rs t tr r - - PowerPoint PPT Presentation

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥

❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❳❛✈✐❡r ❇❊❑❆❊❘❚

▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❡t P❤②s✐q✉❡ ❚❤é♦r✐q✉❡ ✭❚♦✉rs✮

✶✾ ❆♣r✐❧ ✷✵✶✷ ❅ ❱✐❡♥♥❛ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤❊✳ ▼❡✉♥✐❡r ❛♥❞ ❙✳ ▼♦r♦③ P❤②s✳ ❘❡✈✳ ❉ ✭✷✵✶✷✮ ❬❛r❳✐✈✿✶✶✶✶✳✶✵✽✷ ❬❤❡♣✲t❤❪ ❪ ❏❍❊P ✶✷✵✷ ✭✷✵✶✷✮ ✶✶✸ ❬ ❛r❳✐✈✿✶✶✶✶✳✸✻✺✻ ❬❤❡♣✲t❤❪ ❪

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

✶ ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❲❤❛t ✐s ✐t❄ ❍♦✇ t♦ ❞❡s❝r✐❜❡ ❛♥❞ ❤♦✇ t♦ ♣r❡♣❛r❡ ♦♥❡❄ ❲❤② ✐s ✐t ♦❢ t❤❡♦r❡t✐❝❛❧ ✐♥t❡r❡st❄ ❲❤❛t ♠✐❣❤t ❜❡ ❛♥ ❡❞✉❝❛t❡❞ ❣✉❡ss ❢♦r ✐ts ❜✉❧❦ ❞✉❛❧❄

✷ ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤②

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❝✉rr❡♥ts

✸ ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s

❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥s ✭♠❛ss❧❡ss ✈s ♠❛ss✐✈❡ ✜❡❧❞s✮ ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

✹ ❙✉♠♠❛r② ❛♥❞ ♦✉t❧♦♦❦ ❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❚❤❡ ✉♥❞❡r❧②✐♥❣ ♠♦t✐✈❛t✐♦♥ ✐s t❤❡ q✉❡st ❢♦r ❛ ❤♦❧♦❣r❛♣❤✐❝ ❞✉❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ✭❙♦♥✱ ❇❛❧❛s✉❜r❛♠❛♥✐❛♥✱ ▼❝●r❡❡✈②❀ ✷✵✵✽✮✳ ⇒ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❉❡✜♥✐t✐♦♥ ✭s✐♠♣❧✐✜❡❞✮ ❣❛s✿ ❞✐❧✉t❡ ♠❛♥②✲❜♦❞② s②st❡♠ ❉✐❧✉t❡ ⇔ ✐♥t❡r❛❝t✐♦♥ r❛♥❣❡ ≪ ♠❡❛♥ ✐♥t❡r♣❛rt✐❝❧❡ ❞✐st❛♥❝❡ ⇒ ✭❡ss❡♥t✐❛❧❧②✮ t✇♦✲❜♦❞② s❝❛tt❡r✐♥❣ ✭❝♦❧❞✮✿ ❧♦✇✲❡♥❡r❣② s❝❛tt❡r✐♥❣ ▲♦✇✲❡♥❡r❣② ⇔ ✐♥t❡r❛❝t✐♦♥ r❛♥❣❡ ≪ r❡❧❛t✐✈❡✲♠♦♠❡♥t✉♠ ❞❡ ❇r♦❣❧✐❡ ✇❛✈❡❧❡♥❣t❤ ⇒ ✭❛♣♣r♦①✐♠❛t❡❧②✮ ❝♦♥t❛❝t ✐♥t❡r❛❝t✐♦♥

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❉❡✜♥✐t✐♦♥ ✭s✐♠♣❧✐✜❡❞✮ ❋❡r♠✐✿ ❢❡r♠✐♦♥✐❝ ♣❛rt✐❝❧❡s ❈♦❧❞ ♠❛♥②✲❜♦❞② s②st❡♠ ♦❢ ❢❡r♠✐♦♥s ⇒ r❡❧❛t✐✈❡✲♠♦♠❡♥t✉♠ ❞❡ ❇r♦❣❧✐❡ ✇❛✈❡❧❡♥❣t❤ ≈ ❋❡r♠✐ ✇❛✈❡❧❡♥❣t❤ ≈ ≈ ♠❡❛♥ ✐♥t❡r♣❛rt✐❝❧❡ ❞✐st❛♥❝❡ ❚✇♦✲❜♦❞② ❝♦♥t❛❝t ✐♥t❡r❛❝t✐♦♥s ⇒ ✭❛t ❧❡❛st✮ t✇♦ s♣❡❝✐❡s ♦❢ ❢❡r♠✐♦♥s ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ✐♥t❡r❛❝t✐♦♥s ❊①❛♠♣❧❡s✿

❇❈❙ s✉♣r❛❝♦♥❞✉❝t♦rs✿ ❡❧❡❝tr♦♥s ✇✐t❤ s♣✐♥ ✏✉♣✑ ♦r ✏❞♦✇♥✑ ✉❧tr❛❝♦❧❞ ❣❛s❡s✿ ❢❡r♠✐♦♥✐❝ ❛t♦♠s ✭✫ ❡❧❡❝tr✐❝❛❧❧② ♥❡✉tr❛❧ ⇒ ♦❞❞ ♥✉♠❜❡r ♦❢ ♥❡✉tr♦♥s✮ ✐♥ t✇♦ ❤②♣❡r✜♥❡ st❛t❡s ✭❡✳❣✳ ❛❧❦❛❧✐ 6▲✐ ♦r 40❑✮

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❉❡✜♥✐t✐♦♥ ✭s✐♠♣❧✐✜❡❞✮ ✉♥✐t❛r②✿ ♠❛①✐♠❛❧ ✭♠♦❞✉❧✉s ♦❢✮ s❝❛tt❡r✐♥❣ ✭❛♠♣❧✐t✉❞❡✮ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ✉♥✐t❛r✐t② ❧♦✇✲❡♥❡r❣② t✇♦✲❜♦❞② s❝❛tt❡r✐♥❣ ⇒ s✲✇❛✈❡ ❞♦♠✐♥❛t❡s ❛♥❞ t❤❡ ✉♥✐t❛r✐t② ❜♦✉♥❞ ♦♥ t❤❡ s❝❛tt❡r✐♥❣ ❛♠♣❧✐t✉❞❡ ✐s s❛t✉r❛t❡❞ ✐♥ t❤❡ ✉♥✐t❛r✐t② r❡❣✐♠❡ ⇔ r❡❧❛t✐✈❡✲♠♦♠❡♥t✉♠ ❞❡ ❇r♦❣❧✐❡ ✇❛✈❡❧❡♥❣t❤ ≪ ⑤s❝❛tt❡r✐♥❣ ❧❡♥❣t❤⑤

❘❡♠✐♥❞❡r✿ ❚❤❡ ❧♦✇✲❡♥❡r❣② s❝❛tt❡r✐♥❣ ♠❛tr✐① ❢♦r t❤❡ s✲✇❛✈❡ ✐s S = 1 + 2 i T ≈ λ − ia λ + ia ✇❤❡r❡ λ ❂ r❡❧❛t✐✈❡✲♠♦♠❡♥t✉♠ ❞❡ ❇r♦❣❧✐❡ ✇❛✈❡❧❡♥❣t❤ ✭❃✵✮ ❛♥❞ a ❂ s❝❛tt❡r✐♥❣ ❧❡♥❣t❤✳ ❚❤❡r❡❢♦r❡✱ 0 |T| 1 ✐♥ ❣❡♥❡r❛❧✱ ✇❤✐❧❡ |T|

λ≫|a|

≈ 0 ❛♥❞ |T|

λ≪|a|

≈ 1✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❉❡✜♥✐t✐♦♥ ✭s✐♠♣❧✐✜❡❞✮ ⇒ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s✿ ❝♦❧❞ ♠❛♥②✲❜♦❞② s②st❡♠ ♦❢ ❢❡r♠✐♦♥s ✇✐t❤ ✐♥t❡r❛❝t✐♦♥ r❛♥❣❡ ≪ ✐♥t❡r♣❛rt✐❝❧❡ ❞✐st❛♥❝❡ ≪ ⑤s❝❛tt❡r✐♥❣ ❧❡♥❣t❤⑤

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❍♦✇ t♦ ❞❡s❝r✐❜❡ ❛♥❞ ❤♦✇ t♦ ♣r❡♣❛r❡ ❛ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❇❈❙ ♠♦❞❡❧

▼❛♥②✲❜♦❞② s②st❡♠ ✇✐t❤ t✇♦✲❜♦❞② ❝♦♥t❛❝t ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ t✇♦✲❝♦♠♣♦♥❡♥t ❢❡r♠✐♦♥s ⇐ ⇒ ❇❈❙ ♠♦❞❡❧ S[ ψ ; u] =

• dt
• dx

 

α=↑,↓

ψ∗

α

• i∂t + ∆

2m + µ

• ψα − u0 ψ∗

↓ψ∗ ↑ψ↑ψ↓

  ❆ttr❛❝t✐✈❡ ✭r❡♣✉❧s✐✈❡✮ ✐♥t❡r❛❝t✐♦♥ ❢♦r ♥❡❣❛t✐✈❡ ✭♣♦s✐t✐✈❡✮ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t✳

❚❡r♠✐♥♦❧♦❣②✿ ❲❡ ❝❤♦s❡ t♦ r❡❢❡r t♦ t❤❡ ❛❜♦✈❡ ✜❡❧❞ t❤❡♦r② ❛s ❇❈❙ ♠♦❞❡❧ ❜❡❝❛✉s❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❍❛♠✐❧t♦♥✐❛♥ ✐s s♦♠❡t✐♠❡s ❝❛❧❧❡❞ ✏❇❈❙ ❍❛♠✐❧t♦♥✐❛♥✑ ✐♥ t❤❡ ❝♦♥❞❡♥s❡❞ ♠❛tt❡r ❧✐tt❡r❛t✉r❡✳ ■♥ ♦r❞❡r t♦ ❛✈♦✐❞ ❝♦♥❢✉s✐♦♥ ❧❡t ✉s str❡ss t❤❛t✱ str✐❝t❧② s♣❡❛❦✐♥❣✱ ✐♥ t❤❡ ❇❈❙ t❤❡♦r② ♦❢ s✉♣r❛❝♦♥❞✉❝t✐✈✐t② ♦♥❡ ♠✉st ❢✉rt❤❡r ♠✐♥✐♠❛❧ ❝♦✉♣❧❡ t❤❡ ❢❡r♠✐♦♥s t♦ ❛♥ ❡①t❡r♥❛❧ U(1) ✈❡❝t♦r ❣❛✉❣❡ ✜❡❧❞✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❇❈❙ ♠♦❞❡❧

▲♦✇✲❡♥❡r❣② s❝❛tt❡r✐♥❣ t❤❡♦r② ❢♦r ❛♥ ❛ttr❛❝t✐✈❡ s❤♦rt✲r❛♥❣❡ ♣♦t❡♥t✐❛❧ st❛t❡s t❤❛t t❤❡ s✐❣♥ ♦❢ t❤❡ s❝❛tt❡r✐♥❣ ❧❡♥❣t❤ ❝❛r❛❝t❡r✐s❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ✭s❤❛❧❧♦✇✱ ✐✳❡✳ ♥❡❛r❧② ③❡r♦✲❡♥❡r❣②✮ t✇♦✲❜♦❞② ❜♦✉♥❞ st❛t❡ ✐♥ ✈❛❝✉✉♠✳ ❙❝❛tt ❧❡♥❣t❤ ❇♦✉♥❞ st❛t❡ ■♥t❡r❛❝t✐♦♥

• ❛s

> 0 ∃ ❇✐♥❞✐♥❣ ❉✐♠❡rs < 0 ∄ P❛✐r✐♥❣ ❈♦♦♣❡r ♣❛✐rs = 0 ◆♦ ◆♦ ■❞❡❛❧ ±∞ ❩❡r♦ ❡♥❡r❣② ❋❡s❤❜❛❝❤ r❡s♦♥❛♥❝❡ ❯♥✐t❛r② ❊①♣❡r✐♠❡♥t❛❧❧②✱ t❤❡ ✐♥t❡r❛❝t✐♦♥ str❡♥❣t❤ ✭❛♥❞ t❤✉s t❤❡ s❝❛tt❡r✐♥❣ ❧❡♥❣t❤✮ ❝❛♥ ❜❡ ❝♦♥tr♦❧❧❡❞ ✈❡r② ❛❝❝✉r❛t❡❧② ❜② t✉♥✐♥❣ ❛♥ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞✱ s♦ ❛ ✉♥✐t❛r② ❣❛s ❝❛♥ ❜❡ ♣r❡♣❛r❡❞ ✈✐❛ ✜♥❡ t✉♥✐♥❣✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r

❲❤② ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ♦❢ t❤❡♦r❡t✐❝❛❧ ✐♥t❡r❡st❄

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r

✭▲❡❣❣❡tt✱ ✶✾✽✵✮ ❚❤❡ ❣r♦✉♥❞ st❛t❡ ♦❢ t❤❡ ❇❈❙ ♠♦❞❡❧ ✐s ❛❧✇❛②s ❛ s✉♣❡r✢✉✐❞ ✭✐✳❡✳ s♣♦♥t❛♥❡♦✉s ❜r❡❛❦✐♥❣ ♦❢ r✐❣✐❞ U(1) s②♠♠❡tr②✮✳ ❘❡❣✐♠❡ ✐♥t❡r♣❛rt✐❝❧❡✴s❝❛tt❡r✐♥❣ ❧❡♥❣t❤ P❛✐r ❝♦♥❞❡♥s❛t❡ ❇❊❈ ≫ 1 ❚✐❣❤t❧②✲❜♦✉♥❞ ♣❛✐rs ✭❞✐♠❡rs✮ ❯♥✐t❛r✐t② ≈ 0 ❈r♦ss♦✈❡r ❇❈❙ ≪ −1 ▲♦♥❣✲r❛♥❣❡ ✭❈♦♦♣❡r✮ ♣❛✐rs

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r

❍♦♣❡❢✉❧❧②✱ t❤❡ ✉♥✐t❛r✐t② r❡❣✐♠❡ ♠✐❣❤t ♣r♦✈✐❞❡ ❛♥ ❛♥❛❧②t✐❝❛❧❧② tr❛❝t❛❜❧❡ ♠♦❞❡❧ ♦❢ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ✭✐♥t❡r♣❛rt✐❝❧❡ ❞✐st❛♥❝❡ ≪ ⑤s❝❛tt❡r✐♥❣ ❧❡♥❣t❤⑤✮✱ ❍✐❣❤✲t❡♠♣❡r❛t✉r❡ s✉♣❡r✢✉✐❞✐t② ✭❝r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡ ❝❧♦s❡ t♦ ❋❡r♠✐ t❡♠♣❡r❛t✉r❡✮✱ ❙❡❝♦♥❞✲♦r❞❡r q✉❛♥t✉♠ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ✭❛t ③❡r♦ t❡♠♣❡r❛t✉r❡✮✱ ❡t❝✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❋♦r s✐♠♣❧✐❝✐t②✱ ♦♥❡ ❛ss✉♠❡s ❢r♦♠ ♥♦✇ t♦ ❜❡ ❛t ③❡r♦ ❞❡♥s✐t② ❛♥❞ t❡♠♣❡r❛t✉r❡✳ ⇒ ❚❤❡ ♦♥❧② ❧❡♥❣t❤ ♣❛r❛♠❡t❡r ✐♥ t❤❡ ❇❈❙ ♠♦❞❡❧ ✐s t❤❡ s❝❛tt❡r✐♥❣ ❧❡♥❣t❤✳ ⇒ ❚❤❡r❡ ✐s ♥♦ ❧❡♥❣t❤ ♣❛r❛♠❡t❡r ✐❢ t❤❡ s❝❛tt❡r✐♥❣ ❧❡♥❣t❤ ✐s ③❡r♦ ♦r ✐♥✜♥✐t❡✳ ■♥ ❢❛❝t✱ t❤❡ ✐❞❡❛❧ ❛♥❞ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s ❛♣♣❡❛r t♦ ❜❡ s❝❛❧❡✲✐♥✈❛r✐❛♥t✿ t❤❡② ❛r❡ t❤❡ ✷ r❡♥♦r♠❛❧✐s❛t✐♦♥ ❣r♦✉♣ ✜①❡❞ ♣♦✐♥ts ♦❢ t❤❡ ❇❈❙ ♠♦❞❡❧ ✭◆✐❦♦❧✐❝ ✫ ❙❛❝❤❞❡✈✱ ✷✵✵✻✮✳ ⇒ ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡ ✉s❡❞ ❤❡r❡✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❄ ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❛s ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❈❋❚

❚❤❡r❡❢♦r❡✱ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ✐s ❛ t❛♥t❛❧✐s✐♥❣ ❡①❛♠♣❧❡ ♦❢ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❝♦♥❢♦r♠❛❧ ✜❡❧❞ t❤❡♦r②✿ ✏s✐♠♣❧❡✑ ✐♥t❡r❛❝t✐♦♥s ✭t✇♦✲❜♦❞② ❝♦♥t❛❝t ✐♥t❡r❛❝t✐♦♥s ♦❢ t✇♦✲❝♦♠♣♦♥❡♥t ❢❡r♠✐♦♥s✱ ✐✳❡✳ ❇❈❙ ♠♦❞❡❧✮✱ t❤♦✉❣❤ ❝❤❛❧❧❡♥❣✐♥❣ ✭str♦♥❣❧② ✐♥t❡r❛❝t✐♥❣✱ ✐✳❡✳ ♥♦ s♠❛❧❧ ♣❛r❛♠❡t❡r✮✳ ⇒ ❲❤❛t ♠✐❣❤t ❜❡ ❛♥ ❡❞✉❝❛t❡❞ ❣✉❡ss ❢♦r t❤❡ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s❄ ✭❙♦♥✱ ✷✵✵✽✮

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❇❈❙✲❧✐❦❡ ♠♦❞❡❧

❆ s❡♠✐✲❝❧❛ss✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦♥ t❤❡ ❜✉❧❦ s✐❞❡ s❤♦✉❧❞ ❝♦rr❡s♣♦♥❞ t♦ ❛ ❧❛r❣❡✲◆ ❧✐♠✐t ♦♥ t❤❡ ❜♦✉♥❞❛r② s✐❞❡✳ ❋♦rt✉♥❛t❡❧②✱ ❛ ❧❛r❣❡✲◆ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❇❈❙ ♠♦❞❡❧ ✐s ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❝♦♥❞❡♥s❡❞ ♠❛tt❡r ❧✐tt❡r❛t✉r❡ ✭◆✐❦♦❧✐❝ ✫ ❙❛❝❤❞❡✈✱ ❱❡✐❧❧❡tt❡ ✫ ❙❤❡❡❤② ✫ ❘❛❞③✐❤♦✈s❦②❀ ✷✵✵✻✮✿ ❇❈❙✲❧✐❦❡ ♠♦❞❡❧ ✇✐t❤ N ✏✢❛✈♦rs✑ S[ ψ ; u, N] =

• dt
• dx

 

α=↑,↓

• ψ∗

α ·

• i∂t + ∆

2m + µ

• ψα − u0

N | ψ↑ · ψ↓|2   ✇❤❡r❡ ψα ❞❡♥♦t❡s ❛♥ ✈❡❝t♦r✲❧✐❦❡ ♠✉❧t✐♣❧❡t ♦❢ ✷◆ ♠❛ss✐✈❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❢❡r♠✐♦♥s✳ ❚❤❡ ❧❛r❣❡✲◆ ❧✐♠✐t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♠❡❛♥ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❇❈❙✲❧✐❦❡ ♠♦❞❡❧

❆♣♣❧②✐♥❣ t❤❡ ❣❡♥❡r❛❧ ♦❜s❡r✈❛t✐♦♥ ♦❢ ✭●✉❜s❡r ✫ ❑❧❡❜❛♥♦✈✱ ✷✵✵✸✮ ♦♥ ❞♦✉❜❧❡✲tr❛❝❡ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❈❋❚s t♦ t❤❡ ❇❈❙✲❧✐❦❡ ♠♦❞❡❧✱ ✭❇❡▼❡▼♦✱ ✷✵✶✶✮ ■♥ t❤❡ ❧❛r❣❡✲N ❧✐♠✐t✱ t❤❡ ❢r❡❡ ❡♥❡r❣✐❡s ♦❢ t❤❡ ✐❞❡❛❧ ❛♥❞ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s ❛r❡ r❡❧❛t❡❞ ❜② ❛ ▲❡❣❡♥❞r❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s♦✉r❝❡ ❢♦r t❤❡ ❈♦♦♣❡r ♣❛✐r✱ ❛♥❞ ♠♦❞✉❧♦ ♣r♦♣❡r r❡s❝❛❧✐♥❣s✮✳ ❚❤❡r❡❢♦r❡✱ ▼❛♥② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ✐♥ t❤❡ ♠❡❛♥ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ✐❞❡❛❧ ❋❡r♠✐ ❣❛s✳ ❚❤❡ t✇♦ ✜①❡❞ ♣♦✐♥ts ❤❛✈❡ t❤❡ s❛♠❡ ✐♥✜♥✐t❡ s❡t ♦❢ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts ❛♥❞ s②♠♠❡tr✐❡s✱ ♠♦st ♦❢ ✇❤✐❝❤ ❛r❡ ❜r♦❦❡♥ ❜② 1/N ❝♦rr❡❝t✐♦♥s ✐♥ t❤❡ ✐♥t❡r❛❝t✐♥❣ t❤❡♦r②✳ ❇♦t❤ ✜①❡❞ ♣♦✐♥ts s❤♦✉❧❞ ❝♦rr❡s♣♦♥❞ t♦ ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ s❛♠❡ ❜✉❧❦ t❤❡♦r② ✭❤♦❧♦❣r❛♣❤✐❝ ❞❡❣❡♥❡r❛❝②✮✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s

▼❛♥② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❇❈❙✲❧✐❦❡ ♠♦❞❡❧ ❛r❡ r❡♠✐♥✐s❝❡♥t ❢r♦♠ t❤❡ ❖✭◆✮ ♠♦❞❡❧✱ s♦ ❧❡t ✉s ❝♦♠♣❛r❡ t❤❡s❡ ♠♦❞❡❧s ✐♥ ❞❡t❛✐❧s✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s

▼♦❞❡❧s ❖✭◆✮✲❧✐❦❡ ❇❈❙✲❧✐❦❡ ❙♣❛❝❡✲t✐♠❡ ❘❡❧❛t✐✈✐st✐❝ ◆♦♥✲r❡❧❛t✐✈✐st✐❝ ❑✐♥❡t✐❝ ♦♣❡r❛t♦r − M 2 i∂t +

∆ 2m + µ

❙❝❛❧❡✲❢r❡❡ ▼❛ss❧❡ss M = 0 ❩❡r♦ ❝❤❡♠ ♣♦t µ = 0 ❋✉♥❞❛♠❡♥t❛❧ ✜❡❧❞s ❇♦s♦♥s φ ❋❡r♠✐♦♥s ψ↑ ✱ ψ↓ ◆ ❝♦♠♣♦♥❡♥ts ❘❡❛❧ ❬♦r ❝♦♠♣❧❡①❪ ❈♦♠♣❧❡① ✭✉♣✴❞♦✇♥✮ ◗✉❛rt✐❝ ✐♥t❡r❛❝t✐♦♥ ( φ∗ · φ)2 | ψ↑ · ψ↓|2 ■♥t❡r♥❛❧ s②♠♠❡tr② O(N) ❬⊂ U(N)❪ U(2) ❬⊂ U(1) × Sp(2N)❪ ❈♦♠♣♦s✐t❡ ✜❡❧❞ P❛rt✐❝❧❡ ❞❡♥s✐t② φ∗ · φ ❈♦♣♣❡r ♣❛✐r ψ↑ · ψ↓ ❉✐♠❡♥s✐♦♥ D = d + 2 ✭s♣❛❝❡t✐♠❡✮ d = D − 2 ✭s♣❛❝❡✮ ◆♦♥✲tr✐✈ ✜①❡❞ ♣t ❲✐❧s♦♥✲❋✐s❤❡r ❯♥✐t❛r② ❋❡r♠✐ ❣❛s ❑✐♥❡♠❛t✐❝❛❧ s②♠ ❈♦♥❢♦r♠❛❧ o(d + 2, 2) ❙❝❤rö❞✐♥❣❡r sch(d) ❍✐❣❤❡r✲s♣✐♥ s②♠ ❱❛s✐❧✐❡✈ ❲❡②❧

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s

❊♥❣✐♥❡❡r✐♥❣ s❝❛❧❡ ❞✐♠❡♥s✐♦♥s ♦❢ ❡❧❡♠❡♥t❛r② ✜❡❧❞s ✭ φ ✈s ψα✮✿ ∆❡❧❡♠❡♥t❛r② = (D − 2)/2 = d/2 ❇❛r❡ ❛♥❞ ❞r❡ss❡❞ ✭❧❛r❣❡✲◆ ❛♣♣r♦① ✈s ✈❛❝✉✉♠ ❡①❛❝t✮ s❝❛❧❡ ❞✐♠❡♥s✐♦♥s ♦❢ ❝♦♠♣♦s✐t❡ t✇♦✲❜♦❞② ✜❡❧❞ ✭ φ∗ · φ ✈s ψ↑ · ψ↓✮ ❛t t❤❡ ✜①❡❞ ♣♦✐♥ts✿ ∆❢r❡❡ = 2 ∆❡❧❡♠❡♥t❛r② = D − 2 = d , ∆✐♥t = 2 ✭◆♦♥✮r❡❧❛t✐✈✐st✐❝ s❝❛❧❡ ❛♥❞ s♣❛❝❡✭t✐♠❡✮ ❞✐♠❡♥s✐♦♥s ✭∆+ ∆−✮ r❡❧❛t✐♦♥✿ ∆+ + ∆− = ∆❢r❡❡ + ∆✐♥t = D = d + 2 ∆ ❝♦♠♣♦s✐t❡ ❋✐①❡❞ ♣♦✐♥t ∆❢r❡❡

• ❛✉ss✐❛♥

∆✐♥t ◆♦♥✲tr✐✈✐❛❧ ∆+ ■❘ ∆− ❯❱

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s

D ∆− ∆+ Pr♦♣❡rt② D = 2 ∆❢r❡❡ ∆✐♥t s❛t✉r❛t✐♦♥ ♦❢ ✉♥✐t❛r✐t② ❜♦✉♥❞ ✭❧✐♥❡ ♦❢ ✜①❡❞ ♣ts✮ 2 < D < 4 ∆❢r❡❡ ∆✐♥t ♣❛✐r ♦❢ ❛❞♠✐ss✐❜❧❡ ✜①❡❞ ♣ts ✭❛s②♠♣t♦t✐❝ ❢r❡❡❞♦♠✮ D = 4 ∆❢r❡❡ = ∆✐♥t ∆✐♥t = ∆❢r❡❡ ❢✉s✐♦♥ ♦❢ ✜①❡❞ ♣ts ✭tr✐✈✐❛❧✐t②✮ 4 < D < 6 ✲ ∆❢r❡❡ ♦♥❧② s✐♥❣❧❡ ❛❞♠✐ss✐❜❧❡ ✜①❡❞ ♣t ✭✉♥st❛❜❧❡ ✐♥t✮ D = 6 ✲ ∆❢r❡❡ s❛t✉r❛t✐♦♥ ♦❢ ✉♥✐t❛r✐t② ❜♦✉♥❞ D > 6 ✲ ∆❢r❡❡ ♦♥❧② s✐♥❣❧❡ ❛❞♠✐ss✐❜❧❡ ✜①❡❞ ♣t ✭♥♦♥✲✉♥✐t❛r② ✐♥t✮ D = 3✿ ❢r❡❡✴❝r✐t✐❝❛❧ ❖✭◆✮ ♠♦❞❡❧s

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s

d ∆− ∆+ Pr♦♣❡rt② d = 0 ∆❢r❡❡ ∆✐♥t s❛t✉r❛t✐♦♥ ♦❢ ✉♥✐t❛r✐t② ❜♦✉♥❞ 0 < d < 2 ∆❢r❡❡ ∆✐♥t ♣❛✐r ♦❢ ❛❞♠✐ss✐❜❧❡ ✜①❡❞ ♣ts ✭❛s②♠♣t♦t✐❝ ❢r❡❡❞♦♠✮ d = 2 ∆❢r❡❡ = ∆✐♥t ∆✐♥t = ∆❢r❡❡ ❢✉s✐♦♥ ♦❢ ✜①❡❞ ♣ts ✭tr✐✈✐❛❧✐t②✮ 2 < d < 4 ∆✐♥t ∆❢r❡❡ ♣❛✐r ♦❢ ❛❞♠✐ss✐❜❧❡ ✜①❡❞ ♣ts ✭❛s②♠♣t♦t✐❝ s❛❢❡t②✮ d = 4 ∆✐♥t ∆❢r❡❡ s❛t✉r❛t✐♦♥ ♦❢ ✉♥✐t❛r✐t② ❜♦✉♥❞ d > 4 ✲ ∆❢r❡❡ ♦♥❧② s✐♥❣❧❡ ❛❞♠✐ss✐❜❧❡ ✜①❡❞ ♣t ✭♥♦♥✲✉♥✐t❛r② ✐♥t✮ d = 1 ♦r 3✿ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s ✭r❡♣✉❧s✐✈❡ ♦r ❛ttr❛❝t✐✈❡ ✐♥t❡r❛❝t✐♦♥s✮

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s

❘❡♥♦r♠❛❧✐s❛t✐♦♥ ❣r♦✉♣ ✢♦✇ ♦❢ t❤❡ ❇❈❙ ♠♦❞❡❧

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❙♦✉r❝❡ ♦❢ ✐♥s♣✐r❛t✐♦♥✿ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤②

• r♦✇✐♥❣ ❜♦❞② ♦❢ ❡✈✐❞❡♥❝❡ ✭P❡t❦♦✉✲❙❡③❣✐♥✲❙✉♥❞❡❧❧✱ ✷✵✵✸❀ ●✐♦♠❜✐✲❨✐♥✱

✷✵✶✵❀ ▼❛❧❞❛❝❡♥❛✲❩❤✐❜♦❡❞♦✈✱ ✷✵✶✶✮ t❤❛t t❤❡ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ❢r❡❡✴❝r✐t✐❝❛❧ ❖✭◆✮ ♠♦❞❡❧s s❤♦✉❧❞ ❜❡ ❱❛s✐❧✐❡✈✬s ♠✐♥✐♠❛❧ ❤✐❣❤❡r✲s♣✐♥ ❣r❛✈✐t② ♦♥ AdS4 ✭❙❡③❣✐♥✲❙✉♥❞❡❧❧✴❑❧❡❜❛♥♦✈✲P♦❧②❛❦♦✈ ❝♦♥❥❡❝t✉r❡✱ ✷✵✵✷✴✷✵✵✸✮✳ ▼♦r❡♦✈❡r✱ ❜♦s♦♥✐❝ ❤✐❣❤❡r✲s♣✐♥ ❣r❛✈✐t② ❤❛s ❜❡❡♥ ❝♦♥str✉❝t❡❞ ❢♦r ❛♥② ❞✐♠❡♥s✐♦♥ ❛♥❞ ❢♦r ❛♥② ✐♥t❡r♥❛❧ ❝❧❛ss✐❝❛❧ ❝♦♠♣❛❝t ❣r♦✉♣ ✭❱❛s✐❧✐❡✈✱ ✷✵✵✸✮✳ ❚❤✉s✱ t❤❡ ❜✉❧❦ ❞✉❛❧ ♦❢ ❖✭◆✮✲❧✐❦❡ ♠♦❞❡❧s ✐♥ D ❞✐♠❡♥s✐♦♥s s❤♦✉❧❞ ❜❡ ❱❛s✐❧✐❡✈✬s ❤✐❣❤❡r✲s♣✐♥ ❣r❛✈✐t② ♦♥ AdSD+1✳ ⇒ ❲❤❛t ♠✐❣❤t ❜❡ ❛♥ ❡❞✉❝❛t❡❞ ❣✉❡ss ❢♦r t❤❡ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s❄ ❆ ❤✐❣❤❡r✲s♣✐♥ t❤❡♦r②❄ ✭❙♦♥✱ ✷✵✵✽✮ ■♥ ♦r❞❡r t♦ ♠❛❦❡ t❤✐s ✐❞❡❛ ♠♦r❡ ♣r❡❝✐s❡✱ ❧❡t ✉s r❡✈✐❡✇ t❤❡ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤②✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤②

∆ ❝♦♠♣♦s✐t❡ ❆❞❙ ❜❞② ❝♦♥❞ ✫ s②♠♠❡tr② ❈❋❚ ✜①❡❞ ♣t ∆❢r❡❡ ❯♥❜r♦❦❡♥ ❤✐❣❤❡r✲s♣✐♥

• ❛✉ss✐❛♥

∆✐♥t ❇r♦❦❡♥ ❤✐❣❤❡r✲s♣✐♥ ◆♦♥✲tr✐✈✐❛❧ ∆+ ❙t❛♥❞❛r❞ ✭✏❉✐r✐❝❤❧❡t✑✮ ■❘ ∆− ❊①♦t✐❝ ✭✏◆❡✉♠❛♥♥✑✮ ❯❱ ❙❝❛❧✐♥❣ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❝♦❧❧❡❝t✐✈❡ ✜❡❧❞

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤②

❉✐❝t✐♦♥❛r② ❆❞❙D+1 ❈❋❚D ❇✉❧❦✴❇♦✉♥❞❛r② ❱❛s✐❧✐❡✈ t❤❡♦r② ❯✭◆✮ ♠♦❞❡❧ ❙②♠♠❡tr✐❝ ♣❤❛s❡ ❯♥❜r♦❦❡♥ ❋r❡❡ ❇r♦❦❡♥ ♣❤❛s❡ ❇r♦❦❡♥ ❈r✐t✐❝❛❧ ❋✐❡❧❞✴❖♣❡r❛t♦r ❙②♠♠❡tr✐❝ t❡♥s♦r ❙②♠♠❡tr✐❝ t❡♥s♦r ❢✉♥❞❛♠❡♥t❛❧ ❝♦♠♣♦s✐t❡ ❣❛✉❣❡ ✜❡❧❞✿ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥t✿ ❛❞❥♦✐♥t✲✈❛❧✉❡❞ ❢✉♥❞ ⊗ ❛♥t✐❢✉♥❞ ❊①❛♠♣❧❡s ❙✐♥❣❧❡t s❝❛❧❛r P❛rt✐❝❧❡ ❞❡♥s✐t② U(1) ✈❡❝t♦r ❈❤❛r❣❡ ❝✉rr❡♥t ▼❡tr✐❝ t❡♥s♦r ❊♥❡r❣②✲♠♦♠❡♥t✉♠✲str❡ss ❍✐❣❤❡r✲s♣✐♥ ✜❡❧❞s ❍✐❣❤❡r✲s♣✐♥ ❝✉rr❡♥ts ❍♦❧♦❣r❛♣❤✐❝ ❞✐❝t✐♦♥❛r②

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❙❡t ♦❢ s②♠♠❡tr✐❝ ❝✉rr❡♥ts ♦❢ ❛❧❧ r❛♥❦s ✭❇❡r❡♥❞s✱ ❇✉r❣❡rs✱ ✈❛♥ ❉❛♠❀ ✶✾✽✻✮ JAB

µ1...µs(x) = is φA∗(x) ↔

∂ µ1 · · ·

↔

∂ µs φB(x) (A, B = 1, . . . , N ; µ = 0, . . . , D − 1) ❋❡❛t✉r❡s✿ u(N)✲✈❛❧✉❡❞✱ J∗

AB = JBA

❇✐❧✐♥❡❛r ✐♥ t❤❡ s❝❛❧❛r ✜❡❧❞s φ ❛♥❞ ✐ts ❝♦♥❥✉❣❛t❡ ◆✉♠❜❡r ♦❢ ❞❡r✐✈❛t✐✈❡s ❂ ❘❛♥❦ ❈♦♥s❡r✈❡❞ ✭♦♥✲s❤❡❧❧✮ ❢♦r s 1 ∂µJAB

µ1...µs(x) ≈ 0

✇❤❡r❡ t❤❡ ✇❡❛❦ ❡q✉❛❧✐t② ≈ st❛♥❞s ❢♦r ✏♦♥ t❤❡ ❢r❡❡ ♠❛ss s❤❡❧❧✑✱ ✐✳❡✳ ♠♦❞✉❧♦ φ(x) ≈ 0✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

❚❤❡ ❇❡r❡♥❞s✲❇✉r❣❡rs✲✈❛♥❉❛♠ ❝✉rr❡♥ts ❝❛♥ ❜❡ ♣❛❝❦❡❞ ✐♥ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ JAB(x, q) :=

• s0

1 s! qµ1 . . . qµs JAB

µ1...µs(x)

= φ∗(x − iq) · φ(x + iq) = |φ(x + iq)|2 ✇❤✐❝❤ ✐s ❛ ❜✐✲❧♦❝❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❝❛❧❛r ✜❡❧❞✱ ❝✳❢✳ t❤❡ ❝♦❧❧❡❝t✐✈❡ ✜❡❧❞ ♦❢ ✭❉❛s✱ ❏❡✈✐❝❦✐❀ ✷✵✵✸✮ ❛♥❞ ✭❞❡ ▼❡❧❧♦ ❑♦❝❤✱ ❏❡✈✐❝❦✐✱ ❏✐♥✱ ❘♦❞r✐❣✉❡s❀ ✷✵✶✵✮✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

◆♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❝✉rr❡♥ts

❋♦r ❯✭◆✮ ♠♦❞❡❧✱ ♦♥❡ ✉s✉❛❧❧② ❢♦❝✉s❡s ♦♥ ❝♦♠♣♦s✐t❡ ✜❡❧❞s ✇❤✐❝❤ ❛r❡ s✐♥❣❧❡ts ♦❢ t❤❡ ✐♥t❡r♥❛❧ s②♠♠❡tr② ❣r♦✉♣✱ ❛s t❤❡ ♣❛rt✐❝❧❡ ❞❡♥s✐t② J = φ∗ · φ = J∗ . ❚❤❡ ❣❡♥❡r✐❝ ❯✭◆✮✲s✐♥❣❧❡t ❜✐❧♦❝❛❧ ✭φ∗ ❛♥❞ φ ❛t ❞✐st✐♥❝t ♣♦✐♥ts✮ ❝♦♠♣♦s✐t❡ ✜❡❧❞ ❣❡♥❡r❛t❡s ❛❧❧ t❤❡ tr❛❝❡❧❡ss ❝♦♥s❡r✈❡❞ s②♠♠❡tr✐❝ ❝✉rr❡♥ts✱ ✇❤✐❝❤ ❛r❡ t❤❡ ◆♦❡t❤❡r ❝✉rr❡♥ts ❢♦r t❤❡ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ♦❢ t❤❡ ❢r❡❡ ♠❛ss❧❡ss s❝❛❧❛r ✜❡❧❞ ✭✏s✐♥❣❧❡t♦♥✑✮✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

◆♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❝✉rr❡♥ts

❋♦r ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s✱ ♦♥❡ ✇♦✉❧❞ ✐♥st❡❛❞ ❢♦❝✉s ♦♥ ❝♦♠♣♦s✐t❡ ✜❡❧❞s ✇❤✐❝❤ ❛r❡ ✢❛✈♦r✲s✐♥❣❧❡ts ❜✉t ❢♦r♠ t❤❡ ❛❞❥♦✐♥t ♠✉❧t✐♣❧❡t ♦❢ t❤❡ ✐♥t❡r♥❛❧ s②♠♠❡tr② s✉❜❣r♦✉♣ U(2)✿ J =

• −

ψ∗

↑ ·

ψ↑

• ψ↑ ·

ψ↓

• ψ∗

↓ ·

ψ∗

↑

• ψ∗

↓ ·

ψ↓

• = J†

❜❡❝❛✉s❡ ✐t ✐♥❝❧✉❞❡s t❤❡ ❈♦♦♣❡r ♣❛✐r ✜❡❧❞ ψ↑ · ψ↓✳ ✭❇❡▼❡▼♦✱ ✷✵✶✶✮ ❚❤❡ ❛❜♦✈❡ ❖✭◆✮✲s✐♥❣❧❡t ❜✐❧♦❝❛❧ ✭ψ✬s ❛t ❞✐st✐♥❝t ♣♦✐♥ts✮ ❝♦♠♣♦s✐t❡ ✜❡❧❞ ❣❡♥❡r❛t❡s ❛❧❧ ❯✭✶✮✲♥❡✉tr❛❧ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❝✉rr❡♥ts ♦❢ ❛❧❧ ✐♥t❡❣❡r s♣✐♥s ❣❡♥❡r❛❧✐s✐♥❣ t❤❡ ✉♣✴❞♦✇♥ ♣❛rt✐❝❧❡ ♥✉♠❜❡rs ψ∗

α ·

ψα t♦❣❡t❤❡r ✇✐t❤ ❯✭✶✮✲❝❤❛r❣❡❞ t❡♥s♦rs ❣❡♥❡r❛❧✐s✐♥❣ t❤❡ ❈♦♦♣❡r ♣❛✐r ψ↑ · ψ↓✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❖✭◆✮✲❧✐❦❡ ✈s ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s ❍✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❍✐❣❤❡r✲s♣✐♥ ❝♦♥s❡r✈❡❞ ❝✉rr❡♥ts

◆♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❝✉rr❡♥ts

❋♦r ❇❈❙✲❧✐❦❡ ♠♦❞❡❧s✱ ♦♥❡ ✇♦✉❧❞ ✐♥st❡❛❞ ❢♦❝✉s ♦♥ ❝♦♠♣♦s✐t❡ ✜❡❧❞s ✇❤✐❝❤ ❛r❡ ✢❛✈♦r✲s✐♥❣❧❡ts ❜✉t ❢♦r♠ t❤❡ ❛❞❥♦✐♥t ♠✉❧t✐♣❧❡t ♦❢ t❤❡ ✐♥t❡r♥❛❧ s②♠♠❡tr② s✉❜❣r♦✉♣ U(2)✿ J =

• −

ψ∗

↑ ·

ψ↑

• ψ↑ ·

ψ↓

• ψ∗

↓ ·

ψ∗

↑

• ψ∗

↓ ·

ψ↓

• = J†

❜❡❝❛✉s❡ ✐t ✐♥❝❧✉❞❡s t❤❡ ❈♦♦♣❡r ♣❛✐r ✜❡❧❞ ψ↑ · ψ↓✳ ⇒ ❆ ❜✉❧❦✴❜♦✉♥❞❛r② ❞✐❝t✐♦♥❛r② ✇♦✉❧❞ ✐❞❡♥t✐❢② t❤❡s❡ u(2)✲✈❛❧✉❡❞ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ s②♠♠❡tr✐❝ ✏❝✉rr❡♥ts✑ ♦❢ ❛❧❧ ✐♥t❡❣❡r s♣✐♥s ❛s t❤❡ ❜♦✉♥❞❛r② ❞❛t❛ ♦❢ ❛ t♦✇❡r ♦❢ u(2)✲✈❛❧✉❡❞ ❤✐❣❤❡r✲s♣✐♥ ❜✉❧❦ ❣❛✉❣❡ ✜❡❧❞s✳ ❖❢ ✇❤✐❝❤ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❛r❡ t❤❡② ◆♦❡t❤❡r ❝✉rr❡♥ts❄

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❲❤❛t ❛r❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥s❄

❱❛s✐❧✐❡✈ ❜♦s♦♥✐❝ ❤✐❣❤❡r✲s♣✐♥ ❛❧❣❡❜r❛s ❛r❡ ❦♥♦✇♥ t♦ ❜❡ ♠❛①✐♠❛❧ s②♠♠❡tr② ❛❧❣❡❜r❛s ♦❢ ❢r❡❡ r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥s✳ ⇒ ❲❤❛t ❛r❡ ❢r❡❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥s❄ ❲❤❛t ❛r❡ t❤❡✐r ♠❛①✐♠❛❧ s②♠♠❡tr② ❛❧❣❡❜r❛s❄

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❲❤❛t ❛r❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥s❄

• r♦✉♣✲t❤❡♦r❡t✐❝❛❧ ❞❡✜♥✐t✐♦♥s

❋r❡❡ r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥ ❯■❘ ♦❢ t❤❡ P♦✐♥❝❛ré ❛❧❣❡❜r❛ t❤❛t ❝❛♥ ❜❡ ❧✐❢t❡❞ t♦ ❛ ❯■❘ ♦❢ t❤❡ ❝♦♥❢♦r♠❛❧ ❛❧❣❡❜r❛✳ ⇔ ❍❡❧✐❝✐t② r❡♣r❡s❡♥t❛t✐♦♥ ❧❛❜❡❧❡❞ ❜② ③❡r♦ ♠❛ss ❛♥❞ ❜② s♣✐♥ ✭❆♥❣❡❧♦♣♦✉❧♦s✱ ❋❧❛t♦✱ ❋r♦♥s❞❛❧✱ ❙t❡r♥❤❡✐♠❡r✱ ✶✾✽✵✮✳ ❋r❡❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥ ❯■❘ ♦❢ t❤❡ ❇❛r❣♠❛♥♥ ✭❂ ❝❡♥tr❛❧❧② ❡①t❡♥❞❡❞ ●❛❧✐❧❡✐✮ ❛❧❣❡❜r❛ t❤❛t ❝❛♥ ❜❡ ❧✐❢t❡❞ t♦ ❛ ❯■❘ ♦❢ t❤❡ ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛✳ ⇔ ▼❛ss✐✈❡ r❡♣r❡s❡♥t❛t✐♦♥s ❧❛❜❡❧❡❞ ❜② ③❡r♦ ✐♥t❡r♥❛❧ ❡♥❡r❣② ❛♥❞ ❜② s♣✐♥ ✭P❡rr♦✉❞✱ ✶✾✼✼✮✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢r❡❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ s✐♥❣❧❡t♦♥s ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❢r❡❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ✇✐t❤ ③❡r♦ ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧

• i∂t + ∆

2m

• ψ(t, x) = 0

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛

❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛✿ sch(d) = hd

• (d) ⊕ sl(2, R)
• ❙t❛♥❞❛r❞ r❡♣r❡s❡♥t❛t✐♦♥ ❛s ♦r❞❡r✲♦♥❡ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥

✇❛✈❡ ❢✉♥❝t✐♦♥s ψ(t, x)

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ r❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝

❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛✿ sch(d) = hd

• (d) ⊕ sl(2, R)
• hd :

ˆ Pi = −i∂i, ˆ Ki = mxi + it∂i, ˆ m = m,

• (d) :

ˆ Mij = −i(xi∂j − xj∂i), sl(2, R) : ˆ Pt = i∂t, ˆ D = i

• 2 t ∂t + xi∂i + d

2

• ,

ˆ C = i

• t2∂t + t
• xi∂i + d

2 + m 2 x2.

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ ♠❛①✐♠❛❧ ❛❧❣❡❜r❛

❚❤❡♦r❡♠✿ ✭❊❛st✇♦♦❞✱ ✷✵✵✷✮ ❚❤❡ ♠❛①✐♠❛❧ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t❡s✐♠❛❧ s②♠♠❡tr② ❣❡♥❡r❛t♦rs ❢♦r ❛ ❢r❡❡ ♠❛ss❧❡ss s❝❛❧❛r ✜❡❧❞✱ ✐✳❡✳ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ˆ A s✉❝❤ t❤❛t ˆ A = ˆ B ❛♥❞ ♠♦❞✉❧♦ tr✐✈✐❛❧ ❣❡♥❡r❛t♦rs ˆ A = ˆ C✱ ✐s ❣❡♥❡r❛t❡❞ ❛❧❣❡❜r❛✐❝❛❧❧② ❜② t❤❡ ❝♦♥❢♦r♠❛❧ ❑✐❧❧✐♥❣ ✈❡❝t♦rs✳ ❚❤❡ ♠❛①✐♠❛❧ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ s②♠♠❡tr✐❡s ❢♦r ❝♦♥❢♦r♠❛❧ s❝❛❧❛r ✜❡❧❞ ✐♥ ❛ ✢❛t D✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡t✐♠❡ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❣❛✉❣❡ ❛❧❣❡❜r❛ ♦❢ ❱❛s✐❧✐❡✈ ❤✐❣❤❡r✲s♣✐♥ ❣r❛✈✐t② ❛r♦✉♥❞ AdSD+1 ✭❱❛s✐❧✐❡✈✱ ✷✵✵✸✮✳ ⇒ ❲❤❛t ✐s ✐ts ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❛♥❛❧♦❣✉❡❄

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ ♠❛①✐♠❛❧ ❛❧❣❡❜r❛

❚❤❡♦r❡♠✿ ❚❤❡ ♠❛①✐♠❛❧ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t❡s✐♠❛❧ s②♠♠❡tr② ❣❡♥❡r❛t♦rs ❢♦r ❛ ❢r❡❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ♠❛ss✐✈❡ s❝❛❧❛r ✜❡❧❞✱ ✐✳❡✳ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ˆ A s✉❝❤ t❤❛t

• i∂t + ∆

2m

• ˆ

A = ˆ B

• i∂t + ∆

2m

• ❛♥❞ ♠♦❞✉❧♦ tr✐✈✐❛❧ ❣❡♥❡r❛t♦rs ˆ

A = ˆ C(i∂t +

∆ 2m)✱ ✐s ❣❡♥❡r❛t❡❞

❛❧❣❡❜r❛✐❝❛❧❧② ❜② t❤❡ s♣❛❝❡ tr❛♥s❧❛t✐♦♥s ❛♥❞ ❜② t❤❡ ●❛❧✐❧❡❛♥ ❜♦♦sts✳ ⇒ ❚❤✐s ♠❛①✐♠❛❧ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ s②♠♠❡tr✐❡s ❢♦r ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ♣❛rt✐❝❧❡ ♦♥ ❛ ✢❛t d✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❲❡②❧ ❛❧❣❡❜r❛ ♦❢ q✉❛♥t✉♠ ♦❜s❡r✈❛❜❧❡s✳

❘❡♠❛r❦✿ ❚❤✐s ✐s♦♠♦r♣❤✐s♠ ❛❧s♦ ❢♦❧❧♦✇s ❛s ❛ ❝♦r♦❧❧❛r② ❢r♦♠ t❤❡ ❣❡♥❡r❛❧ r❡s✉❧ts ♦♥ ❣❧♦❜❛❧ s②♠♠❡tr✐❡s ♦❢ Sp (2d, R)✲❝♦✈❛r✐❛♥t ✉♥❢♦❧❞❡❞ ❡q✉❛t✐♦♥s ✭❱❛s✐❧✐❡✈✱ ✷✵✵✶✮ ✉♣♦♥ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ ❝♦♦r❞✐♥❛t❡s ✇✐t❤ t❤❡ t✇✐st♦r ✈❛r✐❛❜❧❡s ❛♥❞ ♦❢ t❤❡ t✐♠❡ ❝♦♦r❞✐♥❛t❡ ✇✐t❤ t❤❡ tr❛❝❡ ♦❢ t❤❡ sp(2d, R)✲♠❛tr✐① ❝♦♦r❞✐♥❛t❡s✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ r❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝

❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛✿ sch(d) = hd

• (d) ⊕ sl(2, R)
• ❖❜s❡r✈❛t✐♦♥✿ ✭▼✳ ❱❛❧❡♥③✉❡❧❛✱ ✷✵✵✾✮ ❆❧t❡r♥❛t✐✈❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛s

❞❡❣r❡❡✲t✇♦ ♣♦❧②♥♦♠✐❛❧s ✐♥ t❤❡ ♠♦♠❡♥t❛ ❛♥❞ ●❛❧✐❧❡❛♥ ❜♦♦st ❣❡♥❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ✇❛✈❡ ❢✉♥❝t✐♦♥s s♦❧✉t✐♦♥s ♦❢ ❢r❡❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ (i∂t +

∆ 2m)ψ(t, x) = 0✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ r❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝

❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛✿ sch(d) = hd

• (d) ⊕ sl(2, R)
• hd :

ˆ Pi = ˆ Pi(−t), ˆ Ki = m ˆ Xi(−t), ˆ m = m,

• (d) :

ˆ Mij = ˆ Xi(−t) ˆ P j(−t) − ˆ Xj(−t) ˆ P i(−t), sl(2, R) : ˆ Pt = ˆ P 2(−t) 2m , ˆ D = − ˆ Xi(−t) ˆ Pi(−t) + d 2, ˆ C = m 2 ˆ X2(−t).

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ r❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝

❲❡②❧ ❛❧❣❡❜r❛✿ A(d) = U(hd) ❍✐❣❤❡r✲❞❡r✐✈❛t✐✈❡ ❣❡♥❡r❛t♦rs✿ ❚❤❡ ❲❡②❧ ❛❧❣❡❜r❛ ♦❢ ✐♥✜♥✐t❡s✐♠❛❧ s②♠♠❡tr② ❣❡♥❡r❛t♦rs ❢♦r t❤❡ ❢r❡❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ♣❛rt✐❝❧❡ ✐s ❣❡♥❡r❛t❡❞ ❛❧❣❡❜r❛✐❝❛❧❧② ❜② t❤❡ s♣❛❝❡ tr❛♥s❧❛t✐♦♥ ❛♥❞ t❤❡ ●❛❧✐❧❡❛♥ ❜♦♦st ❣❡♥❡r❛t♦rs✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ r❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝

❲❡②❧ ❛❧❣❡❜r❛✿ A(d) = U(hd) ▼❛①✐♠❛❧ s②♠♠❡tr② ❛❧❣❡❜r❛✿ ✭✐❞❡❛ ♦❢ t❤❡ ♣r♦♦❢✮ ❆❝t✐♥❣ ✇✐t❤ ❛♥② t✐♠❡✲r❡✈❡rs❡❞ ✭❍❡✐s❡♥❜❡r❣ ♣✐❝t✉r❡✮ ♦❜s❡r✈❛❜❧❡ ˆ A ˆ X(−t), ˆ P(−t)

• ♦♥ ❛ t✐♠❡✲❡✈♦❧✈❡❞ ✭❙❝❤rö❞✐♥❣❡r ♣✐❝t✉r❡✮ st❛t❡ ψ(t, x)

✐s ❡q✉✐✈❛❧❡♥t t♦ ❛❝t✐♥❣ ✇✐t❤ ❛♥② ✐♥✐t✐❛❧ ♦❜s❡r✈❛❜❧❡ ˆ A ˆ X(0), ˆ P(0)

• ♦♥ t❤❡

✐♥✐t✐❛❧ st❛t❡ ψ(0, x)✳ ❚❤❡r❡❢♦r❡ ❛♥② q✉❛♥t✉♠ ♦❜s❡r✈❛❜❧❡ ♦❢ A(d) ♠❛♣s s♦❧✉t✐♦♥s ♦♥ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

▲✐❣❤t✲❧✐❦❡ ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥

▼❛✐♥ ✐❞❡❛ ❜❡❤✐♥❞ t❤❡ ❧✐❣❤t✲❧✐❦❡ ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥✿ ❚❤❡ ❦✐♥❡t✐❝ ♦♣❡r❛t♦r ♦❢ ❛ r❡❧❛t✐✈✐st✐❝ t❤❡♦r② − M 2 = −2∂+∂− + ∆ − M 2 ✇❤❡♥ ❛❝t✐♥❣ ♦♥ ❡✐❣❡♥♠♦❞❡s ♦❢ ❛ ❧✐❣❤t✲❧✐❦❡ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ♠♦♠❡♥t✉♠✱ Ψ(x) = e−imx−ψ(x+, xi), ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ❦✐♥❡t✐❝ ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦r ♦❢ ❛ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ t❤❡♦r② i∂t + ∆/2m + µ ✈✐❛ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ x+ = t ❛♥❞ M 2 = −µ/2m✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

▲✐❣❤t✲❧✐❦❡ ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥

▼❛✐♥ ✐❞❡❛ ❜❡❤✐♥❞ t❤❡ ❧✐❣❤t✲❧✐❦❡ ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥✿ ✭●r♦✉♣ t❤❡♦r②✮ ❚❤❡ q✉❛❞r❛t✐❝ ❈❛s✐♠✐r ♦♣❡r❛t♦rs ♦❢ t❤❡ P♦✐♥❝❛ré ❛♥❞ t❤❡ ❇❛r❣♠❛♥♥ ❛❧❣❡❜r❛s ❛r❡ r❡❧❛t❡❞ ˆ P µ ˆ Pµ/2 = − ˆ P+ ˆ P− + ˆ P i ˆ Pi/2 = − ˆ m ˆ Pt + ˆ P i ˆ Pi/2 ✉♣♦♥ t❤❡ st❛♥❞❛r❞ ❧✐❣❤t✲❝♦♥❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ♠❛ss ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥ ♦♣❡r❛t♦rs ˆ P+ = ˆ m , ˆ P− = ˆ Pt . ❚❤❡ ❇❛r❣♠❛♥♥ ✭❙❝❤rö❞✐♥❣❡r✮ ❛❧❣❡❜r❛ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ s✉❜❛❧❣❡❜r❛ ♦❢ t❤❡ P♦✐♥❝❛ré ✭❝♦♥❢♦r♠❛❧✮ ❛❧❣❡❜r❛ t❤❛t ❝♦♠♠✉t❡s ✇✐t❤ ˆ P+ = ˆ m✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛❧ ✈s ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛s ❱❛s✐❧✐❡✈ ✈s ❲❡②❧ ❛❧❣❡❜r❛s ▲✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ❛♥❞ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠

❘✐❣✐❞ s②♠♠❡tr✐❡s✿ r❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝

❊♠❜❡❞❞✐♥❣ ❞✐❛❣r❛♠

• (d + 2, 2)

⊂ ❱❛s✐❧✐❡✈ ❛❧❣❡❜r❛ ✭❞✰✷✱✷✮ ∪ ∪ sch(d) ⊂ ❲❡②❧ ❛❧❣❡❜r❛ ✭❞✮ ✳ ✇❤❡r❡ t❤❡ ❡♠❜❡❞❞✐♥❣s st❛♥❞ ❢♦r✿ ⊂✿ ✜rst✲♦r❞❡r ❣❡♥❡r❛t♦r s✉❜❛❧❣❡❜r❛ ∪✿ ❝❡♥tr❛❧✐s❡r s✉❜❛❧❣❡❜r❛ ♦❢ ˆ P+ = ˆ m✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

❈♦♥❝❧✉s✐♦♥

❙✉♠♠❛r② ❛♥❞ ♦✉t❧♦♦❦

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

❙✉♠♠❛r②

❙♦♠❡ ❤✐♥ts t♦✇❛r❞ ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s ❇♦✉♥❞❛r② s✐❞❡✿

✶ ❙✐♠✐❧❛r✐t✐❡s ❜❡t✇❡❡♥ t❤❡ ❢r❡❡✴❝r✐t✐❝❛❧ O(N) ♠♦❞❡❧s ❛♥❞ t❤❡

✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s✳

✷ ◆♦♥✲r❡❧❛t✐✈✐st✐❝ s②♠♠❡tr✐❡s ✭❙❝❤rö❞✐♥❣❡r ❛♥❞ ❲❡②❧ ❛❧❣✮ ❡♠❜❡❞❞❡❞ ✐♥

r❡❧❛t✐✈✐st✐❝ s②♠♠❡tr✐❡s ✭r❡s♣❡❝t✐✈❡❧②✱ ❝♦♥❢♦r♠❛❧ ❛♥❞ ❱❛s✐❧✐❡✈ ❛❧❣✮✳

✸ ❯♥✐❢♦r♠ tr❡❛t♠❡♥t ♦❢ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥❛❧s ❢♦r r❡❧❛t✐✈✐st✐❝ ✭♦r ♥♦t✮

s❝❛❧❛r t❤❡♦r✐❡s ✇✐t❤ q✉❛rt✐❝ ✭t✇♦✲❜♦❞②✮ ❝♦♥t❛❝t ✐♥t❡r❛❝t✐♦♥✱ ❡✳❣✳ ❖✭◆✮✲❧✐❦❡ ✭♦r ❇❈❙✲❧✐❦❡✮ ♠♦❞❡❧s✳

✹ ◆♦♥✲r❡❧❛t✐✈✐st✐❝ t❤❡♦r✐❡s ❛s ❧✐❣❤t✲❧✐❦❡ ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥ ♦❢

r❡❧❛t✐✈✐st✐❝ t❤❡♦r✐❡s✱ ✐♥ t❤❡ s❡♠✐✲❝❧❛ss✐❝❛❧ ✭♠❡❛♥✲✜❡❧❞✮ r❡❣✐♠❡✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

❙✉♠♠❛r②

❙♦♠❡ ❤✐♥ts t♦✇❛r❞ ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s ❇✉❧❦ s✐❞❡✿

✶ ❙❡③❣✐♥✲❙✉♥❞❡❧❧✴❑❧❡❜❛♥♦✈✲P♦❧②❛❦♦✈ ❝♦♥❥❡❝t✉r❡ ✭✷✵✵✷✴✷✵✵✸✮ ✫ ✐ts

✈❛r✐♦✉s t❡sts ✭P❡t❦♦✉✱ ✷✵✵✸❀ ❙❡③❣✐♥ ❛♥❞ ❙✉♥❞❡❧❧✱ ✷✵✵✸❀ ●✐♦♠❜✐ ❛♥❞ ❨✐♥✱ ✷✵✶✵✮

✷ ◆❡✇t♦♥✲❈❛rt❛♥ ❣r❛✈✐t② ❛s ❧✐❣❤t✲❧✐❦❡ ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥ ♦❢

❊✐♥st❡✐♥✲❈❛rt❛♥ ❣r❛✈✐t② ✭❉✉✈❛❧ ❡t ❛❧✱ ✶✾✽✺❀ ❏✉❧✐❛ ❛♥❞ ◆✐❝♦❧❛✐✱ ✶✾✾✺✮

✸ ❆❞❙✴❈❋❚ ❞✐❝t✐♦♥❛r② ✐♥ t❤❡ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠ ✭▼❡ts❛❡✈✱ ✶✾✾✾✮ ❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

Pr♦♣♦s❛❧

❚♦✇❛r❞s ❛♥ ❡❞✉❝❛t❡❞ ❣✉❡ss ❢♦r t❤❡ ❜✉❧❦ ❞✉❛❧ ♦❢ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s ❆t ❧❡❛st ✐♥ t❤❡ ❧❛r❣❡✲◆ ✭♠❡❛♥ ✜❡❧❞✮ ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠ ♠❛② ❝♦♠♠✉t❡ ✭❝✳❢✳ ●♦❧❞❜❡r❣❡r✱ ❇❛r❜♦♥✲❋✉❡rt❡s✱ ▲✐♥✲❲✉✱ ✷✵✵✽✮✿ ❍❙ ♦♥ ❆❞❙ s♣❛❝❡t✐♠❡ ✭❞✰✸✮ ↔ ❈❋❚ ♦♥ ✢❛t s♣❛❝❡t✐♠❡ ✭❞✰✷✮ ↓↑ ↓↑ ◆❘❍❙ ♦♥ s♣❛❝❡✲t✐♠❡ ✭❞✰✷✮ ↔ ◆❘❈❋❚ ♦♥ ✢❛t s♣❛❝❡✲t✐♠❡ ✭❞✰✶✮ ✳ ✇❤❡r❡ t❤❡ ❛rr♦✇s st❛♥❞ ❢♦r✿ ↔ ❤♦❧♦❣r❛♣❤✐❝ ❞✉❛❧✐t② ✭❙❡③❣✐♥✲❙✉♥❞❡❧❧✲❑❧❡❜❛♥♦✈✲P♦❧②❛❦♦✈ ❧✐❦❡✮ ↓ ❧✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ✭❇❛r❣♠❛♥♥ ❢r❛♠❡✇♦r❦✮ ↑ ❧✐❣❤t✲❧✐❦❡ ♦①②❞❛t✐♦♥ ✭❊✐s❡♥❤❛rt ❧✐❢t✮

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

Pr♦♣♦s❛❧

❚♦✇❛r❞s ❛♥ ❡❞✉❝❛t❡❞ ❣✉❡ss ❢♦r t❤❡ ❜✉❧❦ ❞✉❛❧ ♦❢ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s ❆t ❧❡❛st ✐♥ t❤❡ ❧❛r❣❡✲◆ ✭♠❡❛♥ ✜❡❧❞✮ ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠ ♠❛② ❝♦♠♠✉t❡ ✭❝✳❢✳ ●♦❧❞❜❡r❣❡r✱ ❇❛r❜♦♥✲❋✉❡rt❡s✱ ▲✐♥✲❲✉✱ ✷✵✵✽✮✿ ❍❙ ♦♥ ❆❞❙ s♣❛❝❡t✐♠❡ ✭❞✰✸✮ ↔ ❈❋❚ ♦♥ ✢❛t s♣❛❝❡t✐♠❡ ✭❞✰✷✮ ↓↑ ↓↑ ◆❘❍❙ ♦♥ s♣❛❝❡✲t✐♠❡ ✭❞✰✷✮ ↔ ◆❘❈❋❚ ♦♥ ✢❛t s♣❛❝❡✲t✐♠❡ ✭❞✰✶✮ ✳ ✇✐t❤ d = 1✿ ❢r❡❡✴❝r✐t✐❝❛❧ ❖✭◆✮ ♠♦❞❡❧s d = 1, 3✿ ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s❡s

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

Pr♦♣♦s❛❧

❆ ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ ❤♦❧♦❣r❛♣❤✐❝ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❢❡r♠✐♦♥s ❛t ✉♥✐t❛r✐t② ✐s t❤❡ ❧✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ♦❢ ❛ ❱❛s✐❧✐❡✈ ❤✐❣❤❡r✲s♣✐♥ ❣r❛✈✐t②✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❚❤❡ O(N)✲✐♥✈❛r✐❛♥t s❡❝t♦r ♦❢ t❤❡ ❧❛r❣❡✲N ✐❞❡❛❧✴✉♥✐t❛r② ❋❡r♠✐ ❣❛s ✐♥ d s♣❛t✐❛❧ ❞✐♠❡♥s✐♦♥s ♠✐❣❤t ❜❡ ❞✉❛❧ t♦ t❤❡ ❧✐❣❤t✲❧✐❦❡ ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❱❛s✐❧✐❡✈ ❜♦s♦♥✐❝ t❤❡♦r② ♦♥ AdSd+3 ✇✐t❤ U(2) ✐♥t❡r♥❛❧ s②♠♠❡tr②✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❣r❛✈✐t② ❞✉❛❧ ♦❢ t❤❡ ✏♣❤②s✐❝❛❧✑ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ✭d = 3✮ t✇♦✲❝♦♠♣♦♥❡♥t ✭N = 1✮ ❯❱✲st❛❜❧❡ ✭∆− = 2✮ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ✇♦✉❧❞ ❜❡ t❤❡ ❧✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ♦❢ ❱❛s✐❧✐❡✈ t❤❡♦r② ❞❡s❝r✐❜✐♥❣ ✐♥t❡r❛❝t✐♥❣ u(2)✲✈❛❧✉❡❞ ❤✐❣❤❡r✲s♣✐♥ ❣❛✉❣❡ ✜❡❧❞s ♦♥ AdS6 ✇✐t❤ ❡①♦t✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❜✉❧❦ s❝❛❧❛r ✜❡❧❞ ❞✉❛❧ t♦ t❤❡ ❈♦♦♣❡r ♣❛✐r✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

Pr♦♣♦s❛❧

❇② ❝♦♥str✉❝t✐♦♥✱ s♣❡❝tr✉♠ ♦❢ ✜❡❧❞s✴♦♣❡r❛t♦rs ✭✉♥✮❜r♦❦❡♥ s②♠♠❡tr✐❡s t✇♦✲♣♦✐♥t ❢✉♥❝t✐♦♥s ❛r❡ ♠❛t❝❤❡❞ ✭❛t tr❡❡ ❧❡✈❡❧✱ ✐✳❡✳ ❧❛r❣❡✲◆✮✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

❖♣❡♥ ✐ss✉❡s

▼❛♥② ✐ss✉❡s r❡♠❛✐♥ ♦♣❡♥✿ ❈❧❛r✐❢② t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ ❙❝❤rö❞✐♥❣❡r ❛❧❣❡❜r❛ ✭P❡rr♦✉❞✱ ✶✾✼✼✮ ❡✳❣✳

❤♦❧♦❣r❛♣❤✐❝ ❞✐❝t✐♦♥❛r② ❢r♦♠ ❆❞❙✴❈❋❚ ✐♥ t❤❡ ❧✐❣❤t✲❝♦♥❡ ❢♦r♠❛❧✐s♠ ✭▼❡ts❛❡✈✱ ✶✾✾✾✮✱ st❛t✉s ♦❢ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ♠❛ss❧❡ss r❡♣r❡s❡♥t❛t✐♦♥s✱ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❛♥❛❧♦❣✉❡ ♦❢ ❋❧❛t♦✲❋rø♥s❞❛❧ t❤❡♦r❡♠✳

P❡r❢♦r♠ t❤❡ ❧✐❣❤t✲❧✐❦❡ r❡❞✉❝t✐♦♥ ♦❢ ❱❛s✐❧✐❡✈ ❡q✉❛t✐♦♥s✱ ❡✳❣✳

❡①♣❧✐❝✐t t❤❡♠ ✐♥ ❧✐❣❤t✲❝♦♥❡ ❣❛✉❣❡✱ ❛♥❞✴♦r ❣❡♥❡r❛❧✐s❡ t❤❡ ✇♦r❦s ✭❉✉✈❛❧ ❡t ❛❧✱ ✶✾✽✺❀ ❏✉❧✐❛ ❛♥❞ ◆✐❝♦❧❛✐✱ ✶✾✾✺✮✳

❈❤❡❝❦ t❤❡ ♣r♦♣♦s❛❧ ❜❡②♦♥❞ t✇♦✲♣♦✐♥t ❢✉♥❝t✐♦♥s✳ ❊t❝ ✳✳✳

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s

❚❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ ❤♦❧♦❣r❛♣❤② ❘❡❧❛t✐✈✐st✐❝ ✈s ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤✐❣❤❡r✲s♣✐♥ s②♠♠❡tr✐❡s ❈♦♥❝❧✉s✐♦♥ ❙✉♠♠❛r② Pr♦♣♦s❛❧ ❖✉t❧♦♦❦

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥

❳✳ ❇❡❦❛❡rt ❚♦✇❛r❞s ❛ ❜✉❧❦ ❞✉❛❧ ♦❢ t❤❡ ✉♥✐t❛r② ❋❡r♠✐ ❣❛s