❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦
■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛
◗✉❛♥t✐③❛çã♦ ❇❘❙❚ ❞❡ ❚❡♦r✐❛s ❝♦♠ ❙✐♠❡tr✐❛ ❞❡
●❛✉❣❡ ❙♣✭✷✱
R
✮
❏♦ã♦ ❊❞✉❛r❞♦ ❋r❡❞❡r✐❝♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❱✐❝t♦r ❞❡ ❖❧✐✈❡✐r❛ ❘✐✈❡❧❧❡s
❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s
❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❱✐❝t♦r ❞❡ ❖❧✐✈❡✐r❛ ❘✐✈❡❧❧❡s ✭■❋❯❙P✮ Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ❖✳ ❈✳ ●♦♠❡s ✭■❋❯❙P✮ Pr♦❢✳ ❉r✳ ❆❞✐❧s♦♥ ❏♦sé ❞❛ ❙✐❧✈❛ ✭■❋❯❙P✮ Pr♦❢✳ ❉r✳ ❉❡♥✐s ❉❛❧♠❛③✐ ✭❋❊●✲❯◆❊❙P✮ Pr♦❢✳ ❉r✳ ◆❡❧s♦♥ ❇r❛❣❛ ✭❯❋❘❏✮
❆❣r❛❞❡❝✐♠❡♥t♦s
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡♥✲ ❝❡rr❛r ❡st❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦✳
❊ s❡❣✉✐❞❛✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ♣❛✐s ❘❡②♥❛❧❞♦ ❡ ▲♦✉r❞❡s✱ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❞❛❞♦ ❛♦ ❧♦♥❣♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳
●♦st❛r✐❛ ❞❡ ♠❛♥✐❢❡st❛r ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❛♦ ♣r♦❢✳ ❱✐❝t♦r✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❝✉✐❞❛❞♦s❛ ♦r✐❡♥t❛çã♦ ♥❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆s ♠✐♥❤❛s ✐r♠ãs ❘♦sâ♥❣❡❧❛✱ ❘♦s❛♥❡ ❡ ❘♦s❡✱ ♠❡✉s s♦❜r✐♥❤♦s ❙❛♠✉❡❧✱ ❘❛✲ ❢❛❡❧✱ ●❛❜r✐❡❧❛✱❘❡❜❡❝❛ ❡ ❛♦s ♠❡✉s ❝✉♥❤❛❞♦s ❊r❛❧❞♦ ❡ P❛✉❧♦ ♣❡❧♦ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s✳
❆♦ ❝❛r✐♥❤♦ ❡ ♦ ❛♣♦✐♦ ❞❛ ❏❛❝q✱ ❙❛❤r❛ ❡ ❈❧❛r❛✳ ❆ ✈♦❝ês ♠❡✉ ❝❛r✐♥❤♦ ❛ ❛♠♦r✳
❆ ♠✐♥❤❛ s❡❣✉♥❞❛ ❢❛♠í❧✐❛❀ ●r❛ç❛✱ ❇❡t❡✱❙ã♦③✐♥❤❛✱◆❛❞❞✐❛✱ ❉❛✐❛♥❡✱❉♦❞ô✱ ❚♦t✐✱ ●ê✱▲ú❝✐♦✱ ❋❛t✐♥❤❛✱ ❆❢♦♥s♦✱ ▲ú❝✐❛✱ ❈❛r♠❡♠ ❡ ❛ t♦❞♦s ♦s ♠❡✉s s♦❜r✐✲ ♥❤♦s✱ ❝✉♥❤❛❞♦s ❛ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❛✉sê♥❝✐❛ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s✳
❆♦s ❛♠✐❣♦s ❲✐❧s♦♥ ❙❛❝r❛♠❡♥t♦✱ ▼❛r❝✐❛ ❍❛t❛❡✱ ▼♦♥✐❝❛ ❑❛❝❤✐♦✱ ▲❡♦♥❛r❞♦ ❙✐♦✉✜✱ ▼❛r❝✐❛ ▼✐③♦✐✱ ❋❛❜✐❛♥♦ ❈és❛r ❈❛r❞♦s♦✱ ●✉✐❧❤❡r♠♦ ▲❛③♦✱ ❯❧②ss❡s ❈❛r✲ ✈❛❧❤♦ ❛❣r❛❞❡ç♦ ♦ ❛♣♦✐♦✳
❆♦s ❛♠✐❣♦s ❞♦ ❏♦❤r❡✐ ❈❡♥t❡r ❇✉t❛♥tã ❈✐❞❛ ❇❛❧❞✉✐♥♦✱ ▼❛r❣❛r✐❞❛ ❞❡ ❏❡s✉s✱ ❨❛ss✉②♦✱ ❘♦s❛ ❙❡✐❦✐✱ ❋❛❜✐❛♥❛ ▲✐♠❛✱ ❉r✐✱ ●❡r❝✐✱ ■r❡♥❡✱ ❙♦♥✐❛ ❱✐❧❡❧❛✱ ▲♦✉r❞❡s ❡ t♦❞♦s ♦s ❛q✉✐ ♥ã♦ ❝✐t❛❞♦s ♣♦r ❢❛❧t❛ ❞❡ ❡s♣❛ç♦✳ ❚❛♠❜é♠ ❛♦s ❛♠✐❣♦s ❞♦ ❏♦❤r❡✐ ❈❡♥t❡r P♦ç♦s ❞❡ ❈❛❧❞❛s✳
❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ❢ís✐❝❛ ♠❛t❡♠át✐❝❛ q✉❡ ♠✉✐t♦ ❝♦♥✲ tr✐❜✉ír❛♠ ❡♠ ♠✐♥❤❛ ❡st❛❞✐❛ ♥♦ ❞❡♣❛rt❛♠❡♥t♦✱ ❆♠é❧✐❛✱ ❙✐♠♦♥❡✱ ❏♦ã♦✱ ❙✐❜❡❧❡ ❡ ❇❡t❡✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❡♠♣r❡❣❛♠♦s ❛ té❝♥✐❝❛ ❞❡ ❇❋❱ ♣❛r❛ q✉❛♥t✐③❛r ✉♠❛ t❡♦r✐❛ ❝♦♠ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡ SP(2,R)✳ P❛r❛ ✐ss♦ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛♥❛❧✐s❛♠♦s ♦ ❝r✐tér✐♦ ❞❡ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❡ ●♦✈❛❡rts ♣❛r❛ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❣❛✉❣❡ ♣❛r❛ ❛ t❡♦r✐❛ ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛✱ ❝✉❥♦ ♣r♦♣❛❣❛❞♦r é ❝❛❧❝✉❧❛❞♦ ♥♦s ❣❛✉❣❡s ❝♦✲ ✈❛r✐❛♥t❡✱ ❝❛♥ô♥✐❝♦ ❡ ❞♦ ❝♦♥❡ ❞❡ ❧✉③ ♣♦r ♠❡✐♦ ❞❛ ❞✐s❝r❡t✐③❛çã♦ ❞❛ ✐♥t❡❣r❛❧ ❞❡ tr❛❥❡tór✐❛ ❡ ❡st❛ ♠♦str❛ q✉❡ ❛ ❛çã♦ ❞✐s❝r❡t✐③❛❞❛ ♣❡r❞❡ ❛ ✐♥✈❛r✐â♥❝✐❛ ♣♦r tr❛♥s✲ ❢♦r♠❛çõ❡s ❞❡ ❇❘❙❚❀ ❡ ♣❛r❛ r❡st❛✉r❛r s✉❛ ✐♥✈❛r✐â♥❝✐❛ é ♥❡❝❡ssár✐♦ ♠♦❞✐✜❝❛r ❛s tr❛♥s❢♦r♠❛ç♦❡s ❞❡ ❇❘❙❚✳
❊♠ s❡❣✉♥❞♦ ❧✉❣❛r✱ ❛♣❧✐❝❛♠♦s ❛ té❝♥✐❝❛ ❞❡ ❇❋❱ ♣❛r❛ ✉♠❛ t❡♦r✐❛ ❝♦♠ ❞♦✐s t❡♠♣♦s ❡ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡SP(2,R)✱ ❡♠ s❡❣✉✐❞❛✱ ❛♥❛❧✐s❛♠♦s ♦ ❡❢❡✐t♦ ❞❛ ❞✐s✲ ❝r❡t✐③❛çã♦ ❡ ♠♦str❛♠♦s q✉❡ ❛ ❛çã♦ ❞✐s❝r❡t✐③❛❞❛ ♣❡r❞❡ ❛ ✐♥✈❛r✐â♥❝✐❛ ♣♦r tr❛♥s✲ ❢♦r♠❛çõ❡s ❞❡ ❇❘❙❚✳ ◆❡st❡ ❝❛s♦✱ ❛s ♠♦❞✐✜❝❛çõ❡s ♥❡❝❡ssár✐❛s ✐♥❝❧✉❡♠ t❡r♠♦s ❞❡ ♦r❞❡♠ ∆τ
N ♥❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❇❘❙❚ ❡ ❡st❛s ♣❛ss❛♠ ❛ s❡r ♥✐❧♣♦t❡♥t❡s
❛♣❡♥❛s ♦♥✲s❤❡❧❧✳ ❆♦ ✜①❛r♠♦s ✉♠ t❡♠♣♦ ❢ís✐❝♦ ❞❡ ❞✉❛s ❢♦r♠❛s ❞✐❢❡r❡♥t❡s ♦❜✲ t✐✈❡♠♦s ♦ ♣r♦♣❛❣❛❞♦r ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ❡♠d❞✐♠❡♥sõ❡s ❡ ❞❡ ✉♠
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ ❡♠♣❧♦② t❤❡ ❇❋❱ t❡❝❤♥✐q✉❡ t♦ q✉❛♥t✐③❡ ❛ t❤❡♦r② ✇✐t❤ ❣❛✉❣❡ s②♠♠❡tr② Sp(2,R)✳ ❋✐rst✱ ✇❡ ❛♥❛❧②③❡ t❤❡ ❛❞♠✐ss✐❜✐❧✐t② ❝r✐t❡r✐♦♥ ♦❢ ●♦✈❛❡rts ❢♦r ❣❛✉❣❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ t❤❡♦r② ♦❢ ❛ r❡❧❛t✐✈✐st✐❝ ♣❛rt✐❝❧❡✳ ❚❤❡ ♣r♦♣❛❣❛t♦r ❢♦r t❤❡ r❡❧❛t✐✈✐st✐❝ ♣❛rt✐❝❧❡ ✐s ❝❛❧❝✉❧❛t❡❞ ✐♥ t❤❡ ❝♦✈❛r✐❛♥t✱ ❝❛♥♦♥✐❝❛❧ ❛♥❞ ❧✐❣❤t ❝♦♥❡ ❣❛✉❣❡s✳ ❚❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ ♣❛t❤ ✐♥t❡❣r❛❧ s❤♦✇s t❤❛t t❤❡ ❞✐s❝r❡t✐③❡❞ ❛❝t✐♦♥ ❧♦♦s❡s ✐♥✈❛r✐❛♥❝❡ ❜② t❤❡ ❇❘❙❚ tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚♦ r❡st♦r❡ t❤❡ ✐♥✈❛r✐❛♥❝❡ ✐t ✐s ♥❡❝❡ss❛r② t♦ ✐♥❝❧✉❞❡ ♠♦❞✐✜❡❞ tr❛♥s❢♦r♠❛t✐♦♥s✳
❙❡❝♦♥❞❧②✱ ✇❡ ❛♣♣❧② t❤❡ ❇❋❱ t❡❝❤♥✐q✉❡ t♦ ❛ t❤❡♦r② ✇✐t❤ t✇♦ t✐♠❡s ❛♥❞ ❣❛✉❣❡ s②♠♠❡tr② Sp(2,R✮✳ ❲❡ ❛♥❛❧②③❡ t❤❡ ❡✛❡❝t ♦❢ ❞✐s❝r❡t✐③❛t✐♦♥ ❛♥❞ s❤♦✇ t❤❛t t❤❡ ❞✐s❝r❡t✐③❡❞ ❛❝t✐♦♥ ❧♦♦s❡s t❤❡ ❇❘❙❚ ✐♥✈❛r✐❛♥❝❡✳ ■♥ t❤✐s ❝❛s❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❝❤❛♥❣❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥s ✐♥❝❧✉❞✐♥❣ t❡r♠s ♦❢ ♦r❞❡r ∆τ
N ✱ ✇❤✐❝❤
❜❡❝♦♠❡ ♥✐❧♣♦t❡♥t ♦♥❧② ♦♥✲s❤❡❧❧✳ ❋✐①✐♥❣ t❤❡ ♣❤②s✐❝❛❧ t✐♠❡ ✐♥ t✇♦ ❞✐✛❡r❡♥t ✇❛②s ✇❡ ❣❡t t❤❡ ♣r♦♣❛❣❛t♦r ❢♦r ❛ r❡❧❛t✐✈✐st✐❝ ♣❛rt✐❝❧❡ ✐♥ d❞✐♠❡♥s✐♦♥s ❛♥❞ ❢♦r
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❙✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s ❱✐♥❝✉❧❛❞♦s ✺
✷✳✶ ❋♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❱í♥❝✉❧♦s ❞❡ Pr✐♠❡✐r❛ ❈❧❛ss❡ ❡ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ●❛✉❣❡ ✳ ✳ ✳ ✼ ✷✳✸ ❋✐①❛çã♦ ❞❡ ●❛✉❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ✾ ✷✳✺ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ❚❡♦r✐❛s ❞❡ ●❛✉❣❡ ✳ ✶✶ ✷✳✺✳✶ ▼ét♦❞♦ ❞❡ ❋❛❞❞❡❡✈✲P♦♣♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✺✳✷ ▼ét♦❞♦ ❇❋❱ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✸ ◗✉❛♥t✐③❛çã♦ ❇❘❙❚ ❞❛ P❛rtí❝✉❧❛ ❘❡❧❛t✐✈íst✐❝❛ ✶✺
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✶✳✶ ❈♦♥❞✐çã♦ ❈♦✈❛r✐❛♥t❡ ❞❡ ●❛✉❣❡ λ˙(τ) = f(λ) ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✸✳✶✳✷ ❈♦♥❞✐çã♦ ❞❡ ●❛✉❣❡ ❈❛♥ô♥✐❝♦x0 −τ = 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸✳✶✳✸ ❈♦♥❞✐çã♦ ❞❡ ●❛✉❣❡ ❞♦ ❈♦♥❡ ❞❡ ▲✉③x+−τ = 0 ✳ ✳ ✳ ✳ ✷✵
✸✳✷ ❆❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❛s ❈♦♥❞✐çõ❡s ❞❡ ●❛✉❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸ Pr♦♣❛❣❛❞♦r ♣❛r❛ ❛ P❛rtí❝✉❧❛ ❘❡❧❛t✐✈íst✐❝❛ ♥♦ ●❛✉❣❡ λ˙ =f(λ) ✷✺
✸✳✸✳✶ ■♥✈❛r✐â♥❝✐❛ ❞❛ ❆❝ã♦ ❉✐s❝r❡t✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✸✳✷ Pr♦♣❛❣❛❞♦r ❉✐s❝r❡t✐③❛❞♦ ♣❛r❛ ♦ ●❛✉❣❡ ❈♦✈❛r✐❛♥t❡λ˙ =
f(λ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✹ Pr♦♣❛❣❛❞♦r ♣❛r❛ ❛ P❛rtí❝✉❧❛ ❘❡❧❛t✐✈íst✐❝❛ ♥♦ ●❛✉❣❡ x0−τ = 0 ✸✷
✸✳✺ Pr♦♣❛❣❛❞♦r ♣❛r❛ ❛ P❛rtí❝✉❧❛ ❘❡❧❛t✐✈íst✐❝❛ ♥♦ ●❛✉❣❡x+−τ = 0 ✸✼
✹ ◗✉❛♥t✐③❛çã♦ ❇❘❙❚ ❡ ❙✐♠❡tr✐❛ ❞❡ ●❛✉❣❡ Sp(2,R) ✹✸
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷ ❋♦r♠✉❧❛çã♦ ❈❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✶ P❛rtí❝✉❧❛ ❘❡❧❛t✐✈íst✐❝❛ ▲✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✷✳✷ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❡♠ d−2 ❉✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✹✳✸ ◗✉❛♥t✐③❛çã♦ ❇❘❙❚ ❈♦♠ ❙✐♠❡tr✐❛ ❞❡ ●❛✉❣❡Sp(2,R) ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✹ Pr♦♣❛❣❛❞♦r ♥♦ ❋♦r♠❛❧✐s♠♦ ❇❋❱ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
❙❯▼➪❘■❖ ✐✈
✹✳✹✳✶ ■♥✈❛r✐â♥❝✐❛ ❞❛ ❆çã♦ ❉✐s❝r❡t✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✹✳✷ Pr♦♣❛❣❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✹✳✸ Pr♦♣❛❣❛❞♦r ❞❛ P❛rtí❝✉❧❛ ❘❡❧❛t✐✈íst✐❝❛ ❡♠ d ❉✐♠❡♥sõ❡s ✻✺
✹✳✹✳✹ Pr♦♣❛❣❛❞♦r ♣❛r❛ ♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❡♠d−2❉✐✲
♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✺ ❈♦♥❝❧✉sã♦ ❡ P❡rs♣❡❝t✐✈❛s ✼✵
❆ ◆✐❧♣♦tê♥❝✐❛ ❞❛s ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ❇❘❙❚ ✼✷
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❆ s✐♠❡tr✐❛ ❝♦♥❢♦r♠❡ ❡♠ ✉♠ ❡s♣❛ç♦ ❝♦♠ d ❞✐♠❡♥sõ❡s é r❡❛❧✐③❛❞❛ ♣❡❧♦ ❣r✉♣♦ SO(d,2) ❝✉❥❛ ❛çã♦ ♥❡ss❡ ❡s♣❛ç♦ ♦❝♦rr❡ ❞❡ ❢♦r♠❛ ♥ã♦ ❧✐♥❡❛r✳ ❊♥tr❡t❛♥t♦✱
❡♠ ✶✾✸✻✱ ❉✐r❛❝ ❬✶❪ ✐♥tr♦❞✉③✐✉ ✉♠❛ ❢♦r♠✉❧❛çã♦ ♣❛r❛ ✉♠❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s ♠❛♥✐❢❡st❛♠❡♥t❡ ❝♦✈❛r✐❛♥t❡ ♥❛ q✉❛❧ ❡ss❡ ❣r✉♣♦ ❛❣❡ ❞❡ ❢♦r♠❛ ❧✐♥❡❛r ❡ ❡ss❛ ♥♦✈❛ ❢♦r♠✉❧❛çã♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ✉♠ ❡s♣❛ç♦ ❝♦♠ d+ 2 ❞✐♠❡♥sõ❡s✱ ❝♦♠
❞✉❛s ❝♦♠♣♦♥❡♥t❡s t❡♠♣♦r❛✐s✳ ❊♠ ❢✉♥çã♦ ❞❡ ✉♠❛ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡ ♣r❡s❡♥t❡ ♥❡st❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❉✐r❛❝✱ ❢♦✐ ♣♦ssí✈❡❧ ❞❡s❝r❡✈❡r t❡♦r✐❛s ❡♠ ✉♠ ❡s♣❛ç♦ ❝♦♠
d ❞✐♠❡♥sõ❡s✳
◆❛ ❞é❝❛❞❛ ❞❡ ✼✵✱ ▼❛r♥❡❧✐✉s ❬✷❪ ❛♦ ❡st✉❞❛r ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ✉♠❛ t❡♦r✐❛ ❝♦♥❢♦r♠❡ ❡♠ ✉♠ ❡s♣❛ç♦ ❝♦♠ d+ 2 ❞✐♠❡♥sõ❡s ♠♦str♦✉ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛
❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ✉♠❛ ♣❛rtí❝✉❧❛ ♠❛ss✐✈❛ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❡♠ d
❞✐♠❡♥sõ❡s ❡ ✉♠❛ ♣❛rtí❝✉❧❛ s❡♠ ♠❛ss❛ q✉❡ s❡ ♣r♦♣❛❣❛ ♥✉♠ ❡s♣❛ç♦ AdSd−1✱
✉t✐❧✐③❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❉✐r❛❝ ❬✶❪ ❡ ❛♣❧✐❝❛❞♦ à t❡♦r✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✳ ❆❧é♠ ❞✐ss♦✱ ❞✐✈❡rs❛s ❣❡♥❡r❛❧✐③❛çõ❡s ❢♦r❛♠ ❢❡✐t❛s ❛ ✜♠ ❞❡ ✐♥tr♦❞✉③✐r ❢❡r♠✐♦♥s ♥❛ t❡♦r✐❛ ❬✸✱ ✷✱ ✹❪✳ ❆ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ q✉❛♥t✐③❛çã♦ ❞❡ss❡s ♠♦❞❡❧♦s ♣❡❧❛ té❝♥✐❝❛ ❞❡ ❇❘❙❚ ✭❇❡❝❤✐✱ ❘♦✉❡t✱❙t♦r❛✱❚②❧t✐♥✮❢♦✐ ❢❡✐t❛ ❡♠ ❬✺❪✳
◆❛ ❞é❝❛❞❛ ❞❡ ✾✵✱ ❡st❡ ♠♦❞❡❧♦ ❢♦✐ ❡st✉❞❛❞♦ ♥✉♠ ♦✉tr♦ ❝♦♥t❡①t♦ ♣♦r ▼♦♥✲ t❡s✐♥♦s ❬✻✱ ✼❪ ♦ q✉❛❧ ✐♥tr♦❞✉③✐✉ ✉♠❛ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡ Sl(2,R✮ ❛ ✜♠ ❞❡ ♦❜t❡r ✉♠❛ t❡♦r✐❛ q✉❡ ❞❡s❝r❡✈❡ss❡ ❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✳ ❈♦♠ ✐ss♦✱ ♦❜t❡✈❡ ❛s s♦❧✉çõ❡s ❝❧áss✐❝❛s q✉❡ ❞❡s❝r❡✈❡♠✿ ✉♠❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ❝♦♠ ♠❛ss❛ ❡ ✉♠ ♦s❝✐❧❛✲ ❞♦r ❤❛r♠ô♥✐❝♦ ❛♣ós r❡s♦❧✈❡r ❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❞✉❛s ❡s❝♦❧❤❛s ❞❡ ❣❛✉❣❡ ❞✐❢❡r❡♥t❡s✳
◆♦ ✜♥❛❧ ❞❡ ❞é❝❛❞❛ ❞❡ ✾✵✱ ❝♦♠ ♦s ❛✈❛♥ç♦s ❞❛s t❡♦r✐❛ ❞❛s ❝♦r❞❛s ♥♦ ❡st✉❞♦ ❞♦s ❛s♣❡❝t♦s ♥ã♦ ♣❡rt✉r❜❛t✐✈♦s✱ ❇❛rs ❡ ❝♦❧❛❜♦r❛❞♦r❡s✱ ♦❜s❡r✈❛r❛♠ ❡✈✐❞ê♥❝✐❛s ❞❡ q✉❡ ❛ ❞❡s❝r✐çã♦ ❞❛s t❡♦r✐❛s ❞❡ ✉♥✐✜❝❛çã♦ ♣♦❞❡r✐❛ ✐♥❝❧✉✐r ❞✉❛s ❝♦♦r❞❡♥❛❞❛s t❡♠♣♦r❛✐s ❬✽✱ ✾✱ ✶✵✱ ✶✶✱ ✶✷✱ ✶✸✱ ✶✹✱ ✶✺✱ ✶✻❪✳ ❆ ♣❛rt✐r ❞❡ss❡ ♠♦♠❡♥t♦✱ ❡❧❡s ❡st✉✲ ❞❛r❛♠ ❝♦♠♦ ✐♠♣❧❡♠❡♥t❛r ❡ss❛s t❡♦r✐❛s ❬✶✼✱ ✶✽❪✳ ❊ ❡♠ ✶✾✾✽✱ r❡❢♦r♠✉❧❛r❛♠ ❛ t❡♦r✐❛ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ s❡♠ ♠❛ss❛ ❡ ✐♥tr♦❞✉③✐♥❞♦ ✉♠❛ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡
✷
Sp(2,R)❡ ✉♠❛ s✐♠❡tr✐❛ ❣❧♦❜❛❧SO(d,2)❛❣✐♥❞♦ ❧✐♥❡❛r♠❡♥t❡ ♥✉♠ ❡s♣❛ç♦ ❝♦♠
d+ 2 ❞✐♠❡♥sõ❡s ❬✶✾✱ ✷✵✱ ✷✶❪✳ ❉❡st❛ ❢♦r♠❛✱ ♠♦str❛r❛♠ q✉❡ ❡st❛ t❡♦r✐❛ ❡r❛
❝❛♣❛③ ❞❡ ❞❡s❝r❡✈❡r✱ ❡♠ ❡s♣❛ç♦s ❝♦♠ ❞✐♠❡♥sã♦ ♠❛✐s ❜❛✐①❛✱ s✐st❡♠❛s q✉❡ ❛♣❛✲ r❡♥t❡♠❡♥t❡ ♥ã♦ sã♦ r❡❧❛❝✐♦♥❛❞♦s✱ ❝♦♠♦ ❡q✉✐✈❛❧❡♥t❡s✿ ❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ❝♦♠ ♦✉ s❡♠ ♠❛ss❛✱ ♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ❡ ♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦✳ ❊s✲ s❡s r❡s✉❧t❛❞♦s ♣❡r♠✐t✐r❛♠ ❛ ❡st❡s ❛✉t♦r❡s ♣r♦♣♦r❡♠ q✉❡ ♦ ♠♦❞❡❧♦ ❝♦♠ ❞♦✐s t❡♠♣♦s ❞❡s❝r❡✈❡ss❡ ❞❡ ❢♦r♠❛ ✉♥✐✜❝❛❞❛ ❡st❡s s✐st❡♠❛s✳ ❆s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ q✉❡ ❝♦♥❡❝t❛♠ ❡st❡s ❞✐✈❡rs♦s s✐st❡♠❛s ❢♦r❛♠ ❝❤❛♠❛❞❛s ❞✉❛❧✐❞❛❞❡ ❡ ❡st❛ r❡❢♦r♠✉❧❛çã♦ ❞❡ ❢ís✐❝❛ ❝♦♠ ❞♦✐s t❡♠♣♦s✳ ❆ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡ t♦r♥❛ ❛ t❡♦r✐❛ ✉♥✐tár✐❛ r❡♠♦✈❡♥❞♦ ♦s ❡st❛❞♦ ❞❡ ♥♦r♠❛ ♥❡❣❛t✐✈❛ r❡s✉❧t❛♥t❡s ❞❛ ✐♥tr♦✲ ❞✉çã♦ ❞❛s ❞✉❛s ❝♦♦r❞❡♥❛❞❛s t❡♠♣♦r❛✐s✳ ❊♠ s❡❣✉✐❞❛✱ ❇❛rs ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❣❡♥❡r❛❧✐③❛r❛♠ ❛ ❢♦r♠✉❧❛çã♦ ❛ ✜♠ ❞❡ ✐♥tr♦❞✉③✐r ❢ér♠✐♦♥s ♦ q✉❡ ✐♠♣❧✐❝♦✉ ♥❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s✉r♣❡rs✐♠❡tr✐❛ ♥❛ ❧✐♥❤❛ ♠✉♥❞♦ ❬✷✷❪ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❬✷✸❪✳ ❚❛♠❜é♠ ❢♦r❛♠ r❡❛❧✐③❛❞❛s ♣♦r ❡❧❡s ❛♣❧✐❝❛çõ❡s ❞❛ ❢♦r♠✉✲ ❧❛çã♦ ♣❛r❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s✱ ❜r❛♥❛s ❡ t❡♦r✐❛ ▼ ❬✷✹✱ ✷✺✱ ✷✻❪✳ ❊♠ s❡❣✉✐❞❛✱ ❡❧❡s ❛♣❧✐❝❛r❛♠ ♦ ❢♦r♠❛❧✐s♠♦ ❛ ✜♠ ❞❡ ✐♥❝❧✉✐r ❝❛♠♣♦s ❞❡ ❢✉♥❞♦ ❣r❛✈✐t❛❝✐♦♥❛✐s ❡ ❞❡ ❣❛✉❣❡ ❬✷✼❪ ❡ ✐♥tr♦❞✉③✐r❛♠ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ s❡❣✉♥❞❛ q✉❛♥t✐③❛çã♦✱ ♦✉ s❡❥❛✱ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s✱ ❞❡s❝r✐t♦ ❡♠ ❬✷✽✱ ✷✾✱ ✸✵✱ ✸✶❪✳
❊♠ ✷✵✵✶✱ ❇❛rs ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❝♦♠❡ç❛r❛♠ ❛ ❡st✉❞❛r ❛ r❡❧❛çã♦ ❞❛ ❢♦r♠✉✲ ❧❛çã♦ ❞❡ ❢ís✐❝❛ ❝♦♠ ❞♦✐s t❡♠♣♦s ❬✸✷✱ ✸✸✱ ✸✹✱ ✸✺✱ ✸✻❪ ❝♦♠ ♦s t✇✐st♦rs ♣r♦♣♦st♦s ♣♦r P❡♥r♦s❡ ❬✸✼❪ ❡ ❡♠ ✷✵✵✻✱ ♦❜t✐✈❡r❛♠ ✉♠❛ ❢♦r♠✉❧❛çã♦ q✉❡ ❞❡s❝r❡✈❡ ♦ ♠♦✲ ❞❡❧♦ ♣❛❞rã♦ ❞❛s ♣❛rtí❝✉❧❛s ❝♦♠♦ ✉♠❛ ❡s❝♦❧❤❛ ❞❡ ❣❛✉❣❡ ❞❛ ❢♦r♠✉❧❛çã♦ ❝♦♠ ❞♦✐s t❡♠♣♦s❬✸✽❪✳❖✉tr❛s ❛♣❧✐❝❛çõ❡s ❞❛ t❡♦r✐❛ ❝♦♠ ❞♦✐s t❡♠♣♦s ❢♦r❛♠ ♦❜t✐❞❛s ❡♠ ❬✸✾✱ ✹✵✱ ✹✶✱ ✹✷✱ ✹✸✱ ✹✹✱ ✹✺✱ ✹✻✱ ✹✼✱ ✹✽✱ ✹✾❪ ❡ ❛❧❣✉♠❛s r❡✈✐sõ❡s q✉❡ ❞✐s❝✉t❡♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❢♦r♠❛❧✐s♠♦ ❡stã♦ ❞❡s❝r✐t❛s ❡♠ ❬✺✵✱ ✺✶❪✳ ❊♥tr❡t❛♥t♦✱ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ❡st❛ ❢♦r♠✉❧❛çã♦ ✉t✐❧✐③❛♠ ♦ ♠ét♦❞♦ ❞❡ q✉❛♥✲ t✐③❛çã♦ ❞❡ ♦♣❡r❛❞♦r❡s✳ ◆❡st❡ tr❛❜❛❧❤♦ ✐♥tr♦❞✉③✐♠♦s ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ✐♥t❡❣r❛✐s ❞❡ tr❛❥❡tór✐❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠❛✐s s✐♠♣❧❡s ❝♦♠ ❞♦✐s t❡♠♣♦s✳
✸
◗✉❛♥t♦ à ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❢♦r♠❛❧✐s♠♦ ❇❋❱ ♣❛r❛ ✉♠❛ t❡♦r✐❛ ❞❛ ♣❛r✲ tí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ❡st❛ ❢♦✐ r❡❛❧✐③❛❞❛ ♣♦r ❍❡♥♥❡❛✉① ❡ ❚❡✐t❡❧❜♦✐♠❬✻✶❪✳ ❊♠ s❡✉ tr❛❜❛❧❤♦✱ ♦s ❛✉t♦r❡s ❝❛❧❝✉❧❛r❛♠ ♦ ♣r♦♣❛❣❛❞♦r ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ✉s❛♥❞♦ ✉♠❛ ❡s❝♦❧❤❛ ❞❡ ❣❛✉❣❡ ❝♦✈❛r✐❛♥t❡✳ ◆✉♠ ♦✉tr♦ tr❛❜❛❧❤♦✱ ❚❡✐t❡❧❜♦✐♠ ❬✻✷❪ ♠♦str♦✉ q✉❡ ❞❡✈✐❞♦ à s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡ ❞❛ t❡♦r✐❛ ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈ís✲ t✐❝❛✱ ❛s ú♥✐❝❛s ❡s❝♦❧❤❛s ❞❡ ❣❛✉❣❡ ♣♦ssí✈❡✐s ❡r❛♠ ❛s ❞♦ t✐♣♦ ❝♦✈❛r✐❛♥t❡✳ ❊ss❡ r❡s✉❧t❛❞♦ ❡①❝❧✉✐✉ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❣❛✉❣❡ ❝❛♥ô♥✐❝❛s✳
◆❛ ❞❡❝❛❞❛ ❞❡ ✽✵✱ ●♦✈❛❡rts ❬✻✸✱ ✻✹✱ ✻✺❪ ✐♥tr♦❞✉③✐✉ ✉♠ ♦✉tr♦ ❝r✐tér✐♦ ❞❡ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ♣❛r❛ ❛s ❡s❝♦❧❤❛s ❞❡ ❣❛✉❣❡ ♥♦ ❢♦r♠❛❧✐s♠♦ ❇❋❱✳ ❊♠ s❡✉s tr❛✲ ❜❛❧❤♦s✱ ❡❧❡ ♣r♦♣ôs ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝ó♣✐❛s ❞❡ ●r✐❜♦✈ ❬✻✻❪✳ ❆✐♥❞❛ s♦❜r❡ ♦ tr❛❜❛❧❤♦ ♦r✐❣✐♥❛❧ ❞❡ ❝ó♣✐❛s ❞❡ ●r✐❜♦✈ ♣♦❞❡✲s❡ ❡♥❝♦♥tr❛r ✉♠❛ r❡✈✐sã♦ ❡♠ ❬✻✼❪✳
◆✉♠ ♦✉tr♦ ♠♦♠❡♥t♦✱ ❱❡r❣❛r❛ ❡ ❝♦❧❛❜♦r❛❞♦r❡s ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠ ♠ét♦❞♦ ♣❛r❛ ✐♠♣❧❡♠❡♥t❛r ❛s ❝♦♥❞✐çõ❡s ❞❡ ❣❛✉❣❡ ❝❛♥ô♥✐❝❛s ❬✻✽✱ ✻✾❪ ❡ ✉♠❛ ♦✉tr❛ ❛❜♦r✲ ❞❛❣❡♠ ❢♦✐ ❢❡✐t❛ ♣♦r ■❦❡♠♦r✐❬✼✵❪✳ P❛r❛ ❝❛❧❝✉❧❛r ♦ ♣r♦♣❛❣❛❞♦r✱ ❡❧❡ ❛♥❛❧✐s♦✉ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ t❡♦r✐❛ ❛♣ós ❛ ❞✐s❝r❡t✐③❛çã♦✳ ◆❡ss❛ ❛♥á❧✐s❡✱ ■❦❡♠♦r✐ ♦❜s❡r✲ ✈♦✉ q✉❡ ❛ ❛çã♦ ❞✐s❝r❡t✐③❛❞❛ ♣❡r❞❡ ❛ ✐♥✈❛r✐â♥❝✐❛ ❡♠ r❡❧❛çã♦ às tr❛♥s❢♦r♠❛çõ❡s ❞✐s❝r❡t✐③❛❞❛s ❞❡ ❢♦r♠❛ ✐♥❣ê♥✉❛✳ P❛r❛ s♦❧✉❝✐♦♥❛r ❡ss❡ ♣r♦❜❧❡♠❛✱ ❡❧❡ s✉❣❡r✐✉ ♠♦❞✐✜❝❛r ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ ❞✐s❝r❡t✐③❛❞❛s✳ ❊ss❛ ✐❞é✐❛ t❛♠❜é♠ ❢♦✐ ❛♣❧✐❝❛❞❛ ❛ ✉♠❛ t❡♦r✐❛ ❞❛❞❛ ❡♠ ❬✼✶✱ ✼✷❪✳ ❈♦♠ ❡ss❛ ❛♥á❧✐s❡✱ ♠♦str♦✉ ✉♠ ❝á❧✲ ❝✉❧♦ ❡①♣❧í❝✐t♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ♥✉♠ ❣❛✉❣❡ ❝❛♥ô♥✐❝♦ ❬✼✵❪ s❡♠ ❛s ♠♦❞✐✜❝❛çõ❡s ❢❡✐t❛s ♣♦r ❱❡r❣❛r❛ ❡ ❝♦❧❛❜♦r❛❞♦r❡s✳
❊♠ ♥♦ss♦ tr❛❜❛❧❤♦✱ ✈❛♠♦s ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❞✐s❝✉✐r ♦ ❝r✐tér✐♦ ❞❡ ❛❞♠✐s✲ s✐❜✐❧✐❞❛❞❡ ♣r♦♣♦st♦ ♣♦r ●♦✈❛❡rts ❬✻✸✱ ✻✹✱ ✻✺✱ ✼✸❪✳ ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❛♠♦s ♦ ♣r♦♣❛❣❛❞♦r ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❇❋❱ ❡ s❡❣✉✐♥❞♦ ❛ ❛♥á❧✐s❡ ✐♥tr♦❞✉③✐❞❛ ♣♦r ■❦❡♠♦r✐ ❬✼✵❪✳ ◆✉♠ s❡❣✉♥❞♦ ♠♦♠❡♥t♦✱ ❛♣❧✐❝❛♠♦s ❛s té❝♥✐❝❛s ✉s❛❞❛s ♥♦ ❝❛s♦ ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ♣❛r❛ ♦ ❝❛s♦ ❞❛ t❡♦r✐❛ ❝♦♠ ❞♦✐s t❡♠♣♦s✱ ♦✉ s❡❥❛✱ ❝♦♠ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡ Sp(2,R) ❬✼✹❪✳
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❢❛③❡♠♦s ✉♠❛ r❡✈✐sã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❛ t❡♦r✐❛ ❞❡ ✈í♥❝✉❧♦s ❬✼✺✱ ✼✻❪✱ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛s s✐♠❡tr✐❛s ❞❡ ❣❛✉❣❡ ❡ ❞✐s❝✉t✐♠♦s s✉❛ q✉❛♥t✐③❛çã♦ ♣♦r ♠❡✐♦ ❞❛ ✐♥t❡❣r❛çã♦ ❞❡ tr❛❥❡tór✐❛✳ ❊♠ ✉♠ ♣r✐♠❡✐r♦ ♠♦♠❡♥t♦✱ ❞❡✜♥✐♠♦s ❛ ✐♥t❡❣r❛❧ ❞❡ tr❛❥❡tór✐❛ ♣❛r❛ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❡ ❞❡♣♦✐s ❛♣r❡s❡♥t❛♠♦s ♦s ♠ét♦❞♦s ❞❡ ❋❛❞❞❡❡✈ ❡ ❇❛t❛❧✐♥✱ ❋r❛❞❦✐♥ ❡ ❱✐❧❦♦✈✐s❦✐✳
◆♦ ❈❛♣ít✉❧♦ ✸✱ ❢❛r❡♠♦s ❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ❇❋❱ ♣❛r❛ ♦ ❝❛s♦ ❞❛ ♣❛r✲ tí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛✳ P❛r❛ ✐ss♦✱ ❛♥❛❧✐s❛♠♦s ♦ ❝r✐tér✐♦ ♣r♦♣♦st♦ ♣♦r ●♦✈❛❡rts ❬✻✹✱ ✻✺✱ ✼✸✱ ✻✸❪ s♦❜r❡ ❛ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❛s ❡s❝♦❧❤❛s ❞❡ ❣❛✉❣❡✳ ❊♠ s❡❣✉✐❞❛✱ ❞✐s❝r❡t✐③❛♠♦s ❛ ❡①♣r❡ssã♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡ ❛♥❛❧✐s❛♠♦s s✉❛ ✐♥✈❛r✐â♥ç❛ ❛♣ós ❛ ❞✐s❝r❡t✐③❛çã♦ ❞♦ t❡♠♣♦✳ ❋✐♥❛❧✐③❛♥❞♦ ❡ss❡ ❝❛♣ít✉❧♦✱ ❝❛❧❝✉❧❛♠♦s ❡①♣❧✐❝✐t❛✲ ♠❡♥t❡ ♦s ♣r♦♣❛❣❛❞♦r❡s ♣❛r❛ ❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ♥♦s ❣❛✉❣❡s ❝♦✈❛r✐❛♥t❡✱ ❝❛♥ô♥✐❝♦ ❡ ❞♦ ❝♦♥❡ ❞❡ ❧✉③✳
✹
❡ ❛♥❛❧✐s❛♠♦s s✉❛ ❡str✉t✉r❛ ❞❡ ✈í♥❝✉❧♦s✳ ◆❛ s❡q✉ê♥❝✐❛ ♠♦str❛♠♦s ♦s r❡s✉❧t❛✲ ❞♦s ♦❜t✐❞♦s ♣♦r ❇❛rs ❡ ❝♦❧❛❜♦r❛❞♦r❡s q✉❡ ❞❡s❝r❡✈❡♠ ❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ❡♠ d ❞✐♠❡♥sõ❡s ❡ ♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ❡♠ d−2 ❞✐♠❡♥sõ❡s ❡s♣❛❝✐❛✐s ❬✷✵❪✳
❊♠ s❡❣✉✐❞❛✱ ❝♦♥str✉✐♠♦s ♦ ❢♦r♠❛❧✐s♠♦ ❇❋❱ ❡ ❞✐s❝r❡t✐③❛♠♦s ♦ ♣r♦♣❛❣❛❞♦r✱ ❛♥❛❧✐s❛♠♦s ❛ ✐♥✈❛r✐â♥❝✐❛ ❞❛ ❛çã♦ ❛♣ós ❛ ❞✐s❝r❡t✐③❛ç❛♦ ❡ ✜♥❛❧♠❡♥t❡ ❝❛❧❝✉❧❛♠♦s ♦ ♣r♦♣❛❣❛❞♦r✳
❈❛♣ít✉❧♦ ✷
❙✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s
❱✐♥❝✉❧❛❞♦s
✷✳✶ ❋♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦
❱❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠ s✐st❡♠❛ ❢ís✐❝♦ ❝♦♠ N ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❡s❝r✐t♦ ♣❡❧❛
❛çã♦
S=
✂ τ2
τ1
dτ L(xi,x˙i, τ), i= 1,· · ·N, ✭✷✳✶✮
♦♥❞❡ ▲ é ❢✉♥çã♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s xi✱ ❞❛s ✈❡❧♦❝✐❞❛❞❡s ❣❡♥❡r❛✲
❧✐③❛❞❛s x˙i ❡ ❞♦ t❡♠♣♦ τ✳ ❆s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ❝❧áss✐❝❛s ♦❜t✐❞❛s ♣❡❧❛
✈❛r✐❛çã♦ ❞❡ ✭✷✳✶✮✱ s✉❥❡✐t❛s às ❝♦♥❞✐çõ❡s δxi = 0 ♥♦s ❡①tr❡♠♦s✱ sã♦
∂L ∂xi −
d dτ
∂L ∂x˙i
= 0. ✭✷✳✷✮
P♦❞❡♠♦s r❡❡s❝r❡✈❡r ❡ss❛s ❡q✉❛çõ❡s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
∂2L (∂x˙i∂x˙j)x¨
j = ∂L ∂xi −
∂2L ∂xi∂x˙jx˙
j. ✭✷✳✸✮
❖ t❡r♠♦
∂2L
∂x˙i∂x˙j. ✭✷✳✹✮
é ❝❤❛♠❛❞♦ ❞❡ ♠❛tr✐③ ❤❡ss✐❛♥❛✳ ❙❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡st❛ ♠❛tr✐③ ❢♦r ♥✉❧♦✱ ❞✐③❡✲ ♠♦s q✉❡ ♦ s✐st❡♠❛ é s✐♥❣✉❧❛r ♦✉ ✈✐♥❝✉❧❛❞♦ ❬✼✺✱ ✼✻❪ ❡ ♥ã♦ ♣♦❞❡r❡♠♦s ❞❡t❡r♠✐✲ ♥❛r ✉♥✐✈♦❝❛♠❡♥t❡ ❛s ❛❝❡❧❡r❛çõ❡s ❡♠ ❢✉♥çã♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s ❡
✷✳✶✿ ❋♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦ ✻
❞❡ s✉❛s ✈❡❧♦❝✐❞❛❞❡s✳ ◆❡st❡ ❝❛s♦✱ ❡①✐st❡♠ ❞✐❢❡r❡♥t❡s ❡✈♦❧✉çõ❡s t❡♠♣♦r❛✐s ♣❛r❛ ✉♠❛ ♠❡s♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✳
❆ tr❛♥s✐çã♦ ♣❛r❛ ♦ ❢♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦ é ❢❡✐t❛ ❞❡✜♥✐♥❞♦ ♦ ♠♦♠❡♥t♦ ❝❛♥ô♥✐❝♦ ❝♦♥❥✉❣❛❞♦
pi = ∂L
∂x˙i. ✭✷✳✺✮
P❛r❛ s✐st❡♠❛s ✈✐♥❝✉❧❛❞♦s ♥ã♦ é ♣♦ssí✈❡❧ ❡①♣r❡ss❛r ❛tr❛✈és ❞❛ ❡①♣r❡ssã♦ ✭✷✳✺✮ t♦❞❛s ❛s ✈❡❧♦❝✐❞❛❞❡s x˙i ❝♦♠♦ ❢✉♥çã♦ ❞♦s ♠♦♠❡♥t♦s ❡ ❞❛s ❝♦♦r❞❡♥❛❞❛s✱ ♦✉
s❡❥❛✱ ♥❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s pi sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♠❛s ❡①✐st❡♠ r❡❧❛çõ❡s
φm(x, p) = 0, m= 1,· · · , M, ✭✷✳✻✮
q✉❡ s❡❣✉❡♠ ❞❡ ✭✷✳✺✮ ❡ q✉❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈í♥❝✉❧♦s ♣r✐♠ár✐♦s✳ ❊ss❡s ✈í♥❝✉✲ ❧♦s ❞❡✜♥❡♠ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞❡♥♦♠✐♥❛❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈í♥❝✉❧♦s ♣r✐♠ár✐♦s q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r ΓM✳
❆ ❤❛♠✐❧t♦♥✐❛♥❛ ❝❛♥ô♥✐❝❛ é ❞❛❞❛ ♣♦r
H=pix˙i−L(xi,x˙i, τ), ✭✷✳✼✮
q✉❡ ❞❡✈✐❞♦ à ❡①♣r❡ssã♦ ✭✷✳✻✮ é ✈á❧✐❞❛ s♦♠❡♥t❡ ♥❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈í♥❝✉❧♦s ♣r✐♠á✲ r✐♦s✳ P❛r❛ ❡st❡♥❞❡r♠♦s ❡st❛ ❞❡✜♥✐çã♦ ♣❛r❛ t♦❞♦ ♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ❞❡✜♥✐♠♦s ♦✉tr❛ ❤❛♠✐❧t♦♥✐❛♥❛ HT ❞❛❞❛ ♣♦r
HT =H+cmφm ✭✷✳✽✮
♦♥❞❡ ♦s cm sã♦ ❢✉♥çõ❡s ❛r❜✐trár✐❛s ❞❡ x ❡ p✳
❈♦♠♦ ♦s ✈í♥❝✉❧♦s φm ♣♦❞❡♠ t❡r ❝♦❧❝❤❡t❡s ❞❡ P♦✐ss♦♥ ♥ã♦ ♥✉❧♦s ❝♦♠ ❛❧✲
❣✉♠❛ ✈❛r✐á✈❡❧ ❝❛♥ô♥✐❝❛✱ ❞❡✈❡♠♦s ❝❛❧❝✉❧❛r ♦s ❝♦❧❝❤❡t❡s ❛♥t❡s ❞❡ ❧❡✈❛r♠♦s ❡♠ ❝♦♥t❛ ❛s ❡q✉❛çõ❡s ❞♦s ✈í♥❝✉❧♦s✳ P❛r❛ ❧❡♠❜r❛r♠♦s ❞❡st❡ ❢❛t♦✱ ❉✐r❛❝ ❬✼✺❪ ✐♥✲ tr♦❞✉③✐✉ ❛ ♥♦çã♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❢r❛❝❛✱ ❡s❝r❡✈❡♥❞♦ ♦s ✈í♥❝✉❧♦s ❝♦♠♦
φm ≈0. ✭✷✳✾✮
❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ❣❡r❛❞❛s ♣♦r ❡st❛ ♥♦✈❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
˙
xi ≈ {xi, H}+cm{xi, φm} ˙
pi ≈ {pi, H}+cm{pi, φm}. ✭✷✳✶✵✮
❆ ✜♠ ❞❡ ❝♦♥str✉✐r♠♦s ✉♠❛ t❡♦r✐❛ ❝♦♥s✐st❡♥t❡✱ ❞❡✈❡♠♦s ✐♠♣♦r ❛ ❝♦♥s❡r✲ ✈❛çã♦ t❡♠♣♦r❛❧ ❞♦s ✈í♥❝✉❧♦s
˙
φm ≈0⇒ {φm, H}+cn{φm, φn} ≈0. ✭✷✳✶✶✮
✷✳✷✿ ❱í♥❝✉❧♦s ❞❡ Pr✐♠❡✐r❛ ❈❧❛ss❡ ❡ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ●❛✉❣❡ ✼
✶✳ ❛ ❡q✉❛çã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ s❛t✐s❢❡✐t❛❀
✷✳ ❡ss❛ ❡q✉❛çã♦ ❞❡t❡r♠✐♥❛ ❛s ❢✉♥çõ❡s ❛r❜✐trár✐❛scm ✉♥✐✈♦❝❛♠❡♥t❡❀
✸✳ ❡ss❛ ❡q✉❛çã♦ ♣♦❞❡ ❞❛r ♦r✐❣❡♠ ❛ ♥♦✈♦s ✈í♥❝✉❧♦s✱ ♦s ❝❤❛♠❛❞♦s ✈í♥❝✉❧♦s s❡❝✉♥❞ár✐♦s ❡ ♥❡st❡ ❝❛s♦✱ ❞❡✈❡✲s❡ ✐♠♣♦r ❛ ❝♦♥s✐stê♥❝✐❛ ❞❡ss❡s ✈í♥❝✉❧♦s ♥♦ t❡♠♣♦ ❛té q✉❡ ♥ã♦ ❤❛❥❛ ♠❛✐s ✈í♥❝✉❧♦s✳
◆❛ ❛♥á❧✐s❡✱ ❢❡✐t❛ ♣♦r ❉✐r❛❝ ❬✼✻❪✱ ✉s❛♠♦s ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ q✉❡ ❞✐✈✐❞❡ ♦s ✈í♥✲ ❝✉❧♦s ❡♠ ♣r✐♠❡✐r❛ ❝❧❛ss❡ ❡ s❡❣✉♥❞❛ ❝❧❛ss❡✳ ❯♠ ✈í♥❝✉❧♦✱ ♦✉ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ✉♠❛ ❢✉♥çã♦ ❆✭①✱♣✮ é ❞✐t♦ ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡ s❡ s❡✉s ❝♦❧❝❤❡t❡s ❞❡ P♦✐ss♦♥ ❝♦♠ t♦❞♦s ♦s ✈í♥❝✉❧♦s sã♦ ❢r❛❝❛♠❡♥t❡ ♥✉❧♦s✱ ✐st♦ é
{A, φm} ≈0, ✭✷✳✶✷✮
❡♥q✉❛♥t♦ ♦s ❞❡ s❡❣✉♥❞❛ ❝❧❛ss❡✱ sã♦ ❛q✉❡❧❡s q✉❡ t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❝♦❧❝❤❡t❡s ❞❡ P♦✐ss♦♥ ❞❛ ❢✉♥çã♦ ❆✭①✱♣✮ ❝♦♠ ❛❧❣✉♠ ✈í♥❝✉❧♦✱ ♥ã♦ ❢r❛❝❛♠❡♥t❡ ♥✉❧♦
{A(x, p), φm} 6≈0. ✭✷✳✶✸✮
✷✳✷ ❱í♥❝✉❧♦s ❞❡ Pr✐♠❡✐r❛ ❈❧❛ss❡ ❡ ❚r❛♥s❢♦r♠❛✲
çõ❡s ❞❡ ●❛✉❣❡
❱❛♠♦s ❛♥❛❧✐s❛r ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ s✐st❡♠❛s ❝♦♠ ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡
(x0, p0✮ ❡♠ t = t0✳ ❊st❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❋✭q✱♣✮ s❡rá
❞❛❞❛ ♣♦r
˙
F ={F, H}+cm{F, φm}. ✭✷✳✶✹✮
❉❡✈✐❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ❛r❜✐trár✐❛s ♥❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ❝♦♥✲ ❝❧✉✐♠♦s q✉❡ ♥ã♦ ♣♦❞❡r❡♠♦s ❞❡t❡r♠✐♥❛r ✉♥✐✈♦❝❛♠❡♥t❡ s✉❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧✳ ◆❡st❡ ❝❛s♦ ❡①✐st❡♠ ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ❞❛❞♦s ♣♦r ♣❛r❡s ❞❡ x❡ pq✉❡ ❝♦rr❡s✲
♣♦♥❞❡♠ à ♠❡s♠❛ ❝♦♥✜❣✉r❛çã♦ ❢ís✐❝❛✳ ❊st❡ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ✜s✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s ❞❡✜♥❡♠ ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❙✉r❣❡ ❡♥tã♦ ❛ s❡❣✉✐♥t❡ ♣❡r✲ ❣✉♥t❛✿ ❡①✐st❡ ❛❧❣✉♠❛ tr❛♥s❢♦r♠❛çã♦ q✉❡ ❝♦♥❡❝t❡ ♦s ♣♦♥t♦s ❞❡ ✉♠❛ ♠❡s♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛❄ P❛r❛ r❡s♣♦♥❞❡r♠♦s ❡st❛ q✉❡stã♦ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ ✈❛r✐❛çã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❡ ❞✉❛s ❡s❝♦❧❤❛s ❢✉♥çõ❡s ❛r❜r✐tár✐❛s✳ ◆❛ ❡s❝♦❧❤❛ ❞❛ ❢✉♥çã♦ ❛r❜✐trár✐❛ cm✱ F s❡rá ❞❛❞♦ ♣♦r
✷✳✸✿ ❋✐①❛çã♦ ❞❡ ●❛✉❣❡ ✽
❏á ♥❛ ❡s❝♦❧❤❛ ❞❡ ✉♠❛ ♦✉tr❛ ❢✉♥çã♦ um✱ ♦❜t❡♠♦s ♦ ✈❛❧♦r ❞❡ ❋
F′(τ) =F(0) +{F, H}∆τ+um{F, φm}∆τ. ✭✷✳✶✻✮
❈❛❧❝✉❧❛♥❞♦ ❛ ✈❛r✐❛çã♦ ❞❛ ❢✉♥çã♦ ❋ ♦❜t❡♠♦s
δF =F(τ)−F′(τ) =δvm{F, φm}, ✭✷✳✶✼✮
♦♥❞❡ δvm = (cm
−um)∆τ✳
❆ ♣❛rt✐r ❞❡st❛ ❡①♣r❡ssã♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♦s ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡ ❣❡r❛♠ ❛s tr❛♥s❢♦r♠❛çõ❡s ❝❛♥ô♥✐❝❛s q✉❡ ❝♦♥❡❝t❛♠ ♦s ❞✐✈❡rs♦s ❝♦♥❥✉♥✲ t♦s ❞❡ ♣♦♥t♦s ❞❡♥tr♦ ❞❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❊st❛s tr❛♥❢♦r♠❛çõ❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ ❤❛♠✐❧t♦♥✐❛♥❛s✳
✷✳✸ ❋✐①❛çã♦ ❞❡ ●❛✉❣❡
❈♦♠♦ ❢♦✐ ❞✐s❝✉t✐❞♦ ♥❛s s❡çõ❡s ❛♥t❡r✐♦r❡s✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡ ❛❝❛rr❡t❛ q✉❡ ♦s ❡st❛❞♦s ❢ís✐❝♦s ♣♦❞❡♠ s❡r ❞❡s❝r✐t♦s ♣♦r ♠❛✐s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ❝❛♥ô♥✐❝❛s✱ ❝❛✉s❛♥❞♦ ❛♠❜✐❣ü✐❞❛❞❡s ♥❛ t❡♦r✐❛✳ P❛r❛ ❡❧✐♠✐♥❛r♠♦s t❛✐s ❛♠❜✐❣ü✐❞❛❞❡s✱ ✐♠♣♦♠♦s ❝❡rt❛s ❝♦♥❞✐çõ❡s ❡①tr❛s à ♥♦ss❛ t❡♦r✐❛✱ ❝❤❛♠❛❞❛s ❞❡ ❝♦♥❞✐çõ❡s ❞❡ ❣❛✉❣❡✳ ❆s ❝♦♥❞✐çõ❡s ❞❡ ❣❛✉❣❡ ♣♦❞❡♠ s❡r✿
✶✳ ❝♦♥❞✐çõ❡s ❈❛♥ô♥✐❝❛s ❞❡ ●❛✉❣❡✿ q✉❛♥❞♦ ✐♠♣♦♠♦s r❡str✐çõ❡s às ✈❛r✐á✈❡✐s ❝❛♥ô♥✐❝❛s ❞♦ s✐st❡♠❛✳ ❊①✳✿ ●❛✉❣❡ ❞❡ ❈♦✉❧♦♠❜ ♥❛ ❡❧❡tr♦❞✐♥â♠✐❝❛❀ ✷✳ ❝♦♥❞✐çõ❡s ❈♦✈❛r✐❛♥t❡s ❞❡ ●❛✉❣❡✿ q✉❛♥❞♦ r❡str✐♥❣✐♠♦s ❛ ❞❡r✐✈❛❞❛ t❡♠✲
♣♦r❛❧ ❞♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✳ ❊①✳✿ ●❛✉❣❡ ❞♦ t❡♠♣♦ ♣ró♣r✐♦ ♥❛ t❡♦r✐❛ ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛✳
P❛r❛ q✉❡ ❛s ❛♠❜✐❣ü✐❞❛❞❡s s❡❥❛♠ ❡❧✐♠✐♥❛❞❛s✱ ♥♦ss❛s ❝♦♥❞✐çõ❡s ❞❡ ❣❛✉❣❡ ❞❡✈❡♠ s❛t✐s❢❛③❡r
✶✳ ❛ ❛❝❡ss✐❜✐❧✐❞❛❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ ❣❛✉❣❡✱ ♦✉ s❡❥❛✱ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ❝❛♥ô♥✐❝❛s✱ ❞❡✈❡ ❡①✐st✐r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❣❛✉❣❡ q✉❡ ❧❡✈❡ ♦ ❝♦♥❥✉♥t♦ ✐♥✐❝✐❛❧ ❛ ♦✉tr♦ q✉❡ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ❣❛✉❣❡✳
✷✳ ❛ ✜①❛çã♦ ❞❡ ❣❛✉❣❡ ❞❡✈❡ s❡r ❝♦♠♣❧❡t❛✱ ♦✉ s❡❥❛✱ ❞❛❞❛ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ❣❛✉❣❡ χ✱ ❡st❛ ❞❡✈❡ s❛t✐s❢❛③❡r ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦
{χ, φm} 6≈0. ✭✷✳✶✽✮
✷✳✹✿ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ✾
P♦r ♦✉tr♦ ❧❛❞♦✱ ❛♥❛❧✐s❛♥❞♦ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦ ♣r♦❝❡ss♦ ❞❡ ✜①❛çã♦ ❞❡ ❣❛✉❣❡✱ ♦❜s❡r✈❛♠♦s q✉❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❣❛✉❣❡ ❞❡✜♥❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ q✉❡ ✐♥t❡r❝❡♣t❛ ❛s ór❜✐t❛s ❞❡ ❣❛✉❣❡ s♦♠❡♥t❡ ✉♠❛ ✈❡③✳ ❖s r❡s✉❧t❛❞♦s ❛❝✐♠❛ ❣❛r❛♥t❡♠ s♦♠❡♥t❡ q✉❡ ❛ ✜①❛çã♦ é ❝♦♠♣❧❡t❛ ❧♦❝❛❧♠❡♥t❡✳
✷✳✹ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠
▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛
◆❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❋❡②♥♠❛♥ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❬✺✷❪✱ ♦ ♦❜❥❡t♦ ❢✉♥❞❛♠❡♥t❛❧ é ❛ ❛♠♣❧✐t✉❞❡ ❞❡ tr❛♥s✐çã♦ ♦✉ ♣r♦♣❛❣❛❞♦r✱ q✉❡ ♠❡❞❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ s✐st❡♠❛ ♣❛ss❛r ❞❡ ✉♠ ❡st❛❞♦ ❛ ♦✉tr♦✱ ♦✉ s❡❥❛✱ ❞❡s❝r❡✈❡ ♦ ♣r♦❝❡ss♦ q✉â♥t✐❝♦✳ ❱❛♠♦s ❝♦♥str✉✐r ❡st❡ ♣r♦♣❛❣❛❞♦r ❛ ♣❛rt✐r ❞❛ ❢♦r♠✉❧❛çã♦ ✉s✉❛❧ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❡♠ q✉❡ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ✉♠ s✐st❡♠❛ é ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r
ˆ
H|Ψ;ti=i~d
dt|Ψ;ti, ✭✷✳✶✾✮
♦♥❞❡ |Ψ;ti r❡♣r❡s❡♥t❛ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ♥✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t✳ ❆
s♦❧✉çã♦ ❞❡st❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦
|Ψ;ti=U(t, t0)|Ψ;t0i, ✭✷✳✷✵✮
❡♠ q✉❡ ♦ ♦♣❡r❛❞♦r U(t, t0) é ♦ ❝❤❛♠❛❞♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧✳
❙❡ Hˆ ♥ã♦ ❢♦r ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦
♦♣❡r❛❞♦r ❡✈♦❧✉çã♦ ♣♦r
U(t, t0) = exp−
i
~((t−t0) ˆH), ✭✷✳✷✶✮ q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s
✶✳ U(t3, t2)U(t2, t1) =U(t3, t1)
✷✳ U(t2, t1)† =U−1(t2, t1) =U(t1, t2)✳
P♦❞❡♠♦s ❡♥tã♦ ❞❡✜♥✐r ♦ ♣r♦♣❛❣❛❞♦r ♥♦ ❡s♣❛ç♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❝♦♠♦
Z(xN, x0;tN, t0) = hxN|U(t)|xoi, ✭✷✳✷✷✮
♦♥❞❡ ♦s ❛✉t♦✲❡st❛❞♦s ❞❛ ♣♦s✐çã♦ |xii sã♦ ♦rt♦♥♦r♠❛✐s✱ ❝♦♠♣❧❡t♦s q✉❡ s❛t✐s❢❛✲
✷✳✹✿ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ✶✵
hx|x′i = δ(x−x′)
✂
dx|xihx| = 1 ✭✷✳✷✸✮
hx|pi = exp i
~px
❱❛♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ♣r♦♣❛❣❛❞♦r ❡♠ t❡r♠♦s ❞❡ ✐♥t❡❣r❛✐s ❞❡ tr❛❥❡tór✐❛✳ P❛r❛ ✐st♦✱ ❞✐✈✐❞✐♠♦s ♦ ✐♥t❡r✈❛❧♦ ❡♥tr❡ ♦ ✐♥st❛♥t❡ ✜♥❛❧ ❡ ♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ ❡♠ N s✉❜✐♥t❡r✈❛❧♦s ✐♥✜♥✐t❡s✐♠❛✐s ✐♥t❡r♠❡❞✐ár✐♦s ❞❡ ✈❛❧♦r ǫ = ∆τ
N = τN−τ0
N ✳
P♦rt❛♥t♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ✶ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ♦♣❡r❛❞♦r ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
U(tN, t0) =U(tN, tN−1)U(tN−1, tN−2)...U(t1, t0). ✭✷✳✷✹✮
❖ ♣r♦♣❛❣❛❞♦r ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦
Z(xN, x0;tN, t0) =hxN|U(tN, tN−1)U(tN−1, tN−2)...U(t1, t0)|xoi. ✭✷✳✷✺✮
■♥s❡r✐♥❞♦N−1❝♦♥❥✉♥t♦s ✐♥t❡r♠❡❞✐ár✐♦s ❝♦♠♣❧❡t♦s ❞❡ ❡st❛❞♦s ❡♥tr❡ ❛ ♣♦s✐çã♦
✐♥✐❝✐❛❧ x0 ❡ ✜♥❛❧ xN ♥❛ ❡①♣r❡ssã♦ ✭✷✳✷✺✮ t❡♠♦s
Z(xN, x0;tN, t0) =
✂
dxN−1dxN−2....dx1hxN|U(tN, tN−1)|xN−1i
hxN−1|U(tN−1, tN−2)|xN−2i... ✭✷✳✷✻✮
hx1|U(t1, t0)|x0i. ✭✷✳✷✼✮
❊ ✐♥s❡r✐♥❞♦ ❡♥tr❡ ❡st❡s ◆ ✈❡③❡s ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦ ✂
dp|pihp|= 1. ✭✷✳✷✽✮
P♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ ♣r♦♣❛❣❛❞♦r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
Z(xN, x0;tN, t0) = lim
N→∞
✂
dxN−1dxN−2...dx1dpN−1dpN−2....dp0
hxN|U(tN, tN−1)|pN−1ihpN−1||xN−1i
hxN−1|U(tN−1, tN−2)|pN−1ihpN−1||xN−2i
hxN−1|U(tN−1, tN−2)|pN−1ihpN−1||xN−2i.
✷✳✺✿ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ❚❡♦r✐❛s ❞❡ ●❛✉❣❡ ✶✶
▼❛s ✉s❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦
hxi+1|U(tN, tN−1)|piihpi||xii = hxi+1|exp(
i
~Hǫˆ )|piihpi||xii
= hxi+1|(1−
i
~Hˆ)|piihpi||xii
= (1− i
~hk) exp(
i
~pi(xi+1−xi))✭✷✳✷✾✮
= exp i
~(pi(xi+1−xi)−hi), ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ ♣r♦♣❛❣❛❞♦r ❝♦♠♦
Z(xN, x0;tN, t0) = lim
N→∞
✂ N−1
Y
i=0
dpi N−1
Y
i=1
dxiexp i
~
N−1
X
i=0
(pi(xi+1−xi)
−ǫhi),
=
✂
DxDpexp i
~ ✂
dt(pix˙i−h), ✭✷✳✸✵✮
♦♥❞❡ Dx✱Dp é ❛ ♠❡❞✐❞❛ ❢✉♥❝✐♦♥❛❧ ♥❛ ✐♥t❡❣r❛❧ ❞❡ tr❛❥❡tór✐❛✳
✷✳✺ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠
❚❡♦r✐❛s ❞❡ ●❛✉❣❡
P❛r❛ ✉♠❛ t❡♦r✐❛ ❞❡ ❣❛✉❣❡✱ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ✐♥t❡❣r❛✐s ❞❡ tr❛❥❡tór✐❛ ✜❝❛ ✉♠ ♣♦✉❝♦ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛ ❡♠ ❢✉♥çã♦ ❞❛ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡✳ ❊①✐st❡♠ ❞✉❛s ❢♦r♠❛s ❞❡ ✐♠♣❧❡♠❡♥t❛r ❛ ❢♦r♠✉❧❛çã♦✿ ♦ ♠ét♦❞♦ ❞❡ ❋❛❞❞❡❡✈✲P♦♣♦✈❬✺✸❪ ❡ ♦ ♠ét♦❞♦ ❇❋❱❬✺✽✱ ✺✼❪✳
✷✳✺✳✶ ▼ét♦❞♦ ❞❡ ❋❛❞❞❡❡✈✲P♦♣♦✈
❖ ♣r✐♠❡✐r♦ ♠ét♦❞♦ ❛ ✐♥❝♦r♣♦r❛r ✈í♥❝✉❧♦s ♥❛ ❢♦r♠✉❧❛çã♦ ❞❡ ✐♥t❡❣r❛✐s ❞❡ tr❛❥❡✲ tór✐❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❋❛❞❞❡❡✈❬✺✸❪✱ ♣❛r❛ ♦ q✉❛❧ s❡ ❝♦♥s✐❞❡r❛✈❛♠ ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡✳ P♦st❡r✐♦r♠❡♥t❡✱ ❙❡♥❥❛♥♦✈✐❝❬✼✼❪ ❣❡♥❡r❛❧✐③♦✉ ❡st❡ r❡s✉❧t❛❞♦ ♣❛r❛ ✐♥❝❧✉✐r ✈í♥❝✉❧♦s ❞❡ s❡❣✉♥❞❛ ❝❧❛ss❡✳ ❊♠ ♥♦ss♦s s✐st❡♠❛s ❡st✉❞❛❞♦s ❡①✐s✲ t❡♠ s♦♠❡♥t❡ ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡ ❡✱ ♣♦rt❛♥t♦✱ ✐r❡♠♦s s♦♠❡♥t❡ ❞✐s❝✉t✐r ♦ r❡s✉❧t❛❞♦ ❞❡ ❋❛❞❞❡❡✈✳
❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ❞❡ ◆ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ q✉❡ ♣♦ss✉✐ ▼ ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡ φa✳ ❚❡♠♦s ❡♥tã♦ ❞❡ ✐♥tr♦❞✉③✐r ▼ ✜①❛çõ❡s ❞❡ ❣❛✉❣❡ Ωa✳ ❊st❡
✷✳✺✿ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ❚❡♦r✐❛s ❞❡ ●❛✉❣❡ ✶✷
{φa, φb} ≈ 0
det{Ωa, φb} 6= 0, ✭✷✳✸✶✮
s♦❜r❡ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❞❛❞❛ ♣♦r φa = 0 ❡ Ωa = 0✳ P♦❞❡♠♦s ❛ ♣❛rt✐r ❞❛q✉✐
❡♥✉♥❝✐❛r ♦ t❡♦r❡♠❛ ❞❡ ❋❛❞❞❡❡✈✳ ❚❡♦r❡♠❛
❖ ♣r♦♣❛❣❛❞♦r é ❞❛❞♦ ♣♦r
Z(xN, X0) =
✂
dµexp i
~( ✂
dτ(px˙ −H)), ✭✷✳✸✷✮
❡♠ q✉❡ ❛ ♠❡❞✐❞❛ ❞❛ ✐♥t❡❣r❛❧ é ❞❛❞❛ ♣♦r
dµ=det{Ωa, φa} M
Y
a=1
δ(Ωa)δ(φa) N
Y
i=1
dpidxi. ✭✷✳✸✸✮
✷✳✺✳✷ ▼ét♦❞♦ ❇❋❱
❱❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠ s✐st❡♠❛ ❝♦♠ N ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❡✜♥✐❞♦ ❡♠ ✉♠
❡s♣❛ç♦ ❞❡ ❢❛s❡ ❝♦♠♣♦st♦ ♣♦r ✈❛r✐á✈❡✐s q✉❡ ❝♦♠✉t❛♠ ❡ ♣♦r ✈❛r✐á✈❡✐s ❞❡ ●r❛s✲ s♠❛♥♥✳ ❆s ✈❛r✐á✈❡✐s q✉❡ ❝♦♠✉t❛♠ tê♠ ♣❛r✐❞❛❞❡ ❞❡ ●r❛ss♠❛♥♥ ǫa = 0 ❡ ❛s
✈❛r✐á✈❡✐s ❞❡ ●r❛ss♠❛♥♥ tê♠ ♣❛r✐❞❛❞❡ ǫa = 1✳ ❖s ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡
sã♦ φa(a = 1....M) ❡ s❛t✐s❢❛③❡♠ à s❡❣✉✐♥t❡ á❧❣❡❜r❛
{φa, φb} = fabdφd ✭✷✳✸✹✮
{H, φa} = Vabφb,
♦♥❞❡ fd
ab❡Vab sã♦ ❛s ❢✉♥çõ❡s ❞❡ ❡str✉t✉r❛ ✳
❖ ❢♦r♠❛❧✐s♠♦ é ✐♠♣❧❡♠❡♥t❛❞♦ ❡♠ ❞♦✐s ♣❛ss♦s✳ ❖ ♣r✐♠❡✐r♦ ❝♦♥s✐st❡ ❡♠ ♣r♦♠♦✈❡r ♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ λa ❛ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s ❞♦ ❡s♣❛ç♦
❞❡ ❢❛s❡ ❡ ✐♥tr♦❞✉③✐r ✉♠ ♠♦♠❡♥t♦ ❝❛♥ô♥✐❝♦ ♣❛r❛ ♦ ♠✉❧t✐♣❧✐❝❛❞♦r ❞❡ ❧❛❣r❛♥❣❡
Πa✱ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦
{λa,Πb}=δba. ✭✷✳✸✺✮
✷✳✺✿ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ❚❡♦r✐❛s ❞❡ ●❛✉❣❡ ✶✸
Πa= 0. ✭✷✳✸✻✮
❖s ✈í♥❝✉❧♦s φ ❡ Π ❢♦r♠❛♠ ✉♠ s✐st❡♠❛ ❞❡ ✈í♥❝✉❧♦s ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡✳ ◆♦
❢♦r♠❛❧✐s♠♦ ❇❋❱ ❡st❡ ❝♦♥❥✉♥t♦ é ❞❡♥♦t❛❞♦ ♣♦rGi(i= 1,2, ...2M)q✉❡ s❛t✐s✲
❢❛③
{Gi, Gj} = fijkGk ✭✷✳✸✼✮
{H, Gi} = VijGj.
◆✉♠ s❡❣✉♥❞♦ ♣❛ss♦✱ ✐♥tr♦❞✉③✐♠♦s ♥♦✈♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❛ ✜♠ ❞❡ ❝♦♠✲ ♣❡♥s❛r ♦ ❛✉♠❡♥t♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✳ ❊st❡s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❛❞✐❝✐♦♥❛✐s sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣❛r❡s ❞❡ ❢❛♥t❛s♠❛s ❞❡ ❇❋❱✳ P❛r❛ ❝❛❞❛ ✈í♥❝✉❧♦ ❞❡ ♣r✐✲ ♠❡✐r❛ ❝❧❛ss❡ ✐♥tr♦❞✉③✐♠♦s ✉♠ ♣❛r ❞❡ ❢❛♥t❛s♠❛s (ηi,Pi)✱ ♠❛s ❝♦♠ ♣❛r✐❞❛❞❡
❞❡ ●r❛ss♠❛♥ ♦♣♦st❛ ❛♦ ❝♦rr❡s♣♦❞❡♥t❡ ✈í♥❝✉❧♦ ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡✱ q✉❡ s❛t✐s❢❛③ à s❡❣✉✐♥t❡ á❧❣❡❜r❛
{Pi, ηj} = −δij, ✭✷✳✸✽✮
❡ ♦s ♦✉tr♦s ❝♦❧❝❤❡t❡s ❞❡ P♦✐ss♦♥ ❣❡♥❡r❛❧✐③❛❞♦s ♥✉❧♦s✳ P♦rt❛♥t♦✱ ♥♦ss♦ ♥♦✈♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ é (x, p, λa,Πa;ηi,
Pi)✳ ◆❡st❡ ♥♦✈♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ♦ ♥ú♠❡r♦ ❞❡
❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ é ❞❛❞♦ ♣♦r 2(N −M)✳
◆❡st❡ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❡st❡♥❞✐❞♦✱ ❛ s✐♠❡tr✐❛ ♦r✐❣✐♥❛❧ ❞❡ ❣❛✉❣❡ é s✉❜st✐t✉í❞❛ ♣♦r ✉♠❛ s✐♠❡tr✐❛ ❣❧♦❜❛❧ ❣❡r❛❞❛ ♣❡❧❛ s❡❣✉✐♥t❡ ❝❛r❣❛ ❢❡r♠✐ô♥✐❝❛ QB
QB =ηiGi− 1 2η
jηkfi
jkPi, ✭✷✳✸✾✮
q✉❡ é ❛♥t✐❝♦♠✉t❛t✐✈♦ ❡ ♣♦r ❝♦♥str✉çã♦ s❛t✐s❢❛③
{QB, QB}= 0. ✭✷✳✹✵✮
❊st❛ ❝❛r❣❛ ❢❡r♠✐ô♥✐❝❛ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❝❛r❣❛ ❇❘❙❚✳ ❊st❛ ❝❛r❣❛ ❣❡r❛ ❛s s❡❣✉✐♥t❡s tr❛♥s❢♦r♠❛çõ❡s
δBx = {x, Gi}ηi δBp = {p, Gi}ηi
δBλa = ηa ✭✷✳✹✶✮
δBΠa = 0 δBηi =
1 2f
i jkηjηk
✷✳✺✿ ◗✉❛♥t✐③❛çã♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❚r❛❥❡tór✐❛ ❡♠ ❚❡♦r✐❛s ❞❡ ●❛✉❣❡ ✶✹
P♦❞❡♠♦s ❛❣♦r❛ ❡s❝r❡✈❡r ❛ ❛çã♦ ✐♥✈❛r✐❛♥t❡ ♣♦r ❇❘❙❚
Sef =
✂
dτ( ˙xp−λaΠa˙ + ˙ηiP
i−H− {Ψ, QB}), ✭✷✳✹✷✮
♦♥❞❡ Ψ é ✉♠❛ ❢✉♥çã♦ ❢❡r♠✐ô♥✐❝❛ ❛r❜✐trár✐❛✳
❆s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ♣r♦✈❡♥✐❡♥t❡s ❞❡st❛ ❛çã♦ ❞❡✈❡♠ s❡r s✉♣❧❡♠❡♥✲ t❛❞❛s ♣♦r ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ q✉❡ s❡❥❛♠ ✐♥✈❛r✐❛♥t❡s ♣❡❧❛ s✐♠❡tr✐❛ ❞❡ ❇❘❙❚✳ ❊①✐st❡♠ ✈ár✐❛s ❝♦♥❞✐çõ❡s ✐♥✈❛r✐❛♥t❡s q✉❡ ♣♦❞❡rí❛♠♦s ❛❞♦t❛r✳ ❱❛♠♦s ✐♥tr♦✲ ❞✉③✐r ✉♠ ❝♦♥❥✉♥t♦ ♠✉✐t♦ ✉s❛❞♦ ♥❛ ❧✐t❡r❛t✉r❛✱ ❡ ♣❛r❛ ✐♠♣❧❡♠❡♥tá✲❧♦✱ ✈❛♠♦s ♣r✐♠❡✐r♦ ❞❡❝♦♠♣♦r ♦ ♣❛r ❞❡ ❢❛♥t❛s♠❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
ηi = (Pa, Ca),
Pi = ( ¯Ca,P¯a), ✭✷✳✹✸✮
q✉❡ s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s
{Pa,C¯
a} = −1,
{Ca,P¯a} = −1. ✭✷✳✹✹✮
P♦❞❡♠♦s ❛❣♦r❛ ❞❡✜♥✐r ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ✐♥✈❛r✐❛♥t❡s ❞❡ ❇❘❙❚ ❞❛❞❛s ♣♦r
Πa(τ0) = 0, Πa(τN) = 0, Ca(τ0) = 0, Ca(τN) = 0,
¯
Ca(τ0) = 0, C¯a(τN) = 0, ✭✷✳✹✺✮ x(τ0) = x0, x(τN) =xN.
P♦❞❡♠♦s ❞❡✜♥✐r ❛❣♦r❛ ♦ t❡♦r❡♠❛ ❇❋❱✳ ❚❡♦r❡♠❛ ❇❋❱
❖ ♣r♦♣❛❣❛❞♦r é ❞❛❞♦ ♣♦r
Z(xN, x0) =
✂
Dµexpi
✂
dτ( ˙xp−λaΠa˙ + ¯PaC˙a−C¯aP˙a−H+{Ψ, QB})
,
✭✷✳✹✻✮ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ Ψ ❡ Dµ = DxDpDλDΠDP¯DPDCDC¯ é ❛ ♠❡❞✐❞❛ ❞❡
▲✐♦✉✈✐❧❧❡✳ ❆s ❞✐✈❡rs❛s ❡s❝♦❧❤❛s ❞❡ ❣❛✉❣❡ sã♦ ♦❜t✐❞❛s ♣♦r ❞✐❢❡r❡♥t❡s ❡s❝♦❧❤❛s ❞❛ ❢✉♥çã♦ ❛r❜✐trár✐❛ Ψ✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛
❈❛♣ít✉❧♦ ✸
◗✉❛♥t✐③❛çã♦ ❇❘❙❚ ❞❛ P❛rtí❝✉❧❛
❘❡❧❛t✐✈íst✐❝❛
✸✳✶ ■♥tr♦❞✉çã♦
◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❛♥❛❧✐s❛r ♦ ❝❛s♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛ ❧✐✈r❡✱ ❞❡ ♠❛ss❛ m✱ ♥✉♠ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❡♠ d ❞✐♠❡♥sõ❡s✳ ❆ ❛çã♦ q✉❡ ❛ ❞❡s❝r❡✈❡
é
S =−m
✂ τ2
τ1
dτp−x˙2(τ), ✭✸✳✶✮
❡ τ ♣❛r❛♠❡tr✐③❛ ❛ ❧✐♥❤❛ ♠✉♥❞♦ ❞❛ ♣❛rtí❝✉❧❛ r❡❧❛t✐✈íst✐❝❛✳ ❊ ❛ ♣♦s✐çã♦ ♥♦
❡s♣❛ç♦✲t❡♠♣♦ é ❞❡s❝r✐t❛ ♣♦r xµ(τ) ❝♦♠ µ = 0,1, ...d −1✳ ❆❞♦t❛r❡♠♦s ❛
❝♦♥✈❡♥çã♦ ηµν = diag(−1,1,1....1) ♣❛r❛ ❛ ♠étr✐❝❛✳ ❊st❛ ❛çã♦ é ✐♥✈❛r✐❛♥t❡
♣♦r r❡♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❧✐♥❤❛ ♠✉♥❞♦ ❞❡✜♥✐❞❛ ♣❡❧❛s tr❛♥s❢♦r♠❛çõ❡s
τ →˜τ = ˜τ(τ),
xµ(τ)→x˜µ(˜τ) = xµ(τ), ✭✸✳✷✮
♦✉ s❡❥❛✱ ❛s ❝♦♦r❞❡♥❛❞❛s s❡ tr❛♥s❢♦r♠❛♠ ❝♦♠♦ ❡s❝❛❧❛r❡s✳ ❆s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ sã♦
d dτ
mx˙µ(τ)
p
−x˙2(τ)
!
= 0, ✭✸✳✸✮
s✉❥❡✐t❛s às ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ xµ(τ
1) =xµ1 ❡ xµ(τ2) =xµ2 ✳
P❛ss❛♠♦s à ❢♦r♠✉❧❛çã♦ ❤❛♠✐❧t♦♥✐❛♥❛ ❝❛❧❝✉❧❛♥❞♦ ♦ ♠♦♠❡♥t♦ ❝❛♥ô♥✐❝♦
✸✳✶✿ ■♥tr♦❞✉çã♦ ✶✻
pµ(τ) = ∂L ∂x˙µ, = pmx˙µ
−x˙2(τ)
′
✭✸✳✹✮
❡ ✉t✐❧✐③❛♥❞♦ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s ♦ ✈í♥❝✉❧♦ ♣r✐♠ár✐♦
φ= 1 2(p
2+m2)
≈0. ✭✸✳✺✮
❊♥tã♦ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❝❛♥ô♥✐❝❛ é
H = pµx˙µ−L, = pmx˙µ
−x˙2(τ)x˙
µ
−L,
= 0, ✭✸✳✻✮
❡✱ ♥❡st❡ ❝❛s♦✱ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦ s✐st❡♠❛ s❡rá ❣♦✈❡r♥❛❞❛ ♣❡❧❛ ❤❛♠✐❧t♦✲ ♥✐❛♥❛ t♦t❛❧ HT
HT = H+λ(τ)φ
= λ(τ)φ, ✭✸✳✼✮
♦♥❞❡ λ é ♦ ♠✉❧t✐♣❧✐❝❛❞♦r ❞❡ ▲❛❣r❛♥❣❡✳ ■♠♣♦♥❞♦✲s❡ ❛ ❝♦♥s✐stê♥❝✐❛ t❡♠♣♦r❛❧
❞♦ ✈í♥❝✉❧♦ ✭✸✳✺✮ t❡♠♦s
˙
φ=λ{φ, φ} ≈0, ✭✸✳✽✮
♦ q✉❡ ♣❡r♠✐t❡ ❝♦♥❝❧✉✐r q✉❡ φ é ú♥✐❝♦ ❡ ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡✳ ❘❡❡s❝r❡✈❡♥❞♦ ❛
❛çã♦ ♥❛ ❢♦r♠❛ ❤❛♠✐❧t♦♥✐❛♥❛ ♦❜t❡♠♦s
S =
✂ τ2
τ1
dτ
pµx˙µ− 1
2λ(τ)(p
2+m2)
. ✭✸✳✾✮
❆s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ ❣❡r❛❞♦s ♣♦r ✭✸✳✺✮ sã♦
δxµ = ǫ(τ)pµ,
δpµ = 0, ✭✸✳✶✵✮
✸✳✶✿ ■♥tr♦❞✉çã♦ ✶✼
♦♥❞❡ ǫ(τ)é ♦ ♣❛râ♠❡tr♦ ❞❛ tr❛♥s❢♦r♠❛çã♦✳ ❚♦♠❛♥❞♦ ❛ ✈❛r✐❛çã♦ ❞❛ ❛çã♦
δS =
✂ τ2
τ1
dτ d dτ
ǫ(τ)1 2(p
2
−m2)
= ǫ(τ)1 2(p
2
−m2)
τ2
τ1
, ✭✸✳✶✶✮
❝♦♥❝❧✉✐♠♦s q✉❡ ❛ ✐♥✈❛r✐â♥❝✐❛ ✐♠♣õ❡ ♦ ❛♥✉❧❛♠❡♥t♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❣❛✉❣❡ ♥♦s ❡①tr❡♠♦s✳
❱❛♠♦s ❛❣♦r❛ ❝♦♥s✐❞❡r❛r ♦ ❢♦r♠❛❧✐s♠♦ ❇❋❱✳ ◆✉♠ ♣r✐♠❡✐r♦ ♠♦♠❡♥t♦ ✐♥✲ tr♦❞✉③✐♠♦s ✉♠ ❣r❛✉ ❞❡ ❧✐❜❡r❞❛❞❡ Π❝♦♥❥✉❣❛❞♦ ❛♦ ♠✉❧t✐♣❧✐❝❛❞♦r ❞❡ ▲❛❣r❛♥❣❡ λ✱ s❛t✐s❢❛③❡♥❞♦ ♦ s❡❣✉✐♥t❡ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥
{λ,Π}= 1, ✭✸✳✶✷✮
❡ ♣❛r❛ ♥ã♦ ❛❧t❡r❛r♠♦s ♦ ❝♦♥t❡ú❞♦ ❢ís✐❝♦ ❞❛ t❡♦r✐❛ ✐♠♣♦♠♦s
Π = 0. ✭✸✳✶✸✮
❊st❡ ✈í♥❝✉❧♦ ❛❞✐❝✐♦♥❛❧ ✭✸✳✶✸✮ ❢♦r♠❛ ✉♠ s✐st❡♠❛ ❞❡ ♣r✐♠❡✐r❛ ❝❧❛ss❡ ❝♦♠ ♦ ✈í♥❝✉❧♦ φ✱ q✉❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦rGa a= 1,2❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
G1 = Π, G2 =φ, ✭✸✳✶✹✮
s❛t✐s❢❛③❡♥❞♦ ❛ s❡❣✉✐♥t❡ á❧❣❡❜r❛✿
{Ga, Gb} = 0, a, b= 1,2 ✭✸✳✶✺✮
{H, Ga} = 0, ✭✸✳✶✻✮
♦♥❞❡ H é ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❝❛♥ô♥✐❝❛✳
◆✉♠ s❡❣✉♥❞♦ ♣❛ss♦✱ ✐♥tr♦❞✉③✐♠♦s ✉♠ ♣❛r ❞❡ ❢❛♥t❛s♠❛s (C,P¯) ❡ (P,C¯)
❛ss♦❝✐❛❞♦s ❛♦s ✈í♥❝✉❧♦s ❞❛❞♦s ❡♠ ✭✸✳✶✹✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❛t✐s❢❛③❡♥❞♦ ♦s s❡❣✉✐♥t❡s ❝♦❧❝❤❡t❡s ❞❡ P♦✐ss♦♥✿
{C,P}¯ = {P,C¯}=−1, ✭✸✳✶✼✮
❡ ❝♦♠ ♦s ♦✉tr♦s ❝♦❧❝❤❡t❡s ❞❡ P♦✐ss♦♥ ♥✉❧♦s✳ ❆ ❝❛r❣❛ ❞❡ ❇❘❙❚ é
QB = 1 2C p
2+m2
+PΠ, ✭✸✳✶✽✮
✸✳✶✿ ■♥tr♦❞✉çã♦ ✶✽
δxµ = Cpµ, δpµ= 0, δλ = P, δΠ = 0, δC = 0, δC¯=−Π, δP¯ = −1
2(p
2+m2), δP = 0. ✭✸✳✶✾✮
❱❛♠♦s ❛♥❛❧✐s❛r ❛❧❣✉♠❛s ❡s❝♦❧❤❛s ❞❡ ❣❛✉❣❡ ♥♦ ❢♦r♠❛❧✐s♠♦ ❇❋❱✳
✸✳✶✳✶ ❈♦♥❞✐çã♦ ❈♦✈❛r✐❛♥t❡ ❞❡ ●❛✉❣❡
λ
˙
(
τ
) =
f
(
λ
)
❖ ♣r✐♠❡✐r♦ ❝❛s♦ ❛ s❡r ❛♥❛❧✐s❛❞♦ é ♦ ❞❛ ❝♦♥❞✐çã♦ ❝♦✈❛r✐❛♥t❡ ❞❡ ❣❛✉❣❡ ❡ ❡st❛ é ✐♠♣❧❡♠❡♥t❛❞❛ ♣❡❧❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ ❣r❛s♠❛♥✐❛♥❛
Ψ = ¯Cf(λ) +λP¯, ✭✸✳✷✵✮
♦♥❞❡ f(λ) é ✉♠❛ ❢✉♥çã♦ ❛r❜✐trár✐❛ ❞♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✳ ❈❛❧❝✉✲
❧❛♥❞♦ ♦ ❝♦❧❝❤❡t❡s ❞❡ P♦✐ss♦♥ ❞❡ Ψ❝♦♠ ❛ ❝❛r❣❛ ❞❡ ❇❘❙❚ t❡♠♦s
{Ψ, QB}=−f(λ)Π + ¯CPf ′
(λ) + ¯PP −λ 2(p
2+m2), ✭✸✳✷✶✮
❡♠ q✉❡ f′
(λ)é ❛ ❞❡r✐✈❛❞❛ ❞❡ f ❡♠ r❡❧❛çã♦ ❛ λ✳ ❆ ❛çã♦ ❡❢❡t✐✈❛ é
Sef =
✂ τ2
τ1
dτ(pµx˙µ+ Π ˙λ+ ¯PC˙ + ¯CP −˙ f(λ)Π + ¯CPf ′
(λ)
+ ¯PP −λ 2(p
2+m2)). ✭✸✳✷✷✮
❆s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ sã♦
˙
xµ = λpµ, p˙µ= 0, ˙¯
C = −P¯, C˙ =P, ˙
λ = f(λ), Π =˙ f′(λ)Π + ¯CPf′′(λ)− 1 2(p
2+m2). ✭✸✳✷✸✮
❘❡s♦❧✈❡♥❞♦ ❡st❛s ❡q✉❛çõ❡s ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ f(λ) = 0 ❡ ✉t✐❧✐③❛♥❞♦ ❛s
✸✳✶✿ ■♥tr♦❞✉çã♦ ✶✾
xµ(τ) = xµ1 +∆x µ
∆τ (τ −τ1), p
µ(τ) = ∆x µ
∆τ , ¯
C(τ) = 0, C(τ) = 0,
P(τ) = 0, P(τ) = 0,
λ(τ) =λ0, Π(τ) = 0, ✭✸✳✷✹✮
♦♥❞❡∆xµ=xµ2−xµ1 ❡λ0 é ✉♠❛ ❝♦♥st❛♥t❡✳ ❯♠❛ ❞❛s ❝♦♥s❡q✉ê♥❝✐❛s q✉❡ ♣♦❞❡✲
♠♦s ❝♦♥❝❧✉✐r é q✉❡ ♦ ✈í♥❝✉❧♦ φ é s❛t✐s❢❡✐t♦ ❡♠ q✉❛❧q✉❡r ✐♥st❛♥t❡✳ ❯t✐❧✐③❛♥❞♦
❛ s♦❧✉çã♦ ♣❛r❛ pµ ♦❜t❡♠♦s q✉❡ (∆x)2 =m2(∆τ)2.
✸✳✶✳✷ ❈♦♥❞✐çã♦ ❞❡ ●❛✉❣❡ ❈❛♥ô♥✐❝♦
x
0−
τ
= 0
P❛r❛ ❛♥❛❧✐s❛r♠♦s ❡st❛ ❝♦♥❞✐çã♦ ❞❡ ❣❛✉❣❡ ❡s❝r❡✈❡r❡♠♦s xµ= (x0, ~x✮✱ ♦♥❞❡x0
é ❝♦♠♣♦♥❡♥t❡ t❡♠♣♦r❛❧ ❡ ~x ❝♦rr❡s♣♦♥❞❡ ❛s d−1 ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s✳ ❆
❢✉♥çã♦ ❣r❛s♠❛♥✐❛♥❛ q✉❡ ✐♠♣❧❡♠❡♥t❛ ❡st❡ ❣❛✉❣❡ é
Ψ = 1 β(x
0
−τ) ¯C+λP¯, ✭✸✳✷✺✮
♦♥❞❡ β é ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❈❛❧❝✉❧❛♥❞♦✲s❡ ♦ ❝♦❧❝❤❡t❡ ❞❡
P♦✐ss♦♥ t❡♠♦s
{Ψ, QB}= 1 βΠ(x
0
−τ) + 1
βCCp¯ 0+ ¯PP + λ 2(p
2
0−~p2−m2). ✭✸✳✷✻✮
◆❡st❡ ❝❛s♦✱ ❛ ❛çã♦ ❡❢❡t✐✈❛ é
Sef =
✂ τ2
τ1
dτ(−p0x˙0+~p·~x˙ + Π ˙λ+ ¯PC˙ + ¯CP˙ + 1 βΠ(x
0−τ)
+1
βCCp¯ 0+ ¯PP + λ 2(p
2
0−~p2−m2)).
P❛r❛ ♦❜t❡r♠♦s ❛ ✜①❛çã♦ ❞❡ ❣❛✉❣❡ ❞❡s❡❥❛❞❛ t❡♠♦s q✉❡ ❢❛③❡r ❛ s❡❣✉✐♥t❡ ♠✉✲ ❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ✐♥✈❛r✐❛♥t❡ ♣♦r ❇❘❙❚Π →βΠ˜ ❡ C¯ →βC˜¯ q✉❡ r❡❡s❝r❡✈❡ ❛
❛çã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
Sef =
✂ τ2
τ1
dτ(−p0x˙0+~p·~x˙ +βΠ ˙˜λ+ ¯PC˙ +βC˜¯P˙ + ˜Π(x0−τ)
+ ˜¯CCp0+ ¯PP +
λ 2(p
2
