ativistic degree of freedom , i.e. a set which includes negative energy states b u t only one sign of charge. T h e ‘o th er solu tio n s’ of this subset reduce to th e charge conjugate solutions defined by Uc k — Uc d in th e lim it of vanishing ex tern al fields. If, however, one chooses Uc k = 75Uc d-, Ut k — Ut d an d Up k = Up o , th en one o b tain s a subset of four solutions invariant u n d er C P T which behave according to th e Feshbach-V illars description, a subset containing only positive energy states, two w ith charge e, and two w ith charge — e. T h e ‘o th e r solu tio n s’ of this subset reduce to th e charge conjugate solutions defined by Uc k — 75Uc d
in th e lim it of vanishing ex tern al fields.
Any o th e r choice of Uvk requires th e use of all eight solutions to specify a C P T invariant set.
To o b ta in th e resu lts for th e F V 1 /2 eq u atio n from th e KG 1/2 equation, all th a t is necessary is to w rite th e statio n ary sta te solutions of th e KG 1/2 equation in Feshbach-V illars type n o ta tio n using th e replacem ent ^ k g i/2 = 0 ® —> ^f v i/2 = * ® *o . For exam ple w ith ^ k g i/2 = 0 ® = ^o( v) e~lEt ® 4>o(r),
®c k = 0 o(r)etEt <g> Pc kUc d^o (r), 0o (r ) e tEt <g> tjck^ Uc d^o (r ) » (3.V 7.12) th en one sim ply has for ^f v i/2 = (0 o (r), x o { r ) ) Te ~lEt ® ’J'o(r),
or
Sim ilarly, one can derive
3. The use o f the F V 1 / 2 equation in physical problems 89
^f v i/2P = ( ) e ' Et ®'1p kUp d'Üo( - r ) ,
or ( ^ ( I r ) ) e " iE‘ ® ,' pl<">5^ PD 'I,o ( - r ) - (3 .V /.1 5 ) In general, for a given statio n ary s ta te solution ^ k g i/2 of th e K G l/2 equa
tion (and hence ^ f v i/2 of th e F V 1 /2 equation), th en ( a + /^7 5) ^k g i/2 (respec
tively ( a + /?75)^F V i/2) is also a solution of th e K G l/2 equation (respectively th e F V 1 /2 equation). Hence in th e solution of th e F V 1 /2 eq u atio n for a physi cal problem one can always define th e in itial solution ^ f v i/2 — ( ^ F V i/2)oe~l£;<
(in th e lim it of vanishing ex tern al fields, E > -fm c2), to have positive definite f j ° d 3x = -f 1, sim ply by applying a tra n sfo rm atio n ( a -f ß j s ) to \Po in equation (3 ./V .l) 2 . T h en is norm alised to + 1 . T h e ‘o th e r so lu tio n ’ to th e given physical problem ^f v i/2 = (^ F V i/2)° e+,e<’ ls th en norm alised as follows: if one
chooses th e D irac description (Uvk — Uvd), th en f j ° d 3x = + 1 an d = — 1 are th e conditions for th e o th er solution; and if one chooses th e Feshbach-V illars description {Uc k — 7 5Uc d, Ut k — Ut d a n d Up k — Up d), th en f j ° d 3x = — 1
and — +1 are th e conditions for th e o th er solution. U nder a tran sfo rm atio n -> (q + ß j 5) ^ 0, th en
*0 * 0 * J 7 0( M 2 - |/3|2)* o + *1,(0*a - a * 0 ) y s y ° 9 o • ( 3 V /.1 6 )
For a general tran sfo rm atio n , w here can becom e com plex, one has a differ ent description of th e relativistic degree of freedom th a n th a t given by th e D irac or FVO equations. However, by choosing a and ß such th a t th e sta tio n a ry sta te solution $ = $o e~lEt ® has = + 1 , th en it has been shown th a t C P T invariant subsets of th e eight solutions which obey eith er th e D irac or Feshbach- V illars descriptions exist. These subsets each contain all th a t is required for a physical problem of a single p article m inim ally coupled to a classical external electrom agnetic field.
3. The use o f the F V 1 /2 equation in physical problems 90 T he problem s w ith th e negative energy states of th e D irac eq u atio n and con tin u u m dissolution are well known [Heully, L indgren, L in d ro th an d M ärtensson- Pendrill, (1986)], [Sucher, (1984)]. T he phenom enon of con tin u u m dissolution is due to th e D irac description of th e relativistic degree of freedom , which involves negative energy states. O ne m ust tu rn to QED to overcom e this problem . It will be an in terestin g exercise to a tte m p t to solve th e tw o-electron problem using the F V 1 /2 eq u atio n (initially only a t th e RQ M level) w ith a C P T in v arian t subset of four solutions using th e Feshbach-V illars description of C. T his subset represents th e relativ istic degree of freedom by states of opposite signs of charge, all having positive energy.