UNIVERSIDADE DE BRASÍLIA INSTITUTO DE FÍSICA TESE DE DOUTORADO EM FÍSICA

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❯◆■❱❊❘❙■❉❆❉❊ ❉❊ ❇❘❆❙❮▲■❆

■◆❙❚■❚❯❚❖ ❉❊ ❋❮❙■❈❆

❚❊❙❊ ❉❊ ❉❖❯❚❖❘❆❉❖ ❊▼ ❋❮❙■❈❆

▼♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉ ❡ ❙✐♠❡tr✐❛s✿ ❙♦❧✉çõ❡s

❆♥❛❧ít✐❝❛s ❡ ❉✐♥â♠✐❝❛ ❞❡ ❈❛♠♣♦s ❚ér♠✐❝♦s

P❆❯▲❖ ▼❆●❆▲❍➹❊❙ ▼❆❘❈■❆◆❖ ❉❆ ❘❖❈❍❆

❖❘■❊◆❚❆❉❖❘✿

❆❉❊▼■❘ ❊❯●❊◆■❖ ❉❊ ❙❆◆❚❆◆❆

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❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦✿ ▼♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉ ❡

❙✐♠❡tr✐❛s✿ ❙♦❧✉çõ❡s ❆♥❛❧ít✐❝❛s ❡ ❉✐♥â♠✐❝❛ ❞❡ ❈❛♠♣♦s

❚ér♠✐❝♦s

P❛✉❧♦ ▼✳ ▼✳ ❞❛ ❘♦❝❤❛

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❆❣r❛❞❡❝✐♠❡♥t♦s

❈♦♠❡ç♦ ❛❣r❛❞❡❝❡♥❞♦ ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r t♦❞♦ ♦ s✉♣♦rt❡ ❡ ❡♥t❡♥❞✐♠❡♥t♦ ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❥❡t♦✱ ✐♥❞♦ ❛❧é♠ ❞♦ ❡s♣❡r❛❞♦ ♣❛r❛ s❡✉ tr❛❜❛❧❤♦✳

❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦ ♠❡✉ ♣❛✐✱ ♣♦r t♦❞♦ ♦ s✉♣♦rt❡ ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❥❡t♦✱ ♥ã♦ só ♥♦ s❡♥t✐❞♦ ❛❝❛❞ê♠✐❝♦✱ ♠❛s t❛♠❜é♠ ♥♦ ❞✐❛✲❛✲❞✐❛✳ ❆ ♠✐♥❤❛ ♠ã❡✱ ♣♦r s❡r ♠ã❡ ❡ ❝✉♠♣r✐r ❡st❡ ♣❛♣❡❧ ❞❛ ♠❡❧❤♦r ♠❛♥❡✐r❛ q✉❡ s❡ ♣♦❞❡ ✐♠❛❣✐♥❛r✳ ❆♦ ♠❡✉ ✐r♠ã♦ ❏♦ã♦ P❡❞r♦✱ ♣❡❧❛ ❝♦♠♣❛♥❤✐❛ ❡♠ ✈ár✐❛s ♠❛❞r✉❣❛❞❛s✳ ❆ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ❡♠ ❡s♣❡❝✐❛❧ ♠❡✉s ❛✈ós✱ ♣♦r t♦❞♦ ♦ s✉♣♦rt❡ ❛♦ ❧♦♥❣♦ ❞♦s ❛♥♦s✳

◆ã♦ ♣♦ss♦ ❡sq✉❡❝❡r ❞♦s ❛♠✐❣♦s ❇r✉♥♦✱ ❉❛♥✐❡❧✱ ◆❛tá❧✐❛ ❡ ▲✉❝✐❛♥♦✱ ♣♦r ❡st❛r❡♠ ❧á ❞✉r❛♥t❡ ♦s ❞✐❛s ❞❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛s ❞✐s❝✉ssõ❡s✱ ♣r♦✜ss✐♦♥❛✐s ♦✉ ♥ã♦✱ ❞❡s❡♥✈♦❧✈✐❞❛s ♥♦ ♣❡rí♦❞♦✳ ❆♦s ♦✉tr♦s ❛♠✐❣♦s✱ ❋❡r♥❛♥❞♦✱ ❆♥❞ré ❡ ❱✐❝t♦r ♣♦r ✈ár✐♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝❛♥s♦ ✐♥t❡❧❡❝t✉❛❧✳

❊♠ ❡s♣❡❝✐❛❧✱ ♣♦ré♠✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ▼❛ír❛✱ q✉❡ ❝♦♠❡ç♦✉ ❡st❛ ❥♦r♥❛❞❛ ❝♦♠♦ ♥❛♠♦r❛❞❛ ❡ ❛❣♦r❛ é ❡s♣♦s❛✱ ♠❛s s❡♠♣r❡ ❡st❡✈❡ ❡ s❡♠♣r❡ ❡st❛rá ♥♦ ♠❡✉ ❝♦r❛çã♦✳ ❊st❡ tr❛❜❛❧❤♦ ❝❡rt❛♠❡♥t❡ ♥ã♦ s❡r✐❛ ❡s❝r✐t♦ s❡♠ ♦ s✉♣♦rt❡✳

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❘❡s✉♠♦

◆❡st❛ t❡s❡ sã♦ ❞✐s❝✉t✐❞♦s ❞♦✐s ❛s♣❡❝t♦s ❞♦ ♠♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉ ❛tr❛✈és ❞❛ ó♣t✐❝❛ ❞❡ ❙✐♠❡tr✐❛✿ ❙♦❧✉çõ❡s ❛♥❛❧ít✐❝❛s sã♦ ❡♥❝♦♥tr❛❞❛s ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ s✐st❡♠át✐❝❛ ❞❡ s✐♠❡tr✐❛s ❞❛s ❡q✉❛çõ❡s ❣❡r❛❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ é ❡st✉❞❛❞❛ ❛ ♣❛rt✐r ❞❛ r❡st❛✉r❛çã♦ ❞❛ s✐♠❡tr✐❛ q✉✐r❛❧ ♣♦r ♠❡✐♦ ❞❡ ❡❢❡✐t♦s ❞❡ ❝♦♠♣❛❝t✐✜❝❛çã♦✳

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❆❜str❛❝t

❲✐t❤✐♥ t❤✐s t❤❡s✐s✱ t✇♦ ❛s♣❡❝ts ♦❢ t❤❡ ●r♦ss✲◆❡✈❡✉ ♠♦❞❡❧ ❛r❡ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ ❜❛❝❦❞r♦♣ ♦❢ s②♠♠❡tr② ❛♥❛❧②s✐s✿ ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♠♦❞❡❧ ❛r❡ ♦❜t❛✐♥❡❞ t❤r♦✉❣❤ s②st❡♠❛t✐❝ s②♠♠❡tr② ❛♥❛❧②s✐s ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ♠♦❞❡❧ ❛♥❞ ♣❤❛s❡ tr❛♥s✐st✐♦♥ ✐s st✉❞✐❡❞ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ t❤❡ ❝❤✐r❛❧ s②♠♠❡tr② r❡st♦r❛t✐♦♥ t❤r♦✉❣❤ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥ ❡✛❡❝ts✳

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❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✻

✷ ●r✉♣♦s ❡ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ ✾

✷✳✶ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ ❊s♣❛ç♦s ❚♦♣♦❧ó❣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸ ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹ ●r✉♣♦s ❞❡ ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✺ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✻ ●r✉♣♦ ❞❡ P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸ ❉✐♥â♠✐❝❛ ❞❡ ❈❛♠♣♦s ❚ér♠✐❝♦s ✸✷

✸✳✶ ▼♦❞❡❧♦ ❞❡ ●r♦ss ◆❡✈❡✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❆ ❉✐♥â♠✐❝❛ ❞❡ ❈❛♠♣♦s ❚ér♠✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷✳✶ ❊s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❚ér♠✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✷✳✷ ❚❡r♠♦✲➪❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✸ ❉❈❚ ❡ ❚r❛♥s✐çã♦ ❞❡ ❋❛s❡ ♥♦ ▼♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✸✳✶ ❖s❝✐❧❛❞♦r❡s ❚ér♠✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✸✳✷ ❈❛s♦ ❇♦sô♥✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✸✳✸ ❈❛s♦ ❋❡r♠✐ô♥✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✸✳✹ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❇♦❣♦❧✐✉❜♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

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✸✳✸✳✺ ❖♣❡r❛❞♦r❡s ❚❡r♠❛❧✐③❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✸✳✻ ◆♦t❛çã♦ ▼❛tr✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✸✳✼ ❖ Pr♦♣❛❣❛❞♦r ❋❡r♠✐ô♥✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✸✳✽ ❘❡st❛✉r❛çã♦ ❞❡ ❙✐♠❡tr✐❛ ♥♦ ▼♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸✳✸✳✾ ❘❡st❛✉r❛çã♦ ❞❡ ❙✐♠❡tr✐❛✿ ❚❡♠♣❡r❛t✉r❛ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✸✳✶✵ ❘❡st❛✉r❛çã♦ ❞❡ ❙✐♠❡tr✐❛✿ ❈♦♠♣❛❝t✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✸✳✸✳✶✶ ❘❡st❛✉r❛çã♦ ❞❡ ❙✐♠❡tr✐❛✿ ❈♦♠♣❛❝t✐✜❝❛çã♦ ❊s♣❛❝✐❛❧ ❛ ❚❡♠♣❡r❛t✉r❛

❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷

✹ ▼ét♦❞♦s ❞❡ ❙✐♠❡tr✐❛ ✽✺

✹✳✶ ❊♥❝♦♥tr❛♥❞♦ ❙✐♠❡tr✐❛s ❞❡ ▲✐❡ ❞❡ ❊❉❖s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✹✳✷ Ór❜✐t❛s ❡ ●❡r❛❞♦r❡s ■♥✜♥✐t❡s✐♠❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✹✳✸ ❆s ❊q✉❛çõ❡s ❉❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✹✳✹ ❙✐♠❡tr✐❛s ❞❡ ❊❉Ps ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✹✳✺ ❈♦♦r❞❡♥❛❞❛s ❈❛♥ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✹✳✻ ■♥✈❛r✐❛♥t❡s ❉✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✹✳✼ ❆ ➪❧❣❡❜r❛ ❞♦s ●❡r❛❞♦r❡s ■♥✜♥✐t❡s✐♠❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾ ✹✳✽ ❙✐♠❡tr✐❛s ◆ã♦✲❈❧áss✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

✺ ❙♦❧✉çõ❡s ■♥✈❛r✐❛♥t❡s ❞❛ ❊q✉❛çã♦ ❞❡ ●r♦ss✲◆❡✈❡✉ ✶✷✷

✻ ❈♦♥❝❧✉sõ❡s ❡ P❡rs♣❡❝t✐✈❛s ✶✸✵

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❛ ❛♥á❧✐s❡ s✐st❡♠át✐❝❛ ❞❡ ❛s♣❡❝t♦s ❞❡ s✐♠❡tr✐❛ ❞♦ ♠♦❞❡❧♦ ❞❡ ●r♦ss✲ ◆❡✈❡✉ ✭●◆✮✱ ❞❡s❝r❡✈❡♥❞♦ ❛ ✐♥t❡r❛çã♦ ❞❡ ❝♦♥t❛t♦ ❞❡ q✉❛tr♦ ❢ér♠✐♦♥s ❡ ✐♥tr♦❞✉③✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❡❢❡t✐✈♦ ❞❛ ❝r♦♠♦❞✐♥â♠✐❝❛ q✉â♥t✐❝❛ ✭◗❈❉✮✳ ❆ t❡♦r✐❛ ❛t✉❛❧ s♦❜r❡ ❛ ✐♥t❡r❛çã♦ ❢♦rt❡ ❞♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❞❛ ❢ís✐❝❛ ❞❛s ♣❛rtí❝✉❧❛s ❡❧❡♠❡♥t❛r❡s✱ ❛ ◗❈❉✱ ❢♦✐ ♣r♦♣♦st❛ ♣❛r❛ ❞❡s❝r❡✲ ✈❡r ❛s♣❡❝t♦s ❝♦♥st✐t✉t✐✈♦s ❞❛ ♠❛tér✐❛ ❤❛❞rô♥✐❝❛✱ ♦s q✉❛r❦s ❡ ♦s ❣❧ú♦♥s ❬✶❪✱ ❬✷❪✱ ❬✸❪✳ ❉❡✈❡ ♣r♦✈❡r✱ ❛ss✐♠✱ ❡①♣❧✐❝❛çã♦ s❛t✐s❢❛tór✐❛ ♣❛r❛ ♦s s❡❣✉✐♥t❡s ❛s♣❡❝t♦s ❡①♣❡r✐♠❡♥t❛✐s✿ ❧✐❜❡r❞❛❞❡ ❛ss✐♠♣tót✐❝❛ ❡ ❝♦♥✜♥❛♠❡♥t♦✳ ❖ ♣r✐♠❡✐r♦✱ s❡ ♠❛♥✐❢❡st❛ ❡♠ ❛❧tíss✐♠❛s t❡♠♣❡r❛t✉r❛s ❡ ❡♥❡r✲ ❣✐❛s ✭♦✉ ❝✉rt❛s ❞✐stâ♥❝✐❛s✮✳ ❆tr❛✈és ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ✐♥❡❧ást✐❝♦ ♣r♦❢✉♥❞♦✱ ♦❜s❡r✈❛✲s❡ q✉❡ ♦s q✉❛r❦s s❡ ❝♦♠♣♦rt❛♠ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❧✐✈r❡s ♥❡ss❡ r❡❣✐♠❡ ❬✷❪✳ ❖ s❡❣✉♥❞♦ ❛s♣❡❝t♦ ❡st❛❜❡❧❡❝❡ q✉❡✱ ❡♠ ❜❛✐①❛s ❡♥❡r❣✐❛s ❡ t❡♠♣❡r❛t✉r❛s✱ q✉❛r❦s ❡ ❣❧✉♦♥s ❡stã♦ ❡s♣❛❝✐❛❧♠❡♥t❡ ❝♦♥✜♥❛❞♦s ❡♠ r❡❣✐õ❡s ❞❛ ♦r❞❡♠ ❞❡ ✶ ❢♠✱ ❡♠ ❡st❛❞♦s ❞❡s♣r♦✈✐❞♦s ❞❡ ❝♦r✳ ❊st❡ ❛s♣❡❝t♦ é t❛♠❜é♠ ❝♦♥✜r♠❛❞♦ ♣❡❧❛ ❢❛❧t❛ ❞❡ ♦❜s❡r✈❛çã♦ ❞❡ q✉❛r❦s ❧✐✈r❡s ♥♦s ♣r♦❝❡ss♦s ❢ís✐❝♦s ❝♦✲ ♥❤❡❝✐❞♦s✳ ❊♥tr❡t❛♥t♦✱ é ❛❜r❛♥❣❡♥t❡ ❛❝❡✐t❛çã♦ ❞❡ q✉❡ ❡♠ ❛❧❣✉♠ ❡stá❣✐♦ ❞❛ ❡✈♦❧✉çã♦ ❞♦ ❯♥✐✈❡rs♦✱ q✉❛r❦s ❡ ❣❧ú♦♥s ❡①✐st✐r❛♠ ♥✉♠ ❡st❛❞♦ ♥ã♦ ❝♦♥✜♥❛❞♦✳ ❊st❡ ❡st❛❞♦ é ❝❤❛♠❛❞♦ ❞❡ ♣❧❛s♠❛ ❞❡ q✉❛r❦s ❡ ❣❧✉♦♥s✳ ❈♦♠ ♦ r❡s❢r✐❛♠❡♥t♦ ♦❝♦rr❡ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✱ ♦♥❞❡ ♦s q✉❛r❦s ❡ ❣❧ú♦♥s ❞ã♦ ♦r✐❣❡♠ ❛ ♠❛tér✐❛ ❤❛❞rô♥✐❝❛ ❬✷❪✳

❉❡✈✐❞♦ ❛ ✉♠❛ ❡str✉t✉r❛ ♠❛t❡♠át✐❝❛ ✐♥tr✐❝❛❞❛✱ ❛ ◗❈❉ ❞❡s❝r❡✈❡ ❛s ♦❜s❡r✈❛çõ❡s ❛❝✐♠❛✱

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♠❛s ❛♣❡♥❛s ♣❛r❝✐❛❧♠❡♥t❡✱ ♦✉ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ♣♦✐s r❡s✉❧t❛❞♦s ❛♥❛❧ít✐❝♦s sã♦ ❡①tr❡♠❛✲ ♠❡♥t❡ ❞✐❢í❝❡✐s✱ ❡♠❜♦r❛ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❣✉✐❛r ❝♦rr❡t❛♠❡♥t❡ ♦s ❡①♣❡r✐♠❡♥t♦s✳ ◆♦s ❞♦♠í♥✐♦s ❞❡ ❧✐❜❡r❞❛❞❡ ❛ss✐♠♣tót✐❝❛ ✭❛❧t❛s t❡♠♣❡r❛t✉r❛s ♦✉ ❛❧t❛ ❡♥❡r❣✐❛✮✱ é ♣♦ssí✈❡❧ ✉t✐✲ ❧✐③❛r ♠ét♦❞♦s ♣❡rt✉r❜❛t✐✈♦s ❜❡♠ ❡st❛❜❡❧❡❝✐❞♦s ❬✷❪✲❬✼❪✳ ◆❛ r❡❣✐ã♦ ❞❡ tr❛♥s✐çã♦✱ ♠ét♦❞♦s ♣❡rt✉r❜❛t✐✈♦s sã♦ ❝♦♠♣❧✐❝❛❞♦s✱ ❡ ✉♠ ❡①♣❡❞✐❡♥t❡ ♠✉✐t♦ ✉t✐❧✐③❛❞♦ é ♦ ❝á❧❝✉❧♦ ♥❛ r❡❞❡✱ ✐♠✲ ♣❧❡♠❡♥t❛♥❞♦ ✉♠❛ s✐♠✉❧❛çã♦ ❞♦ s✐st❡♠❛ ❝♦♥✜♥❛❞♦✱ ❡ ♣r♦✈❡♥❞♦ ♣♦r ❡①❡♠♣❧♦ ❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ❞❛ tr❛♥s✐çã♦ ❝♦♥✜♥❛♠❡♥t♦✴❞❡❝♦♥✜♥❛♠❡♥t♦✱ ❞❛ ♦r❞❡♠ ❞❡ ✷✵✵ ▼❡❱ ❬✼❪✳ ❉❡✈✐❞♦ ❛s ❞✐✜❝✉❧❞❛❞❡s ❞❛ ◗❈❉ ❝♦♠ r❡s✉❧t❛❞♦s ❛♥❛❧ít✐❝♦s ♥❛ r❡❣✐ã♦ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛✱ ❡①✐st❡ ✉♠ ❢♦rt❡ ❛♣❡❧♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠♦❞❡❧♦s ❡❢❡t✐✈♦s✱ q✉❡ ♣♦ss❛♠ r❡♣r♦❞✉③✐r ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❛ ♠❛tér✐❛ ❤❛❞rô♥✐❝❛✳ ❖ ♠❛✐s s✐♠♣❧❡s ❞❡ss❡s ♠♦❞❡❧♦s ❞❡s❝r❡✈❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ❝♦♥t❛t♦ ❞❡ q✉❛tr♦ ❢ér♠✐♦♥s✱ ❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉ ✭●◆✮ ❬✽❪✳ ◆❡st❡ ❝❛s♦✱ ♦s ❣❧ú♦♥s✱ ❞❡s❝r✐t♦s ♣♦r ❝❛♠♣♦s ❞❡ ❝❛❧✐❜r❡s ♥ã♦ ❛❜❡❧✐❛♥♦s✱ sã♦ s✉♣r✐♠✐❞♦s✱ ❡♠ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ s✐♠✐❧❛r ❛♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ tr❛t❛♠❡♥t♦ ❞❡ ❋❡r♠✐ à ✐♥t❡r❛çã♦ ❢r❛❝❛✳ ❖ ♠♦❞❡❧♦ ●◆ ♥ã♦ é ♣❡rt✉r❜❛t✐✈❛♠❡♥t❡ r❡♥♦r♠❛❧✐③á✈❡❧ ♣❛r❛ ❞✐♠❡♥sõ❡s ♠❛✐♦r❡s q✉❡ ❉ ❂ ✸❀ ❡ ❛ ❡①♣❛♥sã♦ ♥❛ ♦r❞❡♠ ❞♦♠✐♥❛♥t❡ ❞❡ ✶✴◆✱ ♦♥❞❡ ◆ é ♦ ♥ú♠❡r♦ ❞❡ ❢ér♠✐♦s✱ é ✉s✉❛❧♠❡♥t❡ ❡♠♣r❡❣❛❞❛ ❬✾❪✳

◆❛s ú❧t✐♠❛s ❞é❝❛❞❛s ♦ ♠♦❞❡❧♦ ●◆ ❢♦✐ ❛♥❛❧✐s❛❞♦ ❝♦♠ ❞❡t❛❧❤❡ ♥♦ ❝♦♥t❡①t♦ ❞❡ t❡♠♣❡✲ r❛t✉r❛ ✜♥✐t❛✳ ❆s ♠♦t✐✈❛çõ❡s ❡ ❛ ♥❛t✉r❡③❛ ❞❡ss❡s ❡st✉❞♦s sã♦ ♠ú❧t✐♣❧❛s✱ ❡ s❡ ♣r❡st❛♠ ❛ ♣r♦✈❡r ✐♥❢♦r♠❛çõ❡s ❝♦♠♦ ♠♦❞❡❧♦ ❡❢❡t✐✈♦ ♥ã♦ s♦♠❡♥t❡ ❞❛ ◗❈❉✱ ♠❛s t❛♠❜é♠ ♥♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❢❡r♠✐ô♥✐❝♦s ♥❛ ❢ís✐❝❛ ❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛✳ ❊①❡♠♣❧♦s tí♣✐❝♦s sã♦ ❛s ❛♣❧✐❝❛çõ❡s ❡♠ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡ ❡ ❡♠ ❣r❛❢❡♥♦s ❬✶✵❪✲❬✷✹❪✳ ❙❡❣✉✐♥❞♦ ♠ét♦❞♦s ♣r✐♠❡✐r♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ ❜ós♦♥s✱ ♦ ♠♦❞❡❧♦ ●◆ t❡♠ s✐❞♦ t❛♠❜é♠ ❝♦♥s✐❞❡r❛❞♦ ❡♠ t♦♣♦❧♦❣✐❛s ❝♦♠ ❞✐♠❡♥sõ❡s ❡s♣❛❝✐❛❧♠❡♥t❡ ❝♦♠♣❛❝t✐✜❝❛❞❛s ❛ t❡♠♣❡r❛t✉r❛ ✜♥✐t❛ ❬✷✺❪✲❬✷✽❪✳ ❊♥tr❡t❛♥t♦✱ ❡st❛ ❛♥á❧✐s❡ ♥ã♦ ❢♦✐ ❡st❡♥❞✐❞❛ ♣❛r❛ ♦s ♠ét♦❞♦s ❞❡ t❡♠♣♦ r❡❛❧✱ ❝♦♠♦ ❛ ❉✐♥â♠✐❝❛ ❞❡ ❈❛♠♣♦s ❚ér♠✐❝♦s ✭❉❈❚✮✱ q✉❡ é ✉♠ ❢♦r♠❛❧✐s♠♦ ♣❛r❛ ❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s ❛ t❡♠♣❡r❛t✉r❛ ✜♥✐t❛✱ ❡st❛❜❡❧❡❝✐❞♦ ❛ ♣❛rt✐r ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❣r✉♣♦s ❞❡ s✐♠❡tr✐❛✳ ❯♠ ❞♦s ♦❜❥❡t✐✈♦s ❛q✉✐ ♣r♦♣♦st♦s é tr❛t❛r

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♦ ♠♦❞❡❧♦ ●◆ ♥♦ ❝♦♥t❡①t♦ ❞❛ ❉❈❚✳

❆s ❡q✉❛çõ❡s ❞❡ ●r♦ss✲◆❡✈❡✉ ❛✐♥❞❛ ♥ã♦ t✐✈❡r❛♠ ✉♠ ❡st✉❞♦ s❛t✐s❢❛tór✐♦ ❞❡ s✉❛s s♦❧✉çõ❡s ❛♥❛❧ít✐❝❛s✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❡♠ ✷ ❡ ✸ ❞✐♠❡♥sõ❡s ❡s♣❛❝✐❛✐s✳ ◆♦s ú❧t✐♠♦s ❛♥♦s✱ ❞❡✈✐❞♦ ❛♦ ✐♥t❡r❡ss❡ ♣❡❧♦ ♠♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❝♦♠ ♦s ❣r❛❢❡♥♦s✱ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡st❛s s♦❧✉çõ❡s é s✐❣♥✐✜❝❛t✐✈❛✳ ❆❧❣✉♥s ❛✈❛♥ç♦s ❢♦r❛♠ ❢❡✐t♦s ♥❡st❛ ❧✐♥❤❛ ❬✷✾✱ ✸✵✱ ✸✶❪✱ ♠❛s ❞❡ ♠♦❞♦ ♣r❡❧✐♠✐♥❛r✳ ❉❡✈✐❞♦ ❛ ❡ss❛s ❝❛r❛❝t❡ríst✐❝❛s ❡ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡✱ ♦ ♠♦❞❡❧♦ ●◆ s❡r✈❡ ❝♦♠♦ ✉♠ ♣r♦tót✐♣♦ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ♥❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❞❡ ❝❛♠♣♦s ♣❛r❛ ❢ér♠✐♦♥s✱ ❡ ♥❡ss❛ ♣❡rs♣❡❝t✐✈❛ ❛ ♣r♦❝✉r❛ ♣♦r r❡s✉❧t❛❞♦s ❛♥❛❧ít✐❝♦s ♣❛ss❛ ❛ s❡r ✉♠ ♦✉tr♦ ❛s♣❡❝t♦ ✐♠♣♦rt❛♥t❡ ❬✸✷❪✱ ❬✸✸❪✳ ❯♠ ❞♦s ♣♦ssí✈❡✐s ♣r♦❝❡❞✐♠❡♥t♦s é ❡①♣❧♦r❛r ❛s té❝♥✐❝❛s ❞❡ ❣r✉♣♦s ❞❡ ▲✐❡ ❛♣❧✐❝❛❞❛s ❛ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✳ ❊st❡ t✐♣♦ ❞❡ s♦❧✉çã♦ ❥á ❢♦✐ ♣❛r❝✐❛❧♠❡♥t❡ ❛♥❛❧✐s❛❞♦ ❡ ❝❧❛ss✐✜❝❛❞♦ ♣♦r ❋✉s❤❝❤✐❝❤ ❡ ❩❤❞❛♥♦✈ ❬✸✵❪✱ ❬✸✶❪✱ ♥♦ ❝❛s♦ ❞❡ ✸✰✶ ❞✐♠❡♥sõ❡s✱ ❡ ♣❡❧♦ ♣ró♣r✐♦ ❛✉t♦r ♥♦s ❝❛s♦s ❞❡1 + 1 ❡ 2 + 1 ❞✐♠❡♥sõ❡s❬✷✾❪✳

❊st❡ tr❛❜❛❧❤♦ s❡ ♦r❣❛♥✐③❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ♥♦ ❈❛♣ít✉❧♦ ✷ sã♦ r❡✈✐s❛❞♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ❜ás✐❝♦s s♦❜r❡ ❣r✉♣♦s ❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ◆♦ ❈❛♣ít✉❧♦ ✸✱ s❡ ❝♦♥str♦❡♠ ❛❧❣✉♥s ❛♣❡❝t♦s ❜ás✐❝♦s ❞❛ ❉✐♥â♠✐❝❛ ❞❡ ❈❛♠♣♦s ❚ér♠✐❝♦s✱ ❝✉❧♠✐♥❛♥❞♦ ❝♦♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡st❡s ❝♦♥❝❡✐t♦s ❛♦ ♠♦❞❡❧♦ ❞❡ ●r♦ss✲◆❡✈❡✉✳ ❖ ❈❛♣ít✉❧♦ ✹ é ❞❡❞✐❝❛❞♦ à r❡✈✐sã♦ ❞♦s ♣r✐♥❝✐♣❛✐s ♠ét♦❞♦s ❞❡ s✐♠❡tr✐❛ ❛ s❡r❡♠ ❛♣❧✐❝❛❞♦s✱ ❡♥q✉❛♥t♦ ♥♦ ❈❛♣ít✉❧♦ ✺ s❡ ❛♣r❡s❡♥t❛♠ r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞❡st❡s✳ ❆♦ ❧♦♥❣♦ ❞❡st❛ t❡s❡ sã♦ ❡♠♣r❡❣❛❞❛s ✉♥✐❞❛❞❡s ♥❛t✉r❛✐s✱ ♦♥❞❡

c = ~ ₌ _k_B _{= 1}✳ ❆ ♥♦t❛çã♦ ❞❡ s♦♠❛ ❞❡ ❊✐♥st❡✐♥ ❡stá ✐♠♣❧í❝✐t❛✱ ❛ ♥ã♦ s❡r q✉❡ ❞✐t♦ ♦

❝♦♥trár✐♦✳

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❈❛♣ít✉❧♦ ✷

●r✉♣♦s ❡ ➪❧❣❡❜r❛s ❞❡ ▲✐❡

❖s ♠ét♦❞♦s ♣r♦♣♦st♦s ♥❡st❛ t❡s❡ t❡♠ ❝♦♠♦ ❜❛s❡ ♠❛t❡♠át✐❝❛ ❛ t❡♦r✐❛ ❞❛s s✐♠❡tr✐❛s ❞❡ ▲✐❡ ❡ s♦❧✉çõ❡s ✐♥✈❛r✐❛♥t❡s ❞❡ ❣r✉♣♦❬✸✹❪✳ ❋❛③✲s❡✱ ❡♥tã♦✱ ♥❡❝❡ssár✐♦ ♦ ❡st✉❞♦ ❞♦ ❢❡rr❛♠❡♥t❛❧ r❡❧❛❝✐♦♥❛❞♦ ❛ ❡st❛ t❡♦r✐❛✳ ❆♥t❡s ❞❡ ❡♥tr❛r ❡♠ ❞❡t❛❧❤❡s s♦❜r❡ ❝♦♠♦ s✐♠❡tr✐❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡ ✉t✐❧✐③❛❞❛s ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ❞❡✈❡✲s❡ ♣r✐♠❡✐r♦ ❞♦♠✐♥❛r ♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❜ás✐❝♦s ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞❛s s✐♠❡tr✐❛s✳

◆ã♦ ❡①✐st❡✱ t❛❧✈❡③✱ ❢❡rr❛♠❡♥t❛ ♠❛✐s ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ s✐♠❡tr✐❛s q✉❛♥t♦ ❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s✳ ❊st❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ é ú♥✐❝❛♠❡♥t❡ ❛❞❛♣t❛❞❛ ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞❡ ❝♦♥❥✉♥t♦s ❞❡ tr❛♥s❢♦r♠❛çõ❡s✱ ❡ ❥✉st❛♠❡♥t❡ ❞❡✈✐❞♦ ❛ ❡st❛ ❝❛r❛❝t❡ríst✐❝❛ s❡ t♦r♥♦✉ tã♦ ❝r✉❝✐❛❧ ♣❛r❛ ❛ ❢ís✐❝❛✳ ❖ q✉❡ s❡❣✉❡ é ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ ❞❛s ❡str✉t✉r❛s ❛❧❣é❜r✐❝❛s ♠❛✐s r❡❧❡✈❛♥t❡s ♣❛r❛ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ q✉❡ t❡♠ ❝♦♠♦ ♣❡❞r❛ ❢✉♥❞❛♠❡♥t❛❧ ♦s ❣r✉♣♦s ❞❡ s✐♠❡tr✐❛✳

❊♠ s✉❛ ♣❛rt❡ ✐♥✐❝✐❛❧ ❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③ ❞❡✜♥✐çõ❡s ❡ ❝♦♥❝❡✐t♦s ❞❡ ♠❛♥❡✐r❛ ❛❜str❛t❛✳ ❋❡✐t❛s ❛s ✐♥tr♦❞✉çõ❡s ❝♦♥❝❡✐t✉❛✐s✱ ♦ ❝❛♣ít✉❧♦ t❡♠ ❡♠ s✉❛ ♣❛rt❡ ✜♥❛❧ ✉♠ ❡①❡♠♣❧♦ ❜❛st❛♥t❡ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❡st❛ t❡s❡✱ q✉❛♥❞♦ é ❝♦♥str✉í❞❛ ❛ á❧❣❡❜r❛ ❞♦ ❣r✉♣♦ ❞❡ P♦✐♥❝❛ré✳ ❊st❛ r❡✈✐sã♦ é ❜❛s❡❛❞❛ ♥❛s r❡❢❡rê♥❝✐❛s ❬✶❪✱ ❬✸❪✱ ❬✸✺❪✳

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✷✳✶ ●r✉♣♦s

❯♠ ❣r✉♣♦ é ✉♠ ❝♦♥❥✉♥t♦ {g1, g2, g3, ...} = G ♠✉♥✐❞♦ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ ❝❤❛♠❛❞❛

♣r♦❞✉t♦ ♦✉ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ❣r✉♣♦ ✭❞❡♥♦t❛❞❛ ♣♦r ◦ ✮ t❛❧ q✉❡

✶✳ gi ∈G, gj ∈G⇒gi ◦gj ∈G✳ ❉✐③✲s❡ q✉❡ ❣r✉♣♦s sã♦ ❢❡❝❤❛❞♦s s♦❜ ♦ ♣r♦❞✉t♦✳ ✷✳ gi◦(gj ◦gk) = (gi◦gj)◦gk✳ ❖ ♣r♦❞✉t♦ é ❛ss♦❝✐❛t✐✈♦✳

✸✳ ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ g1 t❛❧ q✉❡ g1 ◦gi =gi◦g1 = gi✳ ❖ ❡❧❡♠❡♥t♦ g1 é ❝❤❛♠❛❞♦ ❞❡

✐❞❡♥t✐❞❛❞❡✳

✹✳ P❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦gk❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦g−k1t❛❧ q✉❡gk◦gk−1 =g−k1◦gk =g1✳

❯♠ ❣r✉♣♦ ♣♦❞❡ s❡r ✜♥✐t♦ ♦✉ ✐♥✜♥✐t♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s✳ ●r✉♣♦s ✐♥✜♥✐t♦s ♣♦❞❡♠ s❡r ❛✐♥❞❛ ❝❧❛ss✐✜❝❛❞♦s ❡♠ ❞✐s❝r❡t♦s ♦✉ ❝♦♥tí♥✉♦s✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♥❛t✉r❡③❛ ❝♦♥tá✈❡❧ ♦✉ ✐♥❝♦♥tá✈❡❧ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✳

❊①❡♠♣❧♦s

✶✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ♣♦ssí✈❡✐s ♣❡r♠✉t❛çõ❡s ❞♦s ♣♦♥t♦s ✶✱ ✷✱ ✸✱ ✹ ❢♦r♠❛ ✉♠ ❣r✉♣♦ ❝♦♠

4! ❡❧❡♠❡♥t♦s✱ ❝❤❛♠❛❞♦ P4✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s ❞❡st❡ ❣r✉♣♦ ✭❛ ❡ ❜✮ é

✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✶✱ ❥✉♥t♦ ❝♦♠ ❛ ❝♦♠♣♦s✐çã♦ ❞❡st❡s ❞♦✐s ❡❧❡♠❡♥t♦s✳

❋✐❣✉r❛ ✷✳✶✿ ❊①❡♠♣❧♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ P4 ❝♦♠ ❛ ❝♦♠♣♦s✐çã♦✳

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✷✳ ❆ ❝♦❧❡çã♦ ❞❡ r♦t❛çõ❡s ❞♦ ❝ír❝✉❧♦ ♣♦r ♠ú❧t✐♣❧♦s ❞❡ 2π

n r❛❞✐❛♥♦s ❢♦r♠❛ ✉♠ ❣r✉♣♦ ❝♦♠

♥ ♦♣❡r❛çõ❡s ❞✐st✐♥t❛s✳ ●r✉♣♦s ✜♥✐t♦s ❝♦♠♦ ❡st❡ sã♦ ❞✐t♦s ❞❡ ♦r❞❡♠ ♥✳

✸✳ ❆ ❝♦❧❡çã♦ ❞❡ r♦t❛çõ❡s ❞❡ ✉♠ ❝ír❝✉❧♦ ♣♦r ✉♠ â♥❣✉❧♦ θ ✭0_≤ θ _≤ 2π✮ é ✉♠ ❡①❡♠♣❧♦

❞❡ ✉♠ ❣r✉♣♦ ❝♦♥tí♥✉♦✳ ❊①✐st❡♠ t❛♥t❛s ♦♣❡r❛çõ❡s g(θ) q✉❛♥t♦ ♣♦♥t♦s ♥♦ ✐♥t❡r✈❛❧♦

[0,2π]✳

✹✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❢♦r♠❛ ✉♠ ❣r✉♣♦ s♦❜ ❛ ❛❞✐çã♦✳ ❆ ✐❞❡♥t✐❞❛❞❡ é ♦ ♥ú♠❡r♦ ③❡r♦ ❡ ♦ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❡ ① é ✲①✳

✺✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ♠❛tr✐③❡s r❡❛✐s n×n ♥ã♦ s✐♥❣✉❧❛r❡s ✭det(g)6= 0✮✱ s♦❜ ❛ ♠✉❧t✐♣❧✐❝❛çã♦

❞❡ ♠❛tr✐③❡s✱ ❢♦r♠❛ ✉♠ ❣r✉♣♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦Gl(n, r)✭●❡♥❡r❛❧ ▲✐♥❡❛r✮✳ ❖ ❝♦♥❥✉♥t♦

❞❡ ♠❛tr✐③❡s n_×n ❝♦♠ ❞❡t❡r♠✐♥❛♥t❡ ✶ t❛♠❜é♠ ❢♦r♠❛ ✉♠ ❣r✉♣♦✱ ❝❤❛♠❛❞♦ Sl(n, r)

✭❙♣❡❝✐❛❧ ▲✐♥❡❛r✮✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ♠❛tr✐③❡s ✉♥✐tár✐❛s n_×n ❢♦r♠❛ ♦ ❣r✉♣♦ U(n)

◆♦t❡ q✉❡✱ ❡♠ ♣r✐♥❝í♣✐♦✱ ❛ ♦♣❡r❛çã♦ ◦ ♥ã♦ é ❝♦♠✉t❛t✐✈❛✳ ❯♠ ❣r✉♣♦ G q✉❡ ♦❜❡❞❡❝❡ gi◦gj =gj ◦gi, ∀gi, gj ∈Gé ❝❤❛♠❛❞♦ ❛❜❡❧✐❛♥♦✱ ♦✉ ❝♦♠✉t❛t✐✈♦✳

✷✳✷ ❊s♣❛ç♦s ❚♦♣♦❧ó❣✐❝♦s

❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❚ é ❝♦♠♣♦st♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✱ ❞❡♥♦t❛❞♦ ♣♦r S✱

s♦❜r❡ ♦ q✉❛❧ s❡ ❝♦❧♦❝❛ ✉♠❛ t♦♣♦❧♦❣✐❛ T✳ ❯♠❛ t♦♣♦❧♦❣✐❛ é ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s

S1, S2, S3,· · · ⊂S q✉❡ ♦❜❡❞❡❝❡ ❛♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿

✶✳ ❖ ❝♦♥❥✉♥t♦ ✈❛③✐♦✭∅✮ ❡ ♦ ❝♦♥❥✉♥t♦ ❙ ♣❡rt❡♥❝❡♠ ❛ T✳

∅ ∈ T , S∈ T

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✷✳ ■♥t❡rs❡❝çõ❡s ✜♥✐t❛s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ T sã♦ ❡❧❡♠❡♥t♦s ❞❡ T✳

✜♥✐t❛_\

i

Si ∈ T

✸✳ ❯♥✐õ❡s ❛r❜✐trár✐❛s ❞❡ ❡❧❡♠❡♥t♦s ❞❡T sã♦ ❡❧❡♠❡♥t♦s ❞❡ T✳

q✉❛❧q✉❡r_[

i

Si ∈ T

❖s ❡❧❡♠❡♥t♦s Si ❞❛ t♦♣♦❧♦❣✐❛ sã♦ ❝❤❛♠❛❞♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✳

❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ q✉❡ ♦❜❡❞❡❝❡ ❛♣❡♥❛s ❛♦s ❛①✐♦♠❛s ✶✲✸ é ❣❡r❛❧ ❞❡♠❛✐s ♣❛r❛ ♥♦ss♦ ♣r♦♣ós✐t♦✳ ■♠♣õ❡✲s❡ ❛✐♥❞❛ ♦ s❡❣✉✐♥t❡ ❛①✐♦♠❛✿

✹✳ ❙❡ p _∈ T, q _∈ T, p ₆= q✱ ❡♥tã♦ ❡①✐st❡♠ Sp ∈ T, Sq ∈ T q✉❡ ♦❜❡❞❡❝❡♠ ❛s ♣r♦✲ ♣r✐❡❞❛❞❡s p_∈Sp, q∈Sq, Sp∩Sq =∅✳

❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ q✉❡ ♦❜❡❞❡❝❡ ❛♦ ❛①✐♦♠❛ ✹ é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❍❛✉s❞♦r✛✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ Sp ❝♦♥t❡♥❞♦p é ❝❤❛♠❛❞♦ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p✳

❊①❡♠♣❧♦

❖ ♣❧❛♥♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ R2 é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❡♠ q✉❡ ♥♦r♠❛❧♠❡♥t❡ s❡ ❡s❝♦❧❤❡

✉♠❛ t♦♣♦❧♦❣✐❛ ✭❝❤❛♠❛❞❛ st❛♥❞❛r❞✱ ♦✉ ♣❛❞rã♦✮ ❢♦r♠❛❞❛ ♣❡❧♦ ✐♥t❡r✐♦r ❞❡ ❝ír❝✉❧♦s ❞❡ ❝❡♥tr♦ ❛r❜✐trár✐♦ ❡ r❛✐♦ ❛r❜✐trár✐♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❆ t♦♣♦❧♦❣✐❛ ❝♦♥s✐st❡ ❞❡st❡s ❝ír❝✉❧♦s ❡ s✉❛s ✐♥t❡rs❡❝çõ❡s ✜♥✐t❛s ❡ ✉♥✐õ❡s ❛r❜✐trár✐❛s✳ ❊st❡ ❡s♣❛ç♦ ❝♦♠ ❡st❛ t♦♣♦❧♦❣✐❛ t❛♠❜é♠ é ✉♠ ❡s♣❛ç♦ ❍❛✉s❞♦r✛✳ ❉♦✐s ♣♦♥t♦s p ❡ q sã♦ s❡♣❛r❛❞♦s ♣♦r ✉♠❛ ❞✐stâ♥❝✐❛ d(p, q)✳ ❈ír❝✉❧♦s

Sp, Sq ❝♦♠ r❛✐♦ d(p,q₃ )✱ ♣♦r ❡①❡♠♣❧♦✱ sã♦ ❝♦♥❥✉♥t♦s s❡♠ ✐♥t❡rs❡❝çã♦ q✉❡ ❝♦♥té♠ p ❡ q r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❉❡✜♥❡♠✲s❡ três ❝♦♥❝❡✐t♦s ❛❞✐❝✐♦♥❛✐s✿

(15)

✶✳ ❖ ❡s♣❛ç♦ ❚ é ❝♦♠♣❛❝t♦ s❡ t♦❞❛ s❡q✉ê♥❝✐❛ ✐♥✜♥✐t❛ ❞❡ ♣♦♥t♦s t1, t2, ...∈ T ❝♦♥té♠

✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ✉♠ ♣♦♥t♦ ❡♠ ❚✳

✷✳ ❯♠ ❝♦♥❥✉♥t♦ ❙ é ❢❡❝❤❛❞♦ s❡ ❝♦♥té♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ❞❡ ❛❝✉♠✉❧❛çã♦✳ ❯♠ ♣♦♥t♦

p é ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ s❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❝♦♥t❡♥❞♦p❝♦♥té♠ ♣❡❧♦

♠❡♥♦s ✉♠ ♣♦♥t♦ ❞❡ ❙ ❞✐❢❡r❡♥t❡ ❞❡p✳ ❙ ❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ❞❡ ❛❝✉♠✉❧❛çã♦

é ❝❤❛♠❛❞♦ ❞❡ ❢❡❝❤♦ ❞❡ ❙✳

✸✳ ❙❡❥❛ φ ✉♠ ♠❛♣❛ ❞♦ ❡s♣❛ç♦ ❚ ❝♦♠ t♦♣♦❧♦❣✐❛ T ♥♦ ❡s♣❛ç♦ ❯ ❝♦♠ t♦♣♦❧♦❣✐❛ U✳ ❖

❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s t1, t2, ...∈ T q✉❡ sã♦ ❧❡✈❛❞♦s ❛♦ ♠❡s♠♦ ♣♦♥t♦ u ∈U

é ❝❤❛♠❛❞♦ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡ u✳ ❖ ♠❛♣❛φ é ❞✐t♦ ❝♦♥tí♥✉♦ s❡ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡

t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❡♠ ❯ é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠ ❚✳

✷✳✸ ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐á✈❡✐s

❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♥s✐st❡ ♥✉♠ ❡s♣❛ç♦ ❍❛✉s❞♦r✛ ✭❚✱ T✮ ♠✉♥✐❞♦ ❞❡ ✉♠❛

❝♦❧❡çã♦Φ ❞❡ ♠❛♣❛s φp ∈Φ

φp : Tp →RN p∈T

◗✉❡ ♦❜❡❞❡❝❡ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✳ φp é ✉♠ ♠❛♣❛ ✶✲✶ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ Tp ✭p ∈ Tp✮ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡

RN✳

✷✳ STp =T✱ ❖✉ s❡❥❛✱ ❛ ✉♥✐ã♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s Tp ❢♦r♠❛ ♦ ❡s♣❛ç♦ ❚✳ ✸✳ ❙❡Tp∩Tq ♥ã♦ ❢♦r ✈❛③✐♦✱ ❡♥tã♦φp(Tp∩Tq)é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠RN✳ φq(Tp∩Tq)

t❛♠❜é♠ é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠RN ❞✐st✐♥t♦ ❞❡_φ_p_(T_p_∩_T_q₎✳ ❖ ♠❛♣❛_φ_p_◦_φ−1

q ❞❡✈❡

s❡r ❝♦♥tí♥✉♦ ❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳

(16)

✹✳ ❖s ♠❛♣❛s φp◦φ−q1 ❡ φq◦φ−p1 sã♦ ❡❧❡♠❡♥t♦s ❞❡ Φ✳

❖ ♣r✐♠❡✐r♦ ❛①✐♦♠❛ ❣❛r❛♥t❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝♦♥tr✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ r❡❢❡rê♥❝✐❛s ❡♠ ✉♠ ♣♦♥t♦ ♣ ❞❛ ✈❛r✐❡❞❛❞❡✳ ❙❡ ♦ ♣♦♥t♦p❢♦r ♠❛♣❡❛❞♦ ♥❛ ♦r✐❣❡♠ ❞❡RN✱ ✉♠ ♣♦♥t♦_q♥❛

✈✐③✐♥❤❛♥ç❛ ❞❡ p❞❡✈❡ s❡r ♠❛♣❡❛❞♦ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠ t❛♠❜é♠✳ ❖ s❡❣✉♥❞♦ ❛①✐♦♠❛

❣❛r❛♥t❡ q✉❡ ❡st❡ s✐st❡♠❛ ♣♦ss❛ s❡r ❡st❛❜❡❧❡❝✐❞♦ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛ ✈❛r✐❡❞❛❞❡✱ ❡♥q✉❛♥t♦ ♦ t❡r❝❡✐r♦ ❛①✐♦♠❛ ❞✐③ r❡s♣❡✐t♦ ❛ ♠❛♣❡❛♠❡♥t♦s ❞♦ t✐♣♦RN _→_RN_{✱ ❞❡s❝r✐t♦s ♣❡❧❛s té❝♥✐❝❛s} ✉s✉❛✐s ❞❡ ❝á❧❝✉❧♦✳

❆ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡ ❞❡ ❱❛r✐❡❞❛❞❡s ❉✐❢❡r❡♥❝✐á✈❡✐s s❡ ❞á ❛♦ ❢❛t♦ ❞❡ q✉❡ ❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ❝❛❞❛ ♣♦♥t♦ ♣♦❞❡ s❡r tr❛♥s♣♦rt❛❞❛ ♣❛r❛ ✉♠ ❡s♣❛ç♦ ❡✉❝❧✐❞❡❛♥♦✳ P♦r ♠❡✐♦ ❞❡ Φ✱ t♦❞♦s ♦s

❝♦♥❝❡✐t♦s ❡ ♠ét♦❞♦s ✉s❛❞♦s ♥♦ ❡st✉❞♦ ❞❡ RN ♣♦❞❡♠ s❡r tr❛♥s❢❡r✐❞♦s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡

❡s♣❛ç♦s ♠❛✐s ❝♦♠♣❧❡①♦s ❡ ❣❡r❛✐s✳

✷✳✹ ●r✉♣♦s ❞❡ ▲✐❡

❯♠ ❣r✉♣♦ ❞❡ ▲✐❡ é ✉♠ ❣r✉♣♦ ❡s♣❡❝✐✜❝❛❞♦ ♣❡❧❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ η✲❞✐♠❡♥s✐♦♥❛❧ ❞❡♥♦t❛❞❛ ♣♦r_M

✷✳ ❯♠❛ ❢✉♥çã♦ φ q✉❡ ❧❡✈❛ ❞♦✐s ♣♦♥t♦s (β, α) ❞❛ ✈❛r✐❡❞❛❞❡ ❡♠ ✉♠ t❡r❝❡✐r♦ ♣♦♥t♦ γ

t❛♠❜é♠ ❞❡♥tr♦ ❞❛ ✈❛r✐❡❞❛❞❡✳

✸✳ ❊♠ t❡r♠♦s ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡♥tr♦ ❞❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

γµ =φµ β1, ..., βη, α1, ..., αη ; µ= 1, ..., η

(17)

❆s ❢✉♥çõ❡s

φ : β_×α_→γ =βα ψ : α→α−1

❉❡✈❡♠ s❡r ❝♦♥tí♥✉❛s C∞_{✳ ❊st❛ ❝♦♥❞✐çã♦ é ♥❛ ✈❡r❞❛❞❡ ❢♦rt❡ ❞❡♠❛✐s✱ ❡ ♥♦ ❝❛s♦ ❣❡r❛❧}

❜❛st❛ q✉❡ s❡❥❛ Ck _♣❛r❛ _k _{✜♥✐t♦✱ ♣♦ré♠ é ✐♠♣♦st❛ ❛ ❝♦♥❞✐çã♦ ♠❛✐s r❡str✐t✐✈❛ ♣❛r❛} q✉❡ ♦s ♠ét♦❞♦s s✉❜s❡q✉❡♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞♦s t❡♥❤❛♠ ✈❛❧✐❞❛❞❡ ❣❛r❛♥t✐❞❛✳

●r✉♣♦s ❞❡ ▲✐❡ t❡♠ ❞♦✐s t✐♣♦s ❞❡ ❡str✉t✉r❛✱ ✉♠❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❡ ✉♠❛ ❡str✉t✉r❛ t♦♣♦❧ó❣✐❝❛✳ ❆❧❣é❜r✐❝❛♠❡♥t❡✱ ❡❧❡s sã♦ ❣r✉♣♦s ❡ ♦❜❡❞❡❝❡♠ ❛ t♦❞♦s ♦s ❛①✐♦♠❛s ❞❡ ❣r✉♣♦✳ ❚♦♣♦❧♦❣✐❝❛♠❡♥t❡ sã♦ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐❛✐s ❡ ❞❡✈❡♠ ♦❜❡❞❡❝❡r ❛ t♦❞♦s ♦s ❛①✐♦♠❛s r❡❧❛✲ ❝✐♦♥❛❞♦s ❛ ❡st❡ t✐♣♦ ❞❡ ❡str✉t✉r❛✳ ❖s ❛①✐♦♠❛s ❞❡ ❣r✉♣♦ sã♦ ❡♥tã♦ tr❛❞✉③✐❞♦s ❝♦♠♦ ❝♦♥❞✐çõ❡s s♦❜r❡φ✱ ✐st♦ é✱

α✳ ❋❡❝❤❛♠❡♥t♦

γµ =φµ(β, α) ; γ, β, α∈ M

β✳ ❆ss♦❝✐❛t✐✈✐❞❛❞❡

φ(β, φ(α, γ)) = φ(φ(β, α), γ)

γ✳ ■❞❡♥t✐❞❛❞❡

φµ(ǫ, α) = αµ =φµ(α, ǫ)

(18)

δ✳ ■♥✈❡rs❛

φµ(α, α−1) =ǫµ=φµ(α−1, α)

●❡r❛❞♦r❡s ■♥✜♥✐t❡s✐♠❛✐s

❙❡❥❛(_T, φ)✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ q✉❡ ❛❣❡ s♦❜r❡ ✉♠ ❡s♣❛ç♦GN ♣♦r ♠❡✐♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛sf(α, x)✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❞❡ tr❛♥s❢♦r♠❛çõ❡s✳

❆❣♦r❛ s❡❥❛ F(p) q✉❛❧q✉❡r ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s p _∈ GN✳ ❯♠❛ ✈❡③ ❞❡✜♥✐❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❙ ♣❛r❛ GN✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♣ ❝♦♠♦ ✉♠❛ N✲✉♣❧❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱

p _→ x1(p), x2(p), ..., xN(p)

❆ ❋✉♥çã♦ F(p) ♣♦❞❡ s❡r ❡s❝r✐t❛✱ ❡♥tã♦✱ ❡♠ ❢✉♥çã♦ ❞♦s ♣❛râ♠❡tr♦s xi_(p) _{♥♦ s✐st❡♠❛ ❞❡} ❝♦♦r❞❡♥❛❞❛s ❙✱ ♦✉ s❡❥❛✱

F(p) = FSx1(p), ..., xN(p) .

❊♠ ✉♠ ♦✉tr♦ s✐st❡♠❛S′ _{❛s ❝♦♦r❞❡♥❛❞❛s ❞❡}_p_{♠✉❞❛rã♦✳ ➱ ♥❛t✉r❛❧ ❡s♣❡r❛r q✉❡ ❛ ❢♦r♠❛ ❞❛}

❢✉♥çã♦ F ♠✉❞❡ ♣❛r❛ ♠❛♥t❡r ♦ ✈❛❧♦r ✜①♦ F(p)✱ ❛ss✐♠ ❡s❝r❡✈❡♠♦s

F(p) = FS′x′1(p), ..., x′N(p) . ✭✷✳✶✮

❙❛❜❡✲s❡ q✉❡ S ❡ S′ _{❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❣r✉♣♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❞❛}

(19)

s❡❣✉✐♥t❡ ❢♦r♠❛

x′i(p) =fi[α, x(p)] .

P❛r❛ r❡❧❛❝✐♦♥❛rFS′

❝♦♠ FS_{✱ ❜❛st❛ ❡♥tã♦ ❡s❝r❡✈❡r}_x′i_(p)_{❡♠ ❢✉♥çã♦ ❞❡} _xi_(p)_{✳ ❚❡♠♦s}

xi_{(p) =} _fi_[α−1_{, x}′_(p)] _. _{✭✷✳✷✮}

❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✳ ✭✷✳✷✮ ♥❛ ❊q✳ ✭✷✳✶✮✱ ♦❜t❡♠♦s

FS′x′1(p), ..., x′N(p) = FSf1(α−1, x′(p)), ..., fN(α−1, x′(p)) . ✭✷✳✸✮

❊st❛ ❡①♣r❡ssã♦ ♥ã♦ ❡stá ❡♠ ✉♠❛ ❢♦r♠❛ ♠✉✐t♦ út✐❧✳ ➱ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ♥♦ss♦ ♣r♦♣ós✐✲ t♦s tr❛t❛r ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♣ró①✐♠❛s ❛ ✐❞❡♥t✐❞❛❞❡✳

P❛r❛ ✉♠❛ ♦♣❡r❛çã♦ ❞♦ ❣r✉♣♦ 0+δαµ _{♣ró①✐♠❛ ❞❛ ✐❞❡♥t✐❞❛❞❡ ✵ ❛ ✐♥✈❡rs❛ é ❞❛❞❛ ♣♦r}

(δα−1₎µ₌₋_δαµ_{✱ ✉♠❛ ✈❡③ q✉❡} ₍₀₊_δαµ_{) (}₀₋_δαµ_{) =} ₀₊_O(δα2₎_{✳ ❊s❝r❡✈❡♠♦s ❡♥tã♦}

xi(p) = fi[₋δα, x′(p)] = fi[0, x′(p)] + ∂f

i_{[β, x}′_(p)]

∂βµ |β=0(−δα) +. . .

= x′i(p)−δα∂f

i_{[β, x}′_(p)]

∂βµ |β=0 . ✭✷✳✹✮

❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✳ ✭✷✳✹✮ ♥❛ ❊q✳ ✭✷✳✸✮ ❡♥❝♦♥tr❛♠♦s

FS′[x′(p)] = FS

x′i(p)−δα∂f

i_{[β, x}′_(p)]

∂βµ |β=0

= FS_x′i_(p)₋_δα∂fi[β, x′(p)]

∂βµ |β=0

∂ ∂x′iF

S_[x′_(p)]_. _{✭✷✳✺✮}

(20)

❊♠ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❛ ✈❛r✐❛çã♦ ❡♠F é ❞❛❞❛ ♣♦r

FS′[x′(p)]₋FSx′i(p) = ₋δα∂f

i_{[β, x}′_(p)]

∂βµ |β=0

∂ ∂x′iF

S_[x′_(p)]

= δα Xµ(x′)FS[x′] . ✭✷✳✻✮

❖ ♦♣❡r❛❞♦r

Xµ = −

∂fi_{[β, x}′_(p)]

∂βµ |β=0

∂ ∂x′i

é ❝❤❛♠❛❞♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞♦ ❣r✉♣♦ ❞❡ ▲✐❡✳ ❆tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ r❡♣❡t✐❞❛ ❞❡st❡s ♦♣❡r❛❞♦r❡s ♣♦❞❡♠♦s ♦❜t❡r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ ❞❡ ▲✐❡ ❣❡r❛❞♦s ♣♦r ❡❧❡s✳

◆♦t❡ q✉❡ ♦s ❣❡r❛❞♦r❡s ❞❡ ❣r✉♣♦s ❞❡ ▲✐❡ só sã♦ ❝❛♣❛③❡s ❞❡ ❣❡r❛r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ ❞❡✈✐❞♦ ❛ ♥❛t✉r❡③❛ ❞❡ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❝♦♥❡①❛ ❞♦s ❣r✉♣♦s ❞❡ ▲✐❡❬✸✺❪✳ ✐st♦ ❢❛③ ❝♦♠ q✉❡ ♦ t❡♦r❡♠❛ ❞❡ ❚❛②❧♦r s❡❥❛ ✈á❧✐❞♦ ❡ ❛♣r♦①✐♠❛çõ❡s ♣ró①✐♠❛s ❛ ✐❞❡♥t✐❞❛❞❡ ♣♦ss❛♠ s❡r ❢❡✐t❛s✳ ❖s ❣❡r❛❞♦r❡s ✐♥✜♥✐t❡s✐♠❛✐s ❞❡ ✉♠ ❣r✉♣♦ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ❞❡ ♠♦❞♦ q✉❡ q✉❛❧q✉❡r ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡st❡s t❛♠❜é♠ é ✉♠ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧✱ ❡ ✉♠ ❡❧❡♠❡♥t♦ ✜♥✐t♦ ❞♦ ❣r✉♣♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♣♦❞❡ s❡r ❡s❝r✐t♦ ❡♠ t❡r♠♦s ❞❡ s❡✉ ❣❡r❛❞♦r ❝♦♠♦

T = exp(εµXµ). ✭✷✳✼✮

✷✳✺ ➪❧❣❡❜r❛s ❞❡ ▲✐❡

❙❡ ✉♠ ❣r✉♣♦ é ❝♦♠✉t❛t✐✈♦ ❡♥tã♦ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✱ ❝♦♠α ❡ β ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦✱

αβα−1 = β .

(21)

❙❡ ♦ ❣r✉♣♦ ♥ã♦ é ❝♦♠✉t❛t✐✈♦✱ ❞❡✜♥❡✲s❡ ♦γ ❝♦♠♦ ✉♠❛ ♠❡❞✐❞❛ ❞♦ q✉❛♥t♦ ♦ r❡s✉❧t❛❞♦ ❞✐❢❡r❡

❞❡β✱ ♦✉ s❡❥❛ ♦ q✉❛♥t♦ ♥ã♦ ❝♦♠✉t❛t✐✈♦ é ♦ ❣r✉♣♦✳

αβα−1 = γβ .

◆♦t❡ q✉❡ γ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❣r✉♣♦

αβα−1β−1 = γ . ✭✷✳✽✮

❙❡ α ❡ β sã♦ ♣ró①✐♠♦s ❞❛ ✐❞❡♥t✐❞❛❞❡✱ ♣♦❞❡♠♦s ❡①♣❛♥❞✐✲❧♦s ❡♠ t❡r♠♦s ❞♦s ❣❡r❛❞♦r❡s

✐♥✜♥✐t❡s✐♠❛✐s ❝♦♠♦

α = I+δαµXµ+

1 2δα

µ_X

µδανXν ,

β = I+δβµXµ+

1 2δβ

µ_X

µδβνXν . ✭✷✳✾✮

❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✳ ✭✷✳✾✮ ♥❛ ❊q✳ ✭✷✳✽✮ ❡ ♠❛♥t❡♥❞♦ ❛♣❡♥❛s t❡r♠♦s ❛té s❡❣✉♥❞❛ ♦r❞❡♠ ♥♦s ❣❡r❛❞♦r❡s✱ ♦❜t❡♠♦s

(αβ)(βα)−1 = I+δαδβ[Xµ, Xν] , ♦♥❞❡[Xµ, Xν] =XµXν −XνXµ é ♦ ❝♦♠✉t❛❞♦r ❡♥tr❡ Xµ ❡ Xν✳

❈♦♠♦ (αβ)(βα)−1 _{é ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❣r✉♣♦✱} _[X

µ, Xν] ❞❡✈❡ ❡st❛r ♥♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦ ❡ ♣♦❞❡ s❡r ❡①♣❛♥❞✐❞♦ ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛ ❜❛s❡✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

[Xµ, Xν] = Cµνλ Xλ, ✭✷✳✶✵✮

♦♥❞❡Cλ

µν sã♦ ❝♦♥st❛♥t❡s ❛ s❡r❡♠ ❡♣❡❝✐✜❝❛❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♥❛t✉r❡③❛ ❞♦ ❣r✉♣♦✳ ❈♦♠♦

(22)

♦s ❣❡r❛❞♦r❡s ❥á ❢♦r♠❛✈❛♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ❛♦ ♠✉♥✐r ♦ ❡s♣❛ç♦ ❞❡ ✉♠ ♣r♦❞✉t♦ ❡♥tr❡ ♦s ✈❡t♦r❡s ✭♦ ❝♦♠✉t❛❞♦r✮ ❝♦♥stró✐✲s❡ ✉♠❛ á❧❣❡❜r❛✳

❯♠❛ á❧❣❡❜r❛ ❝✉❥♦ ♣r♦❞✉t♦ é ❛♥t✐❝♦♠✉t❛t✐✈♦

[Xµ, Xν] =−[Xν, Xµ] ,

❡ ♦❜❡❞❡❝❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐

[Xµ[Xν, Xρ]] + [Xν[Xρ, Xµ]] + [Xρ[Xµ, Xν]] = 0 ,

é ❝❤❛♠❛❞❛ ❞❡ ➪❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❖ ❝♦♠✉t❛❞♦r é ✉♠ ❝❛♥❞✐❞❛t♦ ♥❛t✉r❛❧ ❛ ♣r♦❞✉t♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✱ ✉♠❛ ✈❡③ q✉❡ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❛✉t♦♠❛t✐❝❛♠❡♥t❡ s❛t✐s❢❡✐t❛s✳

❱❡♠♦s✱ ♣♦rt❛♥t♦✱ q✉❡ ♦s ❣❡r❛❞♦r❡s ✐♥✜♥✐t❡s✐♠❛✐s ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❢♦r♠❛♠ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆♣❡s❛r ❞❡ t♦❞♦ ❣r✉♣♦ ❞❡ ▲✐❡ t❡r ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛❞❛ ❛ ❡❧❡✱ ❛ ❝♦rr❡s♣♦♥✲ ❞ê♥❝✐❛ ♥ã♦ é ✶✲✶✳ ❉❡ ❢❛t♦✱ ✈ár✐♦s ❣r✉♣♦s ❞✐❢❡r❡♥t❡s ♣♦❞❡♠ ✈✐r ❛ t❡r ❛ ♠❡s♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛Cλ

µν ❞❡✜♥❡♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ❡str✉t✉r❛ ❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✱

❞❡ ♠♦❞♦ q✉❡ ❞✉❛s á❧❣❡❜r❛s ❝♦♠ ❛s ♠❡s♠❛s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛ sã♦✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ❛ ♠❡s♠❛ á❧❣❡❜r❛✳

✷✳✻ ●r✉♣♦ ❞❡ P♦✐♥❝❛ré

❈♦♠♦ ✉♠ ❡①❡♠♣❧♦ ♣rát✐❝♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❛s té❝♥✐❝❛s ❛❝✐♠❛✱ ❝♦♥str✉✐r❡♠♦s ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ♣❛r❛ ♦ ❣r✉♣♦ ❞❡ P♦✐♥❝❛ré✳ ❉✐t♦ s❡r ♦ ❝♦♥❥✉♥t♦ ❞❡ ✐s♦♠❡tr✐❛s ❞♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐✱ ♦ ❣r✉♣♦ ❞❡ P♦✐♥❝❛ré é ✉♠ ❞♦s ❣r✉♣♦s ❞❡ s✐♠❡tr✐❛ ♠❛✐s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❢ís✐❝❛✳ ◆❡st❛ ❞✐ss❡rt❛çã♦ s❡rá ❢❡✐t♦ ❛♠♣❧♦ ✉s♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡st❡ ❣r✉♣♦ ❡ ❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ r❡❧❛✲ ❝✐♦♥❛❞❛ ❛ ❡❧❡✱ t♦r♥❛♥❞♦ ♥❡❝❡ssár✐♦ ✉♠ tr❛t❛♠❡♥t♦ ❡s♣❡❝✐❛❧ ❞❡st❡ ❣r✉♣♦✳ P♦r s❡r ✉♠ ❣r✉♣♦

(23)

❞❡ tr❛♥s❢♦r♠❛çõ❡s✱ ♣❛r❛ ❞❡✜♥✐r ♦ ❣r✉♣♦ ❞❡✈❡✲s❡ ❛♥t❡s ❞❡✜♥✐r s♦❜r❡ ♦ q✉❡ ♦ ❣r✉♣♦ ❛❣❡✳ P❛r❛ t❛♥t♦✱ ❞❡✜♥❡✲s❡ ♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s✐✱ ♦ ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ❞❡ ❞❡✜♥✐çã♦ ❞♦ ❣r✉♣♦✳

❖ ❡s♣❛ç♦✲t❡♠♣♦ ❞❡ ▼✐♥❦♦✇s❦✐ é ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ♣❧❛♥♦ ♣s❡✉❞♦✲❡✉❝❧✐❞✐❛♥♦ q✉❡ t❡♠ ❡♠ ❛❧❣✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛ ♠étr✐❝❛

gµν =

           

−1 0 0 0 . . . 0 1 0 0 . . . 0 0 1 0 . . . 0 0 0 1 . . . . . . .

           

,

µ, ν = 0. . . d₋1 ,

❈♦♠ ❞ ❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦✳ ❆s ❝♦♦r❞❡♥❛❞❛s ♥❛s q✉❛✐s ❛ ♠étr✐❝❛ t♦♠❛ ❡st❛ ❢♦r♠❛ sã♦ ❡s❝r✐t❛s ❝♦♠♦ (xµ_{) = (x}0_{, . . . , d}₋₁₎ _{❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❝♦✈❛r✐❛♥t❡s s❡ r❡❧❛❝✐♦♥❛♠ ❛s}

❝♦♥tr❛✈❛r✐❛♥t❡s ♣❡❧❛ r❡❣r❛xµ₌_gµν_x

ν✳ ❱❛❧❡ ❛ ♥♦t❛çã♦ ❞❡ s♦♠❛ ❞❡ ❊✐♥st❡✐♥ ❡♠ q✉❡ í♥❞✐❝❡s

r❡♣❡t✐❞♦s sã♦ s♦♠❛❞♦s✳ ◆♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s✐ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s é ❝❤❛♠❛❞❛ ❞❡ ✐♥t❡r✈❛❧♦✳ ❙❡ tr❛t❛♥❞♦ ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ ♦ ✐♥t❡r✈❛❧♦ é ❡s❝r✐t♦ ❝♦♠♦

xµyµ = gµνxµyν .

P❛rt✐❝✉❧❛r♠❡♥t❡✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❛♦ q✉❛❞r❛❞♦ ❞❡ ✉♠ ✈❡t♦r ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

xµxµ = gµνxµyν

= x2₋ x02 . ✭✷✳✶✶✮

◆♦t❡ q✉❡✱ ❛♦ ❝♦♥trár✐♦ ❞♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ ✉s✉❛❧✱ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ♥ã♦ é ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✳ ◆❡st❡ ❡s♣❛ç♦ ♣♦❞❡♠ s❡r ❞❡✜♥✐❞❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s q✉❡ ♠❛♥té♠ ❡st❡

(24)

✐♥t❡r✈❛❧♦ ❝♦♥st❛♥t❡✳ ❊st❛s tr❛♥s❢♦r♠❛çõ❡s✱ ❝❤❛♠❛❞❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③✱ sã♦ ❞❡ ❡s♣❡❝✐❛❧ ✐♥t❡r❡ss❡✱ ✉♠❛ ✈❡③ q✉❡✱ ❝♦♠♦ s❡rá ♠♦str❛❞♦ ❛ s❡❣✉✐r✱ ❝♦♠♣♦❡♠ ✉♠ s✉❜❣r✉♣♦ ❞♦ ❣r✉♣♦ ❞❡ P♦✐♥❝❛ré✳

●r✉♣♦ ❞❡ ▲♦r❡♥t③

❖ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ❝♦♥s✐st❡ ♥❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ❡ ❤♦♠♦❣ê♥❡❛s q✉❡ ♠❛♥té♠ ✐♥✲ ✈❛r✐❛♥t❡ ♦ ✐♥t❡r✈❛❧♦ ❞❛❞♦ ♣❡❧❛ ❊q✳ ✭✷✳✶✶✮✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③ é ❡s❝r✐t❛ ❝♦♠♦

x′µ = Λµ_νxν . ✭✷✳✶✷✮

❆♦ ❝♦♠❜✐♥❛r ❛ ❊q✳ ✭✷✳✶✷✮ ❝♦♠ ❛ ❝♦♥❞✐çã♦ ✐♠♣♦st❛ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❝♦♠♦ ❞❛❞❛ ♥❛ ❊q✳ ✭✷✳✶✶✮ ❞❡✈❡ s❡r ✐♥✈❛r✐❛♥t❡✱ ❛ ♠❛tr✐③Λµ

νxν s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦

gµνΛµρΛνσ = gρσ . ✭✷✳✶✸✮

P❛r❛ ❡①♣❧✐❝✐t❛r ❛ ♥❛t✉r❡③❛ ❞❡ ❣r✉♣♦ ❞❡st❛s tr❛♥s❢♦r♠❛çõ❡s✱ ✈❡r✐✜❝❛♠✲s❡ ❛s q✉❛tr♦ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ s❡❥❛ ✉♠ ❣r✉♣♦✳

✶✳ ❋❡❝❤❛♠❡♥t♦

❙♦❜ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✉s✉❛❧ ❞❡ ♠❛tr✐③❡s✱ ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ é

Λµ_σ = Λ₁µ_νΛ₂µ_ν ,

(25)

❝♦♠ Λ1 ❡Λ2 s❛t✐s❢❛③❡♥❞♦ ❛ ❊q✳ ✭✷✳✶✸✮✳ ❉❡ ❢❛t♦✱

gµνΛ1µρΛ1ν σ = gρσ ,

gµνΛ1µρΛ1ν σΛ ρ

2 δΛ2σγ = gρσΛ2ρδΛ2σ γ ,

= gδγ ,

gµνΛµδΛ ν

γ = gδγ . ✭✷✳✶✹✮

▲♦❣♦✱ ✜❝❛ ❝❧❛r♦ q✉❡ ❞✉❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ s✉❝❡ss✐✈❛s ❛✐♥❞❛ sã♦ ✉♠❛ tr❛♥s✲ ❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③✳

✷✳ ❆ss♦❝✐❛t✐✈✐❞❛❞❡

❆ tr❛♥s❢♦r♠❛çã♦ t❡♠ ❝❛rát❡r ♠❛tr✐❝✐❛❧❀ ❡ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ♠❛tr✐❝✐❛❧ é ❛ss♦❝✐❛t✐✈♦✱ ❡st❛ ❝♦♥❞✐çã♦ é ❛✉t♦♠❛t✐❝❛♠❡♥t❡ s❛t✐s❢❡✐t❛✳

✸✳ ■❞❡♥t✐❞❛❞❡

❆ tr❛♥s❢♦r♠❛çã♦Λ_Iµ_ν =δµ

ν♠❛♥té♠ ❛ ❊q✳ ✭✷✳✶✶✮ ✐♥✈❛r✐❛♥t❡ ❡ t❡♠ ❝♦♠♦ ♣r♦♣r✐❡❞❛❞❡

q✉❡ Λ_·δ =δ_·Λ = Λ✳ P♦rt❛♥t♦✱ ♦ ♣❛♣❡❧ ❞❡ ✐❞❡♥t✐❞❛❞❡ é ❝✉♠♣r✐❞♦ ♣❡❧❛ ♠❛tr✐③δµ ν✳

✹✳ ■♥✈❡rs❛ ❆ ❊q✳ ✭✷✳✶✸✮ ♣♦❞❡ s❡r r❡❡s❝r✐t❛

ΛνρΛνσ = gρσ ,

Λ ρ

ν Λνσ = δρσ .

❉❡st❛ ❢♦r♠❛✱ t♦❞❛ tr❛♥s❢♦r♠❛çã♦ t❡♠ ✐♥✈❡rs❛ ❡s❝r✐t❛ ❝♦♠♦

Λ−1_σν = (Λ)ν_σ . ✭✷✳✶✺✮

❊st❡ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❣❛r❛♥t❡ q✉❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ❢♦r♠❛♠ ✉♠ ❣r✉♣♦ ❝♦♥tí♥✉♦✳

(26)

❱♦❧t❛♥❞♦ ❛ ❛t❡♥çã♦ à á❧❣❡❜r❛ ❞❡ ▲✐❡ r❡❧❛❝✐♦♥❛❞❛ ❛♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③✱ é r❡❧❡✈❛♥t❡ ❡s❝r❡✈❡r ❛s tr❛♥s❢♦r♠❛çõ❡s ❡♠ s✉❛ ❢♦r♠❛ ✐♥✜♥✐t❡s✐♠❛❧

Λµ_ν = δµ_ν +δω_νµ , Λ−1µ_ν = δµν −δωνµ ,

Λµν = gµν − δωµν = Λνµ .

✭✷✳✶✻✮

▲♦❣♦

δωµν = −δωνµ . ✭✷✳✶✼✮

❖✉ s❡❥❛✱ δωµν é ✉♠❛ ♠❛tr✐③ ❛♥t✐ss✐♠étr✐❝❛ ❞❡ ❡♥tr❛❞❛s ♣❡q✉❡♥❛s✱ ❞❡ ♠♦❞♦ q✉❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❛❝✐♠❛ s❡❥❛♠ ♣ró①✐♠❛s ❛ ✐❞❡♥t✐❞❛❞❡✳ ❙❡❥❛ ❞ ❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ s♦❜r❡ ♦ q✉❛❧ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ sã♦ r❡❛❧✐③❛❞❛s✳ P♦r s❡r ❛♥t✐ss✐♠étr✐❝❛✱ δωµν t❡♠ ❛♣❡♥❛s1

2d(d−1)❝♦♠♣♦♥❡♥t❡s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ▲♦❣♦✱ ❛s tr❛♥s❢♦r♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐s

❛❝✐♠❛ sã♦ ❣❡r❛❞❛s ♣♦r 1

2d(d−1)❣❡r❛❞♦r❡s ✐♥✜♥✐t❡s✐♠❛✐s ✐♥❞❡♣❡♥❞❡♥t❡s✳

◆❡♠ t♦❞❛s ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❛♦ s❡ ❝♦♠♣♦r tr❛♥s❢♦r✲ ♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐s✳ ❚♦♠❛♥❞♦ ♦ ❞❡t❡r♠✐♥❛♥t❡ ♥❛ ❊q✳ ✭✷✳✶✸✮✱ t❡♠✲s❡ q✉❡ det (Λ−1_{) =}

det (Λ)✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ det (Λ) = ±1✳ ❆s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ q✉❡ ♦❜❡❞❡❝❡♠ det (Λ) = 1sã♦ ❝❤❛♠❛❞❛s tr❛♥s❢♦r♠❛çõ❡s ♣ró♣r✐❛s✳ ◆♦t❡ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s tr❛♥s❢♦r✲

♠❛çõ❡s ♣ró♣r✐❛s é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♣ró♣r✐❛ ❡ q✉❡ tr❛♥s❢♦r♠❛çõ❡s ❞❛ ❢♦r♠❛Λ = 1 +δω

t❛♠❜é♠ sã♦ ♣ró♣r✐❛s✳ P♦rt❛♥t♦✱ ❝♦♠♣♦s✐çã♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐s só ♣♦❞❡ ❣❡r❛r ✉♠ s✉❜❣r✉♣♦ ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ❡♠ q✉❡ t♦❞❛s ❛s tr❛♥s❢♦r♠❛çõ❡s sã♦ ♣ró♣r✐❛s✱ ❡ ❛ss✐♠ ❝♦♥❡❝t❛❞❛s ❛ ✐❞❡♥t✐❞❛❞❡✳

❯♠ ♦✉tr♦ ♣♦ssí✈❡❧ s✉❜❣r✉♣♦ ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ é ♦ s✉❜❣r✉♣♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ♦rtó❝r♦♥❛s✳ ❆ ❊q✳ ✭✷✳✶✷✮ ✐♠♣❧✐❝❛ ❡♠(Λ0

0) 2

−(Λi

0) 2

= 1✳ P♦rt❛♥t♦ Λ0

0 ≤ −1♦✉Λ00 ≥1✳

(27)

❆s tr❛♥s❢♦r♠❛çõ❡s q✉❡ ♦❜❡❞❡❝❡♠ Λ0

0 ≥ 1 sã♦ tr❛♥s❢♦r♠❛çõ❡s ♦rtó❝r♦♥❛s✳ ◆♦✈❛♠❡♥t❡✱

♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ❞❡❧❛s s❡♠♣r❡ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♦rtó❝r♦♥❛ ❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐s ❝♦♠♦ ❞❡s❝r✐t❛s ❛❝✐♠❛ sã♦ ♦rtó❝r♦♥❛s✳

❆♣❡♥❛s ♦ s✉❜❣r✉♣♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ♣ró♣r✐❛s ❡ ♦rtó❝r♦♥❛s ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞❛ ❝♦♠♣♦s✐çã♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐s ✭❞✐③✲s❡ q✉❡ ❛♣❡♥❛s ❡st❡ s✉❜❣r✉♣♦ é s✉❛✈❡♠❡♥t❡ ❝♦♥❡①♦ ❛ ✐❞❡♥t✐❞❛❞❡✮✳ ❆s tr❛♥❢♦r♠❛çõ❡s q✉❡ ♥ã♦ s❛t✐s❢❛③❡♠ ❡st❡ s✉❜❣r✉♣♦ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❢❛③❡♥❞♦ ✉s♦ ❞❡ ❞✉❛s tr❛♥s❢♦r♠❛çõ❡s ❞✐s❝r❡t❛s✳

• ❆ tr❛♥s❢♦r♠❛çã♦ ❞❡ P❛r✐❞❛❞❡ é ❡s❝r✐t❛ ❝♦♠♦

Pµ ν =            

1 0 0 0 . . . 0 ₋1 0 0 . . . 0 0 ₋1 0 . . . 0 0 0 ₋1 . . . . . . .             ,

❊ é ♦rtó❝r♦♥❛✱ ♣♦ré♠ ✐♠♣ró♣r✐❛✳ ❊st❛ tr❛♥s❢♦r♠❛çã♦ ❧❡✈❛ tr❛♥s❢♦r♠❛çõ❡s ♣ró♣r✐❛s ❡♠ ✐♠♣ró♣r✐❛s✳ ❱❛❧❡ ♥♦t❛r q✉❡ ♥♦ ❝❛s♦ ❞❡ ❞✐♠❡♥sõ❡s í♠♣❛r❡s✱ det(_P) = 1✳ ❊♠

❞✐♠❡♥sõ❡s í♠♣❛r❡s✱ ♥ã♦ ❡①✐st❡ ♠❛♥❡✐r❛ ❢á❝✐❧ ❞❡ s❡ ❞✐st✐♥❣✉✐r tr❛♥s❢♦r♠❛çõ❡s ♣ró♣r✐❛s ❞❡ ✐♠♣ró♣r✐❛s✳

• ❆ tr❛♥s❢♦r♠❛çã♦ ❞❡ r❡✈❡rsã♦ t❡♠♣♦r❛❧ é ❞❛❞❛ ♣♦r

Tµν =

           

−1 0 0 0 . . . 0 1 0 0 . . . 0 0 1 0 . . . 0 0 0 1 . . . . . . .             ,

❡ é ✐♠♣ró♣r✐❛ ❡ ♥ã♦✲♦rtó❝r♦♥❛✳ ▲❡✈❛ tr❛♥s❢♦r♠❛çõ❡s ♦rtó❝r♦♥❛s ❡♠ ♥ã♦ ♦rtó❝r♦♥❛s✳

(28)

P❛r❛ ♦❜t❡r ❣❡r❛❞♦r❡s ✐♥✜♥✐t❡s✐♠❛✐s ♣❛r❛ ♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③✱ é ❝♦♥✈❡♥✐❡♥t❡ tr❛❜❛❧❤❛r ❡♠ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ ❡♠ t❡r♠♦s ❞❡ ♦♣❡r❛❞♦r❡s ❞❡✜♥✐❞♦s ❡♠ ✉♠ ❡s♣❛ç♦ ✐♥✜♥✐t♦✲❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠♦ sã♦ ♦s ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♣♦r ❡①❡♠♣❧♦✳ ❈❛❞❛ ❡❧❡♠❡♥t♦ Λ é ❡♥tã♦ r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ♦♣❡r❛❞♦r U(Λ) t❛❧ q✉❡ s❡❥❛ ♣r❡s❡r✈❛❞❛ ❛ r❡❣r❛

❞❡ ❝♦♠♣♦s✐çã♦ ❞♦ ❣r✉♣♦

U(Λ′Λ) = U(Λ′)U(Λ) . ✭✷✳✶✽✮

❊♥tã♦✱ ♣❛r❛ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✜♥✐t❡s✐♠❛❧ ❡s❝r❡✈❡✲s❡

U(1 +δω) = I + i

2δωµνM

µν _, _{✭✷✳✶✾✮}

❝♦♠Mµν ₌₋_Mνµ _{❞❡s❝r❡✈❡♥❞♦ ♦s ❣❡r❛❞♦r❡s ❞❛ á❧❣❡❜r❛ ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③✳ P❛r❛ ♦❜t❡r} ❛ t❛❜❡❧❛ ❞❡ ❝♦♠✉t❛çõ❡s ❞❡st❛ á❧❣❡❜r❛✱ ✉s❛✲s❡ ❛ ❊q✳ ✭✷✳✶✽✮ ♣❛r❛ ❡s❝r❡✈❡r

U(Λ)−1U(Λ′)U(Λ) = U Λ−1Λ′Λ .

❚r❛t❛✲s❡✱ ❡♥tã♦✱ Λ′ _{❝♦♠♦ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♣ró①✐♠❛ ❛ ✐❞❡♥t✐❞❛❞❡ ❝♦♠♦ ❞❛❞❛ ♥❛ ❊q✳}

✭✷✳✶✾✮✳ ❊①♣❛♥❞✐♥❞♦ ❡ ♠❛♥t❡♥❞♦ ❛♣❡♥❛s t❡r♠♦s ❧✐♥❡❛r❡s ❡♠δω′_{✱ t❡♠✲s❡}

δω_µν′ U(Λ)−1MµνU(Λ) = δω_µν′ Λµ_ρΛν_σMρσ , U(Λ)−1MµνU(Λ) = Λµ_ρΛν_σMρσ .

❚r❛t❛♥❞♦✱ ❡♥tã♦✱ Λ ❝♦♠♦ ♥❛ ❊q✳ ✭✷✳✶✾✮✱ ❡①♣❛♥❞✐♠♦s ♥♦✈❛♠❡♥t❡ ❛té ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡♠ δω ❡ ✐❣✉❛❧❛♠♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s✳ ❚❡♠♦s

I ₋ i

2δωαβM

αβ

Mµν

I+ i

2δωρσM

ρσ

= Mµν+ i

2δωρσ(M

µν_Mρσ

−MρσMµν) , = Mµν+ i

2δωρσ[M

(29)

❊ ♦ ❧❛❞♦ ❞✐r❡✐t♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ✷✳✶✼ ❡ ❛ ❛♥t✐ss✐♠❡tr✐❛ ❞❡Mµν _✜❝❛

i

2δωρσ[M

µν_{, M}ρσ_{] =} _δων

σMµσ +δωµρMρν ,

= δωρσ(gνρMµσ−gσµMρν)

i 2[M

µν_{, M}ρσ_{] =} 1

2[g

νρ_Mµσ

−gσµMρν ₋gνσMµρ+gρµMσν] , [Mµν, Mρσ] = i[gνσMµρ+gσµMρν+gνρMσµ+gµρMνσ] .

❆tr❛✈és ❞❡st❛s ❝♦♠✉t❛çõ❡s é ❣❡r❛❞❛ ❛ á❧❣❡❜r❛ ❞❡ ▲♦r❡♥t③✳

❖ ●r✉♣♦ ❞❡ P♦✐♥❝❛ré

❖ ❣r✉♣♦ ❞❡ P♦✐♥❝❛ré é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ■♥♦♠♦❣ê♥❡♦✳ ❆s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s q✉❡ ♦ ❝♦♠♣õ❡ sã♦ ❞❛ ❢♦r♠❛

x′ = Λµ_νxν+aµ , ✭✷✳✷✵✮

❝♦♠Λ ✉♠❛ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③ ❡aµ _{✉♠ ✈❡t♦r ❝♦♥st❛♥t❡✳ ❱❡r✐✜❝❛♠✲s❡ ❛s} q✉❛tr♦ ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡st❡ t✐♣♦ ❢♦r♠❡♠ ✉♠ ❣r✉♣♦✿

✶✳ ❋❡❝❤❛♠❡♥t♦

❉❛ ❊q✳ ✭✷✳✷✵✮ ♣♦❞❡ ✲s❡ ❝♦♥❝❧✉✐r q✉❡

x′′ = Λ₁µ_νx′ν+aµ₁ ,

= Λ₁µ_νΛ₂ν _ρxρ+ Λ₁µ_νaν₂+aµ₁ , = Λ₁µ_νΛ₂ν _ρxρ+a′₂µ+aµ₁ .

❯♠❛ ✈❡③ ❥á ❡st❛❜❡❧❡❝✐❞♦ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ é ✉♠❛

(30)

tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③✱ s❡❣✉❡ q✉❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ P♦✐♥❝❛ré sã♦ ❢❡❝❤❛❞❛s q✉❛♥t♦ à ❝♦♠♣♦s✐çã♦✳

✷✳ ❆ss♦❝✐❛t✐✈✐❞❛❞❡

❆s ♦♣❡r❛çõ❡s ❡♥✈♦❧✈✐❞❛s ♥❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ P♦✐♥❝❛ré sã♦ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s✳ ❉❡ss❛ ❢♦r♠❛ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛✳

✸✳ ■❞❡♥t✐❞❛❞❡

❖ ♣❛♣❡❧ ❞❡ ✐❞❡♥t✐❞❛❞❡ é ❡①❡r❝✐❞♦ ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦ q✉❡ t❡♠ Λµ

ν =δµν ❡ aµ= 0✳ ✹✳ ■♥✈❡rs❛ ❉❡♥♦t❛✲s❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❛❞❛ ♥❛ ❊q✳ ✭✷✳✷✵✮ ❝♦♠♦

G_·xµ = x′µ ,

= Λµνxν +aµ . ✭✷✳✷✶✮

P❛r❛ ❡♥❝♦♥tr❛r ❛ ✐♥✈❡rs❛✱ ❡s❝r❡✈❡✲s❡ q✉❡

G−1·G·xµ = xµ ,

G−1·xµ = Λ_νµxν−Λ_νµaν

❈♦♠♦ t♦❞❛ ♠❛tr✐③ ❞❡ ▲♦r❡♥t③ t❡♠ ✐♥✈❡rs❛✱ t♦❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ P♦✐♥❝❛ré t❛♠❜é♠ t❡♠ ✐♥✈❡rs❛✳

P❛r❛ s❡ ♦❜t❡r ❛ á❧❣❡❜r❛ ❞❡ P♦✐♥❝❛ré✱ ♥♦✈❛♠❡♥t❡ é ♥❡❝❡ssár✐♦ ♦ tr❛t❛♠❡♥t♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ ✐♥✜♥✐t♦✲❞✐♠❡♥s✐♦♥❛❧✳ P❛r❛ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✜♥✐t❡s✐♠❛❧✱ t❡♠♦s

U(Λ +a) = I+ i

2δωαβM

αβ₊_ε αPα ,

U(Λ +a)−1 = I− i

2δωαβM

αβ ₋_ε αPα .

(31)

❈♦♠ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❛♥á❧♦❣♦ ❛♦ ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③✱ ♦❜té♠✲s❡ ❛ t❛❜❡❧❛ ❞❡ ❝♦♠✉t❛çã♦ ♣❛r❛ ❛ á❧❣❡❜r❛ ❞❡ P♦✐♥❝❛ré

[Pµ, Pν] = 0 ,

[Pµ, Mνρ] = i(gµνPρ−gρνPµ) ,

[Mµν, Mρσ] = i[gνσMµρ+gσµMρν+gνρMσµ+gµρMνσ] .

❉❡st❛ ❢♦r♠❛✱ ❛ á❧❣❡❜r❛ ❞❡ P♦✐♥❝❛ré é ❝♦♥str✉í❞❛ ❡♠ s✉❛ ❢♦r♠❛ ❛❜str❛t❛✳ ❱❛❧❡ ♥♦t❛r q✉❡✱ ❛ss✐♠ ❝♦♠♦ ♦❝♦rr❡✉ ♥♦ ❝❛s♦ ❤♦♠♦❣ê♥❡♦✱ ❛♣❡♥❛s tr❛♥s❢♦r♠❛çõ❡s ♣ró♣r✐❛s ❡ ♦rtó❝r♦♥❛s ♣♦✲ ❞❡♠ s❡r ♦❜t✐❞❛s ♣♦r ❝♦♠♣♦s✐çã♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐s✳ ❆ s❡❣✉✐r ❝♦♥str✉✐r❡♠♦s r❡♣r❡s❡♥t❛çõ❡s ❡①♣❧í❝✐t❛s ❞♦s ❣❡r❛❞♦r❡s ❡ ❡st✉❞❛r❡♠♦s s❡✉s s✐❣♥✐✜❝❛❞♦s✳

❘❡♣r❡s❡♥t❛çã♦ ❊①♣❧í❝✐t❛ ♣❛r❛ ♦s ●❡r❛❞♦r❡s

P♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦s ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ❞❡ ♠♦❞♦ q✉❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛ ♠❡❝â♥✐❝❛ r❡❧❛t✐✈íst✐❝❛ s❡❥❛ ♠❛✐s ✐♠❡❞✐❛t❛✳ ❉❡✜♥❡✲s❡ ❡♥tã♦

Ji =

1 2εijkM

jk _,

Ki = Mi0 .

(32)

❆s r❡❧❛çõ❡s ❞❡ ❝♦♠✉t❛çã♦ s❡ t♦r♥❛♠ ❡♥tã♦

[Ji, Jj] = iεijkJk ,

[Ji, Kj] = iεijkKk ,

[Ki, Kj] = −iεijkJk ,

[Ji, Pj] = iεijkPk ,

[Ji, P0] = 0 ,

[Ki, Pj] = iδijP0 ,

[Ki, P0] = iPi ,

i, j, k = 1. . . d−1 .

❉❡ss❡ ♠♦❞♦ ♦s ❣❡r❛❞♦r❡s Ji ❞❡s❝r❡✈❡♠ r♦t❛çõ❡s ❞♦ t✐♣♦ ❛s ❞♦ Rd−1 ❡ ♦s ❣❡r❛❞♦r❡s Ki ❞❡s❝r❡✈❡♠ r♦t❛çõ❡s q✉❡✱ ♥♦ ❝❛s♦ ❞♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐✱ ♠✐st✉r❛♠ ❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ t✐♣♦ t❡♠♣♦ ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ t✐♣♦ ❡s♣❛ç♦✳ ❘♦t❛çõ❡s ❞❡st❡ t✐♣♦ sã♦ ❝❤❛♠❛❞❛s ❜♦♦sts✱ ♦✉ ✐♠♣✉❧s♦s✳

◆❡st❛ ❞✐ss❡rt❛çã♦✱ ♦s ❝❛s♦s ❞❡ ♠❛✐♦r ✐♥t❡r❡ss❡ sã♦ ♦s ❝❛s♦s ❝♦♠ ❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦✲ t❡♠♣♦ d = 2, 3, 4✳ ◆❡st❛s ❞✐♠❡♥sõ❡s é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ♦s ❣❡r❛❞♦r❡s ❞❡ ▲♦r❡♥t③ ♥♦

❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❝♦♠♦ ✉♠❛ ♠❛tr✐③✱ ❞❡ ♠♦❞♦ ❛ t♦r♥❛r ❛ ❡s❝r✐t❛ ♠❛✐s ❝♦♠♣❛❝t❛✳ ❊♠