Dirac s hole theory versus field theory for a time dependent perturbation. Dan Solomon. Rauland-Borg Corporation 3450 W Oakton Skokie, IL USA

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1 1 Dirac s hole heory versus field heory for a ime dependen perurbaion by Dan Solomon Rauland-Borg Corporaion 345 W Oakon Skokie, IL 676 USA Phone dan.solomon@rauland.com ovember 3, 3

2 Absrac Dirac s hole heory (HT and quanum field heory (QFT are generally considered o be equivalen o each oher. However, i has been recenly shown ha for a ime independen perurbaion differen resuls are obained when he change in he vacuum energy is calculaed. Here we shall exend his discussion o include a ime dependen perurbaion. I will be show ha HT and QFT yield differen resuls for he change in he vacuum energy due o a ime dependen perurbaion. PACS os: 3.65-w, 11.1-z

3 3 I. Inroducion Dirac s hole heory and quanum field heory are generally assumed o be equivalen. Recenly several papers have appeared in he lieraure poining ou ha here are differences beween Dirac s hole heory (HT and quanum field heory (QFT [1 ][][3][4]. The problem was originally examined by Couinho e al[1][]. They calculaed he second order change in he energy of he vacuum sae due o a ime independen perurbaion. They found ha HT and QFT yielded differen resuls. They concluded ha he difference beween HT and QFT was relaed o he validiy of Feynman s belief ha he Pauli Exclusion principle can be disregarded for inermediae saes in perurbaion heory. This belief was based on Feynman s observaion ha erms ha violae he Pauli principle formally cancel ou in perurbaion heory. However Couino e al show ha his is no necessarily he case for HT when applied o an acual problem. This auhor (Solomon [4] found ha his problem was relaed o he way he vacuum sae was defined in QFT. If he definiion of he vacuum sae was modified as described in [4] hen he HT and QFT would yield idenical resuls. The previous work examined his problem for he case of a ime independen perurbaion. In his aricle he differences beween HT and QFT for a ime dependen perurbaion will be examined. In Secion II we derive a formal expression for he quaniy which is he second order change in he vacuum energy in HT. In E hvac Secion III we derive an expression for E qvac which is he second order change in he vacuum energy in QFT. We show ha hese expressions are differen due o he fac ha includes a quaniy called X E hvac, which is no included in E qvac. In

4 4 order, hen, for and E hvac o be equal E qvac X mus be zero for all, possible perurbaions. In order o es his possibiliy an acual problem is worked ou in Secion IV. Here i is shown ha for his paricular problem E < and hvac ( E =. qvac Therefore X, ( and ( E hvac E qvac. This resul is confirmed in Secion V where he value of X, ( is acually calculaed and is shown o be nonzero. I is hen shown in Secion VI ha by modifying he definiion of he QFT vacuum sae he difference beween HT and QFT can be eliminaed. c= 1 II. Hole Theory. In his secion we will derive an expression for he second order change in he vacuum energy due o ime dependen perurbaion in HT. In order o simplify he discussion and avoid unnecessary mahemaical deails we will work in 1-1 dimensional space-ime where he space dimension is aken along he z-axis and use naural unis so ha =. In his case he Dirac equaion for a single elecron is, ( z, ψ i ( = H + qv( z, ψ( z, (.1 where, Hˆ = iσx + mσ z (. z and where m is he mass, q is he elecric charge, and σ and σ are he Pauli spin x z marices, 1 1 σ x = and σ z = (.3 1 1

5 5 We will assume periodic boundary condiions so ha he soluions saisfy ψ ( z, =ψ ( z + L, where L is he 1-dimensional inegraion volume. In his case he orhonormal free field soluions (V is zero of (.1 are given by, ( ( ε i i,r ε prz λ ϕ z, =ϕ z e = u e (.4 where r is an ineger, λ= ± 1 is he sign of he energy, pr = π r L, and where, ( ε =λ ; E r r r E = p + m ; u = p r 1 ( λ E + m r λ E + m LλE ; r = r (.5 The quaniies ( ϕ saisfy he relaionship, z Ĥ ( z ϕ =ε ϕ z (.6 The ( ϕ form an orhonormal basis se and saisfy, z + L ( ( ϕ ( z ϕ ( z dz=δλλ δ rs (.7 L λ,s Suppose a some iniial ime he perurbaion V is zero. A his ime assume all he negaive energy saes ( ϕ ( z, are occupied by a single elecron and all he 1,r posiive energy saes are unoccupied. This is he HT vacuum sae. ow le he perurbaion V be applied for a period of ime and hen removed a some final ime. Therefore V saisfies, V z, = for ; V z, for < < ; V z, = for f (.8 f f

6 6 Each iniial wave funcion ( ϕ λ,r ( z, evolves according o he Dirac equaion ino he final sae ϕ ( z, f. The change in he energy of he sae ( ϕ ( z, from o f is given by, ( δε = ϕ ϕ ε λ (.9 ( z, H ( z, f f,r + L where, o simplify noaion, we define g( z E g z dz. L Le be he sum of he change in energy of he firs +1 vacuum elecrons, i.e., he sum of he δε for λ= 1 and r +. Therefore, 1,r r= E = δε (.1 The change of he HT vacuum energy, Ehvac, is, hen, defined as, E = (.11 hvac E (oe ha he h in he erm E hvac is o indicae ha his is he change in he vacuum energy as deermined by HT. This is o disinguish i from he quaniy inroduced laer, which is he change in he vacuum energy using QFT. The relaionship beween he iniial and final wave funcion is, E qvac, o be ( ϕ ( z, = U(, ϕ z, f f (.1 where U( f, is a uniary operaor. From Thaller [5] a formal expression for U( f, is,

7 7 f f ih 3 U( f, = e 1 iq VI( z, d q VI( z, d VI( z, d + O( q e f + ih (.13 where I + ih V z, e V z, e ih =. From his we can wrie, 1 f f f f 3 ϕ z, =ϕ z, + qϕ z, + q ϕ z, + O q (.14 3 where δε Oq means erms o he hird order of q or higher. Similarly can be expanded as, ( 1 3 δε = qδε + q δε + O q (.15 and, ( 1 3 E = q E + q E + O q (.16 I is shown in Appendix A ha he above relaionships yield, ( 1 ( 1 δε = E = (.17 and, where, and, + ( ( δε = fλ,s; λ,r ελ,s ε λ=± 1s= λ,s; λ,s; f f = V e i ( ( ε λ,s ε d (.18 (.19 ( V ( λ,s; λ,r = ϕλ,s z V z, ϕ ( z (. ex use he above resuls in (.1 along wih (.15 and (.16 o obain,

8 ( + E = fλ,s; 1,r ελ,s ε r= λ =± 1s= 1,r (.1 8 Use (.5 in he above o yield, ( ( + E = fλ,s; 1,r ( λ Es + Er r= λ =± 1s= (. This can be wrien as, E = Y + X ( (.3 where, + Y = f+ 1,s; 1,r ( Es + E r (.4 r= s= and + X = f 1,s; 1,r Es r= s= ( E r (.5 X can hen be wrien as, X = X + X 1,, (.6 where, X f E + 1, = 1,s; 1,r s r r= s= E (.7 and, X, = r= + + ( f 1,s; 1,r E s E r 1 ( f 1,s; 1,r ( Es Er s= + 1 s= (.8

9 9 ow if he dummy indices s and r are swiched in (.7 and we use he fac ha f = f hen we can show ha X1, = X1, = so ha, 1,s; 1,r 1,r; 1,s E = ( X, + Y (.9 Therefore he second order change in he HT vacuum energy is, E = E = X ( + Y ( (.3 hvac, III. Quanum field heory. In his secion we will derive an expression for he change in he vacuum energy due o a ime dependen perurbaion in QFT. We shall work in he Schrödinger picure. In his case he field operaors are ime independen and all changes in he sysem are refleced in he changes of he sae vecors. The field operaors are defined by, ψ ˆ ˆ ˆ ˆ z = a ϕ z ; ψ z = a ϕ + + λ,r z λ=± 1r= λ=± 1r= (3.1 ˆ â where he a are he desrucion(creaion operaors for a paricle in he sae ( ϕ. The operaors a and z ˆ â λ,r saisfy he anicommuaor relaion λ,s + λ,s =δrsδλλ aˆ aˆ aˆ a ˆ ; all oher anicommuaors= (3. The Hamilonian operaor is, ( H ˆ = H ˆ + qv ˆ (3.3 where, + L + ren λ, L λ=± 1r= Hˆ = ψˆ z H ψˆ z dz ξ = aˆ aˆ ε r ξren (3.4 and

10 1 + L = ψ ψ Vˆ ˆ z V z, ˆ z dz (3.5 L ξ ren is a renormalizaion consan defined so ha he energy of he vacuum sae is equal o zero. Following Greiner (see Chap. 9 of [6] define he sae vecor, bare which is he sae vecor ha is empy of all paricles, i.e., âλ,r, bare = for all λ,r (3.6 The vacuum sae vecor is defined as he sae vecor in which all negaive energy saes are occupied by a single paricle. Therefore r= 1,r = aˆ,ba re (3.7 The vacuum sae saisfies he equaion, Ĥ = (3.8 ew saes are produced by acing on he vacuum sae wih he operaors a and ˆ 1,r â + 1,r. The acion of he operaor 1,r â on is o desroy a sae wih negaive energy E r ˆ + 1,r and he acion of a is o creae a sae wih posiive energy + E r. In eiher case he energy of he new sae is increased. In general if we define some sae k by, 1,r 1,r 1,r k = aˆ aˆ aˆ aˆ aˆ aˆ ( j 1,s 1,s 1,s 1 i hen we can wrie, (( 1 j ( 1 i Ĥ k = E + E + + E + E + E + + E k (3.1 r r r s s s

11 11 ow suppose a he iniial ime he sysem is in he vacuum sae. Apply a perurbaion per equaion (.8. Under he acion of he perurbaion he sae evolves ino he perurbed vacuum sae a he final ime. The relaionship beween p f f he final sae p and he iniial sae is given by, f where ˆ = U, (3.11 p f f Û( f, is a uniary operaor. From Sakurai [7] we have, f ˆ f ˆ ˆ ihf ˆ ( ˆ ih ( ˆ 3 + U f, = e 1 iq VI d q VI d VI( d + O( q e (3.1 where, I ˆ ih ( = ˆ ( Vˆ e V e ihˆ (3.13 The change in he energy of he QFT vacuum sae is given by, ˆ E = H Hˆ (3.14 qvac p f p f Use (3.11 and (3.1 in (3.14 along wih he fac ha Hˆ ˆ = H = o obain, 3 Eqvac = q Eqvac + O( q (3.15 where, f f ( E qvac = Vˆ I ( d Hˆ ˆ VI ( d (3.16 Use (3.1 in (3.5 o obain, ( = ( V ˆ a ˆ a ˆ V λ=± 1λ =± 1s= r= λ,s λ,s; (3.17 Use his resul in (3.13 o obain,

12 ( ( i( ε ε I λ,s λ,s; λ=± 1λ =± 1s= r= λ,s ( = ( (3.18 V ˆ a ˆ a ˆ V e where we have used f ihˆ ˆ ih ˆ ˆ,s,r ˆ ˆ λ λ = λ,s ( ( + i( ελ,s ε e a a e a a e. From his we obain, I ( = λ,s λ=± 1λ =± 1s= r= Vˆ d aˆ aˆλ,r fλ,s; (3.19 Use his in (3.16 o yield, ( qvac ˆ ˆ ˆ η,k ˆ ˆ ( η,j; η,k ( λ,s;,r E = a a H a a f f λ (3. λλ,, ηη=±, 1 j,k,r,s= η,j λ,s From (3.7, (3.1, and (3.1 we have he relaionship, aˆ aˆ Ha ˆ ˆ aˆ = δ δ ε ε δ δ δ δ (3.1 η,j η,k λ,s 1, λ + 1, λ λ,s rj sk λη λη oe ha his quaniy is zero if λ =+ 1 or λ = 1. Use his in (3. o obain, E = ε ε f (3. qvac + 1,s 1,r 1,r; + 1,s r= s= where we have used ( f 1,r; + 1,s = f 1,r; + 1,s f+ 1,s; 1,r. Use (.5 in he above o obain, ( E = E + E f ( qvac s r 1,r; + 1,s r= s= (3.3 1 ow compare his wih of equaion (.3. If we examine (.4 we see E hvac ha appears o be equal o Y ( E qvac (Replace wih in equaion (.4. Therefore for ( o be equal o E qvac we mus have ha X ( E hvac, is equal o zero. So he quesion ha mus be deermined is does ( X equal zero for all possible perurbaions. I will be shown in he nex secion ha his in no he case.,

13 13 IV. Calculaing he change in energy. In he previous secions we have derived formal relaionships for he second order change in he vacuum energy of HT and QFT. I has been shown ha he formal expressions are no idenical. In his secion we shall work an acual problem which will clearly demonsrae he differences beween HT and QFT. Le he perurbaion be given by, w w V( z, sin m + m ik z ik z iq V( z, = 4cos( kwz = e + e e dq (4.1 m where m is he mass of he elecron and kw = π w L< m where w is a posiive ineger. I is obvious from he above expression ha V( z, a ±. Under he acion of his elecric poenial each iniial wave funcion ( ϕ ( z,, where, evolves ino he final wave funcion ( z, ϕ where +. Use (4.1 in (.19 o obain, f f ( m f ikwz ikwz i (,s,r iq,s;,r,s z e e +,r z e ε λ ε + λ λ λ λ λ e = ϕ + ϕ dq d (4. m Use (.4 in he above o obain, i λe λe i p p z L + + e λ,s; =,r + m λ,s λ ikwz ikwz iq L ( + m f u u d dz Perform he inegraions over and z o obain, where, ( r s ( r s e e e e dq (4.3 ( w r s m L w r s + δ + λer fλ,s; = π uλ,su δ d q (4.4 +δl + m +λ Es + q

14 14 L if r = δ L ( r = (4.5 if r From he definiion of he dela funcion we have he following relaionship + m λe m r + λes w δ L( ± w+ r s δ dq=δ L( ± w+ r s δ dq +λ E + q +λ E + q m s m s (4.6 Due o he fac ha ( E + E m and k w < m we have ha, + m m ( s w s r s δ λ E +λ E + q dq=δλλ (4.7 Use hese resuls in (4.4 o obain, ( + uλ,r+ wuδ L w r s λ,s; = π δ λλ + uλ,r wu δ L ( w + r s f (4.8 This yields,,r L uλ,r+ wuλ δ w+ r s λ,s; = π δλλ + uλ,r wu δl ( w+ r s f 4 L (4.9 λλ λλ ( L L where we have used ( δ =δ, δ r = Lδ r, and L ( w r s δ + δ L w+ r s =. Use (4.9 in (.18 o obain, ( (,r w,r,r w,r uλ + uλ ελ + ελ δε = 4π L ( ( + u w u ( ε w ε (4.1 I is shown in Appendix B ha, ( pr + kw ( pr kw δε = π λk w Er+ w Er w (4.11

15 15 ow, in order o evaluae we E hvac are ineresed in he quaniy δε for he negaive energy saes in which case λ = 1. Therefore, for negaive energy saes, δε 1,r = π k w Er+ w Er w (4.1 ( pr + kw ( pr kw I is shown in Appendix C ha his quaniy is negaive for all δε. When his resul is 1,r used in (.1 and (.11 i is eviden ha E <. ow use he above resuls o hvac evaluae ( (see equaion (3.3. From (4.9 we have ha f 1,r; + 1,s = because E qvac λ λ. Therefore ( E qvac = so ha ( E E qvac. ow wha accouns for he hvac difference beween HT and QFT. Recall ha (see equaion (.3 consiss of E hvac wo erms Y ( and X (. Due o he fac ha f + = we have, 1,r; 1,s ha Y ( =. Therefore he difference beween ( and E qvac is due o E hvac he erm X which mus be negaive even as., V. Evaluaing X, In his secion we will calculae X,. Refer o (.8 o obain, X, = XA, + XB, (5.1 where, and X X + ( f = E E (5. A, 1,s; 1,r s r r= s= ( f = E E (5.3 B, 1,s; 1,r s r r= s=

16 16 To evaluae X A, use (4.9 in (5. o obain, + u 1,r+ wu 1,r δ L ( w+ r s ( (5.4 = = + + u u δ ( w+ r s A, = π s r r s 1 1,r w 1,r L X 4 L E E ow we will evaluae his expression for. In he above expression r s< so ha for posiive w, L ( w r s δ + = and δ ( w+ r s = L if s r = w. From his we L obain, A, 1,r+ w 1,r r+ w r r= + 1 w X ( 4 L u u ( E E = π (5.5 From (B.3 we have, ( + + ( E ( E pr pr kw m XA, ( = π 1+ ( Er+ w Er (5.6 r= + 1 w r+ w r ow for large r, and, r+ w r r w r w r E E = p + k + m p + m = k (5.7 ( + + r r w Er+ w Er r p p k m Use his in (5.6 o obain, = 1 (5.8 ( = π = π = π X k wk L k A, w w w r= + 1 w (5.9 Apply he same reasoning o X B, o obain, B, ( = π X L k w (5.1 Use hese resuls in (5.1 o obain,

17 17 ( = π X 4 L k, w (5.11 Use his in (.3 along wih he fac ha Y o obain, = E = 4π L k (5.1 hvac w VI. Redefining he QFT vacuum sae. The reason here is a difference beween HT and QFT is due he presence of he erm X, ( in he expression for he second order change in he HT vacuum energy, E hvac. This erm does no appear in he expression for he change of he QFT vacuum energy, E qvac. This is similar o he siuaion ha exiss when a ime independen perurbaion is applied as discussed in [4]. In his case i was shown ha HT and QFT could be made equivalen by redefining he QFT vacuum sae. We will show ha his resul also applies o he case of he ime dependen poenial which has been discussed here. Redefine he QFT vacuum sae as follows: ˆ 1,r r = = a, bare (6.1 According o his definiion he band of vacuum saes wih energies beween m and E are occupied by a single elecron. All posiive energy saes are unoccupied. All saes wih energy less han E are also unoccupied. This differs from he sandard definiion of he vacuum sae given by equaion (3.7. The difference is due o he fac ha he lower edge of he negaive energy band is defined using a limiing procedure.

18 18 ew saes are produced by acing on he vacuum sae wih â + 1,r for all r, â 1,r â 1,r where Er > E, and where E > E r. The acion of he operaor â 1, r or â + 1,r on is o increase he energy. The acion of he operaor â 1,r where E > Er is o add a sae wih energy Er underneah he occupied band wih decreases he energy of he new sae. (For addiional discussion of hese redefined vacuum sae see references [4] and [8]. where Referring o equaion (3.16 and he discussion in Secion III we obain, f f Eqvac = Vˆ I( d Hˆ ˆ VI( d E qvac (6. is he second order change in he energy of he redefined vacuum sae. From his and he discussions leading up o (3. we obain, E = Y ( + X ( qvac, (6.3 where, ˆ ˆ 1,r ˆ+ 1,s ˆ+ 1,s ˆ 1,r 1,r; + 1,s r= s= = (6.4 Y a a H a a f and, ˆ 1,r 1,s 1,s 1,r ( 1,r; 1,s aˆ aˆ H aˆ aˆ f s= + 1 X, = 1 r= + ˆ aˆ 1,raˆ 1,s Haˆ 1,saˆ 1,r ( f 1,r; 1,s s= (6.5 The above can be evaluaed o obain, = ( s + r( 1,r + r= s= ; 1,s (6.6 Y E E f

19 19 and, X ( Es Er ( f 1,r; 1,s s= + 1, = 1 r= + s= ( Es Er ( f 1,r; 1,s (6.7 Compare hese equaions wih (.4 and (.8 o show ha Y = Y and X, = X,. Therefore hole heory and field heory yield idenical resuls when he vacuum sae is defined per equaion (6.1. This resul is consisen wih he resul obained in he case of he ime independen perurbaion as discussed in [4]. VII. Conclusion We have derived expressions for he second order change of he vacuum energy due o a ime dependen perurbaion using HT and QFT. The formal resul for HT is given by (.3 and he resul for QFT is given by (3.3. There is a difference in hese resuls due o he presence of he erm X, which appears in he resul in HT bu no in QFT. In order o undersand he imporance of he erm X, ( we have worked on acual problem which is presened in Secion III. I was shown for his problem ha he change in energy of every negaive energy elecron is negaive. Therefore. However i is shown for his same problem ha ( E < E =. The hvac qvac difference beween E hvac and E is he erm X ( qvac,. I was shown in Secion V ha his erm is nonzero. If he definiion of he QFT vacuum is redefined according o he discussion in Secion VI hen he erm X, HT and QFT give equivalen resuls. Appendix A appears in he QFT resul and

20 Since he Dirac equaion does no affec he normalizaion condiion we have, ( ϕ z, ϕ z, = ϕ z, ϕ z, = 1 (A.1 f f f f Use (.14 in he above o obain, ( ( 1 ( 1 ( ( ( ( ( 1 ( 1 = q,r,r,r,r q ϕλ ϕ λ + ϕλ ϕ λ + ϕ ϕ + ϕ ϕ + ϕ ϕ This yields, (A. ( 1 ( 1 ( ( 1 ( 1 ( ( ( ( q ϕ ϕ = q q ϕ ϕ + ϕ ϕ ϕ ϕ + ϕ ϕ ex use (.14 in (.15 o obain, (A.3 ( 1 ( ( ( ( 1 ( 1,r H,r,r H,r,r H,r ϕλ ϕλ ϕλ ϕ λ + ϕλ ϕλ 3 δε,r = ϕ Hϕ + q + q + O q ε ( ( 1 + ϕ Hϕ + ϕ H ϕ λ (A.4 Use (.6 in he above o obain, ( 1 ( ( (,r,r,r,r ( ϕ ( ( 1 ( 1,r q λ ϕ λ ϕ,r q λ ϕ λ 3 δε λ = ε λ + ε + q ϕ Hϕ + O( q (A.5 ( ( 1 ( ( + ϕ ϕ + ϕ ϕ Use (A.3 in he above o obain, ( 1 ( 1 ( ( 1 ( 1 3,r,r,r,r,r λ λ λ λ λ δε = q ϕ H ϕ q ε ϕ ϕ + O q (A.6 Therefore, ( 1 ( 1 δε = E = (A.7 and

21 1 ( ( 1 1 ( ( 1 δε = ϕ z, H ϕ z, ε ϕ z, ϕ 1 z, (A.8 ( ( f f f f Evaluae his equaion as follows. From (.1, (.13, and (.14 we obain, ( ϕ f 1 z, = i e ih( f V z, e ih ϕ z, d f (A.9 This becomes, ( f 1 ih( f z, i e ϕ = V z, ϕ z, d f (A.1 Define, ( V = ϕ z V z, ϕ z (A.11 ( λ,s; λ,r λ,s Since he ( ϕ form on orhonormal basis we can expand he quaniy V( z, z ϕ ( ( z as a Fourier expansion as follows, (,r + λ λ,s λ=± 1s= V z, ϕ z = ϕ z V λ,s; (A.1 Use his in (A.1 o obain, ( 1 + ( f λ,s λ=± 1s= ( (A.13 ϕ z, = i ϕ z, f f λ,s; where, ( ( f i( ελ,s ε λ,s; λ,s; ( f = V e d (A.14 Use his in (A.8 o obain, + ( ( δε = fλ,s; λ,r ελ,s ε λ=± 1s= (A.15

22 Appendix B We wan o evaluae equaion (4.1 and show ha i resuls in equaion (4.11. Use (.5 o obain, pr( pr + k λ Er± w + m λ Er + m uλ,r± wu = 1+ λler± w λle r λ Er± w + m λ Er + m (B.1 Use E m λ r = p r in he above o obain, u u λ E + m λ E + m 1 ( λer± w m( λer m r± w r λ,r± w = + λler± w λ LE r pr pr ± kw (B. Use his resul o yield, u ( r± w( r r( r w ( λe ( λe L λe λ E + p p ± k + m = λ,r± wu r± w r (B.3 Use his in (4.1 along wih he fac ha λ = 1 o obain pr pr + kw + m 1+ ( Er+ w Er E r we + r δε = π λ pr( pr kw + m + 1+ ( Er w Er Er we r (B.4 This yields, Er+ w k pr + kw 1+ ( Er+ w Er δε E,r r Er+ we λ = π λ r + ( w w (B.5 Some addiional algebraic manipulaion yields, Er+ w kw pr + kw kw pr + kw E r + + δε E,r r Er E λ = π λ r+ w + ( w w (B.6

23 3 Use some simple algebra o obain, pk kw( pr + kw pk kw( pr kw r w r w δε = π λ + + E E E E r r+ w r r w (B.7 Use his resul o yield (4.11. Assume k w Appendix C is posiive. Then i can be shown ha, ( p + k ( p k r w r w E > for all p r (C.1 E r+ w r w Firs consider he case where p is posiive. The relaionship is obviously rue for r kw > r r w p. ow le p > k. In his case boh sides of (C.1 are posiive herefore we can square boh sides o obain, r + w r w > r w r + w p k E p k E (C. From his we obain, This yields, ( r w r w r w r w p + k p k + m > p k p + k + m (C.3 ( p + k > p k r w r w (C.4 which is rue for posiive k and p. If is negaive hen (C.1 becomes, ( pr + kw ( pr kw E This yields, w r w r+ w r > kw pr > (C.5 E

24 ( pr + kw ( pr kw E > (C.6 E r+ w r w which is obviously rue from he previous discussion. 4 References 1. F.A.B. Couinho, D. Kaing, Y. agami, and L. Tomio, Can. J. of Phys., 8, 837 (. (see also quan-ph/139.. F.A.B. Couinho, Y. agami, and L. Tomio, Phy. Rev. A, 59, 64 ( R. M. Cavalcani, quan-ph/ D. Solomon. Can. J. Phys., 81, 1165, (3. 5. B. Thaller, The Dirac Equaion, Springer-Verlag, Berlin ( W. Greiner, B. Muller, and T. Rafelski, Quanum Elecrodynamics of Srong Fields, Springer-Verlag, Berlin ( J.J. Sakurai, Advanced Quanum Mechanics, Addison-Wesley Publishing Company, Inc., Redwood Ciy, California, ( D. Solomon, Can. J. Phys. 76, 111 (1998. (see also quan-ph/9951.