Dirac s hole theory and the Pauli principle: clearing up the confusion.

Transcription

1 Dirac s hole heory and he Pauli rincile: clearing u he conusion. Dan Solomon Rauland-Borg Cororaion 8 W. Cenral Road Moun Prosec IL 656 USA dan.solomon@rauland.com Absrac. In Dirac s hole heory (HT) he vacuum sae is generally believed o be he sae o minimum energy due o he assumion ha he Pauli Exclusion Princile revens he decay o osiive energy elecrons ino occuied negaive energy saes. However recenly aers have aeared ha claim o show ha here exis saes wih less energy han ha o he vacuum[][][3]. Here we will consider a simle model o HT consising o ero mass elecrons in -D sace-ime. I will be shown ha or his model here are saes wih less energy han he HT vacuum sae and ha he Pauli Princile is obeyed. Thereore he conjecure ha he Pauli Princile revens he exisence o saes wih less energy han he vacuum sae is no correc. Keywords: Dirac sea - hole heory - vacuum sae.. Inroducion. I is well known ha here are boh osiive and negaive energy soluions o he Dirac equaion. This creaes a roblem in ha an elecron in a osiive energy sae will quickly decay ino a negaive energy sae in he resence o erurbaions. This o course is no normally observed o occur. This roblem is

2 resumably resolved in Dirac s hole heory (HT) by assuming ha all he negaive energy saes are occuied by a single elecron and hen evoking he Pauli exclusion rincile o reven he decay o osiive energy elecrons ino negaive energy saes. The roosiion ha he negaive energy saes are all occuied urns a one elecron heory ino an -elecron heory where. Due o he ac ha he negaive energy vacuum elecrons obey he same Dirac equaion as he osiive energy elecrons we have o in rincile rack he ime evoluion o an ininie number o saes. Also he vacuum elecrons in heir unerurbed sae are unobservable. All we can do is observe he dierences rom he unerurbed vacuum sae. I is generally assumed ha he HT vacuum sae is he sae o minimum energy. Tha is he energy o all oher saes mus be greaer han ha o he vacuum sae. However a number o aers by he auhor have shown his is no he case ([][][3]). I was shown in hese aers ha saes exis in HT ha have less energy han he vacuum sae. One ossible objecion o his resul is ha i seems o conradic he Pauli exclusion rincile. I is he urose o his aer o show ha his is no he case. I will be shown ha he exisence o saes wih less energy han he HT vacuum sae is erecly consisen wih he Pauli rincile. In his aer we will consider a simle quanum heory consising o non-ineracing ero mass elecrons in a background classical elecric ield. The advanage o such a simle sysem is ha we can easily obain exac soluions or he Dirac equaion or any arbirary elecric oenial. This considerably simliies he analysis. In he ollowing discussion we will assume ha he sysem is in some iniial sae which consiss o an ininie number o elecrons occuying he negaive energy saes (he Dirac sea) along wih a single osiive energy elecron. The sysem will be hen erurbed by an elecric ield. Each elecron will evolve in ime according o he Dirac equaion. Aer some eriod o ime he elecric

3 3 ield is removed and he change in he energy o each elecron can be calculaed. The oal change in he energy o he sysem is he sum o hese changes. I will be shown ha i is ossible o seciy an elecric ield so ha inal energy is less han he energy o he vacuum sae. I will be also shown ha his resul is enirely consisen wih he Pauli exclusion rincile.. The Dirac Equaion In order o simliy he discussion and avoid unnecessary mahemaical deails we will assume ha he elecrons have ero mass and are non-ineracing i.e. hey only inerac wih an exernal elecric oenial. Also we will work in - dimensional sace-ime where he sace dimension is aken along he -axis and use naural unis so ha c. In his case he Dirac equaion or a single elecron in he resence o an exernal elecric oenial is i H where he Dirac Hamilonian is given by where (.) H H qv (.) H is he Hamilonian in he absence o ineracions V is an exernal elecrical oenial and q is he elecric charge. For ero mass elecrons he ree ield Hamilonian is given as H i 3 where 3 is he Pauli marix wih 3. The soluion o (.) can be easily shown o be where and can be wrien as W (.3) (.4) is he soluion o he ree ield equaion i H (.5)

4 4 The quaniy W ih e (.6) is given by W where c and and e ic e ic c saisy he ollowing dierenial equaions c c These relaionshis also saisy c c c c Le energy eigenvalue where c c qv ; qv qv c c c c (.7) (.8) (.9) (.) be he eigenuncions o he ree ield Hamilonian wih. They saisy he relaionshi H (.) e L i ; (.) and where is he sign o he energy is he momenum and L is he dimensional inegraion volume. We assume eriodic boundary condiions so ha he momenum r L where r is an ineger. According o he above deiniions he quaniies and he quaniies. are negaive energy saes wih energy are osiive energy saes wih energy

5 5 The orm an orhonormal basis se and saisy where inegraion rom L o L hen he d (.3) evolve in ime according o is imlied. I he elecric oenial is ero ih i e e (.4) The energy o a normalied wave uncion is given by H V d (.5) In he case where V is ero he energy equals he ree ield energy which is given by H d (.6) ow suose a a normalied wave uncion is seciied and he elecric oenial is ero. ow aly an elecric oenial and hen remove i a some uure ime. The wave uncion has evolved ino he sae which saisies Eq. (.4). In general he alicaion o an elecric oenial will change he ree ield energy o he wave uncion. The change in he ree ield energy rom where o is shown in he Aendix o be given by d V d (.7) J 3 J q (.8) where is given by (.6). The quaniy J may be hough o as he curren densiy o he wave uncion. Recall ha evolves in ime according o he ree ield Dirac equaion. Thereore J wi be called he ree ield curren densiy.

6 6 3. Hole Theory The roosiion ha he negaive energy saes are all occuied urns a one elecron heory ino an -elecron heory where. For an -elecron heory he wave uncion is wrien as a Slaer deerminan [] s... P (3.)! where he n P ( n ) are a normalied and orhogonal se o wave uncions ha obey he Dirac equaion P is a ermuaion oeraor acing on he sace coordinaes and s is he number o inerchanges in P. oe i b are wo wave uncions ha obey he Dirac equaion hen i can be shown ha a b d (3.) Thereore i he n a and in (3.) are orhogonal a some iniial ime hen hey are orhogonal or all ime. The execaion value o a single aricle oeraor O is deined as where oeraor is given by Oe Oo d o (3.3) is a normalied single aricle wave uncion. The -elecron o o n n... O O (3.4) which is jus he sum o one aricle oeraors. The execaion value o a normalied -elecron wave uncion is O... x O d d... d e o This can be shown o be equal o e n o n n (3.5) O O d (3.6)

7 7 Tha is he elecron execaion value is jus he sum o he single aricle execaion values associaed wih each o he individual wave uncions examle he ree ield energy o he -elecron sae is n H n d n n n n. For (3.7) 4. Time varying elecric oenial. Assume a ime he elecric oenial is ero and he sysem is in some iniial sae which is deined in he ollowing discussion. In HT he unerurbed vacuum sae is he sae where each negaive energy wave uncion is occuied by a single elecron and each osiive energy wave uncion is unoccuied. The energy o he vacuum sae is given by summing over he energies o all he negaive energy saes. The Slaer deerminan corresonding o his iniial vacuum sae can be wrien as P... s (4.)! P where we assume he ollowing ordering; 3. The oal ree ield energy o he unerurbed vacuum sae is hen E (4.) vac We can add an addiional elecron rovided i consiss o a combinaion o osiive energy saes so ha i is orhogonal o he vacuum wave uncions. Le he wave uncion ha deines his osiive energy elecron a ime be given by (4.3)

8 8 where he seleced so ha his iniial sae as are consan exansion coeiciens. Assume ha he are is normalied. We can wrie he Slaer deerminan o s... P! P (4.4) Thereore we have a he iniial ime a sysem which consiss o he unerurbed vacuum elecrons. Thereore he oal ree ield energy o he sysem is T vac and a single osiive energy elecron E E (4.5) ow we are no really ineresed in he oal energy bu in he energy wih resec o he unerurbed vacuum sae. Thereore we subrac he vacuum energy Evac rom he above exression o obain T R T vac E E E (4.6) which is jus he energy o he osiive energy elecron. ex consider he change in he energy due o an ineracion wih an exernal elecric oenial. A he iniial ime he elecric oenial is ero and he sysem is in he iniial sae given by (4.4). ex aly an elecric oenial and hen remove i a some laer ime so ha V or ; V or ; V or (4.7) ow wha is he change in he energy o he sysem due o his ineracion wih he elecric oenial? Under he acion o he elecric oenial each negaive energy wave uncion wave uncion evolves ino he inal sae evolves ino. Also he. oe ha er (4.7) he elecric oenial is ero a he iniial ime and he inal ime. Thereore he

9 9 change in he energy is equal o he change in he ree ield energy. For he negaive energy elecrons he change in he ree ield energy o each elecron is (4.8) and he change in he energy o he osiive energy elecron is (4.9) The oal change in he energy o he sysem is hen E E (4.) T vac where vac E (4.) The quaniy Evac energy o he sysem a is he change in energy o he vacuum. Using hese resuls he wih resec o he unerurbed vacuum sae is E E E (4.) T R T R T ow we wan o evaluae he above quaniy. To do his we will use Eq. (.7). For he vacuum elecrons he ree ield curren densiy J 3 is given by J ; q (4.3) Reerring o (.) i is eviden ha obain J ;. Use his in (.7) o.. Tha is he change in he energy o each o he vacuum elecrons is ero. oe ha resul is indeenden o he alied oenial V. This yields Evac (4.4) r ex we have o deermine he change in he energy o he osiive energy elecron. The ree ield curren densiy associaed wih osiive energy elecron is where 3 J q (4.5)

10 Use his in (.7) o obain (4.6) e ih d V d (4.7) ow i is easy o ind a sae J so ha J is non-ero. This can be done by roer selecion he exansion coeiciens. For examle le e e e e L i i i i where boh and are osiive numbers. In his case (4.8) q J cos (4.9) L I is eviden ha he derivae o his quaniy wih resec o is non-ero. When J is non-ero i is ossible o ind a V so ha is an arbirarily large negaive number. For examle le V J g where g is a osiive number. Use his in (4.7) o obain (4.) J g d d (4.) ow he inegraed quaniy is osiive. Thereore as g i is eviden ha. Use his in (4.) along wih (4.4) o obain J E g d d (4.) T Recall ha he energy o he sysem wih resec o he unerurbed vacuum a he inal ime E E E. ow due o he ac ha is given by T R T R T

11 E T can be an arbirarily large negaive number hen T R E can be negaive. Thereore he inal energy o he sysem can be less han ha o he vacuum sae. 5. Discussion. This resul is somewha surrising. I shows ha in HT he unerurbed vacuum sae is no he lowes energy sae and ha i is ossible o exrac an unlimied amoun o energy rom an iniial quanum sae. To review he resuls o he revious secions we sared wih an iniial sysem consising o vacuum elecrons in heir unerurbed sae and a osiive energy elecron as deined by (4.3). We hen aly an elecric oenial. The resul is ha each wave uncion evolves rom is iniial sae in accordance wih he Dirac equaion. We ind ha he change in energy o he vacuum elecrons rom he iniial o inal sae is ero. This is rue or any elecric oenial. However when we consider he change in he energy o he wave uncion he siuaion is dieren. In his case i we se u his wave uncion so ha J is non-ero hen we can easily ind an elecric oenial such ha he change in energy o he wave uncion can be a negaive number wih an arbirarily large magniude. The ne resul is ha he oal energy o he inal sysem is negaive wih resec o he energy o he vacuum sae. This resul is consisen wih ha o revious work [][][3]. In he above examle he energy o he vacuum elecrons doesn change and he energy o he wave uncion which was originally osiive becomes negaive. ow wasn he Pauli rincile suose o reven his? Wha exacly is he Pauli exclusion rincile? In he conex o HT he Pauli rincile is simly he saemen ha no more han one elecron can occuy a given sae a given ime. Equaions (3.) and (3.) are he mahemaical realiaion o his saemen. The Pauli Princile is a resul o he ac ha i he iniial wave uncions in he Slaer deerminan (see Eq. (3.)) are orhogonal hen hese wave uncions will be orhogonal or all ime. This is a consequence o he ac ha he individual

12 wave uncions obey he Dirac equaion (see Eq. (3.)). Thereore wo elecrons canno end u in he same sae. Thereore he calculaions erormed in he aer are consisen wih he Pauli rincile. All he wave uncions are orhogonal or all ime. This means ha he conjecure ha he Pauli rincile revens he exisence o quanum saes wih less energy han ha o he unerurbed vacuum sae is no correc. Aendix. In his secion we will calculae he change in he ree ield energy o a normalied wave uncion. Assume a he iniial ime. A some uure ime ree ield energy o he sae a a given ime is given by he wave uncion is given by he wave uncion is given by Eq. (.4). The (A.) H d W H W d Use (.4) and (.3) in he above o obain c d H d c (A.) From his we obain where J and he sae c c c c J d q and are given by (A.3) are he curren and charge densiy resecively o ; q ˆ ˆ J q 3 (A.4) Using he above deiniions along wih (.5) and (.3) we can readily show ha J J ; Take he derivaive wih resec o ime o (A.3) and use (A.5) o obain (A.5) c c c c J c c c c J q (A.6) d

13 3 Assume reasonable boundary condiions and inegrae by ars o obain c c c c c c c c J d q (A.7) Use (.) o obain Inegrae his rom Reerences V J J d V d (A.8) o o obain Eq. (.7).. D. Solomon. Some dierences beween Dirac s hole heory and quanum ield heory Can. J. Phys. 83 (5) Also arxiv:quanh/567.. D. Solomon. Some new resuls concerning he vacuum in Dirac hole heory. Physc. Scr. 74 (6) 7-. Also arxiv:quan-h/ D. Solomon. Quanum saes wih less energy han he vacuum in Dirac hole heory. arxiv:quan-h/77.