●❯■❆❙ ❉❊ ❖◆❉❆ ❈❖▼❖ ❯▼ ❆▼❇■❊◆❚❊ P❆❘❆ ❚❊❙❚❊❙
❉❊ ❯▼❆ ❊▲❊❚❘❖❉■◆➶▼■❈❆ ❈❖▼ ❱■❖▲❆➬➹❖ ❉❆❙
❙■▼❊❚❘■❆❙ ❉❊ ▲❖❘❊◆❚❩ ❊ ❈P❚
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐❣ê♥✲ ❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❆♣❧✐❝❛❞❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳
❱■➬❖❙❆
GUIAS DE ONDA COMO UM AMBIENTE PARA TESTES DE UMA ELETRODINÂMICA COM VIOLAÇÃO DAS
SIMETRIAS DE LORENTZ E CPT
Dissertação apresentada à Universidade Federal de Viçosa, como parte das exigências do Programa de Pós-Graduação em Física Aplicada, para obtenção do título de Magister Scientiae.
APROVADA: 08 de julho de 2010.
Prof. Daniel Heber Theodoro Franco (Coorientador)
Prof. José Arnaldo Redinz
Prof. Guilherme de Berredo Peixoto Prof. Olivier Piguet
❈❤❛r❧❡s ❙❛♥❞❡rs P❡✐r❝❡ ✭✶✽✸✾✲✶✾✶✹✮
❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r❛ ❡ ❢✉♥❞❛♠❡♥t❛❧♠❡♥t❡✱ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ✖ ♠❡✉ ♣❛✐✱ ❏♦sé P❡❞r♦✱ ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ❆✉❣✉st❛ ❡ ♠✐♥❤❛ ✐r♠ã✱ ❑❛r✐♥❡ ✖ ♣♦r t❡r❡♠ s❡♠♣r❡ ♠❡ ❛♣♦✐❛❞♦✱ ❝♦♠ ❛ ♠❡s♠❛ ✐♥t❡♥s✐❞❛❞❡ ❡ ❝♦♥✜❛♥ç❛✱ ❞❡s❞❡ ❛ ♠✐♥❤❛ ❡♥tr❛❞❛ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡✱ ❝♦♠♦ ❡st✉❞❛♥t❡ ❞❡ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧✱ ❛ss✐♠ ❝♦♠♦ q✉❛♥❞♦ ❞❡❝✐❞✐ ♠✉❞❛r ♣❛r❛ ❛ ❋ís✐❝❛✱ ❡ ❛té ❤♦❥❡✳ ❙❡♠ ❡❧❡s ❡✉ ❥❛♠❛✐s ❝❤❡❣❛r✐❛ ❛ ❡st❡ ♠♦♠❡♥t♦ ❡ ❞✐✜❝✐❧♠❡♥t❡ ❡st❛r✐❛ tã♦ ❝♦♥t❡♥t❡ ❡♠ q✉❡r❡r ✐r ♠✉✐t♦ ♠❛✐s ❛❧é♠✳
❆♦ ♣r♦❢❡ss♦r ❲✐♥❞❡r✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❛♦ ❧♦♥❣♦ ❞❡st❡s ❛♥♦s ❞❡ tr❛❜❛❧❤♦ ❡♠ ❝♦♥✲ ❥✉♥t♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡ ♣❡❧♦ s❡♠♣r❡ ♣r❡s❡♥t❡ ❛♣♦✐♦ à ♠✐♥❤❛ ❛✐♥❞❛ ❥♦✈❡♠ ❥♦r♥❛❞❛ ❝✐❡♥✲ tí✜❝❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❆❢râ♥✐♦✱ ❉❛♥✐❡❧ ❡ ▼❛r❝❡❧♦✱ ♣❡❧♦s ❝✉rs♦s ♠✐♥✐str❛❞♦s ❞✉r❛♥t❡ ♠❡✉ ♠❡str❛❞♦✱ ❡ ♣❡❧♦ q✉❛♥t♦✱ ❛tr❛✈és ❞❡st❡s ❡ ❞❡ ❝♦♥✈❡rs❛s✱ s❡♠♣r❡ ♠❡ s❡r✈✐r❛♠ ❝♦♠♦ ❡♥♦r♠❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ❞❛r s❡♠♣r❡ ♠❡✉ ♠❡❧❤♦r✳ ❚❛♠❜é♠ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉P❋ ❝♦♠ ♦s q✉❛✐s ✜③ ❞✐s❝✐♣❧✐♥❛s ❛♣❡♥❛s ♥❛ ❣r❛❞✉❛çã♦ ♠❛s ❝♦♠ ♦s q✉❛✐s ♠❛♥t❡♥❤♦ ❛♠✐③❛❞❡ ❛té ❤♦❥❡✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ♣ós✲❣r❛❞✉❛çã♦✱ ❝♦♠ ♦s q✉❛✐s ♣❛ss♦ ❜♦❛ ♣❛rt❡ ❞♦s ❞✐❛s✱ ♣❡❧❛ ót✐♠❛ ❝♦♥✈✐✈ê♥❝✐❛ ❡ ❢♦rt❡ ❛♠✐③❛❞❡✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ♣❡❧♦ q✉❛♥t♦✱ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ♠❡ ❛❥✉❞❛r❛♠ ❛♦ ❧♦♥❣♦ ❞♦ ♠❡✉ ♠❡str❛❞♦ ✖ ❡✉ t❡r✐❛ s✐❞♦ ❝❛♣❛③ ❞❡ ♠✉✐t♦ ♠❡♥♦s s❡♠ ❡ss❛s ❛❥✉❞❛s✳
❆♦s ❛♠✐❣♦s ❞♦ ❈❧✉❜❡ ❞❡ ❆str♦♥♦♠✐❛ ❞❡ ❱✐ç♦s❛✱ ♣❡❧❛s ✐♥❝♦♥tá✈❡✐s ❤♦r❛s ♥❛s ♥♦✐t❡s ❡ ♠❛❞r✉❣❛❞❛s ❝♦♥❣❡❧❛♥t❡s ❞♦ ✐♥✈❡r♥♦ ❞❡ ❱✐ç♦s❛✱ ❡s♠✐✉ç❛♥❞♦ ❝❛❞❛ ❝❛♥t♦ ❞♦ ❝é✉ ❡ ♣❡❧♦s ❛❝❛♠♣❛♠❡♥t♦s ♠❛✐s ✐♥s❛♥♦s ♣♦ssí✈❡✐s✳
❆♦s ❛♠✐❣♦s ❞❛ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧ ♣❡❧♦s ✏r❡❡♥❝♦♥tr♦s✑ s❡♠♣r❡ ♥♦stá❧❣✐❝♦s ❡ àq✉❡❧❡s
àq✉❡❧❡s q✉❡ ✈ê❡♠ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦ ❞❡✈❡ s❡ ❡①♣r❡ss❛r ❞❡ ♠♦❞♦ ♠á①✐♠♦ ❛tr❛✈és ❞❛ ❧✐❜❡r❞❛❞❡✳
❆❣r❛❞❡ç♦ ❛ t♦❞♦s ❜r❛s✐❧❡✐r♦s q✉❡ ❝♦♥tr✐❜✉❡♠ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❈✐ê♥❝✐❛ ❡♠ ♥♦ss♦ ♣❛ís ❡ t❛♠❜é♠ àq✉❡❧❡s q✉❡✱ ❛tr❛✈és ❞❛ ❋✉♥❞❛çã♦ ❞❡ ❆♠♣❛r♦ à P❡sq✉✐s❛ ❞♦ ❊st❛❞♦ ❞❡ ▼✐♥❛s ●❡r❛✐s ✭❋❛♣❡♠✐❣✮✱ ❝♦♥❝❡❞❡♠✱ ❛ ♥ós✱ ❡st✉❞❛♥t❡s✱ ♦ ♥❡❝❡ssár✐♦s ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❆ t♦❞♦s✱ ♠❡✉s ♠❛✐s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✐①
◆♦t❛çõ❡s ❡ ❈♦♥✈❡♥çõ❡s ①
❘❡s✉♠♦ ①✐
❆❜str❛❝t ①✐✐✐
■♥tr♦❞✉çã♦ ❡ ▼♦t✐✈❛çã♦ ✶
✶ ❆s♣❡❝t♦s ❋✉♥❞❛♠❡♥t❛✐s ❞♦ ▼P❊ ✼
✶✳✶ ❱✐♦❧❛çã♦ ❞❛ ❙✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ❡ ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥❞✐❞♦ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❈♦♥str✉✐♥❞♦ ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥❞✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ P♦ssí✈❡✐s ❖r✐❣❡♥s ♣❛r❛ ❛ ❱✐♦❧❛çã♦ ❞❛ ❙✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ ❚❡st❛♥❞♦ ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥❞✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷ ❙❡t♦r ❞❡ ❘❛❞✐❛çã♦ ❞♦ ▼P❊✿ ❆s♣❡❝t♦s ●❡r❛✐s ✷✹
✷✳✶ ❆çã♦ ❞♦ ▼♦❞❡❧♦ ❡ ❈❛r❛❝t❡ríst✐❝❛s ●❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❊q✉❛çõ❡s ❞❡ ▼♦✈✐♠❡♥t♦ ❡ Pr♦♣❛❣❛çã♦ ▲✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ❙✐♠❡tr✐❛s ❡ ▲❡✐s ❞❡ ❈♦♥s❡r✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸ ❚❡st❡s ❝♦♠ ❖♥❞❛s ❊❧❡tr♦♠❛❣♥ét✐❝❛s ❈♦♥✜♥❛❞❛s ✹✻
✸✳✶ ●✉✐❛s ❞❡ ❖♥❞❛ ❘❡t❛♥❣✉❧❛r❡s✿ ❙♦❧✉çõ❡s ❊①❛t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✶✳✶ ▼♦❞♦ ❚r❛♥s✈❡rs♦ ❊❧étr✐❝♦✱ Ez ≡0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✶✳✷ ▼♦❞♦ ❚r❛♥s✈❡rs♦ ▼❛❣♥ét✐❝♦✱ Bz ≡0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✷ ❈♦♠❡♥tár✐♦s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✶
✶ ❆ ✏❤✐stór✐❛✑ ❞♦ ❯♥✐✈❡rs♦ s❡❣✉♥❞♦ ✉♠ ❢ís✐❝♦ ❞❡ ♣❛rtí❝✉❧❛s ❡ ❝❛♠♣♦s✿ ❙❡✉ ❝♦♠❡ç♦ ❝♦♠ ❛ s✐♥❣✉❧❛r✐❞❛❞❡ ✐♥✐❝✐❛❧✱ ❞❡s❝r✐t♦ ♣♦r ✉♠❛ ✐♥t❡r❛çã♦ ✉♥✐✜❝❛❞❛✱ ❡✱ ❝♦♥❢♦r♠❡ ❛ t❡♠♣❡r❛t✉r❛ ✭❡♥❡r❣✐❛✮ ❞♦ ❯♥✐✈❡rs♦ ❞✐♠✐♥✉✐✱ ❡st❛ ✐♥t❡r❛çã♦ ✈❛✐ s❡ s❡♣❛r❛♥❞♦ ❛té ❝❤❡❣❛r às q✉❛tr♦ q✉❡ ♦❜s❡r✈❛♠♦s ❤♦❥❡ ✖ ✐♥t❡r❛çõ❡s ●r❛✈✐t❛❝✐♦♥❛❧✱ ♥✉❝❧❡❛r❡s ❋♦rt❡ ❡ ❋r❛❝❛✱ ❡ ❊❧❡tr♦♠❛❣♥ét✐❝❛✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ P❛rtí❝✉❧❛✳ ◆♦t❡ q✉❡ ❛ ✏❢ís✐❝❛ ♠✉❞❛✑✱ ♣♦✐s ♦ â♥❣✉❧♦ ❞❡
❛❝♦♣❧❛♠❡♥t♦ θ ❡♥tr❡ ✈❡t♦r ❞❡ ♦♥❞❛ k ❡ ❝❛♠♣♦ ❞❡ ❢✉♥❞♦kAF é ❛❧t❡r❛❞♦ ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ❖❜s❡r✈❛❞♦r✳ ◆♦t❡ q✉❡ ❛ ✏❢ís✐❝❛ ♥ã♦ ♠✉❞❛✑✱ ♣♦✐s ♦ â♥❣✉❧♦
❞❡ ❛❝♦♣❧❛♠❡♥t♦θ❡♥tr❡ ✈❡t♦r ❞❡ ♦♥❞❛k❡ ❝❛♠♣♦ ❞❡ ❢✉♥❞♦kAF ♥ã♦ é ❛❧t❡r❛❞♦ ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶ ❉❡♣❡♥❞ê♥❝✐❛ ❞❛ ❢r❡q✉ê♥❝✐❛ ω ❝♦♠ ♦ ✭♠ó❞✉❧♦ ❞♦✮ ✈❡t♦r ❞❡ ♦♥❞❛k✭♣❛r❛ ηo=
1✮✳ ❆q✉✐ ✈❡♠♦s q✉❡ ♦s s✐♥❛✐s ❞❡ω♥ã♦ sã♦ ❜❡♠ ❞❡✜♥✐❞♦s ❡✱ ♣❛r❛ ❝❡rt❛s r❡❣✐õ❡s✱
ω− ♣♦❞❡ ❛ss✉♠✐r ✈❛❧♦r❡s ✐♠❛❣✐♥ár✐♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✷ ❉❡♣❡♥❞ê♥❝✐❛ ❞❛ ❢r❡q✉ê♥❝✐❛ ω ❝♦♠ ♦ ✭♠ó❞✉❧♦ ❞♦✮ ✈❡t♦r ❞❡ ♦♥❞❛ k ❡ ❝♦♠ ♦
â♥❣✉❧♦ θ ❡♥tr❡ ❡st❡ ❡ ✈❡t♦r ❞❡ ❢✉♥❞♦ η ✭♣❛r❛ |η| = 1✮✳ ❆q✉✐ ✈❡♠♦s q✉❡ ♦s
s✐♥❛✐s ❞❡ ω sã♦ s❡♠♣r❡ ❜❡♠ ❞❡✜♥✐❞♦s ❡ ❡st❡ é s❡♠♣r❡ r❡❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✷✳✸ ❉❡♣❡♥❞ê♥❝✐❛ ❞❛s ✈❡❧♦❝✐❞❛❞❡s ❞❡ ❣r✉♣♦ vg+ ✭❡sq✉❡r❞❛✮ ❡ vg− ✭❞✐r❡✐t❛✮ ❝♦♠ ♦
✭♠ó❞✉❧♦ ❞♦✮ ✈❡t♦r ❞❡ ♦♥❞❛ k ❡ ❝♦♠ ♦ â♥❣✉❧♦θ ❡♥tr❡ ❡st❡ ❡ ♦ ✈❡t♦r ❞❡ ❢✉♥❞♦
η ✭♣❛r❛ |η|= 1✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✶ ●✉✐❛ ❞❡ ♦♥❞❛ ❞❡ s❡çã♦ tr❛♥s✈❡rs❛❧ ❝♦♥st❛♥t❡ ❛r❜✐trár✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷ ❆♥✐s♦tr♦♣✐❛ ♥♦ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ ❞♦ ❣✉✐❛ ❞❡ ♦♥❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✸✳✺ ❙✉♣r❡ssã♦ ❞❡ ♠♦❞♦s ♥♦ ❣✉✐❛ r❡t❛♥❣✉❧❛r✿ ❆♣❡♥❛s ♠♦❞♦s ❚❊0n sã♦ ♣❡r♠✐t✐❞♦s q✉❛♥❞♦ t❡♠♦s ❛♣❡♥❛s ηx 6= 0❀ ❝❛s♦ ❛ ❛♥✐s♦tr♦♣✐❛ ❡st❡❥❛ t♦❞❛ ❛♦ ❧♦♥❣♦ ❞♦ ❡✐①♦
y✱ηy 6= 0✱ ♦s ú♥✐❝♦s ♠♦❞♦s ♣♦ssí✈❡✐s sã♦ ♦s ❚❊m0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
• ❯t✐❧✐③❛r❡♠♦s ♦ s✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s ♥❛t✉r❛✐s✱ ♦♥❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ c= ~ = 1✱ ❛♦
❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳
• ❮♥❞✐❝❡s ❣r❡❣♦s ✭❝♦♠♦ α, β, µ, ν✮ sã♦ í♥❞✐❝❡s ❡s♣❛ç♦✲t❡♠♣♦r❛✐s✱ ♣♦❞❡♥❞♦ ❛ss✉♠✐r
♦s ✈❛❧♦r❡s 0,1,2 ❡ 3✳ ❮♥❞✐❝❡s ❧❛t✐♥♦s ✭❝♦♠♦ i, j, k✮ sã♦ í♥❞✐❝❡s ❡①❝❧✉s✐✈❛♠❡♥t❡
❡s♣❛❝✐❛✐s✱ ♣♦❞❡♥❞♦ ❛ss✉♠✐r ♦s ✈❛❧♦r❡s1,2❡3✳ ❯t✐❧✐③❛r❡♠♦s ❛ ❝♦♥✈❡♥çã♦ ❞❡ s♦♠❛
❞❡ ❊✐♥st❡✐♥✿ í♥❞✐❝❡s r❡♣❡t✐❞♦s sã♦ s♦♠❛❞♦s✳ P♦r ❡①❡♠♣❧♦✱xµxµ ≡
3
X
µ=0 xµxµ✳
• ❆ ♠étr✐❝❛ ηµν ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ✭❞❡ ▼✐♥❦♦✇s❦✐✮ ❛❞♦t❛❞❛ é ❞✐❛❣♦♥❛❧✱ ❝✉❥♦s ❡❧❡✲ ♠❡♥t♦s sã♦ η00 = 1 ❡ ηij = −δij✱ s❡♥❞♦ δij ♦s ❡❧❡♠❡♥t♦s ❞❡
13×3✳ ❖s ❡❧❡♠❡♥t♦s
❞❛ ♠étr✐❝❛ sã♦ ✐❞ê♥t✐❝♦s ❛♦s ❞❡ s✉❛ ✐♥✈❡rs❛✳
• ◗✉❛♥❞♦ ♥ã♦ ❢♦r ❛♠❜í❣✉♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s q✉❛tr♦ ❝♦♦r❞❡♥❛❞❛s ❡s♣❛ç♦✲t❡♠♣♦r❛❧
xµ s❡rá ❞❡♥♦t❛❞♦ ❛♣❡♥❛s ♣♦r x✳ ❆✐♥❞❛✱ ♥♦s r❡❢❡r✐r❡♠♦s à ❝♦♠♣♦♥❡♥t❡ ✏③❡r♦✑ ❞❡ ✉♠ ✹✲✈❡t♦r ♣♦r ✏❝♦♠♣♦♥❡♥t❡ t❡♠♣♦r❛❧✑ ❡ às ♦✉tr❛s três ♣♦r ✏❝♦♠♣♦♥❡♥t❡s ❡s♣❛❝✐❛✐s✑✱ ❢r❡q✉❡♥t❡♠❡♥t❡ ❞❡♥♦t❛❞❛s ♣♦r ✉♠ ✈❡t♦r ❡♠ ♥❡❣r✐t♦✳ P♦r ❡①❡♠♣❧♦✿
xµ= (xo;x) = (xo;x1, x2, x3)✳
• ❖ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❡♥tr❡ ❞♦✐s ✹✲✈❡t♦r❡s✱ aµ ❡ bµ✱ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r a µbµ ♦✉
a·b✱ s❡♥❞♦ ✐❣✉❛❧ ❛ aobo−a·b✳
• ❖ sí♠❜♦❧♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ǫαβωγ é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ t♦t❛❧♠❡♥t❡ ❛♥t✐✲ s✐♠étr✐❝❛✱ ❞❛❞♦ ♣♦rǫ0123 =−ǫ
0123 = 1✳
• ❉❡✜♥✐♠♦s ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ ♠♦❞♦ q✉❡ ∂µ↔+ikµ✳
●❖▼❊❙✱ ❆♥❞ré ❍❡r❦❡♥❤♦✛✱ ▼✳ ❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❏✉❧❤♦ ❞❡ 2010
●✉✐❛s ❞❡ ♦♥❞❛ ❝♦♠♦ ✉♠ ❛♠❜✐❡♥t❡ ♣❛r❛ t❡st❡s ❞❡ ✉♠❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ ❝♦♠ ✈✐♦❧❛çã♦ ❞❛s s✐♠❡tr✐❛s ❞❡ ▲♦r❡♥t③ ❡ ❈P❚✳ ❖r✐❡♥t❛❞♦r✿ ❲✐♥❞❡r ❆❧❡①❛♥❞❡r ❞❡ ▼♦✉r❛ ▼❡❧♦✳ ❈♦✲♦r✐❡♥t❛❞♦r❡s✿ ❆❢râ♥✐♦ ❘♦❞r✐❣✉❡s P❡r❡✐r❛ ❡ ❉❛♥✐❡❧ ❍❡❜❡r ❚❤❡♦❞♦r♦ ❋r❛♥❝♦✳
❊st✉❞❛♠♦s ✉♠❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ ❡♠ ✭✸✰✶✮ ❞✐♠❡♥sõ❡s✱ ❝♦♠ ✈✐♦❧❛çã♦ ❞❛s s✐♠❡tr✐❛s ❞❡ ▲♦r❡♥t③ ❡ ❈P❚✱ ♦❜t✐❞❛ ❛♦ ❛❞✐❝✐♦♥❛r♠♦s ✉♠ t❡r♠♦ t✐♣♦✲❈❤❡r♥✲❙✐♠♦♥s ❛♦ ♠♦❞❡❧♦ ❞❡ ▼❛①✇❡❧❧ ✉s✉❛❧✱ ❝♦♥t✐❞♦ ♥♦ s❡t♦r ❞❡ r❛❞✐❛çã♦ ✭❡❧❡tr♦♠❛❣♥ét✐❝❛✮ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊s✲ t❡♥❞✐❞♦✳ ❯♠❛ ✈❡③ q✉❡ ❡st❡ t❡r♠♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛ ✉♠ ✏✹✲✈❡t♦r✑ ❞❡ ❢✉♥❞♦ ❝♦♥st❛♥t❡✱
(kAF)µ✱ q✉❡ s❡❧❡❝✐♦♥❛ ✉♠❛ ❞✐r❡çã♦ ✜①❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ✭q✉❡❜r❛♥❞♦ s✉❛ ✐s♦tr♦♣✐❛✮✱
❛s s✐♠❡tr✐❛s ❞❡ ▲♦r❡♥t③ ❡ ❈P❚ ♥ã♦ sã♦ ♠❛✐s ✈❡r✐✜❝❛❞❛s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ r❡✈✐s✐t❛♠♦s ♦ ♠♦❞❡❧♦ ♣❛r❛ ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ♦♥❞❛s ❡❧❡tr♦♠❛❣♥ét✐❝❛s ❝❧áss✐❝❛s✱ s❛t✐s❢❛✲ ③❡♥❞♦ ❡st❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ ♠♦❞✐✜❝❛❞❛✱ ♣r♦♣❛❣❛♥❞♦✲s❡ ♥♦ ❡s♣❛ç♦ ❧✐✈r❡✱ ❡st❛❜❡❧❡❝❡♥❞♦ ✐♠♣♦rt❛♥t❡s r❡s✉❧t❛❞♦s ♣❛r❛ ✉s♦ ❢✉t✉r♦✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❡st❛ r❛❞✐❛çã♦ r❡str✐t❛ ❛ ♣r♦♣❛❣❛r✲s❡ ❞❡♥tr♦ ❞❡ ❣✉✐❛s r❡t❛♥❣✉❧❛r❡s✱ ❛❜❡rt♦s ❛♦ ❧♦♥❣♦ ❞❛ ❞✐r❡çã♦z✳ ◆❡st❛ s✐t✉❛çã♦✱
❡♥❝♦♥tr❛♠♦s q✉❡ ❛s♣❡❝t♦s ❞❡ ❣❡♦♠❡tr✐❛ ❝♦♠❜✐♥❛❞♦s às ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❣❡r❛♠ ❡❢❡✐t♦s q✉❡ ♣♦❞❡♠ ❝♦♥❞✉③✐r ❛ ❢♦rt❡s ❞❡s✈✐♦s ❞♦s r❡s✉❧t❛❞♦s ❞❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ ❝♦♥✈❡♥❝✐✲ ♦♥❛❧✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ q✉ã♦ ♣❡q✉❡♥♦ ♦ ♣❛râ♠❡tr♦ ❞❡ ✈✐♦❧❛çã♦ s❡❥❛✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ s❡
(kAF)µ❛♣♦♥t❛r ❛♦ ❧♦♥❣♦ ❞❛ ❞✐r❡çã♦x✭♦✉y✮✱ ❡♥q✉❛♥t♦(kAF)0 = 0✱ ❛♣❡♥❛s ♠♦❞♦s tr❛♥s✲
✈❡rs♦s ❡❧étr✐❝♦s✱ ❝♦♠ Ez = 0✱ ❞❡✈❡♠ s❡r ♦❜s❡r✈❛❞♦s ♣r♦♣❛❣❛♥❞♦✲s❡ ❛♦ ❧♦♥❣♦ ❞♦ ❣✉✐❛✱ ❡♥q✉❛♥t♦ q✉❡ t♦❞♦s ♠♦❞♦s tr❛♥s✈❡rs♦s ♠❛❣♥ét✐❝♦s✱ Bz = 0✱ ❡stã♦ ❛✉s❡♥t❡s✳ ❊st❛ ❢♦rt❡ s✉♣r❡ssã♦ ❞❡ ♠♦❞♦s t♦r♥❛ ❣✉✐❛s ❞❡ ♦♥❞❛ ❜❛st❛♥t❡ ❛❞❡q✉❛❞♦s ♣❛r❛ t❡st❛r ❡ss❛s ✈✐♦❧❛çõ❡s ❞❡ s✐♠❡tr✐❛✱ ✉s❛♥❞♦✲s❡ ❛♣❛r❛t♦s s✐♠♣❧❡s ❡ ❢❛❝✐❧♠❡♥t❡ r❡♣r♦❞✉tí✈❡✐s✳ ❈♦♥t✉❞♦✱ ❞❛❞❛ ❛
❞❡ ♦✉tr♦ ♠♦❞♦✱ t♦❞♦s ❡①♣❡r✐♠❡♥t♦s ❥á r❡❛❧✐③❛❞♦s t❡r✐❛♠ s✐❞♦ ❢❡✐t♦s ❡♠ r❡❢❡r❡♥❝✐❛✐s ♦♥❞❡
(kAF)0 6= 0✳ ❙❡ ❡st❡ ú❧t✐♠♦ ❢♦r ♦ ❝❛s♦✱ ♥♦ss♦s r❡s✉❧t❛❞♦s ♥♦t❛✈❡❧♠❡♥t❡ ♣r♦✈ê❡♠ ✉♠❛
♠❛♥❡✐r❛ ❛♣r♦♣r✐❛❞❛ ❡ r❡❧❛t✐✈❛♠❡♥t❡ ❢á❝✐❧ ❞❡ s❡ t❡st❛r ❡st❛s q✉❡❜r❛s ❞❡ s✐♠❡tr✐❛✿ ❡st❡s ♠❡s♠♦s ❡①♣❡r✐♠❡♥t♦s✱ s❡ r❡❛❧✐③❛❞♦s ❡♠ r❡❢❡r❡♥❝✐❛✐s ♦♥❞❡ (kAF)0 = 0✱ ❞❡✈❡r✐❛♠✱ s❡♠
❞ú✈✐❞❛s✱ ❞❡t❡❝t❛r ❡ss❡s ❣r❛♥❞❡s ❡❢❡✐t♦s s♦❜r❡ ♦s ❝❛♠♣♦s ❡❧étr✐❝♦ ❡ ♠❛❣♥ét✐❝♦ ❛ss♦❝✐❛❞♦s ❛ ❡st❛s ♦♥❞❛s ❝♦♥✜♥❛❞❛s✳
●❖▼❊❙✱ ❆♥❞ré ❍❡r❦❡♥❤♦✛✱ ▼✳ ❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❏✉❧②✱ 2010
❲❛✈❡❣✉✐❞❡s ❛s ❛ ✇❛② ♦❢ t❡st✐♥❣ ❛♥ ❡❧❡❝tr♦❞②♥❛♠✐❝s ✇✐t❤ ❜r♦❦❡♥ ▲♦r❡♥t③ ❛♥❞ ❈P❚ s②♠♠❡tr✐❡s✳ ❆❞✈✐s❡r✿ ❲✐♥❞❡r ❆❧❡①❛♥❞❡r ❞❡ ▼♦✉r❛ ▼❡❧♦✳ ❈♦✲❛❞✈✐s❡rs✿ ❆❢râ♥✐♦ ❘♦❞r✐❣✉❡s P❡r❡✐r❛ ❛♥❞ ❉❛♥✐❡❧ ❍❡❜❡r ❚❤❡♦❞♦r♦ ❋r❛♥❝♦✳
❲❡ st✉❞② ❛ ❈P❚✲ ❛♥❞ ▲♦r❡♥t③✲♦❞❞ ❡❧❡❝tr♦❞②♥❛♠✐❝s ✐♥ ✭✸✰✶✮ ❞✐♠❡♥s✐♦♥s✱ ♦❜✲ t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ ❛ ❈❤❡r♥✲❙✐♠♦♥s✲❧✐❦❡ t❡r♠ t♦ t❤❡ ✉s✉❛❧ ▼❛①✇❡❧❧ ♠♦❞❡❧✱ ❛s ❞❡s❝r✐❜❡❞ ❜② t❤❡ r❛❞✐❛t✐♦♥ s❡❝t♦r ♦❢ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❊①t❡♥s✐♦♥✳ ❖♥❝❡ t❤✐s t❡r♠ ✐s ♣r♦♣♦r✲ t✐♦♥❛❧ t♦ ❛ ❝♦♥st❛♥t ❜❛❝❦❣r♦✉♥❞ ✏✹✲✈❡❝t♦r✑✱ (kAF)µ✱ ✇✐❝❤ s❡❧❡❝ts ❛ ♣r❡❢❡rr❡❞ ❞✐r❡❝t✐♦♥
✐♥ t❤❡ s♣❛❝❡t✐♠❡ ✭❜r❡❛❦✐♥❣ ✐ts ✐s♦tr♦♣②✮✱ ▲♦r❡♥t③ ❛♥❞ ❈P❚ s②♠♠❡tr✐❡s ♥♦ ❧♦♥❣❡r ❤♦❧❞✳ ❲❡ st❛rt ❜② r❡✈✐s✐t✐♥❣ t❤❡ ♠♦❞❡❧ t♦ ❛♥❛❧②s❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❝❧❛ss✐❝❛❧ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✇❛✈❡s ❝♦♠✐♥❣ ❢r♦♠ t❤✐s ♠♦❞✐✜❡❞ ❡❧❡❝tr♦❞②♥❛♠✐❝s ✇❤❡♥ ♣r♦♣❛❣❛t✐♥❣ ✐♥ ❢r❡❡ s♣❛❝❡✱ ❡s✲ t❛❜❧✐s❤✐♥❣ ✐♠♣♦rt❛♥t r❡s✉❧ts ❢♦r ❢✉t✉r❡ ✉s❡ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ r❛❞✐❛t✐♦♥ ❝♦♥str❛✐♥❡❞ t♦ ♣r♦♣❛❣❛t❡ ✐♥s✐❞❡ ❤♦❧❧♦✇ ❝♦♥❞✉❝t♦r ✇❛✈❡❣✉✐❞❡s✱ ♦♣❡♥ ❛❧♦♥❣ z✳ ❚❤❡r❡✱ ✇❡ ✜♥❞ t❤❛t
❣❡♦♠❡tr✐❝❛❧ ❢❡❛t✉r❡s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ②✐❡❧❞ ❡✛❡❝ts t❤❛t ❝❛♥ str♦♥✲ ❣❧② ❞❡✈✐❛t❡ ❢r♦♠ t❤❡ st❛♥❞❛r❞ r❡s✉❧ts✱ ❞❡s♣✐t❡ t❤❡ s♠❛❧❧♥❡ss ♦❢ t❤❡ ❜r❡❛❦✐♥❣ ♣❛r❛♠❡t❡r ✐ts❡❧❢✳ P❛rt✐❝✉❧❛r❧②✱ ✐❢ (kAF)µ ♣♦✐♥ts ❛❧♦♥❣ x ✭♦r y✮ ❞✐r❡❝t✐♦♥✱ ✇✐t❤ (kAF)0 = 0✱ ♦♥❧②
tr❛♥s✈❡rs❡ ❡❧❡❝tr✐❝ ♠♦❞❡s✱ ✇✐t❤ Ez = 0✱ s❤♦✉❧❞ ❜❡ ♦❜s❡r✈❡❞ ♣r♦♣❛❣❛t✐♥❣ t❤r♦✉❣❤♦✉t t❤❡ ❣✉✐❞❡✱ ✇❤❡r❡❛s ❛❧❧ t❤❡ tr❛♥s✈❡rs❡ ♠❛❣♥❡t✐❝✱Bz = 0✱ s❤♦✉❧❞ ❜❡ ❛❜s❡♥t✳ ❙✉❝❤ ❛ str♦♥❣ ♠♦❞❡ s✉♣♣r❡ss✐♦♥ ♠❛❦❡s ✇❛✈❡❣✉✐❞❡s q✉✐t❡ s✉✐t❛❜❧❡ t♦ ♣r♦❜❡ t❤❡s❡ s②♠♠❡tr② ✈✐♦❧❛t✐✲ ♦♥s ✉s✐♥❣ ❛ s✐♠♣❧❡ ❛♥❞ ❡❛s✐❧② r❡♣r♦❞✉❝✐❜❧❡ ❛♣♣❛r❛t✉s✳ ◆❡✈❡rt❤❡❧❡ss✱ ❜② r❡❝❛❧❧✐♥❣ t❤❡ ♥♦♥✲♦❜s❡r✈❛t✐♦♥ ♦❢ s✉❝❤ ♠❛❝r♦s❝♦♣✐❝ ❡✛❡❝ts ✐♥ t❤✐s ✇✐❞❡❧② ✉s❡❞ ❛♣♣❛r❛t✉s✱ ♦✉r r❡s✉❧ts ✐♥❡✈✐t❛❜❧② ✇✐t♥❡ss ❛❣❛✐♥st t❤❡s❡ ❝❧❛✐♠❡❞ ✈✐♦❧❛t✐♦♥s❀ ♦t❤❡r✇✐s❡✱ ❛❧❧ ♦❢ t❤❡s❡ ❡①♣❡r✐♠❡♥ts ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥ ♣❡r❢♦r♠❡❞ ✐♥ r❡❢❡r❡♥❝❡ ❢r❛♠❡s ✇✐t❤ (kAF)0 6= 0✳ ■❢ t❤✐s ✐s t❤❡ ❝❛s❡✱
❢r❛♠❡s ✇❤❡r❡ (kAF)0 = 0 ✇♦✉❧❞ ✉♥❞♦✉❜t❡❞❧② ❞❡t❡❝t t❤♦s❡ ❧❛r❣❡ ❡✛❡❝ts ✐♥ t❤❡ ❡❧❡❝tr✐❝
❛♥❞ ♠❛❣♥❡t✐❝ ✜❡❧❞s ❛ss♦❝✐❛t❡❞ t♦ t❤❡s❡ ❝♦♥✜♥❡❞ ✇❛✈❡s✳
❆ ◆❛t✉r❡③❛ é ❢❛s❝✐♥❛♥t❡ ❡ ✐st♦ ❝♦st✉♠❛ s❡r r❡✢❡t✐❞♦ ♥♦s ♠♦❞❡❧♦s q✉❡ ❝r✐❛♠♦s ♣❛r❛ ❞❡s❝r❡✈ê✲❧❛✳ ❆ ❘❡❧❛t✐✈✐❞❛❞❡ ❊s♣❡❝✐❛❧ ✖ q✉❡ ❝❡rt❛♠❡♥t❡ ❝♦rr♦❜♦r❛ ❡st❡ ♣♦♥t♦ ❞❡ ✈✐st❛ ✖✱ ❞❡✈✐❞♦ ❛ ❊✐♥st❡✐♥ ❡♠ ✶✾✵✺✱ tr♦✉①❡ ❝♦♥s✐❣♦ ♥ã♦ ❛♣❡♥❛s ♦ ❛❜❛♥❞♦♥♦ ❞♦ ét❡r ❡ ❞❡ ✉♠ r❡❢❡r❡♥❝✐❛❧ ❛❜s♦❧✉t♦✱ ♠❛s t❛♠❜é♠ r❡ss❛❧t♦✉ ♦ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ s✐♠❡tr✐❛ s♦❜r❡ ❛s ❧❡✐s ❞❛ ◆❛t✉r❡③❛✱ ❝♦♠♦ ❡✈✐❞❡♥❝✐❛❞♦ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ❊s♣❡❝✐❛❧ ❬✶❪✱ q✉❡ ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ❝♦♠♦✿
✏❆s ❧❡✐s ❞❛ ◆❛t✉r❡③❛ ❞❡✈❡♠ s❡r ❛s ♠❡s♠❛s ♣❛r❛ q✉❛✐sq✉❡r r❡❢❡r❡♥❝✐❛✐s ✐♥❡r❝✐❛✐s ❝♦♥❡❝t❛❞♦s ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③✑✱
♦ q✉❛❧ r❡♣♦✉s❛ s♦❜r❡ ❛ ❤✐♣ót❡s❡ ✭❛té ❤♦❥❡ ❡♠ ót✐♠♦ ❛❝♦r❞♦ ❝♦♠ ❛ r❡❛❧✐❞❛❞❡✦✮ ❞❡ ✐s♦tr♦♣✐❛ ❡ ❤♦♠♦❣❡♥❡✐❞❛❞❡ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦✳ ❚r❛♥s♣♦♥❞♦ ❡st❡ ♣r✐♥❝í♣✐♦ ❞❡ s✐♠❡tr✐❛ ♣❛r❛ ♦✉tr❛ ❧✐♥❣✉❛❣❡♠✿ s❡ ❢♦r♠✉❧❛r♠♦s ❛s ❧❡✐s ❢ís✐❝❛s ❛tr❛✈és ❞❡ ✉♠❛ ❆çã♦ q✉❡ ❛s ❞❡s❝r❡✈❡✱ ❡♥tã♦ ❡st❛♠♦s ❞✐③❡♥❞♦ q✉❡ ❡st❛ ❆çã♦ ❞❡✈❡ s❡r ✐♥✈❛r✐❛♥t❡ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③❀ s❡ ♦ ❡♥❢♦q✉❡ ❡stá ♥❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦✱ ❡st❛s ❞❡✈❡♠ s❡r ❝♦✈❛r✐❛♥t❡s s♦❜ ▲♦r❡♥t③✳ ❉❡ t♦❞♦ ♠♦❞♦✱ ❛ ❢ís✐❝❛ q✉❡ ❞❡s❝r❡✈❡ ❛ ◆❛t✉r❡③❛ ✭r❡❧❛t✐✈íst✐❝❛✮ ❞❡✈❡ ❡①✐❜✐r t❛❧ s✐♠❡tr✐❛✳ P♦st❡r✐♦r♠❡♥t❡✱ ❡♠ ✶✾✶✺✱ ❊♠♠② ◆♦❡t❤❡r✱ ❛♦ ♣r♦✈❛r ♦ t❡♦r❡♠❛ q✉❡ ❝❛rr❡❣❛ s❡✉ ♥♦♠❡✱ ♥♦s ♣❡r♠✐t✐✉ ♣r❡❝✐s❛r ♦ ❡❢❡✐t♦ ❞❡ s✐♠❡tr✐❛s ❝♦♥tí♥✉❛s s♦❜r❡ s✐st❡♠❛s ❢ís✐❝♦s✿ P❛r❛ ❝❛❞❛ s✐♠❡tr✐❛ ❝♦♥tí♥✉❛ ❞❛ ✭❆çã♦ ❞❛✮ t❡♦r✐❛✱ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ q✉❛♥t✐❞❛❞❡ ✭✏❝❛r❣❛✑✮ ❝♦♥s❡r✈❛❞❛ t❡♠♣♦r❛❧♠❡♥t❡✳ P♦r ❡①❡♠♣❧♦✱ s❡ ✉♠ s✐st❡♠❛ ♠❡❝â♥✐❝♦ ❡①✐❜❡ ✐♥✲ ✈❛r✐â♥❝✐❛ r♦t❛❝✐♦♥❛❧✱ ❣❛r❛♥t✐♠♦s ❛ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r❀ ♥❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ q✉â♥t✐❝❛ ✭◗❊❉✱ ❞♦ ✐♥❣❧ês ◗✉❛♥t✉♠ ❊❧❡❝tr♦❞②♥❛♠✐❝s✮✱ ❛ s✐♠❡tr✐❛ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡ r❡s✉❧t❛ ♥❛ ❝♦♥s❡r✈❛çã♦ ♠✐❝r♦s❝ó♣✐❝❛ ❞❛ ❝❛r❣❛ ❡❧étr✐❝❛❀ ❞❡♥tr❡ ♦✉tr♦s✳ ◆♦ ❡♥✲ t❛♥t♦✱ é ❝♦♠ ❛ ✐❞é✐❛ ❝❡♥tr❛❧ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ❊s♣❡❝✐❛❧ q✉❡ ♦s ❛s♣❡❝t♦s s✐♠étr✐❝♦s ❞❡ ✉♠
♠♦❞❡❧♦ ❞❡✐①❛♠ ❞❡ ❝♦♥t❡♠♣❧❛r ❛♣❡♥❛s ♦ ♣❛♣❡❧ ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ s✐st❡♠❛s ❢ís✐❝♦s ❡♠ ❣❡r❛❧ ♣❛r❛ s❡ t♦r♥❛r❡♠ ✐♥❣r❡❞✐❡♥t❡s ❢✉♥❞❛♠❡♥t❛✐s ♥❛ ♣ró♣r✐❛ ❝♦♥str✉çã♦ ❞♦s ♠♦❞❡❧♦s q✉❡ ✈✐s❛♠ ❞❡s❝r❡✈❡r ❛ ❢ís✐❝❛ ❢✉♥❞❛♠❡♥t❛❧✱ ♣♦✐s✱ ♣♦r s❡r ♦ Pr✐♥❝í♣✐♦ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ❛❧❣♦ tã♦ ♣♦❞❡r♦s♦ ❡ ❣❡r❛❧✱ ❡st❡ é t♦♠❛❞♦ ❝♦♠♦ r❡q✉❡r✐♠❡♥t♦ ❜ás✐❝♦ ♥❛ ❝♦♥tr✉çã♦ ❞❛ ❆çã♦ ❞❛ t❡♦r✐❛✱ ❥✉♥t♦ ❝♦♠ ♦✉tr❛s ✐♠♣♦s✐çõ❡s ❢❡♥♦♠❡♥♦❧ó❣✐❝❛s ✭❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❝❡rt❛s s✐♠❡tr✐❛s ❞✐s❝r❡t❛s✮✳ ❉❡s❞❡ ❛ ❝r✐❛çã♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ❘❡❧❛t✐✈íst✐❝❛ ♣♦r ❉✐r❛❝ ❡ ♦ ✐♥í❝✐♦ ❞❛ ❞❡s❝r✐çã♦ ❞♦ ♠✉♥❞♦ ❞❛s ♣❛rtí❝✉❧❛s ❡❧❡♠❡♥t❛r❡s ❡ s✉❛s ✐♥t❡r❛çõ❡s ✈✐❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❡ ❈❛♠♣♦s ✭❚◗❈✮ ❛té ♦s ❞✐❛s ❛t✉❛✐s✱ ❛ ✐❞é✐❛ ❞❡ q✉❡ ❛s ❧❡✐s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❋ís✐❝❛ ❞❡✈❡♠ s❡r s✐♠étr✐❝❛s s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ t♦r♥♦✉✲s❡ ✉♠ ♣✐❧❛r ❢✉♥❞❛✲ ♠❡♥t❛❧ s♦❜r❡ ♦ q✉❛❧ r❡♣♦✉s❛ ♥♦ss❛ ❢ís✐❝❛✳ ❈♦♠♦ ✈❡r❡♠♦s ❞✐✈❡rs❛s ✈❡③❡s ♣♦st❡r✐♦r♠❡♥t❡✱ ❡st❡ ♣✐❧❛r ✈❡♠ r❡s✐st✐♥❞♦ ❛♦ t❡♠♣♦ ✭❡ às ♥♦✈❛s ❣❡r❛çõ❡s ❞❡ ❢ís✐❝♦s✦✮ s❡♠ ❡①✐❜✐r ♦ ♠❡♥♦r s✐♥❛❧ ❞❡ ❞❡s❣❛st❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡♥❝♦♥tr❛r r❛❝❤❛❞✉r❛s ♥❡st❡ ♣✐❧❛r ♣♦❞❡r✐❛ ✈✐r ❛ s❡r ❛❧❣♦ ✐♥❡s♣❡r❛❞❛♠❡♥t❡ ❢r✉tí❢❡r♦✦
❉❛ ♠❛♥❡✐r❛ ❝♦♠♦ ❛ ✈❡♠♦s ❛t✉❛❧♠❡♥t❡✱ t♦❞❛ ◆❛t✉r❡③❛ ♣♦❞❡ s❡r ❜❡♠ ❞❡s❝r✐t❛ s❡♣❛r❛♥❞♦✲s❡ ❛s ✐♥t❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❡♠ ❊❧❡tr♦♠❛❣♥ét✐❝❛✱ ◆✉❝❧❡❛r❡s ❋♦rt❡ ❡ ❋r❛❝❛✱ ❡ ●r❛✈✐t❛❝✐♦♥❛❧✳ ❆s três ♣r✐♠❡✐r❛s sã♦ ❞❡s❝r✐t❛s ❞❡ ♠❛♥❡✐r❛ ❝♦♥s✐st❡♥t❡ ❛tr❛✈és ❞❡ ✉♠ ú♥✐❝♦ ♠♦❞❡❧♦✱ ❝❤❛♠❛❞♦ ❞❡ ▼♦❞❡❧♦ P❛❞rã♦ ✭▼P✮ ❞❛s P❛rtí❝✉❧❛s ❊❧❡♠❡♥t❛r❡s✱ ❥á ❛ ú❧t✐♠❛ é ♠✉✐t♦ ❜❡♠ ❞❡s❝r✐t❛ ♣❡❧❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ❞❡ ❊✐♥st❡✐♥✳ ❉❛❞♦s ❛str♦❢ís✐❝♦s ✐♥✲ ❞✐❝❛♥❞♦ ❛ ❡①♣❛♥sã♦ ❞♦ ❯♥✐✈❡rs♦ ❡ ❛q✉❡❧❡s r❡❧❛❝✐♦♥❛❞♦s à ❘❛❞✐❛çã♦ ❈ós♠✐❝❛ ❞❡ ❋✉♥❞♦✱ ❛❧✐❛❞♦s ❛♦s ❛t✉❛✐s ♠♦❞❡❧♦s ❝♦s♠♦❧ó❣✐❝♦s✱ s✉❣❡r❡♠ q✉❡ t♦❞♦ ♥♦ss♦ ❯♥✐✈❡rs♦ ✭♦ q✉❡ ✐♥✲ ❝❧✉✐ ♦ ♣ró♣r✐♦ ❡s♣❛ç♦✲t❡♠♣♦✮ t❡✈❡ ❝♦♠❡ç♦ ❡♠ ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ✐♥✐❝✐❛❧✱ ♦ ❇✐❣ ❇❛♥❣ ❬✷❪✳ ❉❡❝♦rr❡♥t❡ ❞✐st♦✱ ❝♦♥❥❡❝t✉r❛✲s❡ q✉❡ ❡♠ ✐♥st❛♥t❡s ✐♠❡❞✐❛t❛♠❡♥t❡ ♣♦st❡r✐♦r❡s ❛♦ ✐♥í❝✐♦ ❞♦ ❈♦s♠♦s✱ ♥❛ ❝❤❛♠❛❞❛ ❡r❛ ❞❡ P❧❛♥❝❦✶✱ ❡ ❞❡✈✐❞♦ à ❛❧tíss✐♠❛ ❡♥❡r❣✐❛ ✭∼ 1019●❡❱✮✱
❡❢❡✐t♦s q✉â♥t✐❝♦s ❞❛ ❣r❛✈✐❞❛❞❡ t♦r♥❛♠✲s❡ ❢♦rt❡s ❡ ❛ tr❛❞✐❝✐♦♥❛❧ ❞❡s❝r✐çã♦ ❞♦ ♠✉♥❞♦ s✉❜❛tô♠✐❝♦ ✈✐❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❡ ❈❛♠♣♦s t♦r♥❛✲s❡ ✐♥✈á❧✐❞❛✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ♥❡❝❡ss✐✲ ❞❛❞❡ ❞❡ ✉♠❛ ❞❡s❝r✐çã♦ q✉â♥t✐❝❛ ❞❛ ●r❛✈✐t❛çã♦ t♦r♥❛✲s❡ ❡ss❡♥❝✐❛❧✳ ◆❡st❛ ❡s❝❛❧❛✱ ❡st❛ ❞❡s❝r✐çã♦ ❡♥✈♦❧✈❡r✐❛ ✉♠ ♠♦❞❡❧♦ ✉♥✐✜❝❛❞♦ ❞❛s ✭❛t✉❛✐s✮ q✉❛tr♦ ✐♥t❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s
✶❆ ❡r❛ ❞❡ P❧❛♥❝❦ ❡stá ❛ss♦❝✐❛❞❛ ❛ ❡s❝❛❧❛ ❞❡ ♠❡s♠♦ ♥♦♠❡✱ ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r ❣r❛♥❞❡③❛s ❝♦♥tr✉✐❞❛s
❛ ♣❛rt✐r ❞❡ ❝❡rt❛s ❝♦♥st❛♥t❡s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ◆❛t✉r❡③❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ❊♥❡r❣✐❛ ❞❡ P❧❛♥❝❦✱ EP❧❛♥❝❦ = p~c5/G∼1019●❡❱❀ ❝♦♠♣r✐♠❡♥t♦ ❞❡ P❧❛♥❝❦✱
ℓP❧❛♥❝❦ = p~G/c3∼10−33❝♠❀ t❡♠♣♦ ❞❡ P❧❛❝❦✱tP❧❛♥❝❦=p~G/c5∼10−43s❀ ❡ t❡♠♣❡r❛t✉r❛ ❞❡ P❧❛♥❝❦✱
TP❧❛♥❝❦=p~c5/Gk2
B∼10
32❑✱ s❡♥❞♦ G ❛ ❝♦♥st❛♥t❡ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡kB ❛ ❝♦♥st❛♥t❡ ❞❡ ❇♦❧t③♠❛♥♥✳
✭q✉❡ t❡r✐❛♠ s❡ s❡♣❛r❛❞♦ ❝♦♥❢♦r♠❡ ❛ t❡♠♣❡r❛t✉r❛ ✭❡♥❡r❣✐❛✮ ❞♦ ❯♥✐✈❡rs♦ ❢♦✐ ❞✐♠✐♥✉✐♥❞♦ ❛té ❛ ♦❜s❡r✈❛❞❛ ❤♦❥❡✱ ❞❛ ♦r❞❡♠ ❞❡ 2,7❑ ✭10−4❡❱✮✱ ❝♦♠♦ ✈✐st♦ ♥❛ ❋✐❣✉r❛ ✶✮✳
❋✐❣✉r❛ ✶✿ ❆ ✏❤✐stór✐❛✑ ❞♦ ❯♥✐✈❡rs♦ s❡❣✉♥❞♦ ✉♠ ❢ís✐❝♦ ❞❡ ♣❛rtí❝✉❧❛s ❡ ❝❛♠♣♦s✿ ❙❡✉ ❝♦♠❡ç♦ ❝♦♠ ❛ s✐♥❣✉❧❛r✐❞❛❞❡ ✐♥✐❝✐❛❧✱ ❞❡s❝r✐t♦ ♣♦r ✉♠❛ ✐♥t❡r❛çã♦ ✉♥✐✜❝❛❞❛✱ ❡✱ ❝♦♥❢♦r♠❡ ❛ t❡♠♣❡r❛✲ t✉r❛ ✭❡♥❡r❣✐❛✮ ❞♦ ❯♥✐✈❡rs♦ ❞✐♠✐♥✉✐✱ ❡st❛ ✐♥t❡r❛çã♦ ✈❛✐ s❡ s❡♣❛r❛♥❞♦ ❛té ❝❤❡❣❛r às q✉❛tr♦ q✉❡ ♦❜s❡r✈❛♠♦s ❤♦❥❡ ✖ ✐♥t❡r❛çõ❡s ●r❛✈✐t❛❝✐♦♥❛❧✱ ♥✉❝❧❡❛r❡s ❋♦rt❡ ❡ ❋r❛❝❛✱ ❡ ❊❧❡tr♦♠❛❣♥ét✐❝❛✳
❏á ❤á ❛❧❣✉♠❛s ❞é❝❛❞❛s t❡♠ ❤❛✈✐❞♦ ✉♠ ❡♥♦r♠❡ ❡s❢♦rç♦ ♣❛r❛ s❡ ♦❜t❡r t❛❧ ❞❡s❝r✐çã♦ ✉♥✐✜❝❛❞❛ ❡✱ ❞❡♥tr❡ ❛s ✈ár✐❛s ♣r♦♣♦st❛s✱ ❛s t❡♦r✐❛s ❞❡ ❈♦r❞❛s ♠♦str❛♠✲s❡ ❝♦♠♦ ❛s ♠❛✐s ♣r♦♠✐ss♦r❛s✳ ◆♦ ❡♥t❛♥t♦✱ t♦❞❛s ❝♦♠♣❛rt✐❧❤❛♠ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❢❛❧t❛ ❞❡ s✉♣♦rt❡ ❡①♣❡r✐✲ ♠❡♥t❛❧✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡✱ ❛ ♣r✐♥❝í♣✐♦✱ só ♣♦❞❡r✐❛♠ s❡r t❡st❛❞❛s ❡♠ ❡s❝❛❧❛s ❞❡ ❡♥❡r❣✐❛ ❛❧tíss✐♠❛s✱ ✐♥❛❝❡ssí✈❡✐s ❛t✉❛❧♠❡♥t❡ ✭♦✉ ❡♠ ❢✉t✉r♦ r❛③♦á✈❡❧✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡s♣❡r❛✲s❡
q✉❡✱ ❡❢❡✐t♦s r❡♠✐♥✐s❝❡♥t❡s ❞❛ ✭❞❡s❝♦♥❤❡❝✐❞❛✮ ❢ís✐❝❛ ❞❛ ❡s❝❛❧❛ ❞❡ P❧❛♥❝❦ ♣♦ss❛♠ s❡ ♠♦s✲ tr❛r ♣r❡s❡♥t❡s ♥❛ ◆❛t✉r❡③❛ q✉❡ t❡♠♦s ❛❝❡ss♦ ❝♦♠♦ ♠✐♥ús❝✉❧♦s ❞❡s✈✐♦s ♥❛s ❧❡✐s q✉❡ ❤♦❥❡ ✐♠❛❣✐♥❛♠♦s s❡r❡♠ ❡①❛t❛s✳ ▼❡s♠♦ s❡♥❞♦ ✈á❧✐❞❛ ❝♦♠ ✉♠ ❛❧t♦ ♥í✈❡❧ ❞❡ ♣r❡❝✐sã♦ ✭❞❛í✱ ❣❡r❛❧♠❡♥t❡ ❛ ❛❞♠✐t✐♠♦s ❝♦♠♦ ❡①❛t❛✮✱ ❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ♣♦❞❡r✐❛ s❡r ✉♠❛ ❞❡st❛s❀ ♦✉tr❛ ♣♦ss✐❜✐❧✐❞❛❞❡ s❡r✐❛ ❛ s✐♠❡tr✐❛ ❈P❚✷ q✉❡✱ ♣❛r❛ ❚◗❈s ♣♦♥t✉❛✐s✱ ❡st❛❜❡❧❡❝❡✲s❡ s♦❜ ❢♦rt❡s ❜❛s❡s t❡ór✐❝❛s ❡ ❡①♣❡r✐♠❡♥t❛✐s✱ ♠❛s q✉❡ ♥ã♦ é ❜❡♠ ❝♦♠♣r❡❡♥❞✐❞♦ ♣❛r❛ ♦❜❥❡t♦s ❡①t❡♥s♦s✱ ❝♦♠♦ ❝♦r❞❛s✳ P♦rt❛♥t♦✱ ❡st✉❞❛r ♠♦❞❡❧♦s ✭❞❡ ❢❛t♦✱ ❢❡♥♦♠❡♥♦❧ó❣✐❝♦s✮ q✉❡ ✐♥✲ ❝♦r♣♦r❛♠ ❛ ♣r❡s❡♥ç❛ ❞❛ ✈✐♦❧❛çã♦ ❞❛s s✐♠❡tr✐❛s ❞❡ ▲♦r❡♥t③ ❡✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ❈P❚ ♥♦s ♣❡r♠✐t❡ ✉♠❛ ❞❡s❝r✐çã♦ q✉❛♥t✐t❛t✐✈❛ ❡ s✐st❡♠át✐❝❛ ❞❡ ❢❡♥ô♠❡♥♦s✱ ❞❡❝♦rr❡♥t❡s ❞❡ ♣♦ssí✲ ✈❡✐s ✈✐♦❧❛çõ❡s ❞❡ss❛s s✐♠❡tr✐❛s ♥❛ ◆❛t✉r❡③❛ q✉❡ ♦❜s❡r✈❛♠♦s ❛ ❡♥❡r❣✐❛s r❡❧❛t✐✈❛♠❡♥t❡ ❜❛✐①❛s ✭∼102●❡❱✮✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ♣♦ssí✈❡❧ ♦❜s❡r✈❛çã♦ ❞❡ss❛s ♣♦❞❡r✐❛ ♥♦s ❞✐③❡r ♠✉✐t♦
❛ r❡s♣❡✐t♦ ❞♦ ❝❛♠✐♥❤♦ ❛ s❡r t♦♠❛❞♦ ❡♠ r✉♠♦ ❛ ✉♠❛ ✏♥♦✈❛ ❢ís✐❝❛✑✱ ❛❧é♠ ❞❛ ❝♦♥✈❡♥❝✐♦♥❛❧✳ ◆❡st❛ ❜✉s❝❛ ♣♦r ♣❡q✉❡♥❛s ✈✐♦❧❛çõ❡s ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✱ ♠✉✐t♦s ♠♦❞❡❧♦s ❡❢❡t✐✈♦s ❥á ❢♦r❛♠ ♣r♦♣♦st♦s✳ ●r❛♥❞❡ ♣❛r❝❡❧❛ ❞❡st❡s sã♦ ♠♦❞❡❧♦s ♣✉r❛♠❡♥t❡ ❝✐♥❡♠át✐❝♦s✱ ♦♥❞❡ t❡♠♦s r❡❧❛çõ❡s ❞❡ ❞✐s♣❡rsã♦ q✉❡ ❝♦♥❞✉③❡♠ à ❡❢❡✐t♦s q✉❡ ♥ã♦ r❡s♣❡✐t❛♠ ❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✭♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❞❛ ❧✉③✱ ❡st❛s r❡❧❛çõ❡s ♣♦❞❡♠ ❛♣r❡s❡♥t❛r ❞❡♣❡♥❞ê♥❝✐❛ s♦❜ ❜♦♦sts ❡✴♦✉ r♦t❛çõ❡s✱ ✐♠♣❧✐❝❛♥❞♦ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ❞❡♣❡♥❞❡ ❞♦ r❡❢❡r❡♥❝✐❛❧✮ ❬✹❪✳ ◆♦ ❡♥t❛♥t♦✱ ❡ss❛s ♣r♦♣♦st❛s ♣♦✉❝♦ ♣♦❞❡♠ ❞✐③❡r s♦❜r❡ ♦s ❛s♣❡❝t♦s ❞✐♥â♠✐❝♦s q✉❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡t❡r♠✐♥❛♠ ❝♦♠♦ s❡ ❞á ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s✳ ❯♠❛ ♣r♦♣♦st❛ ❡♠ ❜❛✐①❛s ❡♥❡r❣✐❛s q✉❡ ❧❡✈❛ t❛♠❜é♠ ❡st❡ ❛s♣❡❝t♦ ❡♠ ❝♦♥t❛ é ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥❞✐❞♦ ✭▼P❊✮✱ q✉❡ ❡♥❣❧♦❜❛ ✭❡♠ s✉❛ ❆çã♦✮ t♦❞❛ ❢ís✐❝❛ ❢✉♥❞❛♠❡♥t❛❧ ❥á ❡st❛❜❡❧❡❝✐❞❛ ✭▼♦❞❡❧♦ P❛❞rã♦ ❡ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✮ ❡ ✐♥❝♦r♣♦r❛ t♦❞♦s ♣♦ssí✈❡✐s t❡r♠♦s q✉❡ ✈✐♦❧❛♠ ❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✭❡✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ❈P❚✮✱ s❡♥❞♦ ♦ ♠♦❞❡❧♦ ♠❛✐s ❣❡r❛❧ ♣❛r❛ ❛ ❞❡s❝r✐çã♦ ❞❡ ✉♠❛ ♣♦ssí✈❡❧ ✏❢ís✐❝❛ ♥♦✈❛✑ ♥❛ ❡s❝❛❧❛ ❞❡ ❡♥❡r❣✐❛ q✉❡ t❡♠♦s ❛❝❡ss♦ ✭♠❛✐♦r❡s ❞❡t❛❧❤❡s ♥♦ ❈❛♣ít✉❧♦ ✶✮✳
❊st❛ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ▼P❊ ❝♦♥tr✐❜✉✐ ♣❛r❛ q✉❡ ❡❧❡ s❡❥❛ ✉♠ ❞♦s ♠♦❞❡❧♦s ♠❛✐s ♣♦♣✉❧❛r❡s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ❡❢❡✐t♦s ❞❛ q✉❡❜r❛ ❞❡ ▲♦r❡♥t③ s♦❜r❡ ❛ ❢ís✐❝❛ ❝♦♥✈❡♥❝✐♦♥❛❧✱
✷❆ s✐♠❡tr✐❛ ❈P❚ é ❣❛r❛♥t✐❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❈P❚✱ q✉❡ ❡st❛❜❡❧❡❝❡ q✉❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❍❛♠✐❧t♦♥✐✲
❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜ ❛ ❛çã♦ ❝♦♥❥✉♥t❛ ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❈♦♥❥✉❣❛çã♦ ❞❡ ❈❛r❣❛ ✭❈✮✱ P❛r✐❞❛❞❡ ✭P✮ ❡ ❘❡✈❡r✲ sã♦ ❚❡♠♣♦r❛❧ ✭❚✮✱ s❡ ❡st❛ ✭✐✮ é ❧♦❝❛❧✱ ♣♦ss✉✐ ▲❛❣r❛♥❣✐❛♥❛ ❤❡r♠✐t✐❛♥❛ ❡ é ✐♥✈❛r✐❛♥t❡ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ♣ró♣r✐❛s ❞❡ ▲♦r❡♥t③ ❡ ✭✐✐✮ é q✉❛♥t✐③❛❞❛ ❝♦♠ ❝♦♠✉t❛❞♦r❡s ♣❛r❛ ❝❛♠♣♦s ❞❡ ❜ós♦♥s ❡ ❛♥t✐❝♦♠✉t❛❞♦r❡s ♣❛r❛ ❝❛♠♣♦s ❞❡ ❢ér♠✐♦♥s ❬✸❪✳
❞❡ ♠♦❞♦ q✉❡ ❛tr❛✈és ❞❡❧❡ ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ t❡st❡s ♥❛ ❜✉s❝❛ ❞❡st❛s ✈✐♦❧❛çõ❡s ❥á ❢♦r❛♠ ♣r♦♣♦st♦s✱ ❡ ♠✉✐t♦s ❥á ❛♥❛❧✐s❛❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✳ ◆♦ s❡t♦r ❢❡r♠✐ô♥✐❝♦ t❡♠♦s✱ ❡♠ ❡s♣❡❝✐❛❧✱ t❡st❡s ❝♦♠ ❡❧étr♦♥s✱ ♠ú♦♥s ❡ ♥❡✉tr✐♥♦s❀ ♥♦ s❡t♦r ❞❡ ❝❛❧✐❜r❡ ❛❜❡❧✐❛♥♦ ❛s ♣r♦♣♦st❛s s✉❣❡r❡♠ ❡❢❡✐t♦s ❞❡ ❜✐rr❡❢r✐♥❣ê♥❝✐❛✱ r❛❞✐❛çã♦ ❈❡r❡♥❦♦✈ ♥♦ ✈á❝✉♦✱ ❞❡❝❛✐♠❡♥t♦ ❞♦ ❢ót♦♥ ❡ ♣❡q✉❡♥❛s ♠♦❞✐✜❝❛çõ❡s ❡♠ ♦♥❞❛s ❝♦♥✜♥❛❞❛s❀ ❞❡♥tr❡ ♦✉tr♦s ❬✹❪ ✭❞❡t❛❧❤❡s s♦❜r❡ ❛s ♣r♦♣♦st❛s ❡①♣❡r✐♠❡♥t❛✐s✱ r❡s✉❧t❛❞♦s ❡ ❧✐t❡r❛t✉r❛ r❡❧❛❝✐♦♥❛❞❛ ♥❛ ❙❡çã♦ ✶✳✹✮✳
❆♣❡s❛r ❞♦s ✈ár✐♦s t❡st❡s ❞❡ ❛❧tíss✐♠❛ ♣r❡❝✐sã♦ ❥á r❡❛❧✐③❛❞♦s✱ ❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ♣❡r♠❛♥❡❝❡ só❧✐❞❛ ❡ s❡♠ q✉❛❧q✉❡r ❡✈✐❞ê♥❝✐❛ ❡①♣❡r✐♠❡♥t❛❧ ❝♦♥trár✐❛ ❛ ✐st♦✳ ❖s ✈❛❧♦r❡s ♠á①✐♠♦s ♣❛r❛ ♦s ♣❛râ♠❡tr♦s ❞❡ ✈✐♦❧❛çã♦ ❡st❛❜❡❧❡❝✐❞♦s ♥❡st❡s t❡st❡s ❡stã♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣r❡✈✐sã♦ t❡ór✐❝❛ ❞❡ q✉❡ ❡st❡s ❞❡✈❡♠ s❡r ♠✉✐t♦ ♣❡q✉❡♥♦s✱ s✉♣r✐♠✐❞♦s ♣♦r ♣♦tê♥❝✐❛s ❞♦ ✐♥✈❡rs♦ ❞❛ ♠❛ss❛ ❞❡ P❧❛♥❝❦ ❬✺❪✱ ❡①✐❣✐♥❞♦ ❡①♣❡r✐♠❡♥t♦s ❛✐♥❞❛ ♠❛✐s s❡♥sí✈❡✐s ♣❛r❛ ✉♠❛ ♣♦ssí✈❡❧ ❝♦♥✜r♠❛çã♦ ❞❡st❛s q✉❡❜r❛s ❞❡ s✐♠❡tr✐❛✳ ❚❡♥t❛♥❞♦ ❡✈✐t❛r ❡st❛ ❡①✐❣ê♥❝✐❛ ❡①♣❡r✐♠❡♥t❛❧✱ ♥♦ss♦ tr❛❜❛❧❤♦ é ♠♦t✐✈❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ q✉❡stã♦✿
✏❙❡r✐❛ ♣♦ssí✈❡❧ ❝♦♥❝❡❜❡r ❡①♣❡r✐♠❡♥t♦s ❡♠ q✉❡ ♦s ❡❢❡✐t♦s ❞❛ ✈✐♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ♣♦❞❡r✐❛♠ s❡r s✉✜❝✐❡♥t❡♠❡♥t❡ ❢♦rt❡s ❞❡ ♠♦❞♦ ❛ ♣❡r♠✐t✐r r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❝♦♥❝❧✉s✐✈♦s ❛ r❡s♣❡✐t♦❄✑
◆❡st❡ tr❛❜❛❧❤♦✱ ♣r❡t❡♥❞❡♠♦s ♠♦str❛r q✉❡ ❛ r❡s♣♦st❛ à ❡st❛ q✉❡stã♦ ♣♦❞❡ s❡r ❛✜r♠❛✲ t✐✈❛✳ ❊♠ ❡s♣❡❝í✜❝♦✱ ♠♦str❛r❡♠♦s ✐st♦ ❛♦ ❡st✉❞❛r ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❡❧❡tr♦♠❛❣✲ ♥ét✐❝❛s ❝♦♥✜♥❛❞❛s ❛ ❣✉✐❛s ❞❡ ♦♥❞❛ ✭r❡t❛♥❣✉❧❛r❡s✮✱ ❞❡s❝r✐t❛s ♣❡❧♦ s❡t♦r ❞❡ r❛❞✐❛çã♦ ❈P❚✲í♠♣❛r ❞❛ ✭♠í♥✐♠❛✮ ◗❊❉ ❡st❡♥❞✐❞❛✳ P❛r❛ ✐ss♦✱ ❡str✉t✉r❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ❝♦♠♦ s❡❣✉❡✿ ♥♦ ❈❛♣ít✉❧♦ ✶✱ ❛❜♦r❞❛r❡♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ❢✉♥❞❛♠❡♥t❛✐s ❛ r❡s♣❡✐t♦ ❞❛ ✈✐♦❧❛✲ çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✭♣♦r ❡①❡♠♣❧♦✱ s✉❛ ✐♠♣❧❡♠❡♥t❛çã♦✮ ❡ ❞♦ ▼P❊ ✭❝♦♠♦ s✉❛ ❝♦♥str✉çã♦✱ ♣♦ssí✈❡✐s ♦r✐❣❡♥s ❡ ♣r❡✈✐sõ❡s ❡①♣❡r✐♠❡♥t❛✐s✮✱ ✜♥❛❧✐③❛♥❞♦✲♦ ❝♦♠ ✉♠ r❡s✉♠♦ ❞♦ st❛t✉s ❡①♣❡r✐♠❡♥t❛❧ ❞♦ s❡t♦r q✉❡ ❡st✉❞❛r❡♠♦s✱ q✉❡ s❡rá ❛❜♦r❞❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ♦♥❞❡ ♥♦s ❝♦♥❝❡♥tr❛r❡♠♦s ❡♠ s✉❛s ❢✉♥❞❛♠❡♥t❛çõ❡s t❡ór✐❝❛s ❡ ❞❛r❡♠♦s ê♥❢❛s❡ ❡s♣❡❝✐❛❧ à s✉❛ ❛♣❧✐❝❛çã♦ ♥♦ ❡st✉❞♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ❧✐✈r❡s ♥♦ ✈á❝✉♦✱ ♦♥❞❡ ♦❜t❡r❡♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❝♦♥s✐stê♥❝✐❛ ❞♦ ♠♦❞❡❧♦ ❡ ❡st❛❜❡❧❡❝❡r❡♠♦s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❛r❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦ q✉❡ s❡rá ❡st✉❞❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✸✱ ♦♥❞❡ tr❛t❛r❡♠♦s ❞❛ ♣r♦♣❛❣❛çã♦ ❞❛ r❛❞✐❛çã♦ ♥♦ ✐♥t❡r✐♦r ❞❡ ❣✉✐❛s ❞❡ ♦♥❞❛ r❡t❛♥❣✉❧❛r❡s✱
❡st✉❞❛♥❞♦ ♦s ♠♦❞♦s ❚r❛♥s✈❡rs♦ ❊❧étr✐❝♦ ✭❚❊✮ ❡ ❚r❛♥s✈❡rs♦ ▼❛❣♥ét✐❝♦ ✭❚▼✮✱ ❡ ✈❡r❡♠♦s q✉❡✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♠❛❣♥✐t✉❞❡ ❞♦ ❝❛♠♣♦ ❞❡ ❢✉♥❞♦kAF ✭❛ss♦❝✐❛❞♦ à q✉❡❜r❛ ❞❡ s✐♠❡✲ tr✐❛s ♥❡st❡ s❡t♦r✮✱ ❡st❡ ✐♠♣❧✐❝❛ ❡♠ s✉♣r❡ssã♦ ♠❛❝r♦s❝ó♣✐❝❛ ❞❡ ✈ár✐♦s ♠♦❞♦s ❞♦ ❡s♣❡❝tr♦ ❞❛ ♦♥❞❛ ❡❧❡tr♦♠❛❣♥ét✐❝❛ ❛í ❝♦♥✜♥❛❞❛✱ t❡♥❞♦ s✐❞♦ ❡st❡s r❡s✉❧t❛❞♦s r❡❝❡♥t❡♠❡♥t❡ ♣✉❜❧✐✲ ❝❛❞♦s ♥♦ ❏♦✉r♥❛❧ ♦❢ ❍✐❣❤ ❊♥❡r❣② P❤②s✐❝s ❬✻❪✳ ❋✐♥❛❧✐③❛♥❞♦✱ ❛♣r❡s❡♥t❛♠♦s ❛s ❈♦♥❝❧✉sõ❡s ❡ P❡rs♣❡❝t✐✈❛s ❋✉t✉r❛s ❞❡st❡ tr❛❜❛❧❤♦✳ ■♥❝❧✉✐♠♦s✱ t❛♠❜é♠✱ ✉♠ ❆♣ê♥❞✐❝❡ ❞❡st✐♥❛❞♦ ❛ ♣r♦✈❛r ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ✉t✐❧✐③❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✸✱ ❛ r❡s♣❡✐t♦ ❞❛ ❢♦r♠❛ ❣❡r❛❧ ❞❡ ♦♥❞❛s ♣r♦♣❛❣❛♥t❡s ♥♦ ✐♥t❡r✐♦r ❞♦ ❣✉✐❛ ❞❡ ♦♥❞❛s✳
❆s♣❡❝t♦s ❋✉♥❞❛♠❡♥t❛✐s ❞♦ ▼P❊
❱✐s❛♥❞♦ ✉♠❛ ♠❛✐♦r ❝♦♠♣r❡❡♥sã♦ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥❞✐❞♦ ❡ ❞❛ ♣ró♣r✐❛ ✈✐✲ ♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✱ ♥❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❛ ❝♦♥str✉çã♦ ❞❡st❡ ♠♦❞❡❧♦✱ ❛❜♦r❞❛♥❞♦ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ ♣❡r❞❛ ❞❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ▲♦r❡♥t③✱ ❡ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s ❢✉♥❞❛♠❡♥t❛✐s✳ ❱❡r❡♠♦s ❛❧❣✉♠❛s ♣♦ssí✈❡✐s ♦r✐❣❡♥s ✭❡♠ ❡s❝❛❧❛s ❞❡ ❡♥❡r❣✐❛ s✉♣❡r✐♦r❡s✮ ♣❛r❛ ♦s t❡r♠♦s ❞❡ ✈✐♦❧❛çã♦ ❡ ✜♥❛❧✐③❛r❡♠♦s ❝♦♠❡♥t❛♥❞♦ s♦❜r❡ ❛❧❣✉♥s ❞♦s t❡st❡s ❥á r❡❛❧✐③❛❞♦s ❛tr❛✈és ❞❡ s✉❛s ♣r❡✈✐sõ❡s✱ ♦♥❞❡ t❛♠❜é♠ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ r❡s✉♠♦ ❞♦ ❛t✉❛❧ st❛t✉s ❡①♣❡r✐♠❡♥t❛❧ ❞♦ s❡t♦r ❞❡ r❛❞✐❛çã♦ ❈P❚✲í♠♣❛r ❞❛ ✭♠í♥✐♠❛✮ ◗❊❉ ❡s✲ t❡♥❞✐❞❛ ✖ ♦ q✉❛❧ s❡rá ♦ ❢♦❝♦ ❞♦s ♣ró①✐♠♦s ❈❛♣ít✉❧♦s ✖ ❝❤❡❣❛♥❞♦✱ ❛ss✐♠✱ à ♠♦t✐✈❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ✭❝♦♠❡♥t❛❞❛ ♥❛ ■♥tr♦❞✉çã♦✮✳
✶✳✶ ❱✐♦❧❛çã♦ ❞❛ ❙✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ❡ ♦ ▼♦❞❡❧♦ P❛✲
❞rã♦ ❊st❡♥❞✐❞♦
❯♠❛ ♣r♦♣♦st❛ ❛♠♣❧❛♠❡♥t❡ ❞✐❢✉♥❞✐❞❛✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛s♣❡❝t♦s ❞❛ ✈✐✲ ♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✭❡✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ❈P❚✮ é ♦ ❝❤❛♠❛❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥❞✐❞♦ ✭▼P❊✮✱ ♣r♦♣♦st♦ ♣♦r ❈♦❧❧❛❞❛② ❡ ❑♦st❡❧❡❝❦ý ❡♠ ✉♠ tr❛❜❛❧❤♦ ❞❡ ✶✾✾✽ ❬✼❪✱ ❝♦♠♦ ❝♦♥t✐♥✉❛çã♦ ♥❛t✉r❛❧ ❞❡ ✉♠❛ ✐♥✈❡st✐❣❛çã♦ s✐st❡♠át✐❝❛✱ r❡❛❧✐③❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ♣❡❧♦s ♠❡s♠♦s ❛✉t♦r❡s ❬✺❪✱ ❞❡ ✉♠❛ ❡①t❡♥sã♦ ♣❛r❛ ♦ ▼♦❞❡❧♦ P❛❞rã♦ q✉❡ ❧❡✈❛ss❡ ❡♠ ❝♦♥t❛ ✉♠❛ ♣♦ssí✈❡❧ ✈✐♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❈P❚ ✖ q✉❡✱ ❛♣❡s❛r ❞❡ ♥ã♦ s❛❜✐❞♦ ♥❛ é♣♦❝❛✱ ♥❡❝❡s✲ s❛r✐❛♠❡♥t❡ ✐♠♣❧✐❝❛ ♥❛ q✉❡❜r❛ ❞❡ ▲♦r❡♥t③ ❬✽✱ ✾❪ ✖✱ ♠♦t✐✈❛❞♦s ♣♦r tr❛❜❛❧❤♦s ❞♦ ✐♥í❝✐♦
❞❛ ❞é❝❛❞❛ ❞❡ ✾✵ ❬✶✵✱ ✶✶❪ q✉❡ ♣r♦♣✉♥❤❛♠ ♠❡❝❛♥✐s♠♦s ❡♠ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ❝❛♣❛③❡s ❞❡ ❝♦♥❞✉③✐r à q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❡ ❈P❚✱ ❛❝♦♠♣❛♥❤❛❞❛ ♣❡❧❛ q✉❡❜r❛ ♣❛r❝✐❛❧✶ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✳ P♦st❡r✐♦r♠❡♥t❡ ❬✶✷❪✱ ❡❢❡✐t♦s ❣r❛✈✐t❛❝✐♦♥❛✐s ❢♦r❛♠ ✐♥❝♦r♣♦r❛❞♦s ❛♦ ▼P❊✱ q✉❡ ❡♥tã♦ ♣❛ss♦✉ ❛ ❡♥❣❧♦❜❛r ❛ ●r❛✈✐t❛çã♦ ✈✐❛ ✉♠❛ ❡①t❡♥sã♦ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✷✱ ❜❡♠ ❝♦♠♦ q✉❡❜r❛s ♥❡st❡ s❡t♦r ❡s♣❡❝í✜❝♦✱ ❛❞✐❝✐♦♥❛♥❞♦✲s❡ t❛♠❜é♠ ❛❝♦♣❧❛♠❡♥t♦s ❡①tr❛s ❡♥tr❡ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡ ♦s ❝❛♠♣♦s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦✳ ❖ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥✲ ❞✐❞♦✱ ♣♦rt❛♥t♦✱ é ❡str✉t✉r❛❞♦ ♣❛r❛ ♠❛♥t❡r ❛ ❢ís✐❝❛ ✐♥✈❛r✐❛♥t❡ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❣❡r❛✐s ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♠❛s ❛ s✐♠❡tr✐❛ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③✱ ❛♣❧✐❝❛❞❛ ❧♦❝❛❧♠❡♥t❡✱ é ♣❡r❞✐❞❛✳
P♦r s❡r ❡♥❝❛r❛❞♦ ❝♦♠♦ ✉♠ ♠♦❞❡❧♦ ❡❢❡t✐✈♦✱ ✐st♦ é✱ ✉♠ ♠♦❞❡❧♦ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❞❡ ✉♠❛ t❡♦r✐❛ ♠❛✐s ❢✉♥❞❛♠❡♥t❛❧ ✭q✉❡ ❞❡s❝r❡✈❡r✐❛ ❛ ◆❛t✉r❡③❛ ❡♠ ❛❧t❛s ❡♥❡r❣✐❛s✱ ✐✳❡✱ ♥❛ ❡s❝❛❧❛ ❞❡ P❧❛♥❝❦✮✱ ♦ ▼P❊ é ❝♦♥str✉í❞♦ ❞❡ ♠♦❞♦ ❛ ❡♥❣❧♦❜❛r t♦❞♦ ♦ ▼P ❡ ❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✖ ❛❜r❛♥❣✐♥❞♦✱ ♣♦rt❛♥t♦✱ t♦❞❛ ❛ ❜❛s❡ ❞❛ ❢ís✐❝❛ ❢✉♥❞❛♠❡♥t❛❧ ❥á ❡st❛❜❡❧❡❝✐❞❛ ✖ ❝♦♠ ♦ ❛❞✐❝✐♦♥❛❧ ❞❡ ✐♥tr♦❞✉③✐r ❛s♣❡❝t♦s q✉❡ ❝♦♥❞✉③❡♠ à q✉❡❜r❛ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✳ ❉❡♥tr❡ s❡✉s ❣r❛♥❞❡s ❛tr❛t✐✈♦s✱ ♣♦❞❡♠♦s ❝✐t❛r ♦ ❢❛t♦ ❞❡ ♦ ▼P❊ ♠❛♥t❡r ✐♥t❛❝t❛s ❝❡rt❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❡s❡❥á✈❡✐s ❞♦ ▼P ❡ s❡r ❝♦♥str✉í❞♦ ❞❡ ♠♦❞♦ ❛ s❡r r❡❧❛t✐✈❛♠❡♥t❡ ❣❡r❛❧ ❡✱ ♣♦r ❝❛✉s❛ ❞✐st♦✱ s❡r ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦s ❞❡t❛❧❤❡s ❞❛✱ ❛té ❡♥tã♦ ❞❡s❝♦♥❤❡❝✐❞❛✱ ❢ís✐❝❛ ♠❛✐s ❢✉♥❞❛♠❡♥t❛❧ ✭♠❛✐♦r❡s ❞❡t❛❧❤❡s ♣♦st❡r✐♦r♠❡♥t❡✮✳ ❈♦♠♦ ♣❛rt❡ ❞❛ ♠♦t✐✈❛çã♦ ♦r✐❣✐♥❛❧✱ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ q✉❡❜r❛ ❞❡ ❈P❚ t❛♠❜é♠ é ✐♥❝❧✉í❞❛✱ s❡♥❞♦ q✉❡✱ ♣♦st❡r✐♦r à ❝r✐❛çã♦ ❞♦ ▼P❊✱ ●r❡❡♥❜❡r❣ ❬✽❪ ♣r♦✈♦✉ q✉❡ ❛ ✈✐♦❧❛çã♦ ❞❡ ❈P❚ ❡♠ t❡♦r✐❛s ✐♥t❡r❛❣❡♥t❡s ♥❡❝❡ss❛r✐✲ ❛♠❡♥t❡ ❝♦♥❞✉③ à ✈✐♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✭♠❛s ❛ r❡❝í♣r♦❝❛ ♥ã♦ é ✈❡r❞❛❞❡✐r❛✮✳ P♦rt❛♥t♦✱ t❡st❡s ❞❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ❈P❚ t❛♠❜é♠ s❡r✈❡♠ ❝♦♠♦ t❡st❡s ❞❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ▲♦r❡♥t③ ✖ ❞✐st♦✱ ♣♦❞❡✲s❡ ♦❜t❡r ❧✐♠✐t❡s ♥♦tá✈❡✐s s♦❜r❡ ♣❛râ♠❡tr♦s q✉❡ ❧❡✈❛♠ à q✉❡❜r❛ ❞❡ ▲♦r❡♥t③✱ ♣♦✐s ❛ s✐♠❡tr✐❛ ❈P❚✱ ❛ss✐♠ ❝♦♠♦ ❛ ❞❡ ▲♦r❡♥t③✱ ❛té ❤♦❥❡✱ ❡st❛❜❡❧❡❝❡✲s❡ ❝♦♠♦ ❡①❛t❛ ❝♦♠ ✉♠ ❛❧tíss✐♠♦ ❣r❛✉ ❞❡ ♣r❡❝✐sã♦✱ ♦ q✉❡ é ❝♦rr♦❜♦r❛❞♦ ♣♦r ❡①♣❡r✐♠❡♥t♦s ❡♥✈♦❧✲
✶❈♦♠♦ ✜❝❛rá ❡✈✐❞❡♥t❡ ♣♦st❡r✐♦r♠❡♥t❡✱ ❛ ✈✐♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✱ ❛ q✉❛❧ ♥♦s r❡❢❡r✐♠♦s s❡♠✲
♣r❡✱ é ❡♠ r❡❧❛çã♦ ❛ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ♣❛rtí❝✉❧❛✱ ❛q✉❡❧❛s q✉❡ ❛t✉❛♠ s♦❜r❡ ❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s ✖ ❛ s✐♠❡tr✐❛ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ♦❜s❡r✈❛❞♦r✱ ✐st♦ é✱ ❛ ✐♥✈❛r✐â♥❝✐❛ s♦❜ ❡s❝♦❧❤❛ ❞❡ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♣❡r♠❛♥❡❝❡ ✐♥t❛❝t❛✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ♣❛rtí❝✉❧❛ s❡rã♦ ❢r❡q✉❡♥t❡♠❡♥t❡ r❡❢❡r✐❞❛s ♣♦r ❛♣❡♥❛s tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ❝♦♥t❡①t♦ ♥ã♦ ♣❡r♠✐t✐rá ❛♠❜✐❣✉✐❞❛❞❡s✳
✷◆♦ ❝❛s♦✱ ❢♦✐ ❛❞♦t❛❞♦ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❝♦♠ ❣❡♦♠❡tr✐❛ ❞♦ t✐♣♦ ❘✐❡♠❛♥♥✲❈❛rt❛♥ ✭♣❡r♠✐t✐♥❞♦ ❡❢❡✐✲
t♦s ❞❡ t♦rsã♦ ♥ã♦✲♥✉❧❛✱ ♣♦r ❡①❡♠♣❧♦✮✱ q✉❡ ✐♥❝❧✉✐✱ ❝♦♠♦ ❝❛s♦ ❧✐♠✐t❡✱ ❛s ❣❡♦♠❡tr✐❛s ❞❡ ❘✐❡♠❛♥♥✱ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✱ ❡ ❞❡ ▼✐♥❦♦✇s❦✐✱ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ❊s♣❡❝✐❛❧✳
✈❡♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❝❡rt♦s t✐♣♦s ❞❡ ♠és♦♥s ♥❡✉tr♦s ❡ s✉❛s ❛♥t✐♣❛rtí❝✉❧❛s ❬✶✸❪✳ ❙❡♥❞♦ ❛♠❜❛s ❛s s✐♠❡tr✐❛s ♠✉✐t♦ ❜❡♠ ❡st❛❜❡❧❡❝✐❞❛s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✱ ♦s ♣❛râ♠❡tr♦s q✉❡ ❝♦♥tr♦❧❛♠ s✉❛s q✉❡❜r❛s ♥♦ ▼P❊ ❞❡✈❡♠ s❡r ♠✉✐t♦ ♣❡q✉❡♥♦s ❡✱ ❞❡ ❢❛t♦✱ s✉❣❡r❡✲s❡ q✉❡ s❡❥❛♠ s✉♣r✐♠✐❞♦s ♣♦r ♣♦tê♥❝✐❛s ❞♦ ✐♥✈❡rs♦ ❞❛ ♠❛ss❛ ❞❡ P❧❛♥❝❦ ❬✺❪✱ ❞❡ ♠♦❞♦ q✉❡ ✜❝❛ ❡♥tã♦ ❝❧❛r♦ q✉❡ ♦ ❛♣♣r♦❛❝❤ ✈✐s❛❞♦ é ❜✉s❝❛r ❛ ♦❜s❡r✈❛çã♦ ❞❡ ❞❡s✈✐♦s ♠✐♥ús❝✉❧♦s ❞❛ ❢ís✐❝❛ ♣❛❞rã♦✱ ✐st♦ é✱ ♣❡q✉❡♥♦s ❡❢❡✐t♦s✱ ✈✐♥❞♦s ❞❛ ❢ís✐❝❛ ❞❛ ❡s❝❛❧❛ ❞❡ P❧❛♥❝❦✱ ♣♦ss✐✈❡❧♠❡♥t❡ ❛❝❡ssí✈❡✐s ✈✐❛ ❡①♣❡r✐♠❡♥t♦s ❞❡ ❛❧tíss✐♠❛ ♣r❡❝✐sã♦✳
✶✳✷ ❈♦♥str✉✐♥❞♦ ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❊st❡♥❞✐❞♦
❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r♠♦s ❢♦r♠❛❧♠❡♥t❡ ♦ ▼P❊ ✭✐st♦ é✱ ❛ ❆çã♦ q✉❡ ♦ ❞❡s❝r❡✈❡ ❡ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s✮✱ ❞✐s❝✉t✐r❡♠♦s✱ ❞❡ ♠♦❞♦ ❛♣❡♥❛s ❡①♣♦s✐t✐✈♦✱ s♦❜r❡ s✉❛ ❝♦♥str✉çã♦ ✭s❡❣✉✐♥❞♦ ❞❡ ♣❡rt♦ ❛ ❧✐♥❤❛ ❛♣r❡s❡♥t❛❞❛ ❡♠ ❬✶✹❪✮✱ ♣♦✐s✱ s❡♥❞♦ ❛ ♣r✐♥❝✐♣❛❧ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ▼P❊ ♣❡r♠✐t✐r q✉❡ ✈✐♦❧❛çõ❡s ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ s❡❥❛♠ ✐♥❝♦r♣♦r❛❞❛s ❛♦ ▼P✱ ♥❛✲ t✉r❛❧♠❡♥t❡ s✉r❣❡ ✉♠❛ q✉❡stã♦ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛✿ ❝♦♠♦ ✐♠♣❧❡♠❡♥t❛r ✐st♦❄ ❆✐♥❞❛ ♣♦❞❡♠♦s ♥♦s ✐♥❞❛❣❛r s♦❜r❡ ♦✉tr♦ ❛s♣❡❝t♦ ♠❛✐s s✉t✐❧✿ ✈✐♦❧❛♠♦s ❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ❞❡ ♣❛rtí❝✉❧❛ ❛♣❡♥❛s ✭q✉❡❜r❛ ♣❛r❝✐❛❧✮ ♦✉ ❞❡ ♦❜s❡r✈❛❞♦r t❛♠❜é♠ ✭q✉❡❜r❛ ❡①♣❧í✲ ❝✐t❛✮❄ ❆ s❡❣✉✐r t❡♥t❛r❡♠♦s ❡s❝❧❛r❡❝❡r ❡st❛s q✉❡stõ❡s✳
◆♦t❛✿ ❆♣❡s❛r ❞❡ ♥ã♦ ❡①♣❧✐❝✐t❛r ❡st❡ ♣♦♥t♦✱ ❛ ❞✐s❝✉ssã♦ ❛ s❡❣✉✐r ✈✐s❛ ❛ ❝♦♥str✉çã♦ ❞♦ ♠♦❞❡❧♦ s❡♠ ❛ ✐♥❝❧✉sã♦ ❞❛ ❣r❛✈✐t❛çã♦✳ ❆ ❣❡♥❡r❛❧✐③❛çã♦ ♣♦❞❡ s❡r ❛r❣✉♠❡♥t❛❞❛✱ ❡✈✐t❛♥❞♦ ❝♦♠♣❧✐❝❛çõ❡s ❡ s❡♠ ❡♥tr❛r ❡♠ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✱ ❡♥t❡♥❞❡♥❞♦✲s❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ❝♦♠♦ s❡♥❞♦ ❛♣❧✐❝❛❞❛s ❧♦❝❛❧♠❡♥t❡✱ ❡♠ ✈✐③✐♥❤❛♥ç❛s ❞❡ ♣♦♥t♦s✱ ❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s s❡♥❞♦✱ ❛❣♦r❛✱ tr❛♥s❢♦r♠❛çõ❡s ❣❡r❛✐s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❬✶✷❪✳
❆ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ❡stá ✐♥tr✐s❡❝❛♠❡♥t❡ ❧✐❣❛❞❛ à ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥❛ ❞❡s❝r✐çã♦ ❞❛ ◆❛t✉r❡③❛✱ ❥á q✉❡ ❡st❡✱ ♣♦r s✐ só✱ ♥ã♦ ♣♦ss✉✐ q✉❛❧q✉❡r r❡❛❧✐❞❛❞❡ ❢ís✐❝❛✳ ❈♦♥s❡❣✉✐♠♦s ❡st❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛✱ ✐st♦ é✱ ❛ ❝♦✈❛r✐â♥❝✐❛ ❞❛s ❧❡✐s ❢ís✐❝❛s✱ ❛❞♦t❛♥❞♦ ✉♠❛ ❡str✉t✉r❛ ♣❛r❛ ♦ ❡s♣❛ç♦✲t❡♠♣♦ ✭▲♦r❡♥t③✐❛♥❛✱ ●❛❧✐❧❡❛♥❛✱ ❡t❝✳✸✮ ❡ r❡♣r❡✲ s❡♥t❛♥❞♦ q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s ❛tr❛✈és ❞❡ ♦❜❥❡t♦s ❣❡♦♠étr✐❝♦s✱ ❝♦♠♦ t❡♥s♦r❡s ❡ ❡s♣✐♥♦r❡s
✸◗✉❛♥❞♦ ❢❛❧❛♠♦s ✏❡str✉t✉r❛✑✱ ❡st❛♠♦s ❜❛s✐❝❛♠❡♥t❡ ♥♦s r❡❢❡r✐♥❞♦ ❛ ❡s♣❡❝✐✜❝❛r ❛ ♠❛♥❡✐r❛ ❞❡ s❡
✐♠♣❧❡♠❡♥t❛r ❛s tr❛♥s❢♦r♠❛çõ❡s q✉❡ ❝♦♥❡❝t❛♠ ❞✐❢❡r❡♥t❡s r❡❢❡r❡♥❝✐❛✐s✳
✖ s❡ s❛❜❡♠♦s ❝♦♠♦ ❡st❡s s❡ tr❛♥s❢♦r♠❛♠✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r q✉❛♥t✐❞❛❞❡s q✉❡ ♥ã♦ s❡♥t❡♠ ❡st❛s tr❛♥s❢♦r♠❛çõ❡s✱ ♦s ❡s❝❛❧❛r❡s✱ ❡ ✉t✐❧✐③á✲❧♦s ♣❛r❛ ❝♦♥str✉✐r ♥♦ss❛s t❡♦r✐❛s ❢ís✐❝❛s ❡♠ ✉♠❛ ❢♦r♠✉❧❛çã♦ ✈✐❛ ❆çã♦✳ ❈♦♠ ✐st♦ ❡♠ ♠❡♥t❡✱ ✈❛♠♦s ❝♦♠❡♥t❛r ❞✉❛s ♣♦s✲ s✐❜✐❧✐❞❛❞❡s ❞❡ ❝♦♠♦ s❡ q✉❡❜r❛r ❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ❬✶✺❪✱ s❡♥❞♦ ❛ s❡❣✉♥❞❛ ❛ ❛❞♦t❛❞❛ ♥♦ ▼P❊✿
✭✐✮ ❈♦♥s✐❞❡r❛r ✉♠❛ ✈✐♦❧❛çã♦ ❡①♣❧í❝✐t❛ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✱ ❞❡ ♠♦❞♦ q✉❡ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥ã♦ sã♦ ♠❛✐s ❝♦♥❡❝t❛❞♦s ✈✐❛ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③✱ ✐st♦ é✱ ❛ ❡str✉t✉r❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ♥ã♦ é ♠❛✐s ▲♦r❡♥t③✐❛♥❛✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ♣❛r❛ ♠❛♥t❡r ❛ ❢ís✐❝❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✉t✐❧✐③❛❞♦✱ ♦ ❡s♣❛ç♦✲ t❡♠♣♦ ❞❡✈❡ ♣♦ss✉✐r ✉♠❛ ♦✉tr❛ ❡str✉t✉r❛ ♦✉✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❞❡✈❡♠♦s ❜✉s❝❛r ✉♠ ❣r✉♣♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♠❛✐s ❣❡r❛✐s q✉❡ ❛s ❞❡ ▲♦r❡♥t③✱ s✉♣❡r❛♥❞♦ ❛ q✉❡❜r❛✱ ♠❛♥t❡♥❞♦ ❛ ❝♦✈❛r✐â♥❝✐❛ ♥❛ t❡♦r✐❛✳ ❊st❛ é ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ♣r❡s❡♥t❡ ♥❛ ❧✐t❡r❛t✉r❛ ✭♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❚❡♦r✐❛s ◗✉â♥t✐❝❛s ❞❡ ❈❛♠♣♦ ◆ã♦✲❈♦♠✉t❛t✐✈❛s ❬✶✻✱ ✶✼❪✮✱ ♠❛s✱ ❛♦ q✉❡ ♣❛r❡❝❡✱ s✉❛ ✐♥t❡r♣r❡t❛çã♦ ❛✐♥❞❛ ♥ã♦ é ❝♦♠♣❧❡t❛♠❡♥t❡ ❝❧❛r❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ às s✉❛s ♣r♦♣r✐❡❞❛❞❡s s♦❜ ❈P❚ ❬✶✹❪✳
✭✐✐✮ ❆❞♦t❛❞❛ ♣❡❧♦ ▼P❊ ❬✼✱ ✺✱ ✶✽❪✱ ❛ ✈✐♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✭❡✱ ♥♦ ❝❛s♦✱ ♣♦ss✐✲ ✈❡❧♠❡♥t❡ t❛♠❜é♠ ❞❡ ❈P❚✮ é ✐♠♣❧❡♠❡♥t❛❞❛ ♠❛♥t❡♥❞♦✲s❡ ❛ ❡str✉t✉r❛ ▲♦r❡♥t③✐❛♥❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ✖ ❞✐❢❡r❡♥t❡s r❡❢❡r❡♥❝✐❛✐s sã♦ ❝♦♥❡❝t❛❞♦s ✈✐❛ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ✖ ❡ ❞♦t❛♥❞♦✲s❡ ♦ ✈á❝✉♦ ❞❡ ✉♠❛ ❡str✉t✉r❛ ♥ã♦✲tr✐✈✐❛❧✳ ❇❛s✐❝❛♠❡♥t❡✱ ♦ q✉❡ ❛❝♦♥t❡❝❡ ❛❣♦r❛ é q✉❡ ♦ ✈á❝✉♦ é ♣❡r♠❡❛❞♦ ♣♦r ❝❛♠♣♦s ❞❡ ❢✉♥❞♦ ♥ã♦✲❞✐♥â♠✐❝♦s ✭✐✳❡✳✱ ❝❛♠♣♦s q✉❡ s❡❧❡❝✐♦♥❛♠ ❞✐r❡çõ❡s ✜①❛s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✮✱ q✉❡ ❛❝♦♣❧❛♠✲s❡ ❝♦♠ ❝❛♠♣♦s ❞✐♥â♠✐❝♦s ✭❝♦♠♦ ♦ ❝❛♠♣♦ ❞❡ r❛❞✐❛çã♦ Aµ✱ ❡s♣✐♥♦r✐❛❧ ψ✱ ❡t❝✳✮✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ❢ís✐❝❛ t♦r♥❛✲s❡ ❞❡♣❡♥❞❡♥t❡ ❞❡ r♦t❛çõ❡s ♦✉ ❜♦♦sts ❛♣❧✐❝❛❞♦s s♦❜r❡ ♦s ❝❛♠♣♦s ❞✐♥â♠✐❝♦s✳ ❊♠ ❧✐♥❤❛s ❣❡r❛✐s✱ ❛ ❢ís✐❝❛ ♠❛♥t❡♠ ❛ ✐♥✈❛r✐â♥❝✐❛ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ❞❡ ♦❜s❡r✈❛❞♦r ✭♠✉❞❛♥ç❛s ♥♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✮✱ ♠❛s ❛ ♣❡r❞❡ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ♣❛rtí❝✉❧❛s ✭♠✉❞❛♥ç❛s ✐♠♣❧❡♠❡♥t❛❞❛s s♦❜r❡ ❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s✱ ♣❛rtí❝✉❧❛s ♦✉ ❝❛♠♣♦s ❞✐♥â♠✐❝♦s✮✳
❆❣♦r❛ q✉❡ ❝♦♠♣r❡❡♥❞❡♠♦s ❝♦♠♦ s❡ ❞á ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ q✉❡❜r❛ ❞❛ s✐♠❡✲ tr✐❛ ❞❡ ▲♦r❡♥t③✱ ❡ ✈✐♠♦s q✉❡ ♥♦ ▼P❊ ❡st❛ s❡ ❞á ✈✐❛ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ♣❛rtí❝✉❧❛✱ ✈❛❧❡
❡①❡♠♣❧✐✜❝❛r ✉♠❛ s✐t✉❛çã♦✱ ♣r❡✈✐st❛ ♣❡❧♦ ♠♦❞❡❧♦✱ ♦♥❞❡ t❡♠♦s ❡❢❡✐t♦s ✐♥t❡r❡ss❛♥t❡s ❞❡st❛ q✉❡❜r❛✳ ❈♦♠♦ ✈❡r❡♠♦s✱ ♥♦ s❡t♦r ❞❡ r❛❞✐❛çã♦ ❞❛ ❆çã♦ ❞♦ ▼P❊✱ ❡①✐st❡ ✉♠ t❡r♠♦ q✉❡ ❣❡r❛ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❞♦ ❝❛♠♣♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ Fµν ❝♦♠ ✉♠ ✏✹✲✈❡t♦r✑ ❞❡ ❢✉♥❞♦ ❝♦♥s✲ t❛♥t❡✱ ❞❡ ♥❛t✉r❡③❛ t✐♣♦✲♠❛ss❛ t♦♣♦❧ó❣✐❝❛✱(kAF)µ✱ q✉❡ s❡❧❡❝✐♦♥❛ ✉♠❛ ❞✐r❡çã♦ ❡s♣❡❝í✜❝❛ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✱ q✉❡❜r❛♥❞♦ s✉❛ ✐s♦tr♦♣✐❛ ❡✱ ❞❛í✱ ❝♦♥❞✉③✐♥❞♦ à ♣❡r❞❛ ❞❛s s✐♠❡tr✐❛s ❞❡ ▲♦r❡♥t③ ✭❞❡ ♣❛rtí❝✉❧❛✮ s♦❜ r♦t❛çõ❡s ❡✴♦✉ ❜♦♦sts ❡ ❞❡ ❈P❚✳ ❯♠ ❞♦s ❡❢❡✐t♦s ❞✐st♦✱ ❡♠ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ❡sq✉❡♠❛t✐③❛❞♦ ♥❛s ❋✐❣✉r❛s ✶✳✶ ❡ ✶✳✷✱ é q✉❡✱ ❛♥❛❧✐s❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❛ss♦❝✐❛❞❛ à ♣r♦♣❛❣❛çã♦ ❞❛ r❛❞✐❛çã♦ ♥♦ ❡s♣❛ç♦ ❧✐✈r❡✱ ❛ ❢r❡q✉ê♥❝✐❛ w❞❛ ♦♥❞❛
❡❧❡tr♦♠❛❣♥ét✐❝❛ ♣♦ss✉✐ ❛❣♦r❛ ✉♠ t❡r♠♦ t✐♣♦ ♠❛ss❛ ❡①✐❜✐♥❞♦ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♠ˆkAF ·k ✭ké ♦ ✈❡t♦r ❞❡ ♦♥❞❛✮✱ ❞❡ ♠♦❞♦ q✉❡✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦✱ ❛ r❛❞✐❛çã♦
♣♦❞❡ ❛♣r❡s❡♥t❛r ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ✏♠❛ss✐✈♦✑ ♦✉ ♥ã♦✲♠❛ss✐✈♦ ❡✱ ❝♦♠ ✐ss♦✱ ♣♦❞❡ ❛♣r❡✲ s❡♥t❛r ✈❡❧♦❝✐❞❛❞❡s ♠❡♥♦r❡s q✉❡c✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❞✐r❡çã♦ ❞❡ ♣r♦♣❛❣❛çã♦✳ ❈❧❛r❛♠❡♥t❡✱
❛ ❢ís✐❝❛ ♦❜s❡r✈❛❞❛ t♦r♥❛✲s❡ ❞✐r❡❝✐♦♥❛❧♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡ ✭❋✐❣✉r❛ ✶✳✶✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t❛♠❜é♠ ✜❝❛ ❡✈✐❞❡♥t❡ q✉❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ♦❜s❡r✈❛❞♦r ♥ã♦ ❛❧t❡r❛♠ ❛ ❢ís✐❝❛ ♦❜s❡r✈❛❞❛ ✭❋✐❣✉r❛ ✶✳✷✮✳
❋✐❣✉r❛ ✶✳✶✿ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ P❛rtí❝✉❧❛✳ ◆♦t❡ q✉❡ ❛ ✏❢ís✐❝❛ ♠✉❞❛✑✱ ♣♦✐s ♦ â♥❣✉❧♦ ❞❡ ❛❝♦♣❧❛✲ ♠❡♥t♦θ ❡♥tr❡ ✈❡t♦r ❞❡ ♦♥❞❛k❡ ❝❛♠♣♦ ❞❡ ❢✉♥❞♦kAF é ❛❧t❡r❛❞♦ ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦✳
❊s♣❡❝✐✜❝❛❞♦ ❝♦♠♦ ✐♠♣❧❡♠❡♥t❛r ❛ ✈✐♦❧❛çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ ✖ ♠❛♥t❡♥❞♦ ❛ t❡♦r✐❛ ❝♦✈❛r✐❛♥t❡ ❞❡ ▲♦r❡♥t③ ❡ ❛❞♦t❛♥❞♦✲s❡ ✉♠ ✈á❝✉♦ ♥ã♦✲tr✐✈✐❛❧ ✖ ❛ ❆çã♦ ❞♦ ▼P❊✱ ❛ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s ❡❢❡t✐✈❛✱ t♦r♥❛ ♥❛t✉r❛❧ ❛ ✐♥❝♦r♣♦r❛çã♦ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❡ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ❝♦♠♦ ❝❛s♦s ❧✐♠✐t❡s ✭q✉❛♥❞♦ ♦s ❝❛♠♣♦s ❞❡ ❢✉♥❞♦ sã♦ t♦♠❛❞♦s ❝♦♠♦ ♥✉❧♦s ❡ ♦ ✈á❝✉♦ t♦r♥❛✲s❡ tr✐✈✐❛❧✮✳ ❆ss✐♠ s❡♥❞♦✱ ❛ ❆çã♦ S ❜❛s❡❛❞❛
