No-Go Theorems in Non-Commutative Quantum Mechanics

No-Go Theorems in

Noncommutative

Quantum Mechanics

Bruno Alexandre Duarte Madureira

Mestrado em Física

Departamento de Física e Astronomia 2017

Orientador

Orfeu Bertolami Neto, Professor Catedrático, Faculdade de Ciências da Universidade do Porto

Todas as correções determinadas pelo júri, e só essas, foram efetuadas .

O Presidente do Júri,

✧❲❡ ❛❧❧ ♠❛❦❡ ❝❤♦✐❝❡s ✐♥ ❧✐❢❡✱ ❜✉t ✐♥ t❤❡ ❡♥❞ ♦✉r ❝❤♦✐❝❡s ♠❛❦❡ ✉s✳✧

✲❆♥❞r❡✇ ❘②❛♥

✧■❢ ■ ❤❛✈❡ s❡❡♥ ❢✉rt❤❡r ✐t ✐s ❜② st❛♥❞✐♥❣ ♦♥ t❤❡ s❤♦✉❧❞❡rs ♦❢ ❣✐❛♥ts✳✧

❆❝❦♥♦✇❧❡❞❣♠❡♥ts

❖✈❡r t❤❡ ❝♦✉rs❡ ♦❢ t❤❡s❡ ❢❡✇ ②❡❛rs✱ ■ ❤❛✈❡ ♠❡t ♣❡♦♣❧❡ ✇❤♦ ❧❡❢t t❤❡✐r ♠❛r❦ ♦♥ ♠② ♦✇♥ ❥♦✉r♥❡②✳ P❡♦♣❧❡ ✇❤♦ ❤❛✈❡ ❤❡❧♣❡❞ ♠❡✱ ♣❡♦♣❧❡ ✇❤♦ ❤❛✈❡ t❛✉❣❤t ♠❡✱ ♣❡♦♣❧❡ ✇❤♦ ❤❛✈❡ st♦♦❞ ❜② ♠② s✐❞❡✳ P❡♦♣❧❡ ✇❤♦ ♠❛❞❡ ♠❡ ✇❤♦ ■ ❛♠ t♦❞❛②✱ ❛♥❞ ✇✐t❤♦✉t ✇❤♦♠ t❤✐s ✇♦r❦ ✇♦✉❧❞ ♥♦t ❜❡ ♣♦ss✐❜❧❡✳ ❚❤✐s ✐s ❛ s♠❛❧❧ tr✐❜✉t❡ t♦ t❤❡♠✳ ❋✐rst ❛♥❞ ❢♦r❡♠♦st✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ♠② ♠♦t❤❡r ●❡♥❡r♦s❛ ❉✉❛rt❡ ❢♦r ❤❡r ✉♥❝♦♥❞✐✲ t✐♦♥❛❧ s✉♣♣♦rt ❛♥❞ ❧♦✈❡✱ ❛♥❞ ❢♦r ❤❡r str❡♥❣t❤✱ ♣❛t✐❡♥❝❡ ❛♥❞ r❡s♦❧✈❡ ✐♥ ♦r❞❡r t♦ s✉♣♣♦rt ✉s ❛❧❧ t❤❡s❡ ②❡❛rs✳ ▼❛② t❤❡ ❢✉t✉r❡ ❤♦❧❞ ♠❛♥② s✉r♣r✐s❡s✱ ❤❡❛❧t❤ ❛♥❞ ❤❛♣♣✐♥❡ss ❢♦r ②♦✉✳ ■ ✇♦✉❧❞ ❛❧s♦ ❧✐❦❡ t♦ t❤❛♥❦ ♠② s✉♣❡r✈✐s♦r✱ Pr♦❢❡ss♦r ❖r❢❡✉ ❇❡rt♦❧❛♠✐✱ ❢♦r t❤✐s ♦♣♣♦rt✉♥✐t② t♦ ✇♦r❦ ✇✐t❤ ❤✐♠✱ ❛♥❞ ❢♦r ❤✐s ❤❡❧♣ ❛♥❞ ❛✈❛✐❧❛❜✐❧✐t②✳ ❆❧t❤♦✉❣❤ ♦✉r ❝♦♥t❛❝t ✇❛s ♥♦t ❛s ❧♦♥❣ ❧❛st✐♥❣ ❛s ■ ✇♦✉❧❞ ✇✐s❤✱ ✐t ✇❛s s✐♥❝❡r❡❧② ❡♥❥♦②❛❜❧❡✱ ❛♥❞ ■ ❤♦♣❡ ❢♦r t❤❡ ❜❡st ✐♥ ❤✐s ❡♥❞❡❛✈♦rs✳ ❆ t❤❛♥❦s ❛❧s♦ t♦ ❛❧❧ t❤❡ t❡❛❝❤✐♥❣ st❛✛✱ ✇❤❡t❤❡r ♦r ♥♦t t❤❡✐r ♣r❡s❡♥❝❡ ✇❛s ❜✐❣ ♦r s♠❛❧❧✳ ❲✐t❤♦✉t t❤❡♠✱ ■ ❢❡❛r ■ ✇♦✉❧❞ ❜❡ ❞❡✈♦✐❞ ♦❢ ❦♥♦✇❧❡❞❣❡✳ ❊✈❡r② s✐♥❣❧❡ ❞r♦♣ ♦❢ ✐t✱ ✇❤❡t❤❡r ✐t ✇❛s ♣❤②s✐❝s✱ ♠❛t❤❡♠❛t✐❝s✱ ♦r ❛♥② ♦t❤❡r s✉❜❥❡❝t✱ r❡❛❧❧②✱ ❛❧❧♦✇❡❞ ♠❡ t♦ ♣✉s❤ ❢♦r✇❛r❞ ✐♥ ♠② st✉❞✐❡s ❛♥❞ ♠❛❦❡ ♠❡ ❛ ❜❡tt❡r ❝r✐t✐❝❛❧ t❤✐♥❦❡r✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ s♣❡❝✐❛❧ t❤❛♥❦s t♦ Pr♦❢❡ss♦r ◆✉♥❡s ❞❛ ❙✐❧✈❛ ❢♦r ❤✐s t❡❛❝❤✐♥❣ ♦❢ t❤❡ ❊r❛s❡r ❚❡❝❤♥✐q✉❡✱ ❛s ✇❡❧❧ ❛s ❤✐s ✐♥t❡r❡st✐♥❣ ✐♥t❡r❛❝t✐♦♥s ✇✐t❤ ✉s✱ ❤✐s st✉❞❡♥ts✳ ▼❛② ②♦✉ ♥❡✈❡r ❝❤❛♥❣❡ ②♦✉r ✇❛②s✳ ■ ❛❧s♦ ✇✐s❤ t♦ t❤❛♥❦ ♠② ❢r✐❡♥❞s ❛♥❞ ❝♦❧❧❡❛❣✉❡s✳ ❆ s♣❡❝✐❛❧ t❤❛♥❦s t♦ ❆♥tó♥✐♦ ❆♥t✉♥❡s✱ ❏♦ã♦ P✐r❡s✱ ❏♦ã♦ ●✉❡rr❛✱ ▼❛r✐❛ ❘❛♠♦s ❛♥❞ ❙✐♠ã♦ ❏♦ã♦ ❢♦r t❤❡✐r ❢r✐❡♥❞s❤✐♣✱ ❝♦♠♣❛♥② ❛♥❞ ❤❡❧♣ ❞✉r✐♥❣ t❤✐s s♠❛❧❧ ❥♦✉r♥❡②✱ ❛♥❞ t♦ ❉✐♦❣♦ ❘✐❜❡✐r♦ ❛♥❞ ▼❛✉rí❝✐♦ ▼❛rt✐♥s ❢♦r t❤❡ ❝♦✉♥t❧❡ss ❤♦✉rs ♦❢ ❞✐str❛❝t✐♦♥ t❤❛t ❦❡♣t ♠② s❛♥✐t② ✐♥t❛❝t✳ ▲❛st❧②✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ t❤❡ ❝♦✉♥t❧❡ss ♣❡♦♣❧❡✱ ❢❛♠✐❧② ❛♥❞ ❢r✐❡♥❞s ✇❤♦ ❣♦ ✉♥♥❛♠❡❞✳ ❋♦r ❜❡tt❡r ♦r ✇♦rs❡✱ ②♦✉ ✇❡r❡ ❛❧❧ ♣❛rt ♦❢ ♠② ❥♦✉r♥❡②✱ ❛♥❞ ♦❢ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤✐s ✇♦r❦✳ ❨♦✉ ❛❧❧ ❤❛✈❡ ♠② t❤❛♥❦s ❛♥❞ ♠② ❣r❛t✐t✉❞❡✳ ✐✐✐

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ sã♦ ❛❜♦r❞❛❞♦s t❡♦r❡♠❛s ❞❡ ✈❡t♦ ✭✧♥♦✲❣♦✧✮ ♥♦ ❝♦♥t❡①t♦ ❞❡ ▼❡❝â♥✐❝❛ ◗✉â♥✲ t✐❝❛ ◆ã♦✲❈♦♠✉t❛t✐✈❛✳ ❖ ♦❜❥❡❝t✐✈♦ ♣r✐♥❝✐♣❛❧ é ✈❡r✐✜❝❛r s❡ t❡♦r❡♠❛s ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ ◆ã♦✲❈❧♦♥❛❣❡♠ ♠❛♥tê♠✲s❡ ✈á❧✐❞♦s q✉❛♥❞♦ ❝♦♥s✐❞❡r❛❞♦s ♥♦ ❊s♣❛ç♦ ❞❡ ❋❛s❡ ◆ã♦✲❈♦♠✉t❛t✐✈♦ ❣❡♥ér✐❝♦✱ t❡♦r❡♠❛s ❡st❡s q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♥♦ ❝♦♥t❡①t♦ ❞❛ ❚❡♦r✐❛ ❞❛ ■♥❢♦r♠❛çã♦ ◗✉â♥t✐❝❛✳ ❙❡rá ❢❡✐t♦ ✉♠ ♣❡q✉❡♥♦ r❡s✉♠♦ ❞❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❲✐❣♥❡r✲❲❡②❧ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✱ s❡❣✉✐❞♦ ❞❡ ✉♠❛ ❞✐s❝✉ssã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ◆ã♦✲❈❧♦♥❛❣❡♠ ♥♦ ❊s♣❛ç♦ ❞❡ ❋❛s❡ ❤❛❜✐t✉❛❧✱ ❛ss✐♠ ❝♦♠♦ ✉♠❛ ❞✐s❝✉ssã♦ ❞❛ s✉❛ ❣❡♥❡r❛❧✐③❛çã♦✳ P♦r ✜♠✱ é ♣r♦✈❛❞♦ q✉❡ ❡st❡s t❡♦r❡♠❛s ❝♦♥t✐♥✉❛♠ ✈á❧✐❞♦s ❡♠ ❊s♣❛ç♦s ❞❡ ❋❛s❡ ◆ã♦✲❈♦♠✉t❛t✐✈♦s✳ ✈

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ◆♦✲●♦ ❚❤❡♦r❡♠s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛r❡ ❛❞❞r❡ss❡❞✳ ❚❤❡ ♠❛✐♥ ❢♦❝✉s ✐s t♦ s❡❡ ✇❤❡t❤❡r t❤❡♦r❡♠s s✉❝❤ ❛s t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠ st✐❧❧ ❤♦❧❞ ✇❤❡♥ ❛ ❣❡♥❡r✐❝ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ P❤❛s❡ ❙♣❛❝❡ ✐s ❝♦♥s✐❞❡r❡❞✳ ❚❤✐s ✐s ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡✱ ❢♦r ✐♥st❛♥❝❡✱ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✳ ❆ ❜r✐❡❢ s✉♠♠❛r② ♦❢ ❲✐❣♥❡r✲❲❡②❧ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✐s ❣✐✈❡♥✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠ ✐♥ t❤❡ st❛♥❞❛r❞ P❤❛s❡ ❙♣❛❝❡✱ ❛s ✇❡❧❧ ❛s ❛ ❞✐s❝✉ss✐♦♥ ♦❢ ✐ts ❣❡♥❡r❛❧✐③❛t✐♦♥✳ ❋✐♥❛❧❧②✱ ✐t ✐s ♣r♦✈❡♥ t❤❛t t❤❡♦r❡♠s ♦❢ t❤✐s t②♣❡ ❤♦❧❞ ♦♥ ❛ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ P❤❛s❡ ❙♣❛❝❡✳ ✈✐✐

❈♦♥t❡♥ts

❆❝❦♥♦✇❧❡❞❣♠❡♥ts ✐✐✐ ❘❡s✉♠♦ ✈ ❆❜str❛❝t ✈✐✐ ✶ ■♥tr♦❞✉❝t✐♦♥ ✶ ✶✳✶ ◆♦✲●♦ ❚❤❡♦r❡♠s ❛♥❞ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ P❤❛s❡ ❙♣❛❝❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✸ ✷✳✶ ❲❡②❧✲❲✐❣♥❡r tr❛♥s❢♦r♠ ❛♥❞ t❤❡ ❲✐❣♥❡r ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ ▼♦②❛❧ ♣r♦❞✉❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✸ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠✉❧❛t✐♦♥ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✹ ▼❛✐♥ Pr♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✹✳✶ ❚r❛❝❡ ♦❢ ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✹✳✷ ■♥✈❛r✐❛♥❝❡ ✉♥❞❡r ♣❤❛s❡ s♣❛❝❡ ✐♥t❡❣r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◆♦♥✲ ❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✾ ✸✳✶ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ❲✐❣♥❡r ❚r❛♥s❢♦r♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✸✳✷ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✹ ◆♦✲●♦ ❚❤❡♦r❡♠s ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✶✺ ✹✳✶ ◆♦✲❈❧♦♥✐♥❣ ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✹✳✷ ◆♦✲●♦ ●❡♥❡r❛❧✐③❛t✐♦♥ ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✺ ❇r✐❞❣✐♥❣ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✶✾ ✺✳✶ ❚r❛♥s❢♦r♠✐♥❣ ❖♣❡r❛t♦rs ✐♥ ◆❈◗▼ t♦ ❖♣❡r❛t♦rs ✐♥ ◗▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✺✳✷ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s T ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✐①

✺✳✸ ❉❡♥s✐t② ▼❛tr✐① ✐♥ ◆❈◗▼ ❛♥❞ ◗▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✺✳✹ ❚r❛♥s❢♦r♠❛t✐♦♥ ♦❢ fQbN C, bPN C  ♦♣❡r❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✻ ◆♦✲●♦ ❚❤❡♦r❡♠s ✐♥ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✸✶ ✻✳✶ ◆♦✲❈❧♦♥✐♥❣ ✐♥ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✻✳✷ ◆♦✲●♦ ●❡♥❡r❛❧✐③❛t✐♦♥ ✐♥ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✼ ❈♦♥❝❧✉s✐♦♥s ✸✼ ❇✐❜❧✐♦❣r❛♣❤② ✸✽

❈❤❛♣t❡r ✶

■♥tr♦❞✉❝t✐♦♥

✶✳✶ ◆♦✲●♦ ❚❤❡♦r❡♠s ❛♥❞ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐♦♥

❉✉r✐♥❣ t❤❡ ✐♥❝❡♣t✐♦♥ ❛♥❞ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✭◗▼✮✱ ◗✉❛♥t✉♠ ■♥❢♦r✲ ♠❛t✐♦♥ ♣♦s❡❞ ❛ s✐❣♥✐✜❝❛♥t ♣r♦❜❧❡♠ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r✐sts✳ ❚❤❡ ❢❛❝t t❤❛t t❤❡ ❛❝t ♦❢ ♠❡❛s✉r❡♠❡♥t ❝❤❛♥❣❡s t❤❡ s②st❡♠ ✐♥ ❛♥❛❧②s✐s ♠❡❛♥t t❤❛t ❝❧❛ss✐❝❛❧ ♣r♦❝❡❞✉r❡s ❢♦r ✐♥❢♦r♠❛✲ t✐♦♥ tr❡❛t♠❡♥t ✇❡r❡ ♥♦ ❧♦♥❣❡r ❛❜❧❡ t♦ ❜❡ ❛♣♣❧✐❡❞ t♦ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐♦♥✳ ❚❤✐s ❧❡❞ t♦ t❤❡ st✉❞② ♦❢ q✉❛♥t✉♠ s②st❡♠s ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ t❤❡ s✉❜s❡q✉❡♥t ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡♦r❡♠s t❤❛t r❡str✐❝t❡❞ t❤❡ ❛❝t✐♦♥s ✉♣♦♥ q✉❛♥t✉♠ st❛t❡s✳ ❚❤❡♦r❡♠s s✉❝❤ ❛s t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠ ✭✇❤✐❝❤ ❡♥s✉r❡❞ t❤❛t ♥♦ r❛♥❞♦♠ st❛t❡ ❝❛♥ ❜❡ ❞✉♣❧✐❝❛t❡❞ ❬✶✱ ✷❪✮✱ t❤❡ ◆♦✲❉❡❧❡t✐♥❣ ❚❤❡♦r❡♠ ✭✇❤✐❝❤ ❡♥s✉r❡❞ t❤❛t ❣✐✈❡♥ t✇♦ ❝♦♣✐❡s ♦❢ ❛ st❛t❡✱ t❤❡r❡ ✐s ♥♦ ✇❛② t♦ ❞❡❧❡t❡ ♦♥❡ ♦❢ t❤❡♠ ❬✸❪✮✱ ✐♠♣❧② t❤❛t ❝❧❛ss✐❝❛❧ ❡rr♦r ❝♦rr❡❝t✐♦♥ t❡❝❤♥✐q✉❡s ❛r❡ ✉s❡❧❡ss✳ ❋♦r ❡①❛♠♣❧❡✱ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t❤❛t✱ ❞✉r✐♥❣ ❛ q✉❛♥t✉♠ ❝♦♠♣✉t❛t✐♦♥✱ ❛ ❞✉♣❧✐❝❛t❡ ♦❢ ❛ st❛t❡ ✐s ❝r❡❛t❡❞ ❛♥❞ ✉s❡❞ ❢♦r ❝♦rr❡❝t✐♥❣ ❡rr♦rs✳ ❚❤✐s ✐s ✈✐t❛❧ ❢♦r ♣r❛❝t✐❝❛❧ q✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣✱ ❛♥❞ ❢♦r ❛ t✐♠❡ ✐t ✇❛s t❤♦✉❣❤t t♦ ❜❡ ❛ ❦❡② ❧✐♠✐t❛t✐♦♥✳ ❋♦rt✉♥❛t❡❧②✱ ✇✐t❤ t❤❡ ❛❞✈❡♥t ♦❢ ✜rst q✉❛♥t✉♠ ❡rr♦r ❝♦rr❡❝t✐♥❣ ❝♦❞❡s✱ ✐♥ ✶✾✾✺✱ ✇❤✐❝❤ ❝✐r✲ ❝✉♠✈❡♥t❡❞ t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠✱ ◗✉❛♥t✉♠ ❈♦♠♣✉t❛t✐♦♥ ❤❛s s❡❡♥ ❛ s❤❛r♣ ✐♥❝r❡❛s❡ ✐♥ ✐♥t❡r❡st✱ ✇✐t❤ t❤❡ ✜rst s♦❧✐❞✲st❛t❡ q✉❛♥t✉♠ ♣r♦❝❡ss♦r ❜❡✐♥❣ ❝r❡❛t❡❞ ❜② r❡s❡❛r❝❤❡rs ❛t ❨❛❧❡ ❯♥✐✈❡rs✐t② ✐♥ ✷✵✵✾✳ ❙✐♥❝❡ t❤❡♥✱ ◗✉❛♥t✉♠ ❈♦♠♣✉t✐♥❣✱ ❛❧❜❡✐t st✐❧❧ ✐♥ ✐ts ❡❛r❧② ②❡❛rs✱ ✐s ❜❡❝♦♠✲ ✐♥❣ ❛ ♠❛❥♦r ❢♦❝✉s ♦❢ st✉❞②✳ ❚❤❡s❡ t②♣❡s ♦❢ t❤❡♦r❡♠s ❛r❡ ❦♥♦✇♥ ❛s ◆♦✲●♦ ❚❤❡♦r❡♠s ❛♥❞ s♦♠❡ ♦❢ t❤❡♠✱ s✉❝❤ ❛s t❤❡ ◆♦✲❈❧♦♥✐♥❣✱ ❛r❡ t❤❡ ♦❜❥❡❝t ♦❢ st✉❞② ♦❢ t❤✐s ✇♦r❦✳ ❍♦✇❡✈❡r✱ ✐♥st❡❛❞ ♦❢ ✇♦r❦✐♥❣ ✐♥ t❤❡ ✉s✉❛❧ ◗▼ P❤❛s❡✲❙♣❛❝❡✱ t❤❡ ❢♦❝✉s ✇✐❧❧ ❜❡ t❤❡ st✉❞② ♦♥ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ P❤❛s❡ ❙♣❛❝❡s✳ ✶

✷ ❈❤❛♣t❡r ✶✳ ■♥tr♦❞✉❝t✐♦♥

✶✳✷ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ P❤❛s❡ ❙♣❛❝❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s

◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✭◆❈◗▼✮ ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✇✐t❤ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s t❤❛t ❛r❡ ❛ ❞❡❢♦r♠❛t✐♦♥ ♦❢ t❤❡ st❛♥❞❛r❞ ❍❡✐s❡♥❜❡r❣✲❲❡②❧ ❛❧❣❡❜r❛✳ ❚❤❡ r❡♣❧❛❝❡♠❡♥t ♦❢ t❤❡ ❛❧❣❡❜r❛ [qi, qj] = 0, [pi, pj] = 0, [qi, pj] = iℏδij, ✭✶✳✶✮ ✇✐t❤ t❤❡ ❛❧❣❡❜r❛

[qi, qj] = iθij, [pi, pj] = iηij, [qi, pj] = iℏδij, ✭✶✳✷✮

✇❤❡r❡ θ ❛♥❞ η ❛r❡ r❡❛❧ ❛♥t✐✲s②♠♠❡tr✐❝ ♠❛tr✐❝❡s ❛♥❞ ℏ′_{= ℏ}  1 + θη ℏ2  , ✭✶✳✸✮ ✐♥❞✉❝❡s ❛ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ s♣❛❝❡ ❞✐r❡❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t✉♠ ❞✐r❡❝t✐♦♥s✱ ♦r ❡✈❡♥ ❛ q✉❛♥t✐✲ ③❛t✐♦♥ ♦❢ ❜♦t❤ ❝♦♥✜❣✉r❛t✐♦♥ ❛♥❞ ♠♦♠❡♥t✉♠ s♣❛❝❡✳ ◆♦♥✲❈♦♠♠✉t❛t✐✈✐t② ❤❛s r❡❝❡♥t❧② ❣❛✐♥❡❞ ✐♥t❡r❡st ❞✉❡ t♦ str✐♥❣ t❤❡♦r②✱ ❛s t❤❡ ❞②♥❛♠✐❝s ♦❢ str✐♥❣s ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛ ❣❛✉❣❡ t❤❡♦r② ✐♥ ❛ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡ ✭s❡❡✱ ❢♦r ❡①❛♠✲ ♣❧❡✱ ❘❡❢s✳ ❬✹✱ ✺❪✮✳ ❙✐♥❝❡ st❛♥❞❛r❞ ◗▼ ✐s t❤❡ ❧♦✇✲❡♥❡r❣②✱ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣❛rt✐❝❧❡s ❧✐♠✐t ♦❢ ♦t❤❡r ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r✐❡s✱ ◆♦♥✲❈♦♠♠✉t❛t✐✈✐t② ♠✐❣❤t ❛♣♣❡❛r ❛s ❛ s♠❛❧❧ ❡✛❡❝t ❛t t❤❡ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❧❡✈❡❧✳ ❇❡❝❛✉s❡ ♦❢ t❤✐s✱ ❛ ❣r❡❛t ❛♠♦✉♥t ♦❢ ✇♦r❦ ❤❛s ❜❡❡♥ ❞❡✈♦t❡❞ t♦ ◆♦♥✲❈♦♠♠✉t❛t✐✈✐t②✱ ✐♥❝❧✉❞✐♥❣ t♦♣✐❝s s✉❝❤ ❛s✿ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ●❡♦♠❡tr② ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❘❡❢✳ ❬✻❪✮❀ t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ ◆♦♥✲❈♦♠♠✉t❛t✐✈✐t② ✐♥ ♣❛rt✐❝❧❡s ❛✛❡❝t❡❞ ❜② ♠❛❣♥❡t✐❝ ✜❡❧❞s❀ ◆♦♥✲❈♦♠♠✉t❛t✐✈✐t② ✐♥ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❘❡❢✳ ❬✼❪✮❀ ♣❛rt✐❝❧❡s ✐♥ ✇❡❧❧ st✉❞✐❡❞ ♣♦t❡♥t✐❛❧s✱ ❜✉t ✐♥ ❛ ◆❈ P❤❛s❡ ❙♣❛❝❡✱ s✉❝❤ ❛s ❛ ♣❛rt✐❝❧❡ ✐♥ ❛ ❝❡♥tr❛❧ ♣♦t❡♥t✐❛❧ ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❘❡❢✳ ❬✽❪✮✱ t❤❡ ●r❛✈✐t❛t✐♦♥❛❧ ◗✉❛♥t✉♠ ❲❡❧❧ ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❘❡❢✳ ❬✾✱ ✶✵✱ ✶✶❪✮✱ t❤❡ ❍❛r♠♦♥✐❝ ❖s❝✐❧❧❛t♦r ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❘❡❢✳ ❬✶✷❪✮ ❛♥❞ t❤❡ ❍②❞r♦❣❡♥ ❆t♦♠❀ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❝♦s♠♦❧♦❣② ❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❝♦♥s✐❞❡r❡❞ ✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❘❡❢s✳ ❬✶✵✱ ✶✸❪✮ ❛s ✇❡❧❧ ❛s ♦t❤❡r ✇♦r❦s✳ ❆ ❝r✉❝✐❛❧ ♣❛rt ♦❢ t❤❡s❡ ✇♦r❦s ❤❛s ❜❡❡♥ ❞❡✈♦t❡❞ t♦ ❝r❡❛t✐♥❣ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ◆❈◗▼ ❜❛s❡❞ ♦♥ t❤❡ ❲❡②❧✲❲✐❣♥❡r ❢♦r♠✉❧❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❘❡❢s✳ ❬✶✹✱ ✶✺❪✮✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❢✉❧ ✐♥ t❤❡ tr❡❛t♠❡♥t ♦❢ t❤❡s❡ ♥❡✇ ✉♥❝❡rt❛✐♥t② r❡❧❛t✐♦♥s ✭s❡❡ ❘❡❢s✳ ❬✶✻✱ ✶✼✱ ✶✽✱ ✶✾❪✮✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ✇✐❧❧ ❢♦❝✉s ♦♥ s❡❡✐♥❣ ✐❢ ◆♦♥✲❈♦♠♠✉t❛t✐✈✐t②✱ ✉s✐♥❣ ❲❡②❧✲❲✐❣♥❡r ❢♦r♠✉❧❛✲ t✐♦♥✱ ❤❛s ❛♥② ✐♥✢✉❡♥❝❡ ✐♥ ◆♦✲●♦ ❚❤❡♦r❡♠s s✉❝❤ ❛s t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠✳

❈❤❛♣t❡r ✷

❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢

◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s

■♥ t❤❡ s❛♠❡ ✇❛② ❛s ❝❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s ❤❛s ✈❛r✐♦✉s ❡q✉✐✈❛❧❡♥t ❢♦r♠✉❧❛t✐♦♥s✱ s✉❝❤ ❛s ◆❡✇✲ t♦♥✐❛♥✱ ▲❛❣r❛♥❣✐❛♥ ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥✱ ◗▼ ❛❧❧♦✇s ❢♦r ❞✐✛❡r❡♥t ❢♦r♠✉❧❛t✐♦♥s✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❜r✐❡✢② ❞❡s❝r✐❜❡ t❤❡ ❲❡②❧✲❲✐❣♥❡r ✭❲❲✮ ❢♦r♠✉❧❛t✐♦♥✳ ❚❤✐s ❈❤❛♣t❡r ❛♥❞ t❤❡ s✉❝❝❡❡❞✐♥❣ ♦♥❡s ❛ss✉♠❡ t❤❡ ❊✐♥st❡✐♥✬s s✉♠♠❛t✐♦♥ ❝♦♥✈❡♥t✐♦♥✳

✷✳✶ ❲❡②❧✲❲✐❣♥❡r tr❛♥s❢♦r♠ ❛♥❞ t❤❡ ❲✐❣♥❡r ❢✉♥❝t✐♦♥

■♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ◗▼✱ t❤❡ ❦❡② ♦❜❥❡❝t ✐s t❤❡ ✇❛✈❡✲❢✉♥❝t✐♦♥❀ t❤❡ ❝♦r❡ ♦❢ ❲❲ ❢♦r♠✉❧❛t✐♦♥ ✐s t❤❡ ❲✐❣♥❡r ❢✉♥❝t✐♦♥✱ ✇❤✐❝❤ ✐s r❡❧❛t❡❞ t♦ t❤❡ ✇❛✈❡❢✉♥❝t✐♦♥ ❜②✿ f(qi, pi) = ˆ ψ∗  − →_{q −}−→y 2  ψ  − →_{q +}−→y 2  e−ipiyiℏ ddy, ✭✷✳✶✮ ✇❤❡r❡ qi ❛r❡ ♣♦s✐t✐♦♥s✱ pi ❛r❡ t❤❡ ♠♦♠❡♥t❛✱ ℏ ✐s t❤❡ r❡❞✉❝❡❞ P❧❛♥❦ ❝♦♥st❛♥t ❛♥❞ d ✐s t❤❡ ♥✉♠❜❡r ♦❢ s♣❛❝✐❛❧ ❞✐♠❡♥s✐♦♥s✳ ❆♥♦t❤❡r ✐♠♣♦rt❛♥t t♦♦❧ ✐s t❤❡ ❲✐❣♥❡r✲❲❡②❧ tr❛♥s❢♦r♠❛t✐♦♥ ✭s❡❡ ❘❡❢✳ ❬✷✵❪✮✱ ✇❤✐❝❤ ♠❛♣s ♦♣❡r❛t♦rs ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ t♦ ❢✉♥❝t✐♦♥s ✐♥ ♣❤❛s❡ s♣❛❝❡ ❛♥❞ ✈✐❝❡✲✈❡rs❛✳ ●✐✈❡♥ ❛♥ ♦♣❡r❛t♦r ˆ A✱ t❤❡ ❲✐❣♥❡r tr❛♥s❢♦r♠ ✐s ❞❡✜♥❡❞ ❛s✿ WAˆ(qi, pi) = ˆ ddy  − →_{q −}−→y 2 | ˆA|− →_{q +}−→y 2  e−ipiyiℏ . ✭✷✳✷✮ ◆♦t❡ t❤❛t t❤❡ ❲✐❣♥❡r tr❛♥s❢♦r♠ ❤❛s t❤❡ ♣r♦♣❡rt✐❡s✿ ✸

✹ ❈❤❛♣t❡r ✷✳ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s W(q_bi) = qi, ✭✷✳✸✮ W(_bpi) = pi, ✭✷✳✹✮ W(Id) = 1. ✭✷✳✺✮ ❲❡ ❝❛♥ ♥♦✇ s❡❡ t❤❛t ✐♥ ❢❛❝t f(qi, pi) = W (ρ) = W (|ψi hψ|) ,b ✭✷✳✻✮ ✇❤❡r❡ bρ = |ψi hψ| ✐s t❤❡ ❞❡♥s✐t② ♠❛tr✐① ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ st❛t❡ |ψi✳ ❚❤✐s ♠❛♣♣✐♥❣ ✐s ♦♥❡✲t♦✲♦♥❡ ❛♥❞ ❛❞♠✐ts ❛♥ ✐♥✈❡rs❡✱ t❤❡ ❲❡②❧ tr❛♥s❢♦r♠✿ W−1(g) = ˆ d2dk (2π)2d ˆ d2dz g· eikizˆi_e−ikizi_, ✭✷✳✼✮ ✇❤❡r❡ ✇❡ ✉s❡❞ z = (qi, pi) ❢♦r t❤❡ ❝♦♦r❞✐♥❛t❡s ✐♥ t❤❡ ♣❤❛s❡ s♣❛❝❡✱ ❛♥❞ bz = (bqi,pbi)❛r❡ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ ♠♦♠❡♥t❛ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡✱ ❛♥❞ 2d ✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ♣❤❛s❡ s♣❛❝❡✳ ❚❤✐s ♠❡❛♥s t❤❛t✱ ❢♦r ❛♥② ♦♣❡r❛t♦r ˆA✭s❡❡ ❘❡❢✳ ❬✼❪✮✿ ˆ A= W−1WAˆ= ˆ _d2d_k (2π)2d ˆ d2dz WAˆ· eikizˆi_e−ikizi_. ✭✷✳✽✮ ◆♦t❡ t❤❛t W_Aˆ_{✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ z = (qi, pi)✱ ✇❤✐❝❤ ❛r❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ bz = (bqi,pbi)✱ ❛♥❞} t❤✉s t❤❡ ❢❛❝t t❤❛t eikizˆi ❢♦r♠s ❛ ❝♦♠♣❧❡t❡ ❜❛s✐s ❢♦r ♦♣❡r❛t♦rs ❛❧❧♦✇s t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥s t♦ ❜❡ ♦♥❡✲t♦✲♦♥❡✳

✷✳✷ ▼♦②❛❧ ♣r♦❞✉❝t

❚❤❡ ❧❛st ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t t♦ ❜❡ ✐♥tr♦❞✉❝❡❞ ✐s t❤❡ s♦✲❝❛❧❧❡❞ ▼♦②❛❧ ♦r st❛r ♣r♦❞✉❝t✱ ❞❡✜♥❡❞ s♦ t❤❛t✿ WAˆ⋆ WBˆ= WA ˆˆB. ✭✷✳✾✮ ■♥ ❣❡♥❡r❛❧✱ ✐t ❝❛♥ ❜❡ ♣r♦✈❡♥ t❤❛t ▼♦②❛❧ ♣r♦❞✉❝t ❤❛s t❤❡ ❢♦r♠ ✭s❡❡ ❘❡❢s✳ ❬✷✶✱ ✷✷❪✮✿

✷✳✸✳ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠✉❧❛t✐♦♥ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✺ A ⋆ B = Aeiℏ2 ←− ∂_ziΩij −→ ∂_zj B = AB+ ∞ X n=1 1 n!  iℏ 2 n A←−∂ziΩij −→ ∂zj n B = AB+ ∞ X n=1 (iℏ)n n!2n  ∂(n)_z α1...zαnA   ∂_z(n) β1...zβnB  Ωα1β1. . .Ωαnβn, ✭✷✳✶✵✮ ✇❤❡r❡ t❤❡ ❛rr♦✇s ♠❡❛♥ t❤❡ ❞❡r✐✈❛t✐✈❡s ❛r❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ❧❡❢t ♦r r✐❣❤t✱ ❛♥❞ Ω = 0 Idd×d −Idd×d 0 ! ✭✷✳✶✶✮ ✐s t❤❡ ♠❛tr✐① ♦❢ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s✱ ✐✳❡✳✱ ❢♦r bz = (bqi,bpi)✱ [_bzi,zbj] = iℏΩij, ✭✷✳✶✷✮ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❍❡✐s❡♥❜❡r❣✲❲❡②❧ ❛❧❣❡❜r❛ ✭❝✳ ❢✳ ❡q✳ ✭✶✳✶✮✮✳ ◆♦t❡ t❤❛t ❜② ❞❡✜♥✐t✐♦♥ Ωij ✐s ❛♥t✐s②♠♠❡tr✐❝ ❜❡❝❛✉s❡ [bzi,zbj] = − [bzj,bzi]✳ ■♥ ❛❞❞✐t✐♦♥✱ t❛❦✐♥❣ t❤❡ ❲❡②❧ tr❛♥s❢♦r♠ ♦❢ ❡q✳ ✭✷✳✾✮✱ ✇❡ s❡❡ t❤❛t✿ W−1WAˆ⋆ WBˆ = W−1WA ˆˆB = A ˆˆB. ✭✷✳✶✸✮

✷✳✸ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠✉❧❛t✐♦♥ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s

■♥ t❤❡ ❲❲ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ◗▼✱ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦❢ ♦♣❡r❛t♦rs ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ✉s✐♥❣ t❤❡ ❲✐❣♥❡r✲❲❡②❧ tr❛♥s❢♦r♠❛t✐♦♥✱ ❛♥❞ t❤❡ ▼♦②❛❧ ❡q✉❛t✐♦♥❬✷✶✱ ✷✷❪ ❛❝❝♦✉♥ts ❢♦r t❤❡ ❞②♥❛♠✐❝❛❧ ❡✈♦❧✉t✐♦♥✱ ❥✉st ❛s t❤❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ❢♦r t❤❡ st❛♥❞❛r❞ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ◗▼✳ ❯s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❲❡②❧ tr❛♥s❢♦r♠✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦❢ ♦♣❡r❛t♦rs ❛r❡ ❣✐✈❡♥ ❜②✿ D b GE= ˆ f(z)g(z) d2dz. ✭✷✳✶✹✮ ❙✐♠✐❧❛r❧②✱ ♦♥❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ▼♦②❛❧ ❡q✉❛t✐♦♥✿

✻ ❈❤❛♣t❡r ✷✳ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ∂f ∂t = H ⋆ f− f ⋆ H iℏ := 1 iℏ{H, f }⋆, ✭✷✳✶✺✮ ✇❤❡r❡ H = WHb✐s t❤❡ ♣❤❛s❡ s♣❛❝❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❛♥❞ ✇❤❡r❡ ✇❡ ✐♥tr♦❞✉❝❡❞ t❤❡ ▼♦②❛❧ ❜r❛❝❦❡ts ❞❡✜♥❡❞ ❛s✿ {A, B}_⋆= A ⋆ B − B ⋆ A. ✭✷✳✶✻✮ ❋✉rt❤❡r♠♦r❡✱ ♦♥❡ ❝❛♥ ❛❧s♦ ♣r♦✈❡ t❤❛t ❢♦r st❛t✐♦♥❛r② s②st❡♠s t❤✐s ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ✭s❡❡ ❘❡❢✳ ❬✷✷❪✮✿ H(z) ⋆ f (z) = E f (z) . ✭✷✳✶✼✮

✷✳✹ ▼❛✐♥ Pr♦♣❡rt✐❡s

✷✳✹✳✶ ❚r❛❝❡ ♦❢ ♦♣❡r❛t♦rs

❆ ✉s❡❢✉❧ ♣r♦♣❡rt② ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ tr❛❝❡ ♦❢ ❛♥ ♦♣❡r❛t♦r ❛♥❞ ✐ts ❲✐❣♥❡r tr❛♥s✲ ❢♦r♠✳ ❚❤❡ ♣r♦♦❢ ✐s str❛✐❣❤t❢♦r✇❛r❞✿ ❚❤❡♦r❡♠✿ ❋♦r ❛♥② ♦♣❡r❛t♦r ˆA✱ TrAˆ= ˆ d2dz WAˆ. ✭✷✳✶✽✮ Pr♦♦❢✿ ❯s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❲❡②❧ tr❛♥s❢♦r♠✱ ✇❡ ❤❛✈❡✿ TrAˆ = TrW−1hWAˆi = Tr ˆ d2d_k (2π)2d ˆ d2dz WAˆeikizˆi_e−ikizi ! = ˆ d2dz WAˆTr ˆ d2dk (2π)2d e ikizˆi_e−ikizi ! , ✭✷✳✶✾✮ ❣✐✈❡♥ t❤❛t ❲✐❣♥❡r tr❛♥s❢♦r♠ ✐s ♥♦t ❛♥ ♦♣❡r❛t♦r✱ ❛♥❞ t❤✉s ✐s ♥♦t ❛✛❡❝t❡❞ ❜② ❛ tr❛❝❡✳ ❚❤❡♥✱ ❜② ♣r♦♣❡rt② ♦❢ t❤❡ tr❛❝❡ ❛♥❞ ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ✐♥ k✱

✷✳✹✳ ▼❛✐♥ Pr♦♣❡rt✐❡s ✼ Tr ˆ _d2d_k (2π)2de ikizˆi_e−ikizi ! = ˆ _d2d_k (2π)2de ikizi_e−ikizi = ˆ d2dk (2π)2d = 1, ✭✷✳✷✵✮ ✇❤❡r❡ ✇❡ ❡✈❛❧✉❛t❡❞ t❤❡ tr❛❝❡ ✐♥ t❤❡ ❡✐❣❡♥❜❛s✐s ♦❢ bz ✐♥ t❤❡ ✜rst st❡♣✱ ❛♥❞ t❤✉s TrAˆ = ˆ d2dz WAˆTr ˆ d2dk (2π)2de ikizˆi_e−ikizi ! = ˆ d2dz WAˆ.

✷✳✹✳✷ ■♥✈❛r✐❛♥❝❡ ✉♥❞❡r ♣❤❛s❡ s♣❛❝❡ ✐♥t❡❣r❛t✐♦♥

❖♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s t❤❛t ✇✐❧❧ ❜❡ ✉s❡❞ ✐s t❤❡ ✐♥✈❛r✐❛♥❝❡ ✉♥❞❡r ♣❤❛s❡ s♣❛❝❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ▼♦②❛❧ Pr♦❞✉❝t✳ ❚❤❡♦r❡♠✿ ❋♦r ❛♥② Ωij t❤❛t ❞❡✜♥❡s ❛ ▼♦②❛❧ Pr♦❞✉❝t ⋆✱ ❛♥❞ ❢♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s A, B ❞❡✜♥❡❞ ✐♥ P❤❛s❡✲❙♣❛❝❡✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s tr✉❡✿ ˆ d2dz A ⋆ B= ˆ d2dz AB. ✭✷✳✷✶✮ Pr♦♦❢✿ ❋✐rst✱ ♦♥❡ ❡①♣❛♥❞s t❤❡ ▼♦②❛❧ Pr♦❞✉❝t ✐♥ ✐ts s❡r✐❡s ❡①♣❛♥s✐♦♥✿ ˆ d2dz A ⋆ B= ˆ d2dz AB+ ∞ X n=1 1 n!  iℏ 2 nˆ d2dzA←−∂ziΩij −→ ∂zj n B. ✭✷✳✷✷✮ ◆♦✇ ♦♥❡ ❤❛s ♦♥❧② t♦ ♣r♦✈❡ t❤❛t t❤❡ ✐♥t❡❣r❛❧s ✐♥ t❤❡ s❡❝♦♥❞ t❡r♠ ❛❧❧ ②✐❡❧❞ ③❡r♦✳ ❚❤✐s ✐s ❞♦♥❡ ❜② ❡①♣❛♥❞✐♥❣←−∂ziΩij −→ ∂zj n ✐♥t♦ ✐♥❞✐✈✐❞✉❛❧ t❡r♠s ✭❝❢✳ ❡q✳ ✷✳✶✵✮ ❛♥❞ t❤❡♥ tr❛♥s❢♦r♠✐♥❣ t❤❡ ✐♥t❡❣r❛♥❞ ✐♥t♦ ❛ t♦t❛❧ ❞❡r✐✈❛t✐✈❡ ♠✐♥✉s ❛ t❡r♠ t❤❛t ✐s s②♠♠❡tr✐❝ ✐♥ t❤❡ ❝❤❛♥❣❡ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ♣❛✐r ♦❢ ✐♥❞❡①❡s i, j ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ❛♥ ❛ss♦❝✐❛t❡❞ Ωij ♠❛tr✐① ❡❧❡♠❡♥t✱ ❛♥❞ s✐♥❝❡ t❤❡ Ω ♠❛tr✐① ✐s ❛♥t✐s②♠♠❡tr✐❝✱ t❤❛t t❡r♠ ②✐❡❧❞s ③❡r♦ ✭♥♦t✐❝❡ t❤❛t ♦♥❡ ❛ss✉♠❡s t❤❛t A✱ B ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ✈❛♥✐s❤ ❛t ✐♥✜♥✐t②✮✿  ∂_z(n) α1...zαnA   ∂_z(n) β1...zβnB  Ωα1β1. . .Ωαnβn=

✽ ❈❤❛♣t❡r ✷✳ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ∂z_α1  ∂_z(n−1) α2...zαnA   ∂_z(n) β1...zβnB  Ωα1β1. . .Ωαnβn− −∂_z(n) α1...zαnA   ∂_z(n+1) α1zβ1...zβnB  Ωα1β1. . .Ωαnβn. ✭✷✳✷✸✮ ❙✐♥❝❡ t❤❡ ✜rst t❡r♠ ✐s ❛ t♦t❛❧ ❞❡r✐✈❛t✐✈❡✱ ˆ d2dz ∂z_α1  ∂(n−1)_z α2...zαnA   ∂_z(n) β1...zβnB  Ωα1β1. . .Ωαnβn= 0. ✭✷✳✷✹✮ ❇❡❝❛✉s❡ ∂z_α1∂z_β1 = ∂z_β1∂z_α1 ❛♥❞ Ωz_α1z_β1 ✐s ❛♥t✐s②♠♠❡tr✐❝ ✐♥ t❤❡ ❡①❝❤❛♥❣❡ ♦❢ ✐♥❞❡①❡s α1↔ β1✱  ∂_z(n) α1...zαnA   ∂_z(n+1) α1zβ1...zβnB  Ωα1β1. . .Ωαnβn= 0, ✭✷✳✷✺✮ ❛♥❞ t❤❡r❡❢♦r❡ ∞ X n=1 1 n!  iℏ 2 nˆ d2dz A←−∂ziΩij −→ ∂zj n B= 0, ✭✷✳✷✻✮ ✇❤✐❝❤ ♠❡❛♥s ˆ d2dz A ⋆ B= ˆ d2dz AB. ✭✷✳✷✼✮

❈❤❛♣t❡r ✸

❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◆♦♥✲

❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s

✸✳✶ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ❲✐❣♥❡r ❚r❛♥s❢♦r♠

■♥ ♦r❞❡r t♦ ❞❡s❝r✐❜❡ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✭◆❈◗▼✮ ✐♥ P❤❛s❡ ❙♣❛❝❡✱ ♦♥❡ ♥❡❡❞s ❛ ♣r♦♣❡r ✇❛② t♦ ♠❛♣ ◆❈◗▼ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡ ✐♥t♦ ❢✉♥❝t✐♦♥s ✐♥ R2d_✳ ❚❤✐s ♠❡❛♥s ✜♥❞✐♥❣ ❛ ♦♥❡✲t♦✲♦♥❡ ❧✐♥❡❛r ♠❛♣ V s♦ t❤❛t✿ ✶✳ V (Id) = 1✱ ✷✳ V (bq) = q✱ ✸✳ V (bp) = p✱ ✹✳ V _{A b}ˆ_B_{= V} _Aˆ_{⋆N CV} _Bb_✱ ❢♦r ❛ ♣❤❛s❡ s♣❛❝❡ ✇✐t❤ ❞❡❢♦r♠❡❞ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s ♦❢ t❤❡ ✉s✉❛❧ ❍❡✐s❡♥❜❡r❣✲❲❡②❧ ❛❧❣❡❜r❛ ✭s❡❡ ❡q✳ ✭✶✳✷✮✮✿

[qi, qj] = iθij, [pi, pj] = iηij, [qi, pj] = iℏ′δij, ✭✸✳✶✮

✇❤❡r❡ θ ❛♥❞ η ❛r❡ r❡❛❧ ❛♥t✐✲s②♠♠❡tr✐❝ ♠❛tr✐❝❡s✳

❍♦✇❡✈❡r✱ t❤✐s ♠❛♣ ✐s ♥♦t ✉♥✐q✉❡✱ ❛♥❞ t❤✉s t❤❡r❡ ❛r❡ s❡✈❡r❛❧ ♣r♦♣♦s❛❧s ❢♦r t❤✐s ♠❛♣✳ ■♥ ❘❡❢✳ ❬✶✹❪ t❤✐s ✐ss✉❡ ✐s ❞✐s❝✉ss❡❞ ❛♥❞ ❞✐✛❡r❡♥t ♠❛♣s ❛r❡ ❝♦♠♣❛r❡❞✳ ■t ✐s ❛r❣✉❡❞ t❤❛t ❛ s✉✐t❛❜❧❡ ♠❛♣ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶✵ ❈❤❛♣t❡r ✸✳ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◆♦♥✲ ❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s WN C  ˆ A(z) = h−d ˆ ddy ddx e−iΠ(z)·yδ(x − R(z))  x+ℏ 2y| ˆA|x − ℏ 2y  b R , ✭✸✳✷✮ ✇❤❡r❡ x, y ❛r❡ ♣♦s✐t✐♦♥s✱ R(z), Π (z) ❛r❡ ❝❛♥♦♥✐❝❛❧❧② ❝♦♥❥✉❣❛t❡❞ ✈❛r✐❛❜❧❡s t❤❛t ❛r❡ r❡❧❛t❡❞ t♦ z ✈✐❛ ❛ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥✱ t♦ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ✭s❡❡ ❘❡❢✳ ❬✷✸❪ ❢♦r ❛ ❞❡t❛✐❧❡❞ ❡①♣❧❛♥❛t✐♦♥✮✳ ■t ✐s ❛❧s♦ ❛r❣✉❡❞ t❤❛t t❤✐s ♠❛♣ ❤❛s ❡①❛❝t❧② t❤❡ s❛♠❡ ♣r♦♣❡rt✐❡s ❛s ✐♥ s❡❝t✐♦♥ ✷✳✹✱ ✐✳❡✳✿ WN C  ˆ A bB= WN C  ˆ A⋆N CWN C  b B, ✭✸✳✸✮ TrAˆ= ˆ d2dz WN C  ˆ A, ✭✸✳✹✮ ˆ d2dz A ⋆N CB= ˆ d2dz AB, ✭✸✳✺✮ ✇✐t❤ A ⋆N CB= Ae iℏ′ 2 ←− ∂_ziΩN C ij −→ ∂_zj B, ✭✸✳✻✮ ✇❤❡r❡ t❤❡ Ω ♠❛tr✐① ✐s ♥♦✇ ΩN C ₌ 1 ℏΘ Idd×d −Idd×d ℏ1N ! , ✭✸✳✼✮ ❛♥❞ ✇❤❡r❡ Θ = (θij) , N = (ηij) ✭✸✳✽✮ ❛r❡ t❤❡ ♠❛tr✐❝❡s ♦❢ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s ❡❧❡♠❡♥ts✳ ❚❤❡s❡ ♠❛♣s ❛❧❧♦✇s ❢♦r ◆❈◗▼ t♦ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ ♣❤❛s❡✲s♣❛❝❡ ❥✉st ❛s ◗▼✳ ❖♥❡ ♥♦✇ ♥❡❡❞s ❛ ✇❛② t♦ ❝♦♥♥❡❝t ✈❛r✐❛❜❧❡s ✐♥ ◆❈◗▼ ❛♥❞ ✈❛r✐❛❜❧❡s ✐♥ ◗▼✳

✸✳✷ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥

❚❤❡ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥✱ ♦r ❙❡✐❜❡r❣✲❲✐tt❡♥ ♠❛♣✱ ✐s ❛ ♥♦♥✲❝❛♥♦♥✐❝❛❧ ❧✐♥❡❛r tr❛♥s✲ ❢♦r♠❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ s❡ts ♦❢ ♣❤❛s❡ s♣❛❝❡ ✈❛r✐❛❜❧❡s ✇✐t❤ ❞✐✛❡r❡♥t ❝♦♠♠✉t❛t✐♦♥ r❡❧❛✲ t✐♦♥s✱ ✉s✉❛❧❧② ❜❡t✇❡❡♥ ❝♦♠♠✉t❛t✐✈❡ ✈❛r✐❛❜❧❡s t❤❛t ♦❜❡② t❤❡ ❍❡✐s❡♥❜❡r❣✲❲❡②❧ ❛❧❣❡❜r❛ ❛♥❞ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ✈❛r✐❛❜❧❡s t❤❛t ♦❜❡② t❤❡ ❛❧❣❡❜r❛ s❤♦✇♥ ❛❜♦✈❡✱ ❛♥❞ t❤❛t ✇❡ r❡♣❡❛t ❢♦r ❝♦♥✲ ✈❡♥✐❡♥❝❡✿

✸✳✷✳ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥ ✶✶ [qi, qj] = iθij, ✭✸✳✾✮ [pi, pj] = iηij, ✭✸✳✶✵✮ [qi, pj] = iℏ′δij. ✭✸✳✶✶✮ ❚❤❡ ❛✐♠ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ ✇♦r❦ ♦✉t t❤✐s tr❛♥s❢♦r♠❛t✐♦♥✱ ❡st❛❜❧✐s❤ ✐ts ♣r♦♣❡rt✐❡s ✐♥ ♦r❞❡r t♦ s♣❡❝✐❢② ❛ ❧✐♥❦ ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s ✐♥ ◆❈◗▼ ✇✐t❤ t❤❡ ♦♥❡s ✐♥ ◗▼✳ ■♥ ♦r❞❡r t♦ r❡❧❛t❡ ❝♦♠♠✉t❛t✐✈❡ ❛♥❞ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ✈❛r✐❛❜❧❡s✱ ❝♦♥s✐❞❡r ❛ ❧✐♥❡❛r tr❛♥s✲ ❢♦r♠❛t✐♦♥ t♦ ❝❤❛♥❣❡ t❤❡ ✈❛r✐❛❜❧❡s z = (qi, pi) t❤❛t ♦❜❡② t❤❡ ❛❜♦✈❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s ✐♥t♦ st❛♥❞❛r❞ ❝♦♠♠✉t❛t✐✈❡ ✈❛r✐❛❜❧❡s zC _{= q}C i , pCi
_{t❤❛t s❛t✐s❢② t❤❡ ❍❡✐s❡♥❜❡r❣✲❲❡②❧ ❝♦♠✲} ♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s✿ qi= AijqCj + BijpCj, ✭✸✳✶✷✮ pi= CijqjC+ DijpCj. ✭✸✳✶✸✮ ❚❤❡♦r❡♠ ✭s❡❡ ❘❡❢✳ ❬✶✹❪✮✿ ❚❤❡ ♠❛tr✐❝❡s A✱ B✱ C ❛♥❞ D ♦❜❡② t❤❡ r❡❧❛t✐♦♥s❤✐♣s✿ ADT− BCT =ℏ ′ ℏIdd×d, ✭✸✳✶✹✮ ABT − BAT ₌ 1 ℏΘ, ✭✸✳✶✺✮ CDT − DCT ₌ 1 ℏN, ✭✸✳✶✻✮ ✇❤❡r❡ Θ = (θij) , N = (ηij) ✭✸✳✶✼✮ ❛r❡ t❤❡ ♠❛tr✐❝❡s ♦❢ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s ❡❧❡♠❡♥ts ❢♦r ♣♦s✐t✐♦♥ ❛♥❞ ♠♦♠❡♥t❛✱ r❡s♣❡❝✲ t✐✈❡❧②✱ ❢♦r t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ✈❛r✐❛❜❧❡s✳

✶✷ ❈❤❛♣t❡r ✸✳ ❲❡②❧✲❲✐❣♥❡r ❋♦r♠❛❧✐s♠ ♦❢ ◆♦♥✲ ❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s Pr♦♦❢ ♦❢ ❡q✳ ✭✸✳✶✹✮✿ iθij = [qi, qj] = AikAjl  q_kC, q_lC+ AikBjl  qC_k, pC_l  −BikAjlqlC, pCk  + BikBjlpCk, pCl  = iℏ(AikBjlδkl− BikAjlδkl) = iℏ AikBkjT − BikATkj
= iℏ ABT− BAT
ij. ✭✸✳✶✽✮ Pr♦♦❢ ♦❢ ❡q✳ ✭✸✳✶✺✮✿ iηij = [pi, pj] = CikCjlqkC, qlC  + CikDjlqkC, pCl  −DikCjlqlC, pCk  + DikDjlpCk, pCl  = iℏ(CikDjlδkl− DikCjlδkl) = iℏ CikDTkj− DikCkjT
= iℏ CDT − DCT
ij. ✭✸✳✶✾✮ Pr♦♦❢ ♦❢ ❡q✳ ✭✸✳✶✻✮✿ iℏ′δij = [qi, pj] = AikCjlqkC, qlC  + AikDjlqkC, pCl  −BikCjlqlC, pCk  + BikDjlpCk, pCl  = iℏ(AikDjlδkl− BikCjlδkl) = iℏ AikDkjT − BikCkjT
= iℏ ADT _{− BC}T
ij. ✭✸✳✷✵✮ ◆♦t❡✱ ❤♦✇❡✈❡r✱ t❤❛t t❤❡s❡ r❡❧❛t✐♦♥s ❛r❡ ♥♦t ❡♥♦✉❣❤ t♦ ❢✉❧❧② ❞❡t❡r♠✐♥❛t❡ t❤❡ ♠❛tr✐❝❡s Θ ❛♥❞ N✱ ❛s t❤❡r❡ ❛r❡ 4d2 _{♣❛r❛♠❡t❡rs ❛♥❞} d 2(3d − 1)✐♥❞❡♣❡♥❞❡♥t ❡q✉❛t✐♦♥s✱ ❛♥❞ t❤✉s t❤❡r❡ ❛r❡ d 2(5d + 1)❢r❡❡ ♣❛r❛♠❡t❡rs✳ ❖♥❡ ❝❛♥✱ ❤♦✇❡✈❡r✱ ❝♦♥s✐❞❡r s✐♠♣❧✐✜❝❛t✐♦♥s✳ ❋♦r ✐♥st❛♥❝❡✱ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ✇❡ ❝❛♥ t❛❦❡ A ❛♥❞ D t♦ ❜❡ t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✳ ❲❡ t❤❡♥ ❤❛✈❡✿

✸✳✷✳ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥ ✶✸ −BCT ₌ℏ′ ℏIdd×d, ✭✸✳✷✶✮ BT− B = 1 ℏΘ, ✭✸✳✷✷✮ C− CT ₌1 ℏN. ✭✸✳✷✸✮ ❚❤❡ ♠♦st ❝♦♠♠♦♥❧② ❝♦♥s✐❞❡r❡❞ ❝❛s❡ ❤❛s Θ = θǫij ❛♥❞ N = ηǫij✱ ✇❤❡r❡ ǫij = ǫijk ✇✐t❤ k6= i, j ✐s ❛♥t✐s②♠♠❡tr✐❝ ✐♥ i, j✳ ❚❤❡♥✱ ✇❡ ❝❛♥ t❛❦❡✿ B= Θ 2ℏ, C= − N 2ℏ. ✭✸✳✷✹✮ ❚❤✉s✱ ✇❡ t❤❡♥ ❡♥❞ ✉♣ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥✿ qi = qiC+ θ 2ℏǫijp C j, ✭✸✳✷✺✮ pi= pCi − η 2ℏǫijq C j . ✭✸✳✷✻✮ ❋✐♥❛❧❧②✱ ♥♦t❡ t❤❛t ℏ′ _{✭❝✳❢✳ ❡qs✳ ✭✸✳✶✶✮ ❛♥❞ ✭✸✳✷✶✮✮ ❛♥❞ ℏ ❛r❡ t❤✉s r❡❧❛t❡❞ ❜②✿} ℏ′_{= ℏ}  1 + θη 4ℏ2  . ✭✸✳✷✼✮

❈❤❛♣t❡r ✹

◆♦✲●♦ ❚❤❡♦r❡♠s ✐♥ ◗✉❛♥t✉♠

▼❡❝❤❛♥✐❝s

■♥ t❤✐s ❈❤❛♣t❡r ✇❡ ♣r♦✈✐❞❡ ❛♥ ♦✈❡r✈✐❡✇ ♦❢ ◆♦✲●♦ ❚❤❡♦r❡♠s ✐♥ ◗▼✳ ❋✐rst ✇❡ st❛rt ❜② ♣r♦✈✐♥❣ t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ◗▼✱ ❛♥❞ t❤❡♥ ♣r♦✈✐❞❡ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ❢♦r ◆♦✲●♦ ❚❤❡♦r❡♠s t❤❛t ✐♥❝❧✉❞❡ t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❛♥❞ ◆♦✲❉❡❧❡t✐♥❣ ❚❤❡♦r❡♠ ❛s s♣❡❝✐❛❧ ❝❛s❡s✳

✹✳✶ ◆♦✲❈❧♦♥✐♥❣ ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s

❚❤❡ ❝♦♥❝❡♣t ♦❢ ❝❧♦♥✐♥❣ ✐s ❛ s✐♠♣❧❡ ♦♥❡✳ ❚❤❡ ✐❞❡❛ ✐s t♦ t❛❦❡ ❛ ❣❡♥❡r✐❝ s②st❡♠ ❛♥❞ ❛♥ ❡♠♣t② ♦♥❡✱ ❛♥❞ ❡✈♦❧✈❡ t❤❡♠ s♦ t❤❛t ♦♥❡ ❡♥❞s ✉♣ ✇✐t❤ t✇♦ ❝♦♣✐❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠✳ ❍♦✇❡✈❡r✱ ❛s ✇❡ ✇✐❧❧ s❡❡ ♥❡①t✱ ❛ss✉♠✐♥❣ ❝❧♦♥✐♥❣ ❢♦r ❛ ❣❡♥❡r✐❝ st❛t❡ r❡str✐❝ts t❤❡ st❛t❡s t❤❛t ❝❛♥ ❜❡ ❝❧♦♥❡❞✱ ❛♥❞ t❤✉s ❝❧♦♥✐♥❣ ❝❛♥♥♦t ❜❡ ❞♦♥❡ ✐♥ ❣❡♥❡r❛❧✐t②✳ ❚❤❡♦r❡♠ ✭◆♦✲❈❧♦♥✐♥❣ ✐♥ ◗▼✮✿ ▲❡t |ψi ❜❡ ❛ ❣❡♥❡r✐❝ q✉❛♥t✉♠ st❛t❡ ❛♥❞ ❧❡t |0i ❜❡ ❛♥ ❡♠♣t② st❛t❡✳ ❚❤❡♥ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❡✈♦❧✈❡ t❤❡s❡ t✇♦ st❛t❡s ✐♥t♦ t✇♦ ❝♦♣✐❡s ♦❢ |ψi ❢♦r ❛♥② q✉❛♥t✉♠ st❛t❡ |ψi ✳ Pr♦♦❢✿ ❚❤❡ ❝❧❛ss✐❝❛❧ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ✐s ❞♦♥❡ ❜② r❡❞✉❝t✐♦ ❛❞ ❛❜s✉r❞✉♠✳ ❆ss✉♠❡ ❝❧♦♥✐♥❣ ✐s ♣♦ss✐❜❧❡✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ❍❛♠✐❧t♦♥✐❛♥✱ bH✱ ✇✐t❤ ❛♥ ❛ss♦❝✐❛t❡❞ ✉♥✐t❛r② t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r✱ bU = eℏi ´ _b Hdt_{✱ s♦ t❤❛t} b

U|ψi_A|0i_B = |ψi_A|ψi_B, ✭✹✳✶✮

❢♦r ❛♥② ❣✐✈❡♥ ✉♥❦♥♦✇♥ st❛t❡ |ψi ❛♥❞ ❛♥ ❡♠♣t② st❛t❡ |0i✳ ✶✺

✶✻ ❈❤❛♣t❡r ✹✳ ◆♦✲●♦ ❚❤❡♦r❡♠s ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❚❤❡♥✱ ✐❢ |φi ✐s ❛♥♦t❤❡r st❛t❡✱ ✐✳❡✳✱

b

U|φi_A|0i_B= |φi_A|φi_B, ✭✹✳✷✮

♦♥❡ ❤❛s✿

hφ|ψi = hφ|ψi_Ah0|0i_B

= (hφ|_Ah0|_B) (|ψi_A|0i_B) = (hφ|_Ah0|_B) bU†Ub(|ψi_A|0i_B) = (hφ|_Ahφ|_B) (|ψi_A|ψi_B)

= hφ|ψi2, ✭✹✳✸✮

✇❤❡r❡ ✐t ✇❛s ✉s❡❞ t❤❛t bU†_Ub _{= Id❛♥❞ h0|0i = 1✳}

❚❤✉s✱ hφ|ψi = 0 ♦r hφ|ψi = 1✱ ✇❤✐❝❤ ❝❛♥♥♦t ❜❡ tr✉❡ ❢♦r ❛❧❧ st❛t❡s|φi ❛♥❞ |ψi✳ ❚❤❡ ❝♦♥tr❛❞✐❝t✐♦♥ ❛r✐s❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✇❡ ❛ss✉♠❡❞ t❤❛t ❝❧♦♥✐♥❣ ✇❛s ♣♦ss✐❜❧❡ ❢♦r ❛♥② ❣✐✈❡♥ st❛t❡✳ ❍❡♥❝❡✱ t❤❡r❡ ❝❛♥ ❜❡ ♥♦ ❝❧♦♥✐♥❣✳

◆♦t❡ t❤❛t ❛ s✐♠✐❧❛r ♣r♦♦❢ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ✐♥ r❡✈❡rs❡❞ ♦r❞❡r✱ ✐✳❡✳ t❤❡r❡ ✐s ♥♦ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r bU′ _{s♦ t❤❛t✿}

b

U′|ψi_A|ψi_B = |ψi_A|0i_B ✭✹✳✹✮

❢♦r ❛♥② ❣❡♥❡r✐❝ q✉❛♥t✉♠ st❛t❡ |ψi✱ t❤✐s ❜❡✐♥❣ t❤❡ ◆♦✲❉❡❧❡t✐♥❣ ❚❤❡♦r❡♠✳ ❚❤✐s ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t✐♠❡✲r❡✈❡rs❡❞ ❞✉❛❧ ♦❢ t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠✱ ❛s ❜② ❞❡✜♥✐t✐♦♥ ♦❢ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r b U(t0, t1) bU(t1, t0) = Id, ✭✹✳✺✮ ❛♥❞ t❤✉s b

U(t0, t1) |ψi_A|0i_B = |ψi_A|ψi_B ⇐⇒ bU(t1, t0) |ψi_A|ψi_B = |ψi_A|0i_B. ✭✹✳✻✮

✹✳✷ ◆♦✲●♦ ●❡♥❡r❛❧✐③❛t✐♦♥ ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ✐♥tr♦❞✉❝❡ ❛ t❤❡♦r❡♠ str♦♥❣❡r t❤❛♥ t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❛♥❞ ◆♦✲❉❡❧❡t✐♥❣ ❚❤❡♦r❡♠s✱ ❛♥❞ ✇❤✐❝❤ ✐♥❝❧✉❞❡s ❜♦t❤ ❛s s♣❡❝✐❛❧ ❝❛s❡s✳ ❚❤❡ ❦❡② r❡s✉❧t ♦❢ t❤❡ t❤❡♦r❡♠ ✐s t❤❛t

✹✳✷✳ ◆♦✲●♦ ●❡♥❡r❛❧✐③❛t✐♦♥ ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✶✼ ♦♥❡ ❝❛♥♥♦t t❛❦❡ ♦♥❡ ♦r ♠♦r❡ ❝♦♣✐❡s ♦❢ q✉❛♥t✉♠ s②st❡♠s ❛♥❞ ♣❛rt✐❛❧❧② s✉♣❡r♣♦s❡ t❤❡♠ ✇✐t❤ ❛ ✜①❡❞ st❛t❡✳ ❚❤✐s ♣r♦♦❢ ✐s s✐♠✐❧❛r t♦ ♦♥❡ ❣✐✈❡♥ ✐♥ ❘❡❢✳ ❬✷✹❪✳ ❚❤❡♦r❡♠✿ ❚❤❡r❡ ✐s ♥♦ ❍❛♠✐❧t♦♥✐❛♥✱ bH✱ ✇✐t❤ ❛♥ ❛ss♦❝✐❛t❡❞ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r✱ bU✱ s♦ t❤❛t ❢♦r ❛ ✜①❡❞ st❛t❡ |φi ❛♥❞ ❢♦r ❛♥② st❛t❡ |ψi t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s tr✉❡✿ b

U|ψi⊗k|0i⊗N −k = |ϕi⊗n|0i⊗N −n, ✭✹✳✼✮ ✇❤❡r❡ |ϕi = α |ψi + β |φi✱ ✇✐t❤ |α|2

+ |β|2= 1✱ ❛♥❞ ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❡ ♥♦t❛t✐♦♥✿

|ψi⊗k= |ψi ⊗ . . . ⊗ |ψi

k times . ✭✹✳✽✮ Pr♦♦❢✿ ❋✐rst✱ ♥♦t❡ t❤❛t ✇❤❡♥ β = 0 ✭✐✳❡✳ t❤❡r❡ ✐s ♥♦ s✉♣❡r♣♦s✐t✐♦♥ ✇✐t❤ ❛♥♦t❤❡r st❛t❡✮✱ ✐❢ k < n✱ ✇❡ ❤❛✈❡ t❤❡ ◆♦✲❈❧♦♥✐♥❣ ❚❤❡♦r❡♠✱ ❛♥❞ ✐❢ k > n✱ ✇❡ ❤❛✈❡ t❤❡ ◆♦✲❉❡❧❡t✐♥❣ ❚❤❡♦r❡♠✳ ❚❤✉s✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ♣r♦✈❡ t❤❡ ❝❛s❡ 0 < β < 1✳ ❚❤❡ ❣♦❛❧ ✐s t♦ s❤♦✇ t❤❛t t❤✐s ✐♠♣❧✐❡s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ✐✳❡✳ t❤❡ r❡❞✉❝t✐♦ ❛❞ ❛❜s✉r❞✉♠ ♠❡t❤♦❞✳ ❙✉♣♣♦s❡ t❤❡r❡ ✐s ❛♥ ❍❛♠✐❧t♦♥✐❛♥✱ bH✱ s♦ t❤❛t b

U|ψi⊗k|0i⊗N −k = |ϕi⊗n|0i⊗N −n ✭✹✳✾✮ ❢♦r ❛♥② st❛t❡ |ψi ❛♥❞ ✇✐t❤ |ϕi = α |ψi + β |φi✳

❚❤❡♥✱ ✐❢ ✇❡ ✐♥st❡❛❞ ✉s❡❞ t❤❡ st❛t❡ eiθ_{|ψi✱ ✇❡ ✇♦✉❧❞ ❤❛✈❡}

b

U eikθ_|ψi⊗k

|0i⊗N −k = |ϕ′_i⊗n_|0i⊗N −n

, ✭✹✳✶✵✮

✇❤❡r❡ |ϕ′_{i = αe}iθ_{|ψi + β |φi✳}

❍♦✇❡✈❡r✱ ❜❡❝❛✉s❡ |ψi ∝ eiθ_{|ψi✱ ❛ss✉♠✐♥❣ t❤❛t st❛t❡s ❛r❡ ♥♦r♠❛❧✐③❡❞✱ ✇❡ t❛❦❡ t❤❡ ❤❡r♠✐t✐❝}

❝♦♥❥✉❣❛t❡ ♦❢ ❡q✳ ✭✹✳✾✮ ❛♥❞ ♠✉❧t✐♣❧② ✐t ❜② ❡q✳ ✭✹✳✶✵✮ t♦ ♦❜t❛✐♥✿ eikθ= hϕ|ϕ′_in . ✭✹✳✶✶✮ ❚❤❡♥✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ |ϕi ❛♥❞ |ϕ′_i✱ hϕ|ϕ′i = eiθ|α|2+ |β|2, ✭✹✳✶✷✮ ❛♥❞ t❤✉s✱ ❜❡❝❛✉s❡ |α|2 _{❛♥❞ |β|}2_{❛r❡ r❡❛❧ ♥✉♠❜❡rs✱ ❡q✳ ✭✹✳✶✶✮ ❝❛♥ ♦♥❧② ❜❡ tr✉❡ ✐❢ β = 0✱ ✇❤✐❝❤} ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤✐s ❝♦♥❝❧✉❞❡s t❤❡ ♣r♦♦❢✳

❈❤❛♣t❡r ✺

❇r✐❞❣✐♥❣ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡

◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ ◗✉❛♥t✉♠

▼❡❝❤❛♥✐❝s

❇❡❢♦r❡ ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠s ✐♥ ◆❈◗▼✱ ♦♥❡ ♥❡❡❞s ❛ ✇❛② t♦ r❡❧❛t❡ st❛t❡s ✐♥ ◗▼ ✇✐t❤ st❛t❡s ✐♥ ◆❈◗▼✳ ❚❤✐s ❝❤❛♣t❡r ✐♥t❡♥❞s t♦ s❤♦✇ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ˆ A−→ W−1_{D◦ W} N C h ˆ Ai ✭✺✳✶✮ ❛❧❧♦✇s ❢♦r ❞❡s❝r✐❜✐♥❣ st❛t❡s ✐♥ ◆❈◗▼ t❤r♦✉❣❤ st❛t❡s ✐♥ ◗▼ ✈✐❛ ❛ ♠♦❞✐✜❡❞ ❍❛♠✐❧t♦♥✐❛♥✱ ❢♦r ❛ ♠❛♣ WN C✱ ❛♥❞ ✇❤❡r❡ D ✐s t❤❡ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r ✈❛r✐❛❜❧❡s ✐♥ ♣❤❛s❡✲s♣❛❝❡ ❞❡s❝r✐❜❡❞ ✐♥ ❈❤❛♣t❡r ✸✳

✺✳✶ ❚r❛♥s❢♦r♠✐♥❣ ❖♣❡r❛t♦rs ✐♥ ◆❈◗▼ t♦ ❖♣❡r❛t♦rs ✐♥

◗▼

❆s ✇❡ s❛✇ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡rs✱ t❤❡ ❲✐❣♥❡r ❚r❛♥s❢♦r♠ ♠❛♣s ◗▼ ♦♣❡r❛t♦rs t♦ ♣❤❛s❡✲ s♣❛❝❡ ❢✉♥❝t✐♦♥s✱ ❛♥❞ ✐ts ✐♥✈❡rs❡✱ t❤❡ ❲❡②❧ tr❛♥s❢♦r♠✱ ♠❛♣s t❤❡s❡ ♣❤❛s❡✲s♣❛❝❡ ❢✉♥❝t✐♦♥s ❛❣❛✐♥ t♦ ♦♣❡r❛t♦rs ✐♥ ◗▼✱ ✐✳❡✳ W : HC−→ C _R2d_{, ✭✺✳✷✮} ✶✾

✷✵ ❈❤❛♣t❡r ✺✳ ❇r✐❞❣✐♥❣ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s W−1: CR2d_{−→ HC}. ✭✺✳✸✮ ❙✐♠✐❧❛r❧②✱ ❛♥② ◆❈ ❲✐❣♥❡r ❚r❛♥s❢♦r♠ ♠❛♣s ◆❈◗▼ ♦♣❡r❛t♦rs t♦ ♣❤❛s❡✲s♣❛❝❡ ❢✉♥❝t✐♦♥s✱ ✐✳❡✳ WN C : HN C −→ CR2d. ✭✺✳✹✮ ■♥ ♦r❞❡r t♦ st✉❞② st❛t❡s ✐♥ ◆❈◗▼✱ ✇❡ ♥♦✇ ✐♥t❡♥❞ t♦ ✉s❡ ❡qs✳ ✭✺✳✸✮ ❛♥❞ ✭✺✳✹✮ t♦ ♠❛♣ ♦♣❡r❛t♦rs ✐♥ ◆❈◗▼ t♦ s♦♠❡ ♦t❤❡r ♦♣❡r❛t♦rs ✐♥ ◗▼ ❜② ✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t WN C ②✐❡❧❞s ❢✉♥❝t✐♦♥s✱ ❛♥❞ W−1 _{❤❛s ❢✉♥❝t✐♦♥s ❛s ❛r❣✉♠❡♥ts✳ ❚❤✐s ♠❡❛♥s✱ ❢♦r ❛♥ ♦♣❡r❛t♦r bA✱ ♦♥❡ ❝❛♥} ❞❡✜♥❡ ❛♥ ♦♣❡r❛t♦r bA❞❡✜♥❡❞ ❛s b A = W−1WN C h b Ai ✭✺✳✺✮ ✐♥ ◗▼✳ ◆♦t❡ t❤❛t t❤✐s ✐s s✐♠♣❧② ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t✱ ❛♥❞ ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ❤❛✈❡ t❤❡ s❛♠❡ ♣❤②s✐❝❛❧ ♠❡❛♥✐♥❣ ❛s ˆA✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ♦♥❡ ❝♦♥s✐❞❡rs ❛ ♣r♦❞✉❝t ♦❢ t✇♦ ♦♣❡r❛t♦rs✱ ˆA ❛♥❞ bB✱ r❡♠❡♠❜❡r✐♥❣ t❤❛t ✭s❡❡ ❡qs✳ ✭✷✳✾✮ ❛♥❞ ✭✸✳✸✮✮ WN C h b A bBi= WN C h b Ai⋆N CWN C h b Bi WhA bbBi= WhAbi⋆CW h b Bi, ✇❡ ❣❡t✿ W−1WN C h b A bBi = W−1WN C h b Ai⋆N CWN C h b Bi 6= W−1WN C h b AiW−1WN C h b Bi, ✭✺✳✻✮ s✐♥❝❡ W−1_{❤❛s t❤❡ ♣r♦♣❡rt②} W−1(f ⋆Cg) = W−1(f ) W−1(g) ✭✺✳✼✮ ❢♦r ❢✉♥❝t✐♦♥s f ❛♥❞ g✳ ❚❤❡r❡❢♦r❡✱ ✐❢ ✇❡ ✜♥❞ ❛ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s T ✱ ✐✳❡ z′_i= (T ◦ z)_i= Sijzj ✭✺✳✽✮

✺✳✷✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s T ✷✶ ❛♥❞ T◦ f (z) = f (z′_{) = f (S} ijzj) , ✭✺✳✾✮ s♦ t❤❛t W−1T◦WN C h ˆ Ai⋆N CWN C h b Bi= = W−1_{T◦ W} N C h ˆ Ai⋆C  T◦ WN C h b Bi, ✭✺✳✶✵✮ ✇❡ ❝❛♥ ❞❡✜♥❡ b O = W−1T◦ WN C h b Oi ✭✺✳✶✶✮ ❢♦r ❛♥② ♦♣❡r❛t♦r bO ✐♥ ◆❈◗▼✱ ❛♥❞ ❣❡t b A bB = W−1T ◦ WN C h ˆ A bBi. ✭✺✳✶✷✮ ❚❤❡ ♣r♦❜❧❡♠ ♥♦✇ ✐s t♦ ❝❛❧❝✉❧❛t❡ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ T ✳

✺✳✷ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s T

❇❡❝❛✉s❡ ♦❢ ❡q✳ ✭✺✳✶✵✮✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s T s♦ t❤❛t ❢♦r ❛♥② t✇♦ ♣❤❛s❡✲s♣❛❝❡ ❢✉♥❝t✐♦♥s f ❛♥❞ g T ◦ (f (z) ⋆N Cg(z)) = (T ◦ f (z)) ⋆C(T ◦ g (z)) . ✭✺✳✶✸✮ ❊①♣❛♥❞✐♥❣ t❤❡ ▼♦②❛❧ ♣r♦❞✉❝ts ✭s❡❡ ❡qs✳ ✭✷✳✶✵✮✮✱ ✇❡ ❣❡t✿ T◦f eiℏ′2 ←− ∂_ziΩN C ij −→ ∂_zj g= (T ◦ f ) eiℏ2 ←− ∂_ziΩC ij −→ ∂_zj (T ◦ g) , ✭✺✳✶✹✮ ✇❤❡r❡ ΩC ₌ 0 Idd×d −Idd×d 0 ! , ΩN C ₌ 1 ℏΘ Idd×d −Idd×d ℏ1N ! , ✭✺✳✶✺✮ ❛s ❞❡✜♥❡❞ ♣r❡✈✐♦✉s❧②✳ ❆ss✉♠✐♥❣ T ✐s ❛ ❧✐♥❡❛r tr❛♥❢♦r♠❛t✐♦♥✱ s✐♠✐❧❛r t♦ ✇❤❛t ✇❛s ❞♦♥❡ ❢♦r t❤❡ ❉❛r❜♦✉① tr❛♥s✲ ❢♦r♠❛t✐♦♥✱ ✇❡ ❝❛♥ ✇r✐t❡✿ zi−→ zi′= (T ◦ z)i= Sijzj, ✭✺✳✶✻✮ ✇✐t❤ i, j = 1, ..., 2d✱ ♦r✱ s❡♣❛r❛t✐♥❣ ♣♦s✐t✐♦♥s ❢r♦♠ ♠♦♠❡♥t❛✱

✷✷ ❈❤❛♣t❡r ✺✳ ❇r✐❞❣✐♥❣ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s qi T −→ q′ i= αijqj+ βijpj, ✭✺✳✶✼✮ pi T −→ p′ i= γijqj+ ζijpj, ✭✺✳✶✽✮ ✇✐t❤ i, j = 1, ..., 2d✱ ✇❤❡r❡ ✇❡ ❞❡✜♥❡❞ S= (Sij) = α β γ ζ ! , ✭✺✳✶✾✮ ✇❤❡r❡ α, β, γ, ζ ❛r❡ t❤❡ ♠❛tr✐❝❡s ✇✐t❤ ❝♦❡✜❝✐❡♥ts αij, βij, γij, ζij✱ r❡s♣❡❝t✐✈❡❧②✳ ❇❡❝❛✉s❡ ✇❡ ❝❛♥ ❡①♣❛♥❞ t❤❡ ▼♦②❛❧ ♣r♦❞✉❝t ✭❡q✳ ✭✷✳✶✵✮✮ ❛s✿ f ⋆ g= f g + ∞ X n=1 (iℏ)n n!2n  ∂_z(n) α1...zαnf   ∂_z(n) β1...zβng  Ωα1β1. . .Ωαnβn, ✭✺✳✷✵✮ ✇❡ ♥❡❡❞ t♦ ❝❛❧❝✉❧❛t❡ t❡r♠s ❧✐❦❡ ∂zif(T ◦ z)✿ ∂zif(T ◦ z) = ∂zif(z ′₎ = ∂z ′ j ∂zi ∂ ∂z′ j f(z′) = Sji∂z′ jf(z ′_{) ✭✺✳✷✶✮} ❚❤❡r❡❢♦r❡✱ (∂zi(T ◦ f )) ∂zj(T ◦ g)
ΩC ij =  ∂z′ kf(z ′₎_S kiSlj  ∂z′ lg(z ′₎_ΩC ij = ∂z′ kf(z ′₎ _∂ z′ lg(z ′₎_S kiΩCijSjlT ✭✺✳✷✷✮ ❚❤✉s✱ ❡①♣❛♥❞✐♥❣ ❡q✳ ✭✺✳✶✸✮ ✉s✐♥❣ ❡q✳ ✭✺✳✷✵✮ ②✐❡❧❞s s✐♠♣❧② ℏSkiΩCijSjlT = ℏ′ΩN Ckl , ✭✺✳✷✸✮ ♦r✱ ✐♥ ♠❛tr✐① ❢♦r♠✱ ℏ_SΩCST _{= ℏ}′_ΩN C_{. ✭✺✳✷✹✮} ◆♦t❡ t❤❛t ℏ ❛♥❞ ℏ′ _{❛♣♣❡❛r s✐♥❝❡ t❤❡ ❛r❣✉♠❡♥t ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ♦❢ t❤❡ ❝♦♠♠✉t❛t✐✈❡} ▼♦②❛❧ ♣r♦❞✉❝t ✐s iℏ 2 ←− ∂ziΩ N C ij −→ ∂zj, ✭✺✳✷✺✮

✺✳✸✳ ❉❡♥s✐t② ▼❛tr✐① ✐♥ ◆❈◗▼ ❛♥❞ ◗▼ ✷✸ ❛♥❞ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❛♥❛❧♦❣♦✉s ✐s iℏ′ 2 ←− ∂ziΩ N C ij −→ ∂zj. ✭✺✳✷✻✮ ❘❡♣❧❛❝✐♥❣ ❡qs✳ ✭✺✳✶✺✮ ❛♥❞ ✭✺✳✶✾✮ ✐♥t♦ ❡q✳ ✭✺✳✷✹✮✱ ✇❡ ❣❡t✿ 1 ℏΘ ℏ′ ℏIdd×d ℏ′ ℏIdd×d 1 ℏN ! = α β γ ζ ! 0 Idd×d −Idd×d 0 ! αT γT βT ζT ! = α β γ ζ ! βT ζT −αT _−γT ! = αβ T _{− βα}T _αζT _{− βγ}T γβT − ζαT _γζT _{− ζγ}T ! . ✭✺✳✷✼✮ ❇✉t t❤❡s❡ ❛r❡ ❡①❛❝t❧② t❤❡ ❡q✉❛t✐♦♥s ✇❡ ♦❜t❛✐♥❡❞ ✇❤❡♥ ❞❡✈❡❧♦♣✐♥❣ t❤❡ ❉❛r❜♦✉① tr❛♥s❢♦r♠ ✐♥ ❈❤❛♣t❡r ✸ ✭❡qs✳ ✭✸✳✷✶✮✱ ✭✸✳✷✷✮ ❛♥❞ ✭✸✳✷✸✮✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ r❡q✉✐r❡❞ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥✱ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥✿ W−1(D ◦ WN C[∗]) ✭✺✳✷✽✮ r❡❧❛t❡s ♦♣❡r❛t♦rs ✐♥ ◆❈◗▼ ✇✐t❤ ♦♣❡r❛t♦rs ✐♥ ◗▼ ❛♥❞ r❡s♣❡❝ts ❡q✳ ✭✺✳✶✷✮✳

✺✳✸ ❉❡♥s✐t② ▼❛tr✐① ✐♥ ◆❈◗▼ ❛♥❞ ◗▼

◆♦✇✱ s✐♥❝❡ ✇❡ ✇❛♥t❡❞ t♦ st✉❞② st❛t❡s ✐♥ ◆❈◗▼✱ ❛♥❞ s✐♥❝❡ t❤❡ ❞❡♥s✐t② ♠❛tr✐① bρψ❛ss♦❝✐✲ ❛t❡❞ ✇✐t❤ ❛ st❛t❡ |ψi ✐s ❛♥ ♦♣❡r❛t♦r ✭✐✳❡✳ bρψ = |ψi hψ|✮✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✺✳✶✮ t♦ r❡❧❛t❡ t❤✐s ❞❡♥s✐t② ♠❛tr✐① ✐♥ ◆❈◗▼ t♦ ❛♥ ♦♣❡r❛t♦r cM ✐♥ ◗▼✿ c M = W−1 _{D◦ W} N CρbN Cψ 
. ✭✺✳✷✾✮ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♥♦✇ ✐s t♦ ♣r♦✈❡ t❤❛t t❤✐s ♦♣❡r❛t♦r ❜❡❤❛✈❡s ❛s ❞❡♥s✐t② ♠❛tr✐① cM = ρ_b= |ψ′_{i hψ}′_|✳ Pr♦♣❡rt✐❡s✿ ✶✳ DAbE= TrM b˜cA❢♦r ❛♥② ♦♣❡r❛t♦r bA✱ ❛♥❞ ✇❤❡r❡ b˜A= W−1_{D◦ W} N C h ˆ Ai. ✷✳ TrMc= 1✭♥♦r♠❛❧✐③❛t✐♦♥✮

✷✹ ❈❤❛♣t❡r ✺✳ ❇r✐❞❣✐♥❣ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ✸✳ cM†_{= cM ✭❤❡r♠✐❝✐t②✮} ✹✳ TrMc2_{= 1✭♣✉r✐t②✮} Pr♦♦❢ ♦❢ ✶✳✿ TrM b˜cA= ˆ d2Dz WM b˜cA ✭✺✳✸✵✮ ❜❡❝❛✉s❡ ♦❢ ❡q✳ ✭✷✳✶✽✮✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ♣r♦♣❡rt② ✭✷✳✾✮ ♦❢ W ✱ ˆ d2dz WM b˜cA = ˆ d2dz WMc⋆CW  b˜ A = ˆ d2dz D◦ WN C  b ρN C_ψ 
⋆C  D◦ WN C h b Ai, ✭✺✳✸✶✮ ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ cM ❛♥❞ b˜A✱ ❛♥❞ t❤❛t W W−1[∗]
= Id✳ ❚❤❡♥✱ TrM b˜cA = ˆ d2dz D◦ WN CρbN Cψ 
⋆C  D◦ WN C h b Ai = ˆ d2dz D◦WN CρbN Cψ  ⋆N CWN C h b Ai, ✭✺✳✸✷✮ ❛s ✇❡ s❛✇ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ❙✐♥❝❡ D ✐s ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✱ ✇❡ ❝❛♥ ✐♥❝♦r♣♦r❛t❡ ✐t ✐♥t♦ t❤❡ ✐♥t❡❣r❛t✐♦♥✱ ❛♥❞ t❤✉s TrM b˜cA = ˆ d2dz WN C  b ρN C_ψ ⋆N CWN C h b Ai = ˆ d2dz WN C h b ρN C_ψ Abi = TrρbN C_ψ Ab=DAbE=DAb˜E, ✭✺✳✸✸✮ ✇❤❡r❡ ✇❡ ✉s❡❞ ❡qs✳ ✭✸✳✸✮ ❛♥❞ ✭✸✳✹✮✳ ◆♦t❡ t❤❛t hAi = DAeE ❜❡❝❛✉s❡ t❤❡ r❡s✉❧t ♦❢ ❛ ♠❡❛s✉r❡♠❡♥t ❝❛♥♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ✇❛② ♦♥❡ ❝❤♦♦s❡s t♦ r❡♣r❡s❡♥t t❤❡ ♦♣❡r❛t♦rs✳ Pr♦♦❢ ♦❢ ✷✳✿ ❙✐♠✐❧❛r❧② t♦ ✶✮✱

✺✳✸✳ ❉❡♥s✐t② ▼❛tr✐① ✐♥ ◆❈◗▼ ❛♥❞ ◗▼ ✷✺ TrMc = ˆ d2dz WMc = ˆ d2dz D◦ WN C  b ρN C_ψ 
, ✭✺✳✸✹✮ ❜② ❞❡✜♥✐t✐♦♥ ♦❢ cM ❛♥❞ ✉s✐♥❣ ❡q✳ ✭✷✳✶✽✮✳ ❆❣❛✐♥✱ s✐♥❝❡ D ✐s ❛ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s✱ ✇❡ ❝❛♥ ✐♥❝♦r♣♦r❛t❡ ✐t ✐♥t♦ t❤❡ ✐♥t❡❣r❛t✐♦♥✱ ❛♥❞ t❤✉s TrMc = ˆ d2dz WN C  b ρN C_ψ  = Tr ρ_bN C ψ
= 1. ✭✺✳✸✺✮ Pr♦♦❢ ♦❢ ✸✳✿ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ cM✱ c M† = W−1 D◦ WN CρbN Cψ 
† = " ˆ d2dk (2π)2d ˆ d2dz D◦ WN C  b ρN C_ψ 
∗· eikizˆi_e−ikizi #† = ˆ _d2d_k (2π)2d ˆ d2dz D◦ WN CρbN Cψ 
∗ · e−ikizˆi_eikizi_, ✭✺✳✸✻✮ ✇❤❡r❡ ✇❡ ✉s❡❞ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r t❤❡ ❲❡②❧ tr❛♥s❢♦r♠ ✭s❡❡ ❡q✳ ✭✷✳✼✮✮ ✐♥ t❤❡ s❡❝♦♥❞ ❧✐♥❡✳ ◆♦t❡ t❤❛t ✇❡ ❛❧s♦ ✉s❡❞ ✐♥ t❤❡ ❧❛st ❧✐♥❡ t❤❡ ❢❛❝t t❤❛t ♦♥❧② eikizˆi ✐s ❛♥ ♦♣❡r❛t♦r✱ ❛♥❞ t❤❛t ♣♦s✐t✐♦♥ ❛♥❞ ♠♦♠❡♥t❛ ♦♣❡r❛t♦rs ✐♥ ◗▼ ❛r❡ ❤❡r♠✐t✐❛♥✳ ❈❤❛♥❣✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ✈❛r✐❛❜❧❡s ✈✐❛ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ zi→ −zi✱ ✇❡ ❣❡t c M† ₌ ˆ _d2d_k (2π)2d ˆ d2dz D◦ WN C  b ρN C_ψ 
∗· e−ikizˆi_eikizi = (−1)2d ˆ d2dk (2π)2d ˆ d2dz D◦ WN C  b ρN C_ψ 
∗· eikiˆzi_e−ikizi = W−1 D◦ WN C  b ρN C_ψ 
∗. ✭✺✳✸✼✮ ❇❡❝❛✉s❡ t❤❡ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥ ✐s r❡❛❧✱ ✇❡ ❤❛✈❡ D◦ WN C  b ρN C_ψ 
∗= D ◦ WN C  b ρN C_ψ 
∗. ✭✺✳✸✽✮

✷✻ ❈❤❛♣t❡r ✺✳ ❇r✐❞❣✐♥❣ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❇❡❝❛✉s❡ bρN C ψ ✐s ❤❡r♠✐t✐❛♥✱ WN CρbN Cψ  = WN C h b ρN C,†_ψ i = WN C  b ρN C_ψ 
∗, ✭✺✳✸✾✮ ❛♥❞ t❤✉s c M† = W−1 D◦ WN C  b ρN C_ψ 
∗ = W−1 D◦ WN C  b ρN C_ψ 
= M .c ✭✺✳✹✵✮ Pr♦♦❢ ♦❢ ✹✳✿ ❆s ❜❡❢♦r❡✱ TrMc2 = ˆ d2dz WMc2 = ˆ d2dz D◦ WN CρbN Cψ 
⋆C D◦ WN CρbN Cψ 
= ˆ d2dz D◦ WN CρbN Cψ  ⋆N CWN CρbN Cψ 
= ˆ d2dz WN CρbN Cψ  ⋆N CWN CρbN Cψ  = ˆ d2dz WN C  b ρN C_ψ ρ_bN C_ψ  = Tr ρbN C_ψ ρ_bN C_ψ
= 1 ◆♦t❡ t❤❛t t❤✐s ✐s ❡①❛❝t❧② Pr♦♣❡rt② ✶ ✇✐t❤ b˜A= cM✳ ❙✐♥❝❡ cM ♦❜❡②s t❤❡ ♣r♦♣❡rt✐❡s ❛❜♦✈❡✱ ✐t ❜❡❤❛✈❡s ❛s ❞❡♥s✐t② ♠❛tr✐① ❛ss♦❝✐❛t❡❞ ✇✐t❤ s♦♠❡ st❛t❡ |ψ′_{i✐♥ ◗▼✱ ✐✳❡✳} c M =ρ_b= |ψ′_{i hψ}′_{| . ✭✺✳✹✶✮} ❋✉rt❤❡r♠♦r❡✱ ❜❡❝❛✉s❡ ♦❢ ♣r♦♣❡rt② ✶✳✱ E= Trρ_bN C_ψ HbN C  = Trρ b_bHC  , ✭✺✳✹✷✮

✺✳✹✳ ❚r❛♥s❢♦r♠❛t✐♦♥ ♦❢ fQbN C, bPN C  ♦♣❡r❛t♦rs ✷✼ ✇❤❡r❡ bHC = W−1  D◦ WN C h b HN C i ✐s ❝♦rr❡s♣♦♥❞✐♥❣ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✐♥ ◗▼✳ ❲❡ ❛❧s♦ ♦❜t❛✐♥ |hψN C|φN Ci|2= Tr ρbN Cψ ρbN Cφ
= Tr (ρ_bψρbφ) = |hψ|φi|2, ✭✺✳✹✸✮ ❛♥❞ t❤✉s t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ♦❢ st❛t❡s ✐s ♣r❡s❡r✈❡❞✳ ❍❡♥❝❡✱ ✇❡ ♦❜t❛✐♥ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ st❛t❡s ✐♥ ◆❈◗▼ ❛♥❞ st❛t❡s ✐♥ ◗▼✳

✺✳✹ ❚r❛♥s❢♦r♠❛t✐♦♥ ♦❢ f



Q

b

N C

, b

P

N C



♦♣❡r❛t♦rs

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ s❤♦✇ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ fQbN C, bPN C  ♦♣❡r❛t♦rs ❜② ❡q✳ ✭✺✳✷✽✮ ❢♦r t❤❡ WN C ♠❛♣ ✐♥ ❘❡❢✳ ❬✶✹❪✳ ❈♦♥s✐❞❡r t❤❡ ◆❈ ❲✐❣♥❡r ❚r❛♥s❢♦r♠ ✐♥ ❡q✳ ✭✸✳✷✮✿ WN C  ˆ A= ℏ−d ˆ ddx ddy e−iPC(z)·yδ x− QC_(z)
 x+ℏ 2y| ˆA|x − ℏ 2y  QC , ✇✐t❤ t❤❡ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥ ✭s❡❡ ❡qs✳ ✭✸✳✷✺✮ ❛♥❞ ✭✸✳✷✻✮✮✿ Qi = QCi + θij 2ℏP C j , Pi = PiC− ηij 2ℏQ C j, ✇❤❡r❡ QC i ❛♥❞ PiC ❛r❡ ❝♦♠♠✉t❛t✐✈❡ ♣♦s✐t✐♦♥ ❛♥❞ ♠♦♠❡♥t❛ ✈❛r✐❛❜❧❡s ❛♥❞ Qi ❛♥❞ Pi ❛r❡ t❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ✈❛r✐❛❜❧❡s✳ ❋♦r θ, η ≪ ℏ✱ ℏ′_{≈ ℏ❛♥❞ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❡❛s✐❧② ✐♥✈❡rt✐❜❧❡✳} QC_i = Qi− θij 2ℏPj ✭✺✳✹✹✮ P_iC = Pi+ ηij 2ℏQj. ✭✺✳✹✺✮ ❚❤✉s✱ WN C  b Qi  = h−d ˆ ddx ddy e−iPC·yδ x− QC
 x+ℏ 2y| bQi|x − ℏ 2y  d QC

✷✽ ❈❤❛♣t❡r ✺✳ ❇r✐❞❣✐♥❣ ◆♦♥✲❈♦♠♠✉t❛t✐✈❡ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s = h−d ˆ ddx ddy e−iPC·yδ x− QC
 x+ℏ 2y|dQ C i+ θij 2ℏdP C j|x − ℏ 2y  d QC = h−d ˆ ddx ddy e−iPC·yδ x− QC
_x iδ _ℏ 2y  +θij 2ℏ d dQC j δ _ℏ 2y ! . ✭✺✳✹✻✮ ■♥t❡❣r❛t✐♥❣ ❜② ♣❛rts ♦♥ t❤❡ s❡❝♦♥❞ t❡r♠✱ ❛♥❞ ❤❛✈✐♥❣ ❜♦✉♥❞❛r② t❡r♠ ✈❛♥✐s❤✱ ✇❡ ❣❡t WN C  b Qi  = ℏ−d ˆ ddx ddy e−iPC·yδ x− QC_(z)
 xiδ _ℏ 2y  +θij 2ℏδ _ℏ 2y  P_jC  = ˆ ddx δ x− QC_(z)
 xi+ θij 2ℏP C j  =  QC i + θij 2ℏP C j  = Qi, ✭✺✳✹✼✮ ❛s ❡①♣❡❝t❡❞ ❢r♦♠ ❛ ◆❈ ❲✐❣♥❡r tr❛♥s❢♦r♠✳ ❚❤❡♥✱ ✉s✐♥❣ ❡q✳ ✭✷✳✼✮✱ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✭✺✳✷✽✮ ❜❡❝♦♠❡s✿ W−1D◦ WN C h b Qi i = W−1(D ◦ Qi) = W−1  QC_i +θij 2ℏP C j  = = ˆ d2dk d2dk✬ (2π)2d ˆ ddQCddPC  QC_i +θij 2ℏP C j 

· eikldQCi+ik′ldPiCe−iklQCi−ik′lPiC_. ✭✺✳✹✽✮

❚❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❡❛❝❤ t❡r♠ ✐s ❞♦♥❡ ❜② s✐♠♣❧❡ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ♦❢ t②♣❡ ´ dx xe−x_✳ ❋♦r t❤❡ ✜rst t❡r♠✿ ˆ d2dk d2dk✬ (2π)2d ˆ

ddQCddPCQ_iC· eiklQdCi+ikl′dPiCe−iklQCi−ikl′PiC =

= ˆ d2d_{k d}2d_k✬ (2π)2d e ikldQCi+ik′ldPiC ˆ ddPCki −xe−x− e−x
|∞−∞e−ik ′ lP C i = = ˆ _d2d_{k d}2d_k✬ (2π)2d kie iklQdCi+ik′lPdiC ˆ dd_PC_e−ik′ lP C i = = ˆ d2dk d2dk✬ (2π)2d kie iklQdCi+ik′ldPiC = dQC i . ✭✺✳✹✾✮ ❙✐♠✐❧❛r❧② ❢♦r t❤❡ s❡❝♦♥❞ t❡r♠✱

✺✳✹✳ ❚r❛♥s❢♦r♠❛t✐♦♥ ♦❢ fQbN C, bPN C  ♦♣❡r❛t♦rs ✷✾ ˆ d2d_{k d}2d_k✬ (2π)2d ˆ

ddQCddPCP_iC· eikldQiC+ikl′dPiCe−iklQCi−ik′lP C i = dPC i . ✭✺✳✺✵✮ ❚❤✉s✱ W−1D◦ WN C h b Qi i = dQC i + θij 2ℏdP C j . ✭✺✳✺✶✮ ◆♦t❡ t❤❛t t❤❡s❡ ♦♣❡r❛t♦rs ♦❜❡② t❤❡ ❍❡✐s❡♠❜❡r❣ ❛❧❣❡❜r❛ ❜② ❞❡✜♥✐t✐♦♥ ✭s❡❡ ❘❡❢✳ ❬✼❪✮✳ ❚❤❡ s❛♠❡ ♣r♦❝❡ss ❝❛♥ ❜❡ r❡♣❡❛t❡❞ ❢♦r W−1_{D◦ W} N C h b Pi i ✱ ②✐❡❧❞✐♥❣✿ WN C  b Pi  = ℏ−d ˆ ddx ddy e−iPC·yδ x− QC
 x+ℏ 2y| bPi|x − ℏ 2y  d QC = Pi, ✭✺✳✺✷✮ ❛♥❞ t❤✉s W−1D◦ WN C h b Pi i = W−1(D ◦ Pi) = = W−1P_iC−ηij 2ℏQ C j  = dPC i − ηij 2ℏdQ C i . ✭✺✳✺✸✮ ◆♦t❡ t❤❛t t❤✐s ✐s s✐♠♣❧② t❤❡ ❉❛r❜♦✉① tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦rs✱ ✇❤✐❝❤ ✐s ✇❤❛t ✐s ❡①♣❡❝t❡❞✳ ❍♦✇❡✈❡r✱ ❢♦r ❛ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥✱ fQ, bb P✱ t❤✐s ♠✐❣❤t ♥♦t ❜❡ tr✉❡✳ ■♥ t❤❡ ❝❛s❡ fQ, bb P= X n,m,i,j αnmQbniPbjm, ✭✺✳✺✹✮ ✇❡ ❤❛✈❡ WN C  fQ, bb P= = X n,m,i,j αnmijh−d ˆ ddx ddy e−iPC·yδ x− QC
 x+ℏ 2y| bQ n iPbjm|x − ℏ 2y  d QC = X n,m,i,j αnmijQi⋆N C. . . ⋆N CQi n times ⋆N CPj⋆N C. . . ⋆N CPj m times