Aspectos clássicos de teorias de segunda ordem

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✡✡✡ ✪✪

✪ ✱✱

✱✱ ✑✑✑ ✟✟ ❡

❡ ❡ ❅

❅_❅ ❧

❧ ❧ ◗

◗◗ ❍ ❍P P P ❳❳ ❳ ❤❤ ❤ ❤

✭ ✭ ✭

✭✏✟✏

■❋❚

■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❚❡ór✐❝❛

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛

❚❊❙❊ ❉❊ ❉❖❯❚❖❘❆▼❊◆❚❖ ■❋❚✕❚✳✵✵✺✴✵✾

❆s♣❡❝t♦s ❝❧áss✐❝♦s ❞❡ t❡♦r✐❛s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠

P❡❞r♦ ❏♦sé P♦♠♣❡✐❛

❖r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ❇r✉t♦ ▼❛① P✐♠❡♥t❡❧ ❊s❝♦❜❛r

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❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣❡❧♦ ❡t❡r♥♦ ✐♥❝❡♥t✐✈♦ ❡ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ♣❛✐s✱ à ❈❛r❧❛ ❡ à ❆♥♥❛ ❙♦♣❤✐❛✱ ♣♦r t✉❞♦✳

❆♦ Pr♦❢✳ P✐♠❡♥t❡❧ ♣♦r s✉❛ ♦r✐❡♥t❛çã♦✱ ♣❛❝✐ê♥❝✐❛✱ ❡s❢♦rç♦ ❡ s✐♥❝❡r✐❞❛❞❡ t❛♥t♦ ❡♠ ❛ss✉♥t♦s ❛❝❛❞ê♠✐❝♦s✱ ❝♦♠♦ ❡♠ ❛ss✉♥t♦s ♣❛rt✐❝✉❧❛r❡s✳

❆♦ Pr♦❢✳ ▼❛r❝ ❞❡ ▼♦♥t✐❣♥② ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ ❞✉r❛♥t❡ ♦ ❞♦✉t♦r❛❞♦✲s❛♥❞✉í❝❤❡ ♥♦ ❈❛♥❛❞á ❡ à s✉❛ ❢❛♠í❧✐❛ q✉❡ tã♦ ❜❡♠ r❡❝❡❜❡✉ ❛ ♠✐♠ ❡ à ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ❆♦ Pr♦❢✳ ❑❤❛♥♥❛✱ q✉❡ t❛♠❜é♠ ♠✉✐t♦ ❝♦♥tr✐❜✉✐✉ ❝♦♠ s✉❛ ❡①♣❡r✐ê♥❝✐❛ ❞✉r❛♥t❡ ❡st❡ ♣❡rí♦❞♦✳

❆ t♦❞♦ ♦ ♣❡ss♦❛❧ ❞♦ ■❋■ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❛♣♦✐♦ ❛♦ ♠❡✉ ❛♣r✐♠♦r❛♠❡♥t♦ ♣r♦✜s✲ s✐♦♥❛❧✳

❆♦s ❛♠✐❣♦s ❈áss✐✉s✱ ▲é♦ ❡ ❘♦❞r✐❣♦ ♣❡❧❛s t❛♥t❛s ❞✐s❝✉ssõ❡s ❡ ❝♦❧❛❜♦r❛çõ❡s✳ ❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ■❋❚ ♣❡❧♦ ❛♣♦✐♦ ♣r❡st❛❞♦✳

❆♦s t❛♥t♦s ❛♠✐❣♦s q✉❡ ♥ã♦ ❢♦r❛♠ ❝✐t❛❞♦s✱ ♠❛s q✉❡ ❞❡ ❢♦r♠❛ ❛❧❣✉♠❛ ❢♦r❛♠ ❡sq✉❡❝✐❞♦s✱ ❡ q✉❡✱ ❞❡ ♠♦❞♦ ❞✐r❡t♦ ♦✉ ✐♥❞✐r❡t♦✱ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦ ❈◆Pq ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞♦ ❞♦✉t♦r❛❞♦✲s❛♥❞✉í❝❤❡✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❢♦r❛♠ ❡st✉❞❛❞♦s ❛s♣❡❝t♦s ❝❧áss✐❝♦s ❡ s❡♠✐❝❧áss✐❝♦s ❞❡ t❡♦r✐❛s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ Pr✐♠❡✐r❛♠❡♥t❡ ❢♦✐ ❢❡✐t❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ ❣❛✉❣❡ ♣❛r❛ s✐st❡♠❛s ❡♠ q✉❡ ❛ ▲❛❣r❛♥❣✐❛♥❛ ❞♦ ❝❛♠♣♦ ❞❡ ❣❛✉❣❡ ❞❡♣❡♥❞❡ ❞❡ s✉❛s ❝♦♠✲ ♣♦♥❡♥t❡s ❡ ❞❡ s✉❛s ❞❡r✐✈❛❞❛s ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ❡ ❛♣❧✐❝❛çõ❡s ❞❡st❛ ❡str✉t✉r❛ ♣❛r❛ ♦s ❝❛s♦sU(1)✲ ❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ P♦❞♦❧s❦② ✲ ❡SU(N)✲ ▲❛✲ ❣r❛♥❣✐❛♥❛ ❡❢❡t✐✈❛ ❞❡ ❆❧❡❦s❡❡✈✱ ❆❜✉③♦✈ ❡ ❇❛✐❦♦✈✳ ❖ ✉s♦ ❞❡st❛ ❡str✉t✉r❛ ♣❛r❛ ♦ ❝❛s♦ ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③SO(3,1)❢♦✐ ❢❡✐t♦ ❡♠ s❡♣❛r❛❞♦ ❡ ✉♠❛ ❛♥á❧✐s❡ ❞❛ s♦❧✉çã♦ ❡stát✐❝❛ ❡ ✐s♦tró♣✐❝❛ ♥♦ r❡❣✐♠❡ ❧✐♥❡❛r ❢♦✐ ❝♦♥❞✉③✐❞❛ ♣❛r❛ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r✳ ◆❛ s❡q✉ê♥❝✐❛✱ ❛♥❛❧✐s♦✉✲s❡ ❝♦♠♦ ❛ t❡♦r✐❛ ❞❡ P♦❞♦❧s❦② ♣♦❞❡r✐❛ s❡r ✈✐♥❝✉❧❛❞❛ ❞❡ três ♠❛♥❡✐r❛s ❞✐st✐♥t❛s✱ ❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞✐s♣♦♥í✈❡✐s ♥❛ ❧✐t❡r❛t✉r❛✳ ❋✐♥❛❧♠❡♥t❡✱ ❡st✉❞♦✉✲s❡ r❡❣✐♠❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ❞❛ ❡❧❡tr♦❞✐♥â♠✐❝❛ ❞❡ P♦❞♦❧✲ s❦②✱ ✉t✐❧✐③❛♥❞♦✲s❡ ♦ ❢♦r♠❛❧✐s♠♦ ❣❛❧✐❧❡❛♥♦ ❡♠ ✺ ❞✐♠❡♥sõ❡s✳ ❖s ❧✐♠✐t❡s ❡❧étr✐❝♦ ❡ ♠❛❣♥ét✐❝♦ ❞❛ t❡♦r✐❛ ❢♦r❛♠ ❛♥❛❧✐s❛❞♦s✳

P❛❧❛✈r❛s ❈❤❛✈❡s✿ ❚❡♦r✐❛ ❞❡ ❣❛✉❣❡✱ ❚❡♦r✐❛s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ♥❛ ❉❡r✐✈❛❞❛✱ ●r❛✈✐t❛çã♦✱ ❊❧❡tr♦❞✐♥â♠✐❝❛ ●❡♥❡r❛❧✐③❛❞❛ ❞❡ P♦❞♦❧s❦②✱ ❈♦✈❛r✐â♥❝✐❛ ●❛❧✐❧❡❛♥❛✳ ➪r❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✿ ❚❡♦r✐❛ ❞❡ ❈❛♠♣♦s✱ ●r❛✈✐t❛çã♦✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ❝❧❛ss✐❝❛❧ ❛♥❞ s❡♠✐❝❧❛ss✐❝❛❧ ❛s♣❡❝ts ♦❢ s❡❝♦♥❞ ♦r❞❡r t❤❡♦r✐❡s ✇❡r❡ ❛♥❛❧②③❡❞✳ ❋✐rst t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❛ ❣❛✉❣❡ t❤❡♦r② ❢♦r s②st❡♠s ✇❤♦s❡ ▲❛❣r❛♥❣✐❛♥ ❢♦r t❤❡ ❣❛✉❣❡ ✜❡❧❞ ❞❡♣❡♥❞s ♦♥ t❤❡ ✜❡❧❞ ✐ts❡❧❢ ❛♥❞ ✐ts ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ✇❛s ♣r♦♣♦s❡❞✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤✐s str✉❝t✉r❡ ❢♦r t❤❡U(1)❣r♦✉♣ ✲ P♦❞♦❧s❦② ❣❡♥✲ ❡r❛❧✐③❡❞ ❡❧❡❝tr♦❞②♥❛♠✐❝s ✲ ❛♥❞ ❢♦rSU(N)✲ ❆❧❡❦s❡❡✈✱ ❆❜✉③♦✈ ❡ ❇❛✐❦♦✈ ❡✛❡❝t✐✈❡ ▲❛❣r❛♥❣✐❛♥ ✲ ✇❡r❡ ♠❛❞❡✳ ❚❤❡ ▲♦r❡♥t③ ❣r♦✉♣ SO(3,1) ✇❛s ❛❧s♦ ❛♥❛❧②③❡❞ ❛♥❞ ❛♥ ✐s♦tr♦♣✐❝ ❛♥❞ st❛t✐❝ s♦❧✉t✐♦♥ ✐♥ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✇❛s ♦❜t❛✐♥❡❞ ❢♦r ❛ ♣❛r✲ t✐❝✉❧❛r ❝❛s❡✳ ◆❡①t ✐t ✇❛s ❛♥❛❧②③❡❞ ❤♦✇ P♦❞♦❧s❦② t❤❡♦r② ❝♦✉❧❞ ❜❡ ❝♦♥str❛✐♥❡❞ ✐♥ t❤r❡❡ ❞✐✛❡r❡♥t ✇❛②s ✉s✐♥❣ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ❛✈❛✐❧❛❜❧❡ ♦♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❋✐✲ ♥❛❧❧② t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❧✐♠✐ts ♦❢ P♦❞♦❧s❦② ❡❧❡❝tr♦❞②♥❛♠✐❝s ✇❡r❡ ❛♥❛❧②③❡❞ ♦♥ t❤❡ ✺✲❞✐♠❡♥s✐♦♥❛❧ ❣❛❧✐❧❡❛♥ ❢♦r♠❛❧✐s♠✳ ❚❤❡ ❡❧❡❝tr✐❝ ❛♥❞ ♠❛❣♥❡t✐❝ ❧✐♠✐ts ✇❡r❡ ♦❜t❛✐♥❡❞✳

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❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ à ❧❛ ❯t✐②❛♠❛ ✼

✷✳✶ ❆ ❢♦r♠✉❧❛çã♦ ❞❡ ❯t✐②❛♠❛ ✲ ♦ ❝❛s♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✶✳✶ ■♥✈❛r✐â♥❝✐❛ ▲♦❝❛❧ ❡ ♦ ❈❛♠♣♦A ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷✳✶✳✷ ❊q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ♣❛r❛A❧✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✶✳✸ ❈❛♠♣♦s ❡♠ ■♥t❡r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✹ ❖ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ❡ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❚❡♦r✐❛ ❞❡ ❣❛✉❣❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✶ ❆ ▲❛❣r❛♥❣✐❛♥❛ ❞♦ ❝❛♠♣♦ ❞❡ ❣❛✉❣❡ ❧✐✈r❡ ❡♠ s❡❣✉♥❞❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳✷ ❚❡♦r✐❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✲ ♦ ❝❛s♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸ ❱✐♥❝✉❧❛♥❞♦✲s❡ ❛ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❞❡ P♦❞♦❧s❦② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸✳✶ ❆ ❝♦rr❡♥t❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡ ❛ ♠❛ss❛ ❞♦ ❢ót♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸✳✷ ■♥t❡r❢❡r♦♠❡tr✐❛ ❞❡ í♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✸✳✸ ❖ át♦♠♦ ❞❡ ❍✐❞r♦❣ê♥✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✹ ❆rt✐❣♦s ❡ ♣r❡♣r✐♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✹✳✶ ❚❡♦r✐❛ ❞❡ ❣❛✉❣❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✹✳✷ ❋♦r♠✉❧❛çã♦ ❞❡ ❣❛✉❣❡ ♣❛r❛ ❣r❛✈✐❞❛❞❡ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✷✳✹✳✸ ❈♦♠♦ ✐♥✈❡st✐❣❛r ❛ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❞❡ P♦❞♦❧s❦②❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✸ ❋♦r♠❛❧✐s♠♦ ❝♦✈❛r✐❛♥t❡ ❞❡ ●❛❧✐❧❡✐ ✾✵

✸✳✶ ❖s ❧✐♠✐t❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ▼❛①✇❡❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✸✳✶✳✶ ❆ ❛❜♦r❞❛❣❡♠ ❞❡ ▲é✈②✲▲❡❜❧♦♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✸✳✶✳✷ ❆ ❛❜♦r❞❛❣❡♠ ✺✲❞✐♠❡♥s✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✸✳✷ ❆ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ P♦❞♦❧s❦② ❡ s❡✉s ❧✐♠✐t❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ✳ ✳ ✳ ✳ ✶✵✺ ✸✳✷✳✶ ❖ ❧✐♠✐t❡ ♠❛❣♥ét✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻ ✸✳✷✳✷ ❖ ❧✐♠✐t❡ ❡❧étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵

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✹ ❈♦♥❝❧✉sõ❡s ✶✶✹

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

➱ ❝♦♠✉♠ s❡ ❞✐③❡r q✉❡ ♦s ❞♦✐s ♣✐❧❛r❡s ❞❛ ❋ís✐❝❛ ▼♦❞❡r♥❛ ❞❡s❡♥✈♦❧✈✐❞❛ ❛♦ ❧♦♥❣♦ ❞♦ sé❝✉❧♦ ❳❳ sã♦ ❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❡ ❈❛♠♣♦s ✭❚◗❈✮ ❡ ❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✭❘●✮✳ ❆ ♣r✐♠❡✐r❛ ❞❡s❝r❡✈❡ ❝♦♠ ❜❛st❛♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❛ ❋ís✐❝❛ ❡♠ ❡s❝❛❧❛s ♠✐❝r♦s❝ó♣✐❝❛s✱ ❞❡s❞❡ s❡✉s ❝♦♥st✐t✉✐♥t❡s ♠❛✐s ❡❧❡♠❡♥t❛r❡s ❡ s✉❛s ✐♥t❡r❛çõ❡s✱ s❡♥❞♦ ✉♠ ❞♦s ❣r❛♥❞❡s s✉❝❡ss♦s ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❞❛ ❋ís✐❝❛ ❞❡ P❛rtí❝✉❧❛s ❊❧❡♠❡♥t❛r❡s✳ ❆ s❡❣✉♥❞❛✱ s❡♥❞♦ ✉♠❛ t❡♦r✐❛ ❞❡ ❣r❛✈✐t❛çã♦✱ ❞❡s❝r❡✈❡ ❜❡♠ ♦s ❢❡♥ô♠❡♥♦s ❡♠ ❡s❝❛❧❛s ♠❛❝r♦s❝ó♣✐❝❛s✱ ❞❛♥❞♦ ❝♦♥t❛ ❞❡ ❢❡♥ô♠❡♥♦s q✉❡ ♦❝♦rr❡♠ ❡♠ ❞✐♠❡♥sõ❡s ❞❛ ♦r❞❡♠ ❞❡ ❝❡♥tí♠❡tr♦s ❛ ❡s❝❛❧❛s ❝ós♠✐❝❛s✱ s❡♥❞♦ ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❞❛ ❈♦s♠♦❧♦❣✐❛ ✉♠ ❞♦ ❣r❛♥❞❡s ❢r✉t♦s ❞❡st❛ t❡♦r✐❛✳ ❚❛♥t♦ ❛ ❚◗❈✱ q✉❛♥t♦ ❛ ❘●✱ t✐✈❡r❛♠ s❡✉s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❛ss♦❝✐❛❞♦s ❛♦ ❡st✉❞♦ ❞❡ ✉♠ ♦✉tr♦ ❝♦♥❝❡✐t♦✱ q✉❡ s❡ t♦r♥♦✉ ♣❛r❛❞✐❣♠❛ ❞❛ ❋ís✐❝❛ ❞♦ sé❝✉❧♦ ❳❳✿ ❙✐♠❡tr✐❛✳

❉✉❛s s✐♠❡tr✐❛s ♥♦r♠❛❧♠❡♥t❡ ❝♦rr❡♠ ♣♦r trás ❞❡ ✉♠❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s✳ ❯♠❛ é ❛ s✐♠❡tr✐❛ ❞♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦✱ q✉❡ ❞✐③ r❡s♣❡✐t♦ à s✐♠❡tr✐❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡♠ q✉❡ ♦ s✐st❡♠❛ ❢ís✐❝♦ é ❝♦♥s✐❞❡r❛❞♦✱ s❡♥❞♦ ❡st❡ ú❧t✐♠♦ ♥♦r♠❛❧♠❡♥t❡ ❞❡s❝r✐t♦ ♣♦r ❝❛♠♣♦s q✉❡ s❡❥❛♠ r❡♣r❡s❡♥t❛çõ❡s ✐rr❡❞✉tí✈❡✐s ❞♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦ ❬✶❪✳ ❊st❛ s✐♠❡tr✐❛ ❣❡r❛❧♠❡♥t❡ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡s❝r❡✈❡r ❝❛♠♣♦s ❧✐✈r❡s✱ s❡♠ ✐♥t❡r❛çã♦ ❝♦♠ ♦✉tr♦s ❝❛♠♣♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦ ❝❛♠♣♦ ❞❡ ❉✐r❛❝✱ Pr♦❝❛✱ ❑❧❡✐♥✲●♦r❞♦♥✱ s❡♠ ✐♥t❡r❛çã♦ ❝♦♠ ❝❛♠♣♦s ❝♦♠♦ ♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦✱ ❣r❛✈✐t❛❝✐♦♥❛❧✱ ❡t❝✳✳

❆♦ ❧♦♥❣♦ ❞♦ sé❝✉❧♦ ❳❳✱ ❛ ❝♦♥str✉çã♦ ❞❛s t❡♦r✐❛s ❞❡ ❝❛♠♣♦s ❢♦✐ ❢❡✐t❛ ❝♦♠ ♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ s❡♥❞♦ ♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦ ♥❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✳ ❊st❛ ❡s❝♦❧❤❛ s❡ ❥✉st✐✜❝❛ ✉♠❛ ✈❡③ q✉❡ ❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③ é ❛q✉❡❧❛ q✉❡ ❞❡s❝r❡✈❡ ❛❞❡q✉❛❞❛♠❡♥t❡ s✐st❡♠❛s ❡♠ r❡❣✐♠❡s r❡❧❛t✐✈íst✐❝♦s❀ ❞❡✈❡✲s❡ ❝♦♥s✐❞❡r❛r ❛✐♥❞❛ q✉❡✱ ❝♦♠ ❡st❛ s✐♠❡tr✐❛✱ é ♣♦ssí✈❡❧ t❛♠❜é♠ s❡ ❛♥❛❧✐s❛r ♦s r❡❣✐♠❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s✱ ♦ q✉❡ ♥♦r♠❛❧♠❡♥t❡ é ❢❡✐t♦ q✉❛♥❞♦ s❡ ❝♦♥s✐❞❡r❛ ♦ ❧✐♠✐t❡c→ ∞✳ ❋♦✐ ❛♣❡♥❛s ❡♠ ✶✾✻✼

❬✶❪✱ ❝❡r❝❛ ❞❡ ✻✵ ❛♥♦s ❞❡♣♦✐s q✉❡ ❛ ❘❡❧❛t✐✈✐❞❛❞❡ ❘❡str✐t❛ ❤❛✈✐❛ s✐❞♦ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❊✐♥st❡✐♥✱ q✉❡ ♦ q✉❡st✐♦♥❛♠❡♥t♦ s♦❜r❡ ❡st❛ ♣r❡s❝r✐çã♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ r❡❣✐♠❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ✭✐✳❡✳ c → ∞✮

❢♦✐ ❢❡✐t❛ ♣♦r ▲é✈②✲▲❡❜❧♦♥❞✳ ❊♠ s✉❛ ♦♣✐♥✐ã♦✱ ❛ ❢♦r♠❛ ❛❞❡q✉❛❞❛ ❞❡ ❡st✉❞❛r ❡st❡s r❡❣✐♠❡s ❞❡✈❡r✐❛♠ ❧❡✈❛r ❡♠ ❝♦♥t❛ ❛ s✐♠❡tr✐❛ ❞❡ ●❛❧✐❧❡✐ ❝♦♠♦ ♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦ ♥♦ ❧✉❣❛r ❞❛ s✐♠❡tr✐❛ ❞❡ ▲♦r❡♥t③✳ ❊♠ ❬✶❪ ▲é✈②✲▲❡❜❧♦♥❞ ♠♦str♦✉✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ é ♣♦ssí✈❡❧ ❞❡s❝r❡✈❡r ✉♠ ❝❛♠♣♦ ❞❡ ❉✐r❛❝ ❝♦♠ s✐♠❡tr✐❛

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❞❡ ●❛❧✐❧❡✐ ✭❧❡✈❛♥❞♦ ❛♦ q✉❡st✐♦♥❛♠❡♥t♦ s♦❜r❡ ♦ s♣✐♥ s❡r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ♣✉r❛♠❡♥t❡ r❡

❧❛t✐✈íst✐❝❛✮ ❡ ♠❛✐s t❛r❞❡✱ ❡♠ ✶✾✼✸ ❬✷❪✱ ❡♠ ✉♠ tr❛❜❛❧❤♦ ❝♦♠ ▲❡ ❇❡❧❧❛❝✱ ✈❡r✐✜❝♦✉ q✉❡ ❡①✐st❡♠ ❞♦✐s ❧✐♠✐t❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ♣❛r❛ ♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ❝♦♥t❡♠♣❧❛♥❞♦ ❛ s✐♠❡tr✐❛ ❞❡ ●❛❧✐❧❡✐✳ ❊st❡s ❧✐♠✲ ✐t❡s ❢♦r❛♠ ❝❤❛♠❛❞♦s ❞❡ ❧✐♠✐t❡ ❡❧étr✐❝♦ ❡ ❧✐♠✐t❡ ♠❛❣♥ét✐❝♦ ❡ s❡ ❝❛r❛❝t❡r✐③❛♠ ♣❡❧❛s ❞♦♠✐♥â♥❝✐❛s ❞♦s ❡❢❡✐t♦s ❡❧étr✐❝♦ ❡ ♠❛❣♥ét✐❝♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊st❡s tr❛❜❛❧❤♦s ❛❜r✐r❛♠ ❝❛♠✐♥❤♦ ♣❛r❛ ♥♦✈♦s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ●❛❧✐❧❡✐ ❡♠ ✺ ❞✐♠❡♥sõ❡s✱ ❝♦♠♦ ♦s tr❛✲ ❜❛❧❤♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ❚❛❦❛❤❛s❤✐ ❬✸✱ ✹❪ ♥❛ ❞é❝❛❞❛ ❞❡ ✶✾✽✵✱ r❡❝✉♣❡r❛♥❞♦ ✉♠❛ ✐❞é✐❛ ❥á ♣r♦♣♦st❛ ♣❡❧♦ ♠❡♥♦s ✷✵ ❛♥♦s ❛♥t❡s ❬✺❪✳ ❊st❡ ❢♦r♠❛❧✐s♠♦ ❣❛❧✐❧❡❛♥♦ ❡♠ ✺ ❞✐♠❡♥sõ❡s ♣♦ss✉✐ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ✐♥t❡r❡ss❛♥t❡✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ❛♣r❡s❡♥t❛r ✉♠❛ ❡str✉t✉r❛ ❝♦✈❛r✐❛♥t❡ ♣❛r❛ ❛s ❡q✉❛çõ❡s✱ ❛s q✉❛✐s s❡ ❛ss❡♠❡❧❤❛♠ ♠✉✐t♦ ❡♠ ❢♦r♠❛ àq✉❡❧❛ ♦❜t✐❞❛ ♥♦ ❝❛s♦ ❧♦r❡♥t③✐❛♥♦✳ ❊st❛ s❡♠❡❧❤❛♥ç❛ ♣❡r♠✐t✐✉ ❛ ❝♦♥✲ str✉çã♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ r❡❣✐♠❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ❞❡ ❞✐✈❡rs♦s s✐st❡♠❛s ❢ís✐❝♦s✱ ❛ ♣❛rt✐r ❞❛s ❡q✉❛çõ❡s ♥❛s s✉❛s ❢♦r♠❛s r❡❧❛t✐✈íst✐❝❛s ❬✻✱ ✼❪✳

❆ s✐♠❡tr✐❛ ❞♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦✱ s❡❥❛ ❡❧❛ ❛ ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ♦✉ ●❛❧✐❧❡✐✱ s❡❥❛ ❡♠ ✹ ♦✉ ✺ ❞✐✲ ♠❡♥sõ❡s✱ ♥♦ ❡♥t❛♥t♦✱ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡s❝r❡✈❡r s✐st❡♠❛s ❡♠ ✐♥t❡r❛çã♦ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r ❝♦♠♣❧❡①♦ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦✳ P❛r❛ ✐ss♦✱ ✉♠ ♦✉tr♦ ❣r✉♣♦ ❞❡ s✐♠❡tr✐❛ é ♥❡❝❡ssár✐♦✿ ♦ ❣r✉♣♦ ❞❡ s✐♠❡tr✐❛ ❞❡ ❝❛❧✐❜r❡✱ ♦✉ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡✳ ❆s t❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❢♦r❛♠ ✉♠ ❞♦s tó♣✐❝♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❋ís✐❝❛ ▼♦❞❡r♥❛ ♥♦ sé❝✉❧♦ ❳❳✳ ❙✉❛ ❝♦♥str✉çã♦ t❡✈❡ ✐♥í❝✐♦ ❝♦♠ tr❛❜❛❧❤♦s ❞❡ ❲❡②❧ ❬✽❪✱ q✉❡♠✱ ✐♥s♣✐r❛❞♦ ♣❡❧♦s tr❛❜❛❧❤♦s ❞❡ ❊✐♥st❡✐♥ ❡♠ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✱ ❛❝r❡❞✐t❛✈❛ q✉❡ ❛ ✐♥t❡r❛çã♦ ❡❧❡tr♦✲ ♠❛❣♥ét✐❝❛ ❡st❛r✐❛ ❛ss♦❝✐❛❞❛ à s✉❜st✐t✉✐çã♦ ❞❡ ✉♠❛ s✐♠❡tr✐❛ ❞❡ ❢❛s❡ ❣❧♦❜❛❧ ♣♦r ✉♠❛ ❧♦❝❛❧✳ ❆ ❢♦r♠❛ ✜♥❛❧ ♣❛r❛ ❛ t❡♦r✐❛ ❞❡ ❣❛✉❣❡ ❬✾❪ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❯t✐②❛♠❛ ❬✶✵❪✱ q✉❛s❡ ✸✵ ❛♥♦s ❞❡♣♦✐s ❞♦ tr❛❜❛❧❤♦ ❞❡ ❲❡②❧✱ ❡ ✐♥❝♦r♣♦r❛✈❛ ♦s ❞♦✐s ❝❛s♦s ❞❡ t❡♦r✐❛ ❞❡ ❣❛✉❣❡ ❝♦♥❤❡❝✐❞♦s à é♣♦❝❛✱ q✉❛✐s s❡❥❛♠ ❛ t❡♦r✐❛ ❞❡ ▼❛①✇❡❧❧ ❡ ❛ ❞❡ ❨❛♥❣✲▼✐❧❧s ❬✶✶❪✱ ❡ ❛✐♥❞❛ ❛♣r❡s❡♥t❛✈❛ ❛ ❣r❛✈✐t❛çã♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ t❡♦r✐❛ ❞❡ ❣❛✉❣❡✳ ❊st❛ ú❧t✐♠❛✱ ♣♦r s✉❛ ✈❡③✱ ♣♦ss✉✐ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ✐♥t❡r❡ss❛♥t❡✿ ✉♥❡ ❡♠ ✉♠❛ ♠❡s♠❛ ❡str✉t✉r❛ ❛s s✐♠❡tr✐❛s ❞♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦ ❡ ❞❡ ✐♥t❡r❛çã♦✳

➱ ✐♥t❡r❡ss❛♥t❡ ♣❡r❝❡❜❡r q✉❡ ❛s t❡♦r✐❛s ❞❡ ❣r❛♥❞❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ ♥❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛ sã♦ ❛q✉❡❧❛s ❝♦♥str✉í❞❛s ❝♦♠ ▲❛❣r❛♥❣✐❛♥❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥❛ ❞❡r✐✈❛❞❛ ❞♦s ❝❛♠♣♦s ❡♥✈♦❧✈✐❞♦s1_✳ ■st♦ ♣♦ss✉✐ s✉❛s r❛③õ❡s✱ ♣♦✐s t❡♦r✐❛s ❝✉❥❛s ▲❛❣r❛♥❣✐❛♥❛s sã♦ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❛♣r❡s❡♥t❛♠ sér✐♦s ♣r♦❜❧❡♠❛s✱ ❞❡♥tr❡ ♦s q✉❛✐s s❡ ❞❡st❛❝❛♠ ❛ ♥ã♦ ✉♥✐t❛r✐❡❞❛❞❡ ❞❛ t❡♦r✐❛ ❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ❡st❛❞♦s ❢❛♥t❛s♠❛s ❝♦♠ ♥♦r♠❛ ♥❡❣❛t✐✈❛ ❬✶✸❪✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❜♦❛s ♣r♦♣r✐❞❛❞❡s ❞❡st❛s t❡♦r✐❛s t❛♠❜é♠ ❢♦r❛♠ ♦❜t✐❞❛s✱ ♦ q✉❡ ❛✐♥❞❛ ♠♦t✐✈❛ ♦ s❡✉ ❡st✉❞♦ ❤♦❥❡ ❡♠ ❞✐❛ ✲ ✈❡❥❛ r❡❢❡rê♥❝✐❛s ❡♠ ❬✶✹❪✳ ❖ q✉❡ é ✐♠♣♦rt❛♥t❡ ❛t❡♥t❛r é q✉❡ ❛ ❡s❝♦❧❤❛ ♣♦r ▲❛❣r❛♥❣✐❛♥❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ã♦ sã♦ ❛♠♣❛r❛❞❛s✴♠♦t✐✈❛❞❛s ♣♦r ❛r❣✉♠❡♥t♦s ❞❡ s✐♠❡tr✐❛✱ ♦✉ s❡❥❛✱ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r t❡♦r✐❛s ❝♦♠ ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r q✉❡ ♣r❡s❡r✈❡♠ ❛s s✐♠❡tr✐❛s s♦❜r❡ ♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦ ❡ ♦ ❣r✉♣♦ ❞❡ ✐♥t❡r❛çã♦✳

❚❡♦r✐❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ♣♦ss✉❡♠ s❡✉ ✐♥t❡r❡ss❡ ❡♠ ❋ís✐❝❛✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ 1_{❆ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ é ✉♠❛ ❡①❝❡ssã♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❛ ▲❛❣r❛♥❣✐❛♥❛ ❞❡♣❡♥❞❡ ❞❡ t❡r♠♦s ❞❡ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛} ❞♦ t❡♥s♦r ♠étr✐❝♦❀ t♦❞❛✈✐❛ ❡st❡s t❡r♠♦s ♣♦❞❡♠ s❡r ❛❣r✉♣❛❞♦s ♥❛ ❢♦r♠❛ ❞❡ ✉♠ t❡r♠♦ ❞❡ s✉♣❡r❢í❝✐❡ ✲ ❛ ❝❤❛♠❛❞❛ ▲❛❣r❛♥❣✐❛♥❛ ✧●❛♠❛✲●❛♠❛✧ ❬✶✷❪✱ ❞❡ ❢♦r♠❛ q✉❡ ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ❝♦♥t✐♥✉❛♠ s❡♥❞♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳

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❞❡ t❡♦r✐❛s ❡❢❡t✐✈❛s✱ q✉❡ sã♦ ❛q✉❡❧❛s q✉❡ ❡①♣❧♦r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❡s❝❛❧❛s ❡♠ s✐st❡♠❛s ❢ís✐❝♦s ❬✶✺❪✳ ❙ã♦ t❡♦r✐❛s q✉❡ s❡r✈❡♠ ♣❛r❛ ❞❡s❝r❡✈❡r ✉♠ ❞❡t❡r♠✐♥❛❞♦ s✐st❡♠❛ ❡♠ ✉♠❛ ❞❛❞❛ ❡s❝❛❧❛ ❞❡ ❡♥❡r✲ ❣✐❛✴❝♦♠♣r✐♠❡♥t♦✴♠♦♠❡♥t♦✳ ❊st❡ t✐♣♦ ❞❡ ❝♦♥str✉çã♦ ❣❡r❛❧♠❡♥t❡ ♥ã♦ é ❛❞❡q✉❛❞♦ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ s✐st❡♠❛ ❡♠ q✉❛❧q✉❡r s✐t✉❛çã♦✳ ❈♦♥s✐❞❡r❡✲s❡ ✉♠ ❡①❡♠♣❧♦ r❡t✐r❛❞♦ ❞❡ ❘❡❢✳❬✶✻❪ ♣❛r❛ ✐❧✉str❛r ❡st❛ s✐t✉❛çã♦✱ q✉❛❧ s❡❥❛ ♦ ❞❡ ❞♦✐s ❝♦r♣♦s ✐♥t❡r❛❣✐♥❞♦ ❣r❛✈✐t❛❝✐♦♥❛❧♠❡♥t❡✱ s❡♥❞♦ ✉♠ ❞♦s ❝♦r♣♦s ♠✉✐t♦ ♠❛✐♦r ❞♦ q✉❡ ♦ s❡❣✉♥❞♦ ✲ ❡✳❣✳ ✉♠❛ ♠❛çã ✐♥t❡r❛❣✐♥❞♦ ❣r❛✈✐t❛❝✐♦♥❛❧♠❡♥t❡ ❝♦♠ ❛ ❚❡rr❛✱ ❡♠ s✉❛ s✉♣❡r❢í❝✐❡✳ P❛r❛ ❡s❝❛❧❛s ❞❡ ❞✐stâ♥❝✐❛s q✉❡ ✈❛r✐❛♠ ♠✉✐t♦ ♣♦✉❝♦✱ ❝♦♠♣❛r❛❞❛s ❛♦ r❛✐♦ ❞❛ ❚❡rr❛✱ ✉♠ ♣♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧ q✉❡ ♠♦str❛✲s❡ ❜❛st❛♥t❡ ❛❞❡q✉❛❞♦ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ♠♦✈✐♠❡♥t♦ ❞❡st❛ ♠❛çã é ❞❛❞♦ ♣♦r

Vef f =mgh, ✭✶✳✶✮

❡♠ q✉❡m é ❛ ♠❛ss❛ ❞❛ ♠❛çã✱hé ❛ ❛❧t✉r❛ ❝♦♠ r❡s♣❡✐t♦ à s✉♣❡r❢í❝✐❡ t❡rr❡str❡ ❡g é ❛❝❡❧❡r❛çã♦ ❞❛

❣r❛✈✐❞❛❞❡✱ s❡♥❞♦ ❡st❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ q✉❡ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞❛ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✳ ➚ ♠❡❞✐❞❛ ❡♠ q✉❡ h❝♦♠❡ç❛ ❛ t♦♠❛r ✈❛❧♦r❡s ❝❛❞❛ ✈❡③ ♠❛✐♦r❡s✱ ♦ ♣♦t❡♥❝✐❛❧ ❛❝✐♠❛ ♣r❡❝✐s❛ s❡r ❝♦rr✐❣✐❞♦✱ ♣♦r

❡①❡♠♣❧♦ ♣❡❧❛ s♦♠❛ ❞❡ ✉♠ t❡r♠♦ ❞♦ t✐♣♦

Vef f =mgh−mgR

h2

R2, ✭✶✳✷✮

❡♠ q✉❡Ré ♦ r❛✐♦ ❞❛ ❚❡rr❛✳ ➱ s❛❜✐❞♦ q✉❡ ♦ ♣♦t❡♥❝✐❛❧ ♠❛✐s ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❞❡s❝r❡✈❡r ❡st❛ s✐t✉❛çã♦

é ❛q✉❡❧❡ ❞❛❞♦ ♣❡❧♦ ♣♦t❡♥❝✐❛❧ ♥❡✇t♦♥✐❛♥♦✱

V =−m GM

h+R, ✭✶✳✸✮

❡♠ q✉❡ M é ❛ ♠❛ss❛ ❞❛ ❚❡rr❛ ❡G❛ ❝♦♥st❛♥t❡ ✉♥✐✈❡rs❛❧ ❞❛ ❣r❛✈✐t❛çã♦✳ P♦ré♠✱ ♦s ♣♦t❡♥❝✐❛✐s ❞❡

❊q✳✭✶✳✶✮ ❡ ❊q✳ ✭✶✳✷✮ ❛❝✐♠❛ ❞❡s❝r❡✈❡♠ ❝♦♠ ❜♦❛ ❡①❛t✐❞ã♦ ❞✐✈❡rs♦s ❝❛s♦s ❡ ♣♦❞❡♠ ❢❛❝✐❧✐t❛r ♠✉✐t♦ ❛ ❞❡s❝r✐çã♦ ❞♦ s✐st❡♠❛✳ ◆❛ ✈❡r❞❛❞❡✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦s ✈❛❧♦r❡s ❞❡M ❡R✭❛q✉✐ ♥ã♦ ♠❛✐s ♣❛r❛ ♦ ❝❛s♦

❞❛ ❚❡rr❛✱ ♠❛s ♣❛r❛ ❛❧❣✉♠ ♦✉tr♦ ❝♦r♣♦ ❝❡❧❡st❡✮✱ ♦ ♣♦t❡♥❝✐❛❧ ❞❛❞♦ ♣♦r ❊q✳✭✶✳✸✮ ❛✐♥❞❛ ♣♦❞❡r✐❛ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ♣♦t❡♥❝✐❛❧ ❡❢❡t✐✈♦✱ ❡♠ q✉❡ ❛ t❡♦r✐❛ ♠❛✐s ❢✉♥❞❛♠❡♥t❛❧ s❡r✐❛ ❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✳

➱ ❡♠ ❝♦♥t❡①t♦s ❝♦♠♦ ♦ ❞♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ q✉❡ ❛s t❡♦r✐❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ♣♦ss✉❡♠ ✐♥t❡r❡ss❡✱ ♦✉ s❡❥❛✱ ♣r♦❜❧❡♠❛s ❡♠ q✉❡ ❡①✐st❡♠ ❡s❝❛❧❛s ❞❡ ❡♥❡r❣✐❛ ❜❡♠ ❞❡✜♥✐❞❛s✳ ◆❡st❡s ❝❛s♦s✱ ♦s ♣r♦❜❧❡♠❛s ❥á ❝✐t❛❞♦s✱ ❝♦♠♦ ❛ ♥ã♦ ✉♥✐t❛r✐❡❞❛❞❡ ❞❛ t❡♦r✐❛✱ ♥ã♦ ♣♦ss✉❡♠ t❛♥t❛ r❡❧❡✈â♥❝✐❛ ❝♦♠♦ ♣❛r❛ ✉♠❛ t❡♦r✐❛ ❢✉♥❞❛♠❡♥t❛❧✱ ✉♠❛ ✈❡③ q✉❡ ❡❧❛s ♥ã♦ s❡r✐❛♠ ❛❞❡q✉❛❞❛s ♣❛r❛ ❞❡s❝r❡✈❡r t♦❞♦ ♦ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛✱ ♠❛s ❛♣❡♥❛s ♣❛rt❡ ❞❡❧❡✳ ❖ ✐♥t❡r❡ss❡ ♥❡st❡ tr❛❜❛❧❤♦ ❡stá ✈♦❧t❛❞♦ ♣❛r❛ ❛s t❡♦r✐❛s ❝✉❥❛s ▲❛❣r❛♥❣✐❛♥❛s ❞❡♣❡♥❞❡♠ ❞❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞♦ ❝❛♠♣♦✱ ❡ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ ❛s ▲❛❣r❛♥❣✐❛♥❛s q✉❡ ❞❡s❝r❡✈❡♠ ♦ ❝❛♠♣♦ ❞❡ ❣❛✉❣❡✳ ❊①❡♠♣❧♦s ❞❡st❡ ❝❛s♦ sã♦ ❡♥❝♦♥tr❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✱ ❞❡♥tr❡ ♦s q✉❛✐s ❞❡st❛❝❛♠✲s❡ ❛ ▲❛❣r❛♥❣✐❛♥❛ ❡❢❡t✐✈❛ ❞❡ ❆❧❡❦s❡❡✈✱ ❆r❜✉③♦✈ ❡ ❇❛✐❦♦✈ ❬✶✼❪✱ q✉❡ ❞❡s❝r❡✈❡♠ ♦ r❡❣✐♠❡ ✐♥❢r❛✈❡r♠❡❧❤♦ ❞♦ ❣❧ú♦♥ ♥❛ ◗❈❉✱ ❛s t❡♦r✐❛s ❞❡ ❣r❛✈✐t❛çã♦ ❞♦ t✐♣♦ R2_{✱ ♠♦t✐✈❛❞❛s ♣❡❧❛} ♥ã♦ r❡♥♦r♠❛❧✐③❛❜✐❧✐❞❛❞❡ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✭♥♦ ❝♦♥t❡①t♦ ❞❛ q✉❛♥t✐③❛çã♦ ❞❛ t❡♦r✐❛✮ ❬✶✽❪✱ ❡ ❛ ❊❧❡tr♦❞✐♥â♠✐❝❛ ●❡♥❡r❛❧✐③❛❞❛ ❞❡ P♦❞♦❧s❦② ❬✶✾❪✱ q✉❡ ❢♦✐ ♣r♦♣♦st❛ ❝♦♠♦ ✉♠❛ ❢♦r♠❛ ♣❛r❛ s❡ ❝♦♥tr♦❧❛r ❛s ❞✐✈❡r❣ê♥❝✐❛s ✉❧tr❛✈✐♦❧❡t❛s ❞♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦✱ ❡♠ ✉♠❛ é♣♦❝❛ ❡♠ q✉❡ ❛ r❡♥♦r♠❛❧✐③❛çã♦ ❛✐♥❞❛

(11)

♥ã♦ ❡r❛ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❡st❛❜❡❧❡❝✐❞♦ ❡♠ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s✱ ❡♥tr❡ ♦✉tr❛s ❬✷✵❪✳ ❆ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❞❡ P♦❞♦❧s❦② ♣♦ss✉✐ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡ss❡ ❛q✉✐✱ ♣♦r s❡r ❛ t❡♦r✐❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ♥♦ ❝❛♠♣♦ ❞❡ ❣❛✉❣❡ ♠❛✐s s✐♠♣❧❡s q✉❡ ♣♦❞❡ s❡r ❝♦♥str✉í❞❛ ❡ ❛ q✉❡ ♠❛✐s ♣♦ss✉✐ ♠❛✐s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ❬✶✹✱ ✷✶❪ ❡✱ ♣♦r ❡st❛ r❛③ã♦✱ s❡rá r❡✈✐s✐t❛❞❛ ♥♦s ❞✐❢❡r❡♥t❡s ❡st✉❞♦s r❡❛❧✐③❛❞♦s ❛q✉✐✳

❆ ✐♥t❡♥çã♦ ❛q✉✐ é ❡①♣❧♦r❛r ❛s♣❡❝t♦s r❡❧❛❝✐♦♥❛❞♦s às s✐♠❡tr✐❛s ❞♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦ ❡ ❞♦ ❣r✉♣♦ ❞❡ ❣❛✉❣❡ ♣❛r❛ t❡♦r✐❛s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❢♦❝❛♥❞♦ ❛♣❧✐❝❛çõ❡s ❝♦♠♦ t❡♦r✐❛s ❡❢❡t✐✈❛s✳ ◆❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❡st❛ t❡s❡ s❡rá ❢❡✐t❛ ✉♠❛ r❡✈✐sã♦ ❞❛ t❡♦r✐❛ ❞❡ ❣❛✉❣❡ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ❛ ▲❛❣r❛♥❣✐❛♥❛ ❞♦ ❝❛♠♣♦ ❞❡ ❣❛✉❣❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❞❡st❡✱ ❝♦♠♦ ❡st✉❞❛❞♦ ♣♦r ❯t✐②❛♠❛✳ ❊♠ s❡❣✉✐❞❛✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡st❛ ❡str✉t✉tr❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠❛ t❡♦r✐❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ♥♦ ❝❛♠♣♦ ❞❡ ❣❛✉❣❡✳ ❈♦♠♦ ❡①❡♠♣❧♦s✱ ❛ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❞❡ P♦❞♦❧s❦② ❡ ❛ ▲❛❣r❛♥❣✐❛♥❛ ❞❡ ❆❧❡❦s❡❡✈✱ ❆r❜✉③♦✈ ❡ ❇❛✐❦♦✈ s❡rã♦ ❛♥❛❧✐s❛❞❛s✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s❡ ✈❡r✐✜❝❛r s❡ ❡❧❛s ❛t❡♥❞❡♠ ♦s r❡q✉✐s✐t♦s ❞❡ t❡♦r✐❛s ❞❡ ❣❛✉❣❡✳ ◆❛ s❡q✉ê♥❝✐❛✱ ❝♦♥s✐❞❡r❛r✲s❡✲á ❛ ❛♣❧✐❝❛çã♦ ❞❡st❛ ❡str✉t✉r❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ♣❛r❛ ♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ❤♦♠♦❣ê♥❡♦✱ ♥♦ ❡st✉❞♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✱ q✉❡ é ♦ ❝❛s♦ ❡♠ q✉❡ ❛ s✐♠❡tr✐❛ ❞♦ ❣r✉♣♦ ❝✐♥❡♠át✐❝♦ ❡ ❛ ❞♦ ❣r✉♣♦ ❞❡ ✐♥t❡r❛çã♦ s❡ ✐♥t❡r❧✐❣❛♠✳ ◆❡st❡ ❝❛s♦ s❡rã♦ ❛♥❛❧✐s❛❞❛s ❛❧❣✉♠❛s ▲❛❣r❛♥❣✐❛♥❛s ✐♥✈❛r✐❛♥t❡s ❞❡ ❣❛✉❣❡ ❡ ❞❡♥tr❡ ❡❧❛s✱ ✉♠❛ s❡rá ❡s❝♦❧❤✐❞❛ ❝♦♠♦ ❡st✉❞♦ ❞❡ ❝❛s♦ ✲ ❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ s❡rã♦ ♦❜t✐❞❛s ❡ ♦❜t❡r✲s❡✲á ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ r❡❣✐♠❡ ❧✐♥❡❛r ♥♦ ❝❛s♦ ❡stát✐❝♦ ❡ ✐s♦tró♣✐❝♦✳ ❯♠ ❞♦s ♣r♦❜❧❡♠❛s ❞❛s t❡♦r✐❛s ❡❢❡t✐✈❛s é ❞❡t❡r♠✐♥❛r ❛s ❝♦♥st❛♥t❡s ✐♥tr♦❞✉③✐❞❛s ♥♦s t❡r♠♦s ❡❢❡t✐✈♦s q✉❡✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ❞❡t❡r♠✐♥❛♠ ❛s ❡s❝❛❧❛s ❞❡ ❡♥❡r❣✐❛ ❞❡ ✈❛❧✐❞❛❞❡ ❞❛ t❡♦r✐❛✳ ■ss♦ ♥♦r♠❛❧♠❡♥t❡ é ❢❡✐t♦ ♣♦r ❛❥✉st❡s ❛ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❬✶✻❪✳ ◆❛ s❡q✉ê♥❝✐❛ ❞♦ ❡st✉❞♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✱ s❡rã♦ ❛♥❛❧✐s❛❞❛s três ❢♦r♠❛s ❞✐❢❡r❡♥t❡s ❞❡ ✈✐♥❝✉❧❛r ❛ ❝♦♥st❛♥t❡ ❞❡ P♦❞♦❧s❦② ❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✳

◆❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞❡st❛ t❡s❡✱ ❛ ✐♥t❡♥çã♦ é ❡st✉❞❛r r❡❣✐♠❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ❞❛ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❞❡ P♦❞♦❧s❦②✱ ♥ã♦ ♣❡❧❛ ♣r❡s❝r✐çã♦ ✉s✉❛❧ ❞❡ s❡ t♦♠❛r ♦ ❧✐♠✐t❡c → ∞✱ ♠❛s ♣❡❧❛ ♠✉❞❛♥ç❛ ❞♦ ❣r✉♣♦

❝✐♥❡♠át✐❝♦✱ ❞♦ ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ♣❛r❛ ♦ ❣r✉♣♦ ❞❡ ●❛❧✐❧❡✐✳ ❖ ❢♦r♠❛❧✐s♠♦ ❣❛❧✐❧❡❛♥♦ ❡♠ ✺ ❞✐♠❡♥sõ❡s s❡rá ❡♥tã♦ ✐♥tr♦❞✉③✐❞♦ ❡✱ ♣❛r❛ ❡❢❡✐t♦ ❞❡ ❝♦♠♣❛r❛çã♦✱ ♦s r❡s✉❧t❛❞♦s ❞❡ ▲❡ ❇❡❧❧❛❝ ❡ ▲é✈②✲▲❡❜❧♦♥❞ ♣❛r❛ ♦s ❧✐♠✐t❡s ♥ã♦ r❡❧❛t✐✈íst✐❝♦s ❞♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ❡♠ ✹ ❞✐♠❡♥sõ❡s ❬✷❪ ❡ ♦s r❡s✉❧t❛❞♦s ❡q✉✐✈❛❧❡♥t❡s ♦❜t✐❞♦s ♥♦ ❢♦r♠❛❧✐s♠♦ ❡♠ ✺ ❞✐♠❡♥sõ❡s ❬✼❪ t❛♠❜é♠ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s✱ ♦s q✉❛✐s s❡r✈✐rã♦ ❞❡ ❣✉✐❛ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ❝❛s♦ ❞❡ P♦❞♦❧s❦②✳

P♦r ✜♠✱ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛s ❝♦♥❝❧✉sõ❡s ❡ ❛s ♣❡rs♣❡❝t✐✈❛s ❞❡ ❢✉t✉r♦s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s✳ ❆ ♥♦t❛çã♦ ❡♠♣r❡❣❛❞❛ ♥♦ ❝❛♣ít✉❧♦ ✶ s❡rá✿ í♥❞✐❝❡s ❞♦ ❣r✉♣♦ ❞❡ ❣❛✉❣❡ s❡rã♦ ✐❞❡♥t✐✜❝❛❞❛s ♣♦r í♥❞✐❝❡s ❧❛t✐♥♦s❀ í♥❞✐❝❡s ❞❡ ❡s♣❛ç♦✲t❡♠♣♦ ✭✹ ❞✐♠❡♥sõ❡s✮ sã♦ ❞❛❞♦s ♣♦r ❧❡tr❛s ❣r❡❣❛s✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ♦s í♥❞✐❝❡s ❞❡ ❡s♣❛ç♦✲t❡♠♣♦ sã♦ ✐❞❡♥t✐✜❝❛❞♦s ♣♦r ❧❡tr❛s ❣r❡❣❛s ✭µ, ν = 1,2, ...,5✮ ❡ ❧❡tr❛s ❧❛t✐♥❛s

✭i, j= 1,2,3✮✳

(12)

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❏✳ ▼✳ ▲é✈②✲▲❡❜❧♦♥❞ ✲ ❈♦♠♠✉♥✳ ▼❛t❤✳ P❤②s✳ ✻✱ ✷✽✻ ✭✶✾✻✼✮✳

❬✷❪ ▼✳ ▲❡ ❇❡❧❧❛❝ ❡ ❏✳ ▼✳ ▲é✈②✲▲❡❜❧♦♥❞ ✲ ◆✉♦✈✳ ❈✐♠✳ ❇ ✶✹✱ ✷✶✼ ✭✶✾✼✸✮✳

❬✸❪ ❨✳ ❚❛❦❛❤❛s❤✐ ✲ ❋♦rts❝❤r✳ P❤②s✳ ✸✻✱ ✻✸ ✭✶✾✽✽✮❀ ❨✳ ❚❛❦❛❤❛s❤✐ ✲ ❋♦rts❝❤r✳ P❤②s✳ ✸✻✱ ✽✸ ✭✶✾✽✽✮✳ ❬✹❪ ▼✳ ❖♠♦t❡✱ ❙✳ ❑❛♠❡❢✉❝❤✐✱ ❚✳ ❚❛❦❛❤❛s❤✐✱ ❨✳ ❖❤♥✉❦✐ ✲ ❋♦rts❝❤r✳ P❤②s✳ ✸✼✱ ✾✸✸ ✭✶✾✽✾✮✳ ❬✺❪ ●✳ P✐♥s❦✐ ✲ ❏✳ ▼❛t❤✳ P❤②s✳ ✾✱ ✶✾✷✼ ✭✶✾✻✽✮✳

❬✻❪ ❆✳ ❊✳ ❙❛♥t❛♥❛✱ ❋✳ ❈✳ ❑❤❛♥♥❛ ❡ ❨✳ ❚❛❦❛❤❛s❤✐ ✲ Pr♦❣✳ ❚❤❡♦r✳ P❤②s✳ ✾✾✱ ✸✷✼ ✭✶✾✽✽✮✳

❬✼❪ ❊✳ ❙✳ ❙❛♥t♦s✱ ▼✳ ❞❡ ▼♦♥t✐❣♥②✱ ❋✳ ❈✳ ❑❤❛♥♥❛ ❡ ❆✳ ❊✳ ❙❛♥t❛♥❛ ✲ ❏✳ P❤②s✳ ❆ ✸✼✱ ✾✼✼✶ ✭✷✵✵✹✮✳ ❬✽❪ ❍✳ ❲❡②❧ ✲ ❙✐t③✉♥❣s❜❡r✳ Pr❡✉ss✳ ❆❦❛❞✳ ❇❡r❧✐♥ ✹✻✺ ✭✶✾✶✽✮❀ ❍✳ ❲❡②❧ ✲ ❩❡✐t✳ ❢✳ P❤②s✐❦ ✸✸✵✱ ✺✻

✭✶✾✷✾✮✳

❬✾❪ ▲✳ ❖✬❘❛②❢❡❛rt❛✐❣❤ ✲ ❚❤❡ ❉❛✇♥✐♥❣ ♦❢ ●❛✉❣❡ ❚❤❡♦r②✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss ✭✶✾✾✼✮✳ ❬✶✵❪ ❘✳ ❯t✐②❛♠❛ ✲ P❤②s✳ ❘❡✈✳ ✶✵✶✱ ✶✺✾✼ ✭✶✾✺✻✮✳

❬✶✶❪ ❈✳ ◆✳ ❨❛♥❣ ❡ ❘✳ ▲✳ ▼✐❧❧s ✲ P❤②s✳ ❘❡✈✳ ✾✻✱ ✶✾✶ ✭✶✾✺✹✮✳ ❬✶✷❪ P✳ ❆✳ ▼✳ ❉✐r❛❝ ✲ Pr♦❝✳ ❘♦②✳ ❙♦❝✳ ❆ ✷✹✻✱ ✸✸✸ ✭✶✾✺✽✮✳ ❬✶✸❪ ❆✳ P❛✐s✱ ●✳ ❯❤❧❡♥❜❡❝❦✱ P❤②s✳ ❘❡✈✳ ✼✾✱ ✶✹✺ ✭✶✾✺✵✮✳

❬✶✹❪ ❏✳ ❇❛r❝❡❧♦s✲◆❡t♦✱ ❈✳❆✳P✳ ●❛❧✈ã♦✱ ❈✳P✳ ◆❛t✐✈✐❞❛❞❡ ✲ ❩✳ P❤②s✳ ❈ ✺✷✱ ✺✺✾ ✭✶✾✾✶✮❀ ▲✳❱✳ ❇❡❧✈❡❞❡r❡✱ ❈✳P✳ ◆❛t✐✈✐❞❛❞❡✱ ❈✳❆✳P✳ ●❛❧✈ã♦ ✲ ❩✳ P❤②s✳ ❈ ✺✻✱ ✻✵✾ ✭✶✾✾✷✮✳

❬✶✺❪ ❯✳ ✈❛♥ ❑♦❧❝❦ ✲ ❛r❳✐✈✿ ♥✉❝❧✲t❤✴✾✾✵✷✵✶✺✈✶ ✭✶✾✾✾✮✳

❬✶✻❪ ❯✳ ✈❛♥ ❑♦❧❝❦✱ ▲✳ ❏✳ ❆❜✉✲❘❛❞❞❛❞ ❡ ❉✳ ▼✳ ❈❛r❞❛♠♦♥❡ ✲ ❛r❳✐✈✿ ♥✉❝❧✲t❤✴✵✷✵✺✵✺✽✈✶ ✭✶✾✾✾✮✳ ❬✶✼❪ ❆✳ ■✳ ❆❧❡❦s❡❡✈✱ ❇✳ ❆✳ ❆r❜✉③♦✈ ❡ ❱✳ ❆✳ ❇❛✐❦♦✈ ✲ ❚❤❡♦r✳ ▼❛t❤✳ P❤②s✳ ✺✷✱ ✼✸✾ ✭✶✾✽✷✮❀ ❆✳ ■✳

❆❧❡❦s❡❡✈ ❡ ❇✳ ❆✳ ❆r❜✉③♦✈ ✲ ❚❤❡♦r✳ ▼❛t❤✳ P❤②s✳ ✺✾✱ ✸✼✷ ✭✶✾✽✹✮✳

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❬✶✽❪ ❑✳ ❙✳ ❙t❡❧❧❡ ✲ ●❡♥✳ ❘❡❧✳ ●r❛✈✳ ✾✱ ✸✺✸ ✭✶✾✼✽✮❀ ■✳ ▲✳ ❇✉❝❤❜✐♥❞❡r ❡ ❙✳ ▲✳ ▲②❛❤♦✈✐❝❤ ✲ ❈❧❛ss✳ ◗✉❛♥t✉♠ ●r❛✈✳ ✹ ✶✹✽✼ ✭✶✾✽✼✮ ▲✳ ◗✉❡r❡❧❧❛ ✲ ❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ▲✐è❣❡ ✭✶✾✾✽✮❀ ❆✳ ❆❝❝✐♦❧②✱ ❆✳ ❆③❡r❡❞♦ ❡ ❍✳ ▼✉❦❛✐ ✲ ❏✳ ▼❛t❤✳ P❤②s✳ ✹✸✱ ✹✼✸ ✭✷✵✵✷✮✳

❬✶✾❪ ❇✳ P♦❞♦❧s❦② ✲ P❤②s✳ ❘❡✈✳ ✻✷✱ ✻✽ ✭✶✾✹✷✮❀ ❇✳ P♦❞♦❧s❦② ❡ ❈✳ ❑✐❦✉❝❤✐ ✲ P❤②s✳ ❘❡✈✳ ✻✺✱ ✷✷✽ ✭✶✾✹✹✮❀ ❇✳ P♦❞♦❧s❦② ❡ P✳ ❙❝❤✇❡❞ ✲ ❘❡✈✳ ▼♦❞✳ P❤②s✳ ✷✵✱ ✹✵ ✭✶✾✹✽✮✳

❬✷✵❪ ❚✳ ■t♦❤ ❡ P✳ ❖❤ ✲ P❤②s✳ ❘❡✈✳ ❉ ✻✸✱ ✵✷✺✵✶✾ ✭✷✵✵✵✮❀ ❆✳ ❇✐❧❛❧✲ ◆✉❝❧✳ P❤②s✳ ❇ ✻✶✽✱ ✷✶ ✭✷✵✵✶✮❀ ❚✳ P✳ ❙♦t✐r✐♦✉✱ ❱✳ ❋❛r❛♦♥✐ ✲ ❛r❳✐✈✿✵✽✵✺✳✶✼✷✻ ✭✷✵✵✽✮✳

❬✷✶❪ ❉✳❏✳ ▼♦♥t❣♦♠❡r② ✲ P❤②s✳ ❘❡✈✳ ✻✾✱ ✶✶✼ ✭✶✾✹✻✮❀ ❆✳❊✳❙✳ ●r❡❡♥ ✲ P❤②s✳ ❘❡✈✳ ✼✷ ✻✷✽ ✭✶✾✹✼✮❀ ❆✳❊✳❙✳ ●r❡❡♥ ✲ P❤②s✳ ❘❡✈✳ ✼✸✱ ✷✻ ✭✶✾✹✽✮❀ ❈✳❆✳P✳ ●❛❧✈ã♦ ❡ ❇✳▼✳ P✐♠❡♥t❡❧ ✲ ❈❛♥✳ ❏✳ P❤②s✳✻✻✱ ✹✻✵ ✭✶✾✽✽✮❀ ❏✳ ❋r❡♥❦❡❧ ✲ P❤②s✳ ❘❡✈✳ ❊ ✺✹ ✺✽✺✾ ✭✶✾✾✻✮❀ ❇✳▼✳ P✐♠❡♥t❡❧ ❡ ❘✳●✳ ❚❡✐①❡✐r❛ ✲ ◆✉♦✈✳ ❈✐♠✳ ❇ ✶✶✸✱ ✽✵✺ ✭✶✾✾✽✮❀ ❏✳ ❋r❡♥❦❡❧ ✲ ■♥t✳ ❏✳ ▼♦❞✳ P❤②s✳ ✶✸✱ ✸✶✺ ✭✶✾✾✾✮❀ ▼✳❈✳ ❇❡rt✐♥✱ ❇✳▼✳ P✐♠❡♥t❡❧ ❡ P✳❏✳ P♦♠♣❡✐❛ ✲ ❆♥♥✳ P❤②s✳ ✸✷✸✱✺✷✼ ✭✷✵✵✽✮❀ ❍✳ ❚♦rr❡s✲❙✐❧✈❛ ✲ ❘❡✈✳ ❈❤✐❧✳ ■♥❣✳ ✶✻✱ ✻✺ ✭✷✵✵✽✮✳

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❈❛♣ít✉❧♦ ✷

❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ à ❧❛ ❯t✐②❛♠❛

❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ❝❤❛♠❛❞❛s t❡♦r✐❛s ❞❡ ❣❛✉❣❡ ♦❝♦rr❡✉ ❛♦ ❧♦♥❣♦ ❞❡ ♣r❛t✐❝❛♠❡♥t❡ ✻✵ ❛♥♦s✱ t❡♥❞♦ ❡st❡ ❞✉❛s ❢❛s❡s ❜❡♠ ❞❡✜♥✐❞❛s ❬✶❪✳ ❆ ♣r✐♠❡✐r❛ ❢❛s❡✱ q✉❡ ❝✉❧♠✐♥♦✉ ❝♦♠ ♦ q✉❡ s❡ ❝❤❛♠❛ ❤♦❥❡ ❞❡ t❡♦r✐❛ ❞❡ ❣❛✉❣❡ ❛❜❡❧✐❛♥❛✱ ❢♦✐ ❝♦♥str✉í❞❛ ❛♦ ❧♦♥❣♦ ❞❡ ❝❡r❝❛ ❞❡ ✶✶ ❛♥♦s ♣♦r ❲❡②❧✳ ■♥s♣✐r❛❞♦ ♣❡❧♦ tr❛❜❛❧❤♦ ❞❡ ❊✐♥st❡✐♥ ❡♠ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ❬✷❪✱ ❡♠ q✉❡ ♦ ♠❡s♠♦✱ ❛♦ s✉❜st✐t✉✐r tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❣❡r❛✐s ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❡r❛ ❝❛♣❛③ ❞❡ ❞❡s❝r❡✈❡r ❛ ✐♥t❡r❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧✱ ❲❡②❧ t❡♥t♦✉ ♠♦str❛r q✉❡ ❛ ✐♥t❡r❛çã♦ ❡❧❡tr♦♠❛❣♥ét✐❝❛ t❛♠❜é♠ ♣♦❞❡r✐❛ s❡r ✐♥tr♦❞✉③✐❞❛ ❝♦♠ ❛ s✉❜st✐t✉✐çã♦ ❞❡ ✉♠❛ ♦✉tr❛ s✐♠❡tr✐❛ ❣❧♦❜❛❧ ♣♦r ✉♠❛ s✐♠❡tr✐❛ ❧♦❝❛❧ ✲ ♥♦ ❝❛s♦✱ ❛ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❢❛s❡✳ ❖ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦ ♥❡st❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ♣♦r ❲❡②❧ ❡♠ ✶✾✶✽ ❬✸❪✱ ♠❛s ❛✐♥❞❛ ♥ã♦ ❝♦♥t❡♠♣❧❛✈❛ ♦s ♦❜❥❡t✐✈♦s ❞❡ ❢♦r♠❛ s❛t✐s❢❛tór✐❛✳ ❖ ❡st✉❞♦ ❢♦✐ s❡♥❞♦ ❛♠❛❞✉r❡❝✐❞♦ ❛té q✉❡✱ ❡♠ ✶✾✷✾ ❬✹❪✱ ✉♠❛ ❢♦r♠❛ ✜♥❛❧ ♣❛r❛ ❛ ✐♥t❡♥çã♦ ❞❡ ❲❡②❧ ❢♦✐ ❛t✐♥❣✐❞❛✱ ❡♥❝❡rr❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❢❛s❡ ❞❡ ❝♦♥str✉çã♦ ❞❛s t❡♦r✐❛s ❞❡ ❣❛✉❣❡✳

❖ ✉s♦ ❞❡ s✐♠❡tr✐❛s ♣❛r❛ ❞❡s❝r❡✈❡r ✐♥t❡r❛çõ❡s ♥✉❝❧❡❛r❡s✱ q✉❡ ♠❛r❝❛ ♦ s❡❣✉♥❞♦ ♣❡rí♦❞♦ ❞❡ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦✱ ❢♦✐ ♦❝♦rr❡r ✷✺ ❛♥♦s ♠❛✐s t❛r❞❡✱ ❝♦♠ ♦ tr❛❜❛❧❤♦ ❞❡ ❨❛♥❣ ❡ ▼✐❧❧s ❬✺❪✳ ❆ ✐❞é✐❛ ❞❡❧❡s ❡r❛ t♦r♥❛r ❧♦❝❛❧ ❛ s✐♠❡tr✐❛ ❞❡ ✐s♦s♣✐♥✱ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❲❡②❧ ❡♠ s❡✉ ❛rt✐❣♦ ❞❡ ✶✾✷✾✳ ❊st❡ ❢♦✐ ♦ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦ ❡♠ t❡♦r✐❛ ❞❡ ❣❛✉❣❡ ♥ã♦ ❛❜❡❧✐❛♥❛✱ q✉❡ ♥❛ ♠❡s♠❛ é♣♦❝❛ ❡st❛✈❛ s❡♥❞♦ ✐♥✈❡st✐❣❛❞❛✱ ❡♠ ❝♦♥t❡①t♦s ❞✐❢❡r❡♥t❡s✱ ♣♦r P❛✉❧✐ ❬✻❪ ❡ ❙❤❛✇ ❬✼❪✳ ◆❡st❡ ♠❡s♠♦ ♣❡rí♦❞♦✱ ❯t✐②❛♠❛ t❛♠❜é♠ ❢❛③✐❛ s❡✉s ❡st✉❞♦s ♥♦ t❡♠❛✱ t❡♥t❛♥❞♦ ❡st❛❜❡❧❡❝❡r r❡❧❛çõ❡s ❡♥tr❡ ❛ ❣r❛✈✐t❛çã♦ ♥❛ ❢♦r♠✉❧❛çã♦ ❞❡ t❡tr❛❞❛s ❡ ♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦✳ P♦ré♠✱ ❛♦ ❡♥tr❛r ❡♠ ❝♦♥t❛t♦ ❝♦♠ ♦ tr❛❜❛❧❤♦ ❞❡ ❨❛♥❣ ❡ ▼✐❧❧s ❞✉r❛♥t❡ ✉♠❛ ✈✐s✐t❛ ❛ Pr✐♥❝❡t♦♥✱ ❯t✐②❛♠❛ ❡♥❣❛✈❡t♦✉ s❡✉ tr❛❜❛❧❤♦✱ ❡♠ ❢✉♥çã♦ ❞❛s s❡♠❡❧✲ ❤❛♥ç❛s ❞❡ s❡✉s r❡s✉❧t❛❞♦s ❝♦♠ ♦s ❞♦ ♣r✐♠❡✐r♦✳ ❋♦✐ só ❛❧❣✉♠ t❡♠♣♦ ❞❡♣♦✐s q✉❡ ❯t✐②❛♠❛ ♣❡r❝❡❜❡✉ q✉❡ s✉❛ ❝♦♥str✉çã♦ ❡r❛ ✈á❧✐❞❛ ♣❛r❛ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ q✉❛❧q✉❡r✱ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ ❝❛s♦ ❞❡ ❨❛♥❣ ❡ ▼✐❧❧s✱ ❡ q✉❡ ❛✐♥❞❛ ❝♦♥t❡♠♣❧❛✈❛ ♦ ❝❛s♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳ ❊st❡ tr❛❜❛❧❤♦ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✾✺✻ ❬✽❪ ❡ é ❝♦♥s✐❞❡r❛❞♦ ♣♦r ❛❧❣✉♥s ❛✉t♦r❡s ❬✶❪ ❝♦♠♦ ❛ ❢♦r♠❛ ✜♥❛❧ ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞❡ t❡♦r✐❛s ❞❡ ❣❛✉❣❡✳ ➱ ❡st❛ ❝♦♥str✉çã♦ q✉❡ s❡rá ♠♦str❛❞❛ ❛ s❡❣✉✐r✳

(15)

✷✳✶ ❆ ❢♦r♠✉❧❛çã♦ ❞❡ ❯t✐②❛♠❛ ✲ ♦ ❝❛s♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠

❊♠ s❡✉ tr❛❜❛❧❤♦ ❞❡ ✶✾✺✻ ❬✽❪✱ ❯t✐②❛♠❛ q✉❡r✐❛ ❡♥t❡♥❞❡r ❛ r❡❧❛çã♦ ❡♥tr❡ ❝❛♠♣♦s ❡♠ ✐♥t❡r❛çã♦ ❡ ❛ ❡①✐❣ê♥❝✐❛ ❞❡ ✐♥✈❛r✐â♥❝✐❛ ❞❛ ▲❛❣r❛♥❣✐❛♥❛ ❛ss♦❝✐❛❞❛ ❛ ❡ss❡s ❝❛♠♣♦s✱ s♦❜ tr❛♥❢♦r♠❛çõ❡s ❛ ♣❛râ♠❡tr♦s ❞❡♣❡♥❞❡♥t❡s ❞♦ ♣♦♥t♦✳ P❛r❛ ✐ss♦ ❡❧❡ ❡st❛❜❡❧❡❝❡✉ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ❤✐♣ót❡s❡s✿ ❝♦♥s✐❞❡r❡✲s❡ ✉♠ s✐st❡♠❛ ❞❡ ❝❛♠♣♦s QA₍_x₎ _{✐♥✈❛r✐❛♥t❡ s♦❜ ✉♠ ❣r✉♣♦ ❞❡ tr❛♥s❢♦r♠❛çã♦} _G _{✭❞❡ ▲✐❡✮ ❞❡♣❡♥❞❡♥t❡ ❞❡}

n ♣❛râ♠❡tr♦s ǫc_{❀ s✉♣♦♥❤❛✲s❡ q✉❡ ♦ ❣r✉♣♦} _G _{s❡❥❛ s✉❜st✐t✉í❞♦ ♣♦r ✉♠ ♦✉tr♦ ❣r✉♣♦}_G′ _{❡♠ q✉❡ ♦s}

♣❛râ♠❡tr♦sǫc_{✭❞♦ ❣r✉♣♦}_G_{✮ sã♦ tr♦❝❛❞♦s ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s ❛r❜✐trár✐❛s}_ǫc₍_x_{)❀ ❝♦♥s✐❞❡r❡✲s❡} ❛✐♥❞❛ q✉❡ ♦ s✐st❡♠❛ ♣❡r♠❛♥❡ç❛ ✐♥✈❛r✐❛♥t❡ ♣♦r ❡st❡ ❣r✉♣♦ ♠❛✐s ❣❡r❛❧ G′ _{❛tr❛✈és ❞❛ ✐♥tr♦❞✉çã♦ ❞❡}

✉♠ ♥♦✈♦ ❝❛♠♣♦A(x)✳ ❆ ♣❛rt✐r ❞❡st❛s ❝♦♥s✐❞❡r❛çõ❡s✱ ❝✐♥❝♦ ♣❡r❣✉♥t❛s ❞❡✈❡r✐❛♠ s❡r r❡s♣♦♥❞✐❞❛s✿

✭✶✮ ◗✉❡ t✐♣♦ ❞❡ ❝❛♠♣♦A(x)❞❡✈❡ s❡r ✐♥tr♦❞✉③✐❞♦ ♣❛r❛ ❣❛r❛♥t✐r ❛ ✐♥✈❛r✐â♥❝✐❛❄ ✭✷✮ ❈♦♠♦ ❡st❡ ❝❛♠♣♦ s❡ tr❛♥s❢♦r♠❛ s♦❜G′_❄

✭✸✮ ◗✉❡ ❢♦r♠❛ t♦♠❛ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦ ❝❛♠♣♦A(x)❡ ♦ ❝❛♠♣♦ ♦r✐❣✐♥❛❧QA₍_x_)❄

✭✹✮ ❈♦♠♦ ❞❡t❡r♠✐♥❛r ❛ ♥♦✈❛ ▲❛❣r❛♥❣✐❛♥❛L′₍_{Q, A}₎_{❛ ♣❛rt✐r ❞❛ ▲❛❣r❛♥❣✐❛♥❛ ♦r✐❣✐♥❛❧}_L₍_Q_)❄

✭✺✮ ◗✉❡ t✐♣♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ sã♦ ♣❡r♠✐t✐❞❛s ♣❛r❛A❄

❊st❡ ❢♦✐ ♦ r♦t❡✐r♦ ♣r♦♣♦st♦ ♣♦r ❯t✐②❛♠❛ ❡ ❛s r❡s♣♦st❛s ♣❛r❛ ❡st❛s ♣❡❣✉♥t❛s s❡❣✉❡♠ ❛❜❛✐①♦✳

✷✳✶✳✶ ■♥✈❛r✐â♥❝✐❛ ▲♦❝❛❧ ❡ ♦ ❈❛♠♣♦

A

❈♦♥s✐❞❡r❡✲s❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ ❞❡s❝r✐t♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❛♠♣♦sQA₍_x₎_,₍_A_{= 1}_,₂_{, ..., N}_{)✱ ❡ ♣♦r} ✉♠❛ ❞❡♥s✐❞❛❞❡ ▲❛❣r❛♥❣✐❛♥❛L(QA_{, ∂}

µQA)✱ ❡♠ q✉❡∂µQA= ∂Q

A

∂xµ✱ ❡ ❝✉❥❛s ❡q✉❛çõ❡s ❞❡ ❝❛♠♣♦ sã♦ ∂L

∂QA −∂µ

_∂L

∂(∂µQA)

= 0. ✭✷✳✶✮

P♦st✉❧❡✲s❡ q✉❡ ✐♥t❡❣r❛❧ ❞❡ ❛çã♦✱I=´

ΩLd4x,s❡❥❛ ✐♥✈❛r✐❛♥t❡ s♦❜ ❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦ ✐♥✜♥✐t❡s✲ ✐♠❛❧✿

QA → QA+δQA,

δQA = T(a), BAǫaQB, ✭✷✳✷✮

❡♠ q✉❡ǫa_{sã♦ ♣❛râ♠❡tr♦s ✐♥✜♥✐t❡s✐♠❛✐s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡}_x₍_a_{= 1}_,₂_{, ..., n}₎_❡_T A

(a), Bsã♦ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s ✭❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦✮✳ ❆ss✉♠❛✲s❡ ❛✐♥❞❛ q✉❡ ❛ tr❛♥s❢♦r♠❛çã♦ ✭✷✳✷✮ é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡G

❞❡♣❡♥❞❡♥t❡ ❞♦sn♣❛râ♠❡tr♦sǫa_{✳ ❆ss✐♠✱ ❞❡✈❡ ❤❛✈❡r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥st❛♥t❡s}_fb

a c✱ ❛s ✏❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛✑✱ ❞❡✜♥✐❞❛s ♣♦r

T(a), T(b) A

B=T A

(a), CT(b), BC −T(b), CAT(a), BC =fa bc T(c), BA, ✭✷✳✸✮ ❛s q✉❛✐s s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s

fm

a bfm cl +fb cmfm al +fc amfm bl = 0, ✭✷✳✹✮

(16)

fa bc =−fb ac , ✭✷✳✺✮ ❡st❛ ú❧t✐♠❛ ✈✐♥❞❛ ❞❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ✭✷✳✸✮✱ ❡♥q✉❛♥t♦ ❛ ❛♥t❡r✐♦r s❡❣✉❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐ ♣❛r❛ ♦ ❝♦♠✉t❛❞♦r✳

❆ss✉♠✐♥❞♦✲s❡ q✉❡ ❛ ❛çã♦I s❡❥❛ ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✜♥✐t❡s✐♠❛❧ ♥♦ ❝❛♠♣♦ ✭✷✳✷✮ ❡♠

q✉❛❧q✉❡r ❞♦♠í♥✐♦Ω❞♦ ❡s♣❛ç♦✲t❡♠♣♦1 _{❡ ❧❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥t❛ q✉❡ ♦s}_ǫa _{sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❡♠✲s❡}

δL= 0⇒ ∂L ∂QAT

A

(a), BQB+

∂L ∂(∂µQA)

T A

(a), B∂µQB= 0. ✭✷✳✻✮ ❆q✉✐ ❛❞♠✐t❡✲s❡ ❧í❝✐t❛ ❛ tr♦❝❛ ❞❛ ♦r❞❡♠ ♥❛ ♦♣❡r❛çã♦ ✈❛r✐❛çã♦ ♣❡❧❛ ❞❡r✐✈❛çã♦ ♦r❞✐♥ár✐❛✳

❆ ❝♦♥❞✐çã♦δL= 0 ♣♦❞❡ s❡r ❡s❝r✐t❛✱ ✉t✐❧✐③❛♥❞♦✲s❡ ❛ r❡❣r❛ ❞❡ ▲❡✐❜♥✐③✱ ❞❡ ❢♦r♠❛ ❛ s❡ ♦❜t❡r ❛ ❧❡✐ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❡ ✉♠❛ ❝♦rr❡♥t❡Jµ

a✱

∂µJµa = 0, Jµa ≡

∂L ∂(∂µQA)

T(a), BA Q

B_, _{✭✷✳✼✮}

q✉❡ é ✈á❧✐❞❛ q✉❛♥❞♦ ❛ ❡q✉❛çã♦ ❞❡ ❝❛♠♣♦ é s❛t✐s❢❡✐t❛✳

❆❣♦r❛ ❝♦♥s✐❞❡r❡✲s❡ ❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✜♥✐t❡s✐♠❛❧✱ ❝♦♠ ♣❛râ♠❡tr♦ ❞❡♣❡♥❞❡♥t❡ ❞♦ ♣♦♥t♦✿

δQA₍_x_{) =} _T A

(a), Bǫa(x)QB, ✭✷✳✽✮

ǫa₍_x_{) =} _{❢✉♥çã♦ ❛r❜✐trár✐❛ ✐♥✜♥✐t❡s✐♠❛❧(}_a_{= 1}_,₂_{, ..., n}₎_, ❡♠ q✉❡T A

(a), B♣❡r♠❛♥❡❝❡♠ ❝♦♠♦ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s✳ ◆❡st❡ ❝❛s♦✱

δL = {(2.6)}aǫ

a₍_x_{) +} ∂L

∂(∂µQA)

T A

(a), BQB∂µǫa(x), ✭✷✳✾✮ ❆❞♠✐t✐♥❞♦✲s❡ q✉❡ ❛ ❡①♣r❡ssã♦ ❊q✳✭2.6✮ ♣❡r♠❛♥❡ç❛ ✈á❧✐❞❛ ♠❡s♠♦ q✉❛♥❞♦ǫa₌_ǫa₍_x_{)✱ ✐✳❡✳} _{₍₂_.₆₎_}

a= 0✱ ❡♥tã♦ δL ♥ã♦ s❡ ❛♥✉❧❛ ❞❡✈✐❞♦ à ♣r❡s❡♥ç❛ ❞♦ ú❧t✐♠♦ t❡r♠♦ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡✱ ♣♦rt❛♥t♦✱ ♦

s✐st❡♠❛ ♣❡r❞❡ s✉❛ ✐♥✈❛r✐â♥❝✐❛✳

❯♠❛ ❢♦r♠❛ ❞❡ ♣r❡s❡r✈❛r ❛ ✐♥✈❛r✐â♥❝✐❛ ❞❛ ▲❛❣r❛♥❣✐❛♥❛ s♦❜ ✭✷✳✽✮✱ ❢♦rç❛♥❞♦δL= 0✱ é ✐♥tr♦❞✉③✐r

✉♠ ♥♦✈♦ ❝❛♠♣♦

A′J₍_x₎_, _J _{= 1}_,₂_{, ..., M,} q✉❡ s❡ tr❛♥s❢♦r♠❡ ❝♦♠♦

δA′J ₌ _U J

(a), KA′Kǫa(x) +CJ µ, a∂µǫa, ✭✷✳✶✵✮ ♥❛ ❡s♣❡r❛♥ç❛ ❞❡ q✉❡ ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ ❝❛♥❝❡❧❡ ♦ s❡❣✉♥❞♦ t❡r♠♦ ❞♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❞❡ ❊q✳✭✷✳✾✮✳ ◆❛ ❊q✳ ✭✷✳✶✵✮✱ ♦s ❝♦❡✜❝✐❡♥t❡sU ❡C sã♦ ❝♦♥st❛♥t❡s ❛ s❡r❡♠ ❞❡t❡r♠✐♥❛❞❛s✳

P♦st✉❧❡✲s❡ ❛❣♦r❛ q✉❡ ❛ ❛çã♦I′ ₌´

Ωd

4_xL_′₍_QA_{, ∂}

µQA, A′J)✱ s❡❥❛ ✐♥✈❛r✐❛♥t❡ s♦❜ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❊q✳✭✷✳✶✵✮ ❡ ❞❡ ❊q✳✭✷✳✽✮✳ ❯s❛♥❞♦✲s❡ ♦ ❢❛t♦ ❞❡ q✉❡ ❛s ❢✉♥çõ❡s ǫa₍_x₎ _{sã♦ ❛r❜✐trár✐❛s✱ ❡ ♣♦r ✐ss♦}

1_{❊st❛ r❡❧❛çã♦ ✐♥❞❡♣❡♥❞❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡}_QA_❡_∂

µQA✱ ♦✉ s❡❥❛δL= 0✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ❝❛rát❡r ❞♦ ❝❛♠♣♦ ✭❡s❝❛❧❛r✱ ✈❡t♦r✐❛❧✱ ❡s♣✐♥♦r✐❛❧✱ ❡t❝✳✮✳

(17)

❛ss✉♠❡✲s❡ q✉❡ǫa₍_x₎_{❡ s✉❛s ❞❡r✐✈❛❞❛s}_∂

µǫasã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❝❛❞❛ ❝♦❡✜❝✐❡♥t❡ ❞❡ǫa(x)❡∂µǫa ❞❡✈❡ s❡ ❛♥✉❧❛r ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ♦ q✉❡ ❧❡✈❛ ❛✿

∂L′ ∂QAT

A

(a), BQ

B₊ ∂L′

∂(∂µQA)

T(a), BA∂µQB+

∂L′ ∂A′JU

J

(a), KA′

K_{= 0}_, _{✭✷✳✶✶✮}

∂L′ ∂(∂µQA)

T A

(a), BQB+

∂L′ ∂A′JC

J µ

, a = 0. ✭✷✳✶✷✮

❖ ♥ú♠❡r♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❞❡ A′J _{❞❡✈❡ s❡r ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❡q✉❛çõ❡s}2 _{❞❡ ❊q✳✭✷✳✶✷✮✱ ♥♦ ❝❛s♦✱}

M = 4n✳ ❊st❛ ❡①✐❣ê♥❝✐❛ é ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ s❡ ❞❡t❡r♠✐♥❡ ✉♥✐✈♦❝❛♠❡♥t❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ L′

❝♦♠ r❡❧❛çã♦ ❛A′J_✳

❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ t♦r♥❛r ❡①♣❧í❝✐t❛ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡L′ _{❝♦♠ r❡❧❛çã♦ ❛}_A′J_{❡①✐❣❡✲s❡ q✉❡}_CJ µ , a s❡❥❛ ♥ã♦✲s✐♥❣✉❧❛r✱ ❞❡ ❢♦r♠❛ q✉❡ ❡①✐st❛ ✉♠❛ ✐♥✈❡rs❛ ❞❡C t❛❧ q✉❡

CJ µ , a C−

1a

µ,K = δ J K

C−1a

µ,JC J ν

, b = δabδµν. ❊st❛ ✐♥✈❡rs❛ ❞❡C é ✉s❛❞❛ ♣❛r❛ s❡ ❞❡✜♥✐r

Aaµ≡ C−1

a

µ,JA′ J_, ❝♦♠ ♦ q✉❛❧ s❡ r❡❡s❝r❡✈❡ ❊q✳✭✷✳✶✷✮✿

∂L′ ∂Aa µ

+ ∂L′

∂(∂µQA)

T(a), BAQ B _{= 0}_.

❊st❛ ❡q✉❛çã♦ é s❛t✐s❢❡✐t❛ s❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ L′ _{❡♠ r❡❧❛çã♦ ❛} _∂

µQA ❡ Aaµ ♦❝♦rr❡r ❛tr❛✈és ❞❛ ❝♦♠❜✐♥❛çã♦

∇µQA≡∂µQA−T(a), BA Q B_Aa

µ =∂µQA−QB C−1

a

µ,JA′

J_. _{✭✷✳✶✸✮}

❆❣♦r❛ ❛ ♥♦✈❛ ▲❛❣r❛♥❣✐❛♥❛ t❡♠ ❛ ❢♦r♠❛

L′(QA, ∂µQA, A′J) =L′′(QA,∇µQA). ❡ ❛ s✉❛ ✈❛r✐❛çã♦✱δL′′₌ ∂L′′

∂QAδQA+ ∂L

′′

∂∇µQAδ∇µQ

A_, _{❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞❡}_ǫa₍_x₎_{❡ ❛} ❧❡✐ ❞❡ tr❛♥s❢♦r♠❛çã♦ ♣❛r❛Aa

µ✱ ❧❡✈❛ ❛

∂L′′ ∂QAT

A

(a), BQB+

∂L′′ ∂∇µQA

T A

(a), B∇µQB+ + ∂L′′

∂∇µQA

QB_Ab ν

n

[T(a), T(b)]ABδµν−S(a)c, νµ, bT A

(c), B

o

= 0, ✭✷✳✶✹✮

2_{◗✉❛tr♦ ❡q✉❛çõ❡s ♣❛r❛ ♦s í♥❞✐❝❡s ❞❡ ❡s♣❛ç♦✲t❡♠♣♦✱ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s} _n _{í♥❞✐❝❡s q✉❡ r♦t✉❧❛♠ ❛s ❢✉♥çõ❡s ❞❛} tr❛♥s❢♦r♠❛çã♦ ✭✷✳✽✮✳

(18)

❆ ❧❡✐ ❞❡ tr❛♥s❢♦r♠❛çã♦ ♣❛r❛Aa

µ é ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❊q✳✭✷✳✶✵✮✱

δAc

ν =S(a)c, µν, bA b

µǫa(x) +∂νǫc, ✭✷✳✶✺✮

♦♥❞❡ s❡ ❞❡✜♥✐✉

S₍_a₎c, µ_{ν, b}≡ C−1c

ν,JU J

(a), KC K µ , b .

◆♦t❡✲s❡ q✉❡ ♦s t❡r♠♦s ❝♦♠∂µǫa ❛♥✉❧❛r❛♠✲s❡ ❡①❛t❛♠❡♥t❡ ❡♠ ❊q✳✭✷✳✶✹✮✱ ❝♦♥❢♦r♠❡ ❞❡s❡❥❛❞♦ q✉❛♥❞♦ s❡ ✐♥tr♦❞✉③✐✉ ♦ ❝❛♠♣♦A′J_✳

P❛r❛ s❡ ❝♦♠♣r❡❡♥❞❡r ♠❡❧❤♦r ♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ❡♠ ❊q✳✭✷✳✶✹✮ é ✐♠♣♦rt❛♥t❡ ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦r✲ t❛♠❡♥t♦ ❞❡ L′′_{q✉❛♥❞♦ s❡ t♦♠❛ ♦ ❧✐♠✐t❡ ❞♦ ❝❛♠♣♦}_A_{♥✉❧♦✳ ◆❡ss❛ ❝✐r❝✉♥stâ♥❝✐❛✱}

L′′₍_QA_,_∇

µQA) =L′′(QA, ∂µQA).

❋✐s✐❝❛♠❡♥t❡ ❡s♣❡r❛✲s❡ q✉❡✱ ❛♦ t♦♠❛r ♦ ❝❛♠♣♦ A ✐❣✉❛❧ ❛ ③❡r♦✱ ❛ s✐t✉❛çã♦ ♦r✐❣✐♥❛❧✱ ❡♠ q✉❡ ❛ ▲❛✲

❣r❛♥❣✐❛♥❛ ❞♦ ❝❛♠♣♦ ❡r❛ L(QA_{, ∂}

µQA)✱ s❡❥❛ r❡❝✉♣❡r❛❞❛✳ ❈♦♥❝❧✉✐✲s❡ ❛ss✐♠ q✉❡ L′′(QA, ∂µQA) =

L(QA_{, ∂}

µQA)✱ ♠♦t✐✈❛♥❞♦ ❛ ✐❞❡♥t✐✜❝❛çã♦

L′′₍_QA_,_∇

µQA) =L(QA,∇µQA).

❉❡ss❛ ❡①♣r❡ssã♦ ✈ê✲s❡ q✉❡✱ ❛♦ s❡ ✐♥tr♦❞✉③✐r ♦ ❝❛♠♣♦A✱ ❞❡✈❡✲s❡ s✉❜st✐t✉✐r ❛ ❞❡r✐✈❛❞❛ ♦r❞✐♥ár✐❛ ♥❛

▲❛❣r❛♥❣✐❛♥❛ ❞❡ ♣❛rt✐❞❛L(QA_{, ∂}

µQA)♣❡❧❛ ✏❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡✑✱∇µQA✳ ❖ ❝❛rát❡r ❝♦✈❛r✐❛♥t❡ ❞❡st❡ ♦❜❥❡t♦ s❡rá ✈❡r✐✜❝❛❞♦ ❛❞✐❛♥t❡✳ ❊st❡ t✐♣♦ ❞❡ ❝♦♥str✉çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♣r❡s❝r✐çã♦ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ♠í♥✐♠♦✱ s❡♠❡❧❤❛♥t❡ à q✉❡ ❛♣❛r❡❝❡ ❡♠ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧ ✭❘●✮✳ ◆❛ ❘●✱ ♦ ❛❝♦♣❧❛♠❡♥t♦ ♠í♥✐♠♦ é ❡♠♣r❡❣❛❞♦ ♣❛r❛ ❛ ❞❡s❝r✐çã♦ ❞❛ ✐♥t❡r❛çã♦ ❞♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❝♦♠ ♦ ❝❛♠♣♦ ❞❡ ♠❛tér✐❛✳ ◆♦ ♣r❡s❡♥t❡ ❝❛s♦✱ ❡st❛ ♣r❡s❝r✐çã♦ ❞❡s❝r❡✈❡ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦ ❝❛♠♣♦Q❡ ♦ ❝❛♠♣♦A✳

❋❛③❡♥❞♦ ✉s♦ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ♠í♥✐♠♦✱ ∂µ → ∇µ✱ ❞❡✈❡✲s❡ ❡①✐❣✐r q✉❡ ❛ ❊q✳ ✭✷✳✻✮✱ q✉❡ ❛❣♦r❛ s❡ t♦r♥❛

∂L ∂QAT

A

(a), BQB+

∂L ∂∇µQA

T A

(a), B∇µQB= 0,

❝♦♥t✐♥✉❡ ✈á❧✐❞❛✳ ❈♦♥❢r♦♥t❛♥❞♦✲s❡ ❡st❛ ❡①♣r❡ssã♦ ❝♦♠ ❊q✳✭✷✳✶✹✮✱ ❢❛③❡♥❞♦L′′₌_L_{✱ ❝♦♥❝❧✉✐✲s❡ q✉❡} ∂L

∂∇µQA

QBAbν

n

[T(a), T(b)]ABδµν−S c, ν

(a)µ, bT A

(c), B

o = 0,

❡ ♦s ❝♦❡✜❝✐❡♥t❡sS₍_a₎c, ν_{µ, b}✜❝❛♠ ❞❡t❡r♠✐♥❛❞♦s3

S₍_a₎c, ν_{µ, b}=fa bc δνµ. ✭✷✳✶✻✮ ◆❡st❛ ú❧t✐♠❛ ❡①♣r❡ssã♦ ✉t✐❧✐③♦✉✲s❡ ❛ ❞❡✜♥✐çã♦ ❞❛s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛✳

3_{❖❜s❡r✈❡✲s❡ q✉❡ ❛♦ tr❛❜❛❧❤❛r ❝♦♠}_Ab

ν❡♠ ❧✉❣❛r ❞❡A′J✱ tr♦❝❛✲s❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ s❡ ❡♥❝♦♥tr❛r ❛s ❝♦♥t❛♥t❡sU(a)J, K ❡CK µ_{, b} ♣❡❧♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡r♠✐♥❛r ❛♣❡♥❛sS₍_a₎c, ν_{µ, b}✳

(19)

❆ ♣❛rt✐r ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡S✱ ❛ ❧❡✐ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡Aa

µ t♦r♥❛✲s❡

δAc

ν=fa bc Abνǫa(x) +∂νǫc, ✭✷✳✶✼✮ ❡ ❞❛í ✜❝❛ ❡①♣❧í❝✐t♦ ♦ ❝❛rát❡r ❝♦✈❛r✐❛♥t❡ ❞❛ ❞❡r✐✈❛❞❛∇µQA✱ ❝✉❥❛ ❧❡✐ ❞❡ tr❛♥s❢♦r♠❛çã♦ é

δ∇µQA=T(a), BA ∇µQBǫa(x).

✷✳✶✳✷ ❊q✉❛çõ❡s ❞❡ ❝❛♠♣♦ ♣❛r❛

A

❧✐✈r❡

P❛r❛ ❞✐s❝✉t✐r q✉❛✐s ❡q✉❛çõ❡s sã♦ ♣❡r♠✐t✐❞❛s ♣❛r❛ ♦ ❝❛♠♣♦A✱ ❛❞♠✐t❡✲s❡✱ ♥❛ ❝♦♥str✉çã♦ ❞❡ ❯t✐②❛♠❛✱

q✉❡ ❛ ▲❛❣r❛♥❣✐❛♥❛ ♣❛r❛A❧✐✈r❡ ❝♦♥t❡♥❤❛ s♦♠❡♥t❡ ❞❡r✐✈❛❞❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ✐✳❡✳

L0(Aaµ, ∂νAaµ), ∂νAaµ≡

∂Aa µ

∂xν ,

❡ q✉❡ ❡st❛ ▲❛❣r❛♥❣✐❛♥❛ s❡❥❛ ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦ ❞❛❞❛ ♣♦r ❊q✳✭✷✳✶✺✮✳ ❆ss✐♠✱ ❛ ✈❛r✐❛çã♦ ❞❡L0✱

δL0= ∂L0

∂Aa µ

δAaµ+

∂L0

∂ ∂νAaµ

δ ∂νA

a µ

= 0,

❧❡✈❛ ❛♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s✱ q✉❛♥❞♦ s❡ ❝♦♥s✐❞❡r❛ ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞❡ǫ❡ s✉❛s ❞❡r✐✈❛❞❛s4_✿

∂L0

∂Aa µ

fa c bAbµ+

∂L0

∂ ∂νAaµ

f

a

c b∂νAbµ ≡ 0, ✭✷✳✶✽✮

∂L0

∂Aa µ

+ ∂L0

∂(∂µAcν)

fc

a bAbν ≡ 0, ✭✷✳✶✾✮

∂L0

∂ ∂νAaµ

+

∂L0

∂(∂µAaν) ≡

0. ✭✷✳✷✵✮

❆ ♣❛rt✐r ❞❛ ❊q✳✭✷✳✷✵✮ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡L0 ❝♦♠ r❡s♣❡✐t♦ à ❞❡r✐✈❛❞❛ ❞❡A ♦❝♦rr❡ ❛tr❛✈és ❞❛ ❝♦♠❜✐♥❛çã♦

Aa

[µ,ν] ≡∂µAaν−∂νAaµ. ✭✷✳✷✶✮ ❈♦♠ ❡st❡ r❡s✉❧t❛❞♦✱ ❛ ❊q✳✭✷✳✶✾✮ é r❡❡s❝r✐t❛ ❝♦♠♦ s❡♥❞♦5_✿

∂L0

∂Aa µ

− ∂L0

∂Ac

[ν,µ] 2f

c

a bAbν = 0. ✭✷✳✷✷✮

❆ ♣❛rt✐r ❞❡st❛ ❡q✉❛çã♦✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ▲❛❣r❛♥❣✐❛♥❛ ❝♦♠ r❡s♣❡✐t♦ ❛ Aa

[µ,ν] ❞❡✈❡ ♦❝♦rr❡r ❛tr❛✈és ❞❡ ✉♠ ♥♦✈♦ ♦❜❥❡t♦✿

4_{P❛r❛ ❡s❝r❡✈❡r ❊q✳✭✷✳✷✵✮ ❡♠♣r❡❣❛✲s❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s✐♠❡tr✐❛ ❞❡} ∂2ǫa ∂xν∂xµ✳

5_{❊♥t❡♥❞❛✲s❡ ❛q✉✐ q✉❡}_L

0 Aaµ, ∂νAaµ

→L′

0

Aa µ, Aa_[µ,ν]

✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ ♥ã♦ ❝❛rr❡❣❛r ❛ ♥♦t❛çã♦✱ ❛ ❧✐♥❤❛ s❡rá ♦♠✐t✐❞❛✳ ❊ss❡ ♠❡s♠♦ t✐♣♦ ❞❡ ❝♦♥str✉çã♦ s❡rá ❢❡✐t♦ ❛❞✐❛♥t❡✳

(20)

Fa

µν =∂µAaν−∂νAaµ− 1 2f

a

b c AbµAcν−AcµAbν

. ✭✷✳✷✸✮

❋✐♥❛❧♠❡♥t❡✱ ❛♦ s✉❜st✐t✉✐r ❊q✳✭✷✳✶✾✮ ❡♠ ❊q✳✭✷✳✶✽✮ ♦❜té♠✲s❡ 1 2 ∂L0 ∂Fa νµ fa

c bFbνµ= 0. ✭✷✳✷✹✮

P❛r❛ ♦❜t❡r ❡st❡ r❡s✉❧t❛❞♦ é ♥❡❝❡ssár✐♦ ♦ ✉s♦ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐ ♣❛r❛ ❛s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛ ✭❊q✳✭✷✳✹✮✮✳ ❆ ❡①♣r❡ssã♦ ❊q✳✭✷✳✷✹✮ ❞❡✜♥❡ ❝♦♠♦ ❞❡✈❡ s❡r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡L0 ❝♦♠ r❡s♣❡✐t♦ ❛Faνµ ✲ s♦♠❡♥t❡ ❛s ❧❛❣r❛♥❣✐❛♥❛s q✉❡ s❛t✐s❢❛③❡♠ ❡st❛ ❝♦♥❞✐çã♦ sã♦ ❝♦♥s✐st❡♥t❡s ❝♦♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ✐♥✈❛r✐â♥❝✐❛ ❞❛ ▲❛❣r❛♥❣✐❛♥❛✳ P♦ré♠✱ ❛✐♥❞❛ ♣♦❞❡ ❡①✐st✐r ✉♠❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡L0❝♦♠ r❡s♣❡✐t♦ ❛A✱L0(Aaµ, Faµν)✳ P❛r❛ s❡ ❞❡t❡r♠✐♥❛r ❝♦♠♦ ❞❡✈❡ s❡r ❡st❛ ❞❡♣❡♥❞ê♥❝✐❛✱ ❞❡✈❡✲s❡ ❝♦♥❤❡❝❡r ❛ ❧❡✐ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡

Fa

νµ✱ ❛ q✉❛❧ é ♦❜t✐❞❛ ✉t✐❧✐③❛♥❞♦✲s❡ ❊q✳✭✷✳✶✼✮✿

δFaµν =ǫc(x)fc baFbµν. ✭✷✳✷✺✮ ❆ ✈❛r✐❛çã♦ ❞❛ ▲❛❣r❛♥❣✐❛♥❛✱

δL0= ∂L0

∂Aa µ F

δAaµ+

∂L0

∂ Fa µν A

δ Faµν

= 0,

❧❡✈❛ ❛♦ s❡❣✉✐♥t❡ ♣❛r ❞❡ ❡q✉❛çõ❡s q✉❛♥❞♦ s❡ ❝♦♥s✐❞❡r❛ ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞❡∂µǫa ❡ǫc(x)✱ r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✿ ∂L0 ∂Aa µ F = 0 ∂L0 ∂Aa µ F

fc baAbµ+

∂L0

∂ Fa µν A

fc baFbµν= 0.

❆ ♣r✐♠❡✐r❛ ❞❡ss❛s ❡q✉❛çõ❡s ✐♥❢♦r♠❛ q✉❡L0 ♥ã♦ ❞❡✈❡ t❡r ❞❡♣❡♥❞ê♥❝✐❛ ❡①♣❧í❝✐t❛ ❝♦♠ r❡❧❛çã♦ ❛

A ✲ t♦❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♠A❞❡✈❡ ♦❝♦rr❡r s♦♠❡♥t❡ ❛tr❛✈és ❞❡ F✳ ❆ s❡❣✉♥❞❛ ❡q✉❛çã♦ ♥❛❞❛ ♠❛✐s é

❞♦ q✉❡ ❛ ❊q✳✭✷✳✷✹✮✱ q✉❛♥❞♦ s❡ ❧❡✈❛ ❡♠ ❝♦♥t❛ ♦ r❡s✉❧t❛❞♦ ❞❛ ♣r✐♠❡✐r❛✳

✷✳✶✳✸ ❈❛♠♣♦s ❡♠ ■♥t❡r❛çã♦

❖ s✐st❡♠❛ ❝♦♠♣❧❡t♦✱ ❝♦♠♣♦st♦ ❞♦s ❝❛♠♣♦A❡Q❛❣♦r❛ ♣❛ss❛ ❛ s❡r ❞❡s❝r✐t♦ ♣❡❧❛ ▲❛❣r❛♥❣✐❛♥❛ t♦t❛❧✿ LT =LlivreA +LlivreQ +Lintera¸c˜ao=L0(F) +L(Q,∇Q).

❈♦♥s✐❞❡r❛♥❞♦✲s❡ q✉❡ ❛ ✐♥t❡❣r❛❧ ❞❡ ❛çã♦ ❞❡✈❡ s❡r ✐♥✈❛r✐❛♥t❡ ♣❡❧❛s tr❛♥s❢♦r♠❛çõ❡s ❡♠A❡Q✱ s❡❣✉❡

q✉❡

0 = δLT

δQAδQ

A₊δLT

δAa µ

fb caǫbAcµ−∂µ

δLT

δAa µ

ǫa+

+∂µ

_∂L

T

∂∇µQA

δQA₊ ∂L0