resen tatio n . A gain this derivation is m ath em atically correct, an d again a derivation a t a m ore fu n d am en tal level can be presented.
H aving come this far, it is necessary to estab lish th e validity and usefulness of th e Feshbach-V illars linearisation procedure. Up to th is po in t I h ad generally assum ed th a t a priori the Feshbach-V illars lin earisatio n p ro ced u re was valid, based upo n th e lite ra tu re [Feshbach and V illars, (1958)], [Greiner, (1990)] and th e in itial work of, com m unications from , and discussions w ith B. A. Robson*. However these references only show th a t th e Feshbach-V illars lin earisatio n proce d ure tran sfo rm s th e KGO equation into an eq u atio n in H am iltonian form. T hey do no t show a priori why this m ath em atically will work, why a priori th is is a n a tu ra l m a th e m atica l m eth o d of co n stru ctin g a ‘re la tiv istic ’ QM wave equation, th a t th e Feshbach-V illars linearisation procedure will work in th e sp in -1/2 case, or th a t th e Feshbach-V illars linearisation procedure indeed produces an equation which proves convenient for th e stu d y of a single p article m inim ally coupled to a classical ex tern al electrom agnetic field. T h e first th ree of these deficiencies are rectified in section 2.VI, th e section labelled ‘T h e Indefinite M etric Form alism in R Q M ’**. T his is accom plished using th e m ath em atics of indefinite inner p ro d u ct
* See th e in tro d u ctio n ch ap ter to th is thesis
2. D erivations and analysis o f the F V 1 /2 equation 24 spaces an d discussing th e requirem ents of a ‘re la tiv istic ’ QM. T h e Feshbach- V illars lin earisatio n procedure is shown to identically satisfy these requirem ents. C onserved cu rren ts are co n stru cted for th e various equations. It is now possible to present in section 2.V II th e th ird derivation of th e F V 1 /2 eq uation (which I achieved in M arch 1993). T he m ajo r difference from th e second derivation is th a t now th e F V 1 /2 equation is derived a priori using th e requirem ents of a ‘re la tiv istic ’ QM an d th e m ath em atics of in n er p ro d u c t spaces.
T h ere exist o th er relativistic wave equations which are linearisations of K lein-G ordon type equations. T he m otivations b eh in d these equations an d th e equations them selves are com pared w ith th e Feshbach-V illars type equations in section 2.V III.
T he form alism developed proves convenient to stu d y th e ‘re la tiv istic ’ QM of th e F V 1 /2 equation, an d to show it is analogous to th e FVO equation. O p erato rs an d ex p ectatio n values for th e F V 1 /2 equation are discussed in section 2.IX. Q uestions of covariance, bilinear covariants, th e g am m a m a trix algebra etc. are addressed in section 2.X. T he sp in -1/2 Feshbach-V illars type eq u atio n has eight solutions an d these are briefly com pared in section 2.XI w ith th e solutions of th e FVO an d D irac equations in the free p article case to illu stra te an im p o rta n t difference betw een th e equations.
