No-Go Theorems in

Noncommutative

Quantum Mechanics

Bruno Alexandre Duarte MadureiraMestrado em FísicaDepartamento de Física e Astronomia2017

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,

Porto, ______/______/_________

❲ ♠ ♦s ♥ t ♥ t ♥ ♦r ♦s ♠ s

♥r ②♥

s♥ rtr t s ② st♥♥ ♦♥ t s♦rs ♦ ♥ts

r s t♦♥

♥♦♠♥ts

r t ♦rs ♦ ts ②rs ♠t ♣♦♣ ♦ t tr ♠r ♦♥ ♠② ♦♥

♦r♥② P♦♣ ♦ ♣ ♠ ♣♦♣ ♦ tt ♠ ♣♦♣ ♦ st♦♦ ②

♠② s P♦♣ ♦ ♠ ♠ ♦ ♠ t♦② ♥ t♦t ♦♠ ts ♦r ♦ ♥♦t

♣♦ss s s s♠ trt t♦ t♠

rst ♥ ♦r♠♦st ♦ t♦ t♥ ♠② ♠♦tr ♥r♦s rt ♦r r ♥♦♥

t♦♥ s♣♣♦rt ♥ ♦ ♥ ♦r r str♥t ♣t♥ ♥ rs♦ ♥ ♦rr t♦ s♣♣♦rt s

ts ②rs ② t tr ♦ ♠♥② sr♣rss t ♥ ♣♣♥ss ♦r ②♦

♦ s♦ t♦ t♥ ♠② s♣rs♦r Pr♦ss♦r r rt♦♠ ♦r ts ♦♣♣♦rt♥t②

t♦ ♦r t ♠ ♥ ♦r s ♣ ♥ t② t♦ ♦r ♦♥tt s ♥♦t s ♦♥

st♥ s ♦ s t s s♥r② ♥♦② ♥ ♦♣ ♦r t st ♥ s ♥♦rs

t♥s s♦ t♦ t t♥ st tr ♦r ♥♦t tr ♣rs♥ s ♦r s♠

❲t♦t t♠ r ♦ ♦ ♦ ♥♦ r② s♥ r♦♣ ♦ t tr t s

♣②ss ♠t♠ts ♦r ♥② ♦tr st r② ♦ ♠ t♦ ♣s ♦rr ♥ ♠② sts

♥ ♠ ♠ ttr rt t♥r

♥ ♣rtr s♣ t♥s t♦ Pr♦ss♦r ♥s ♦r s t♥ ♦ t rsr

♥q s s s ♥trst♥ ♥trt♦♥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♦ Prs ♦ã♦ rr r ♠♦s ♥ ♠ã♦ ♦ã♦ ♦r tr r♥s♣ ♦♠♣♥② ♥

♣ r♥ ts s♠ ♦r♥② ♥ t♦ ♦♦ r♦ ♥ rí♦ rt♥s ♦r t ♦♥tss

♦rs ♦ strt♦♥ tt ♣t ♠② s♥t② ♥tt

st② ♦ t♦ t♥ t ♦♥tss ♣♦♣ ♠② ♥ r♥s ♦ ♦ ♥♥♠

♦r ttr ♦r ♦rs ②♦ r ♣rt ♦ ♠② ♦r♥② ♥ ♦ t ♦♣♠♥t ♦ ts ♦r

❨♦ ♠② t♥s ♥ ♠② rtt

s♠♦

st tr♦ sã♦ ♦r♦s t♦r♠s t♦ ♥♦♦ ♥♦ ♦♥t①t♦ â♥ â♥

t ã♦♦♠tt ♦t♦ ♣r♥♣ é rr s t♦r♠s ♦♠♦ ♦ ♦r♠

ã♦♦♥♠ ♠♥tê♠s á♦s q♥♦ ♦♥sr♦s ♥♦ s♣ç♦ s ã♦♦♠tt♦

♥ér♦ t♦r♠s sts q sã♦ ♠♣♦rt♥ts ♥♦ ♦♥t①t♦ ♦r ♥♦r♠çã♦ â♥t

rá t♦ ♠ ♣q♥♦ rs♠♦ ♦r♠çã♦ ❲♥r❲② â♥ â♥t s♦

♠ sssã♦ ♦ t♦r♠ ã♦♦♥♠ ♥♦ s♣ç♦ s t ss♠ ♦♠♦ ♠

sssã♦ s ♥r③çã♦ P♦r ♠ é ♣r♦♦ q sts t♦r♠s ♦♥t♥♠ á♦s ♠

s♣ç♦s s ã♦♦♠tt♦s

strt

♥ ts ♦r ♦♦ ♦r♠s ♥ t ♦♥t①t ♦ ♦♥♦♠♠tt ♥t♠ ♥s

r rss ♠♥ ♦s s t♦ s tr t♦r♠s s s t ♦♦♥♥ ♦r♠

st ♦ ♥ ♥r ♦♥♦♠♠tt Ps ♣ s ♦♥sr s s ♦ rt

♠♣♦rt♥ ♦r ♥st♥ ♥ t ♦♥t①t ♦ ♥t♠ ♥♦r♠t♦♥ ♦r② r s♠♠r②

♦ ❲♥r❲② ♦r♠t♦♥ ♦ ♥t♠ ♥s s ♥ ♦♦ ② t sss♦♥ ♦ t

♦♦♥♥ ♦r♠ ♥ t st♥r Ps ♣ s s sss♦♥ ♦ ts ♥r③t♦♥

♥② t s ♣r♦♥ tt t♦r♠s ♦ ts t②♣ ♦ ♦♥ ♦♥♦♠♠tt Ps ♣

♦♥t♥ts

♥♦♠♥ts

s♠♦

strt

♥tr♦t♦♥

♦♦ ♦r♠s ♥ ♥t♠ ♥♦r♠t♦♥

♦♥♦♠♠tt Ps ♣ ♥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♦♣rts

r ♦ ♦♣rt♦rs

♥r♥ ♥r ♣s s♣ ♥trt♦♥

❲②❲♥r ♦r♠s♠ ♦ ♦♥ ♦♠♠tt ♥t♠ ♥s

♦♥♦♠♠tt ❲♥r r♥s♦r♠

r♦① tr♥s♦r♠t♦♥

♦♦ ♦r♠s ♥ ♥t♠ ♥s

♦♦♥♥ ♥ ♥t♠ ♥s

♦♦ ♥r③t♦♥ ♥ ♥t♠ ♥s

r♥ ♦♥♦♠♠tt ♥t♠♥s ♥ ♥t♠♥s

r♥s♦r♠♥ ♣rt♦rs ♥ t♦ ♣rt♦rs ♥

♥ ♦ rs T

①

♥st② tr① ♥ ♥

r♥s♦r♠t♦♥ ♦ f(QNC , PNC

)♦♣rt♦rs

♦♦ ♦r♠s ♥ ♦♥♦♠♠tt ♥t♠ ♥s

♦♦♥♥ ♥ ♦♥♦♠♠tt ♥t♠ ♥s

♦♦ ♥r③t♦♥ ♥ ♦♥♦♠♠tt ♥t♠ ♥s

♦♥s♦♥s

♦r♣②

♣tr

♥tr♦t♦♥

♦♦ ♦r♠s ♥ ♥t♠ ♥♦r♠t♦♥

r♥ t ♥♣t♦♥ ♥ ♦♣♠♥t ♦ ♥t♠ ♥s ♥t♠ ♥♦r

♠t♦♥ ♣♦s s♥♥t ♣r♦♠ ♦r ♥♦r♠t♦♥ ♦rsts t tt t t ♦

♠sr♠♥t ♥s t s②st♠ ♥ ♥②ss ♠♥t tt ss ♣r♦rs ♦r ♥♦r♠

t♦♥ trt♠♥t r ♥♦ ♦♥r t♦ ♣♣ t♦ ♥t♠ ♥♦r♠t♦♥ s t♦ t

st② ♦ q♥t♠ s②st♠s ♥ ♥♦r♠t♦♥ ♥ t ssq♥t ♦♣♠♥t ♦ t♦r♠s tt

rstrt t t♦♥s ♣♦♥ q♥t♠ stts

♦r♠s s s t ♦♦♥♥ ♦r♠ ♥sr tt ♥♦ r♥♦♠ stt ♥

♣t ❬ ❪ t ♦t♥ ♦r♠ ♥sr tt ♥ t♦ ♦♣s ♦ stt

tr s ♥♦ ② t♦ t ♦♥ ♦ t♠ ❬❪ ♠♣② tt ss rr♦r ♦rrt♦♥ t♥qs r

sss ♦r ①♠♣ t s ♠♣♦ss tt r♥ q♥t♠ ♦♠♣tt♦♥ ♣t ♦

stt s rt ♥ s ♦r ♦rrt♥ rr♦rs s s t ♦r ♣rt q♥t♠ ♦♠♣t♥

♥ ♦r t♠ t s t♦t t♦ ② ♠tt♦♥

♦rt♥t② t t ♥t ♦ rst q♥t♠ rr♦r ♦rrt♥ ♦s ♥ r

♠♥t t ♦♦♥♥ ♦r♠ ♥t♠ ♦♠♣tt♦♥ s s♥ sr♣ ♥rs ♥

♥trst t t rst s♦stt q♥t♠ ♣r♦ss♦r ♥ rt ② rsrrs t ❨

❯♥rst② ♥ ♥ t♥ ♥t♠ ♦♠♣t♥ t st ♥ ts r② ②rs s ♦♠

♥ ♠♦r ♦s ♦ st②

s t②♣s ♦ t♦r♠s r ♥♦♥ s ♦♦ ♦r♠s ♥ s♦♠ ♦ t♠ s s t

♦♦♥♥ r t ♦t ♦ st② ♦ ts ♦r ♦r ♥st ♦ ♦r♥ ♥ t s

Ps♣ t ♦s t st② ♦♥ ♦♥♦♠♠tt Ps ♣s

♣tr ♥tr♦t♦♥

♦♥♦♠♠tt Ps ♣ ♥t♠ ♥s

♦♥♦♠♠tt ♥t♠ ♥s s ♥ ①t♥s♦♥ ♦ ♥t♠ ♥s

t ♦♠♠tt♦♥ rt♦♥s tt 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 ♥ts②♠♠tr ♠trs ♥

ℏ′ = ℏ

(1 +

θη

ℏ2

),

♥s ♦rrt♦♥ t♥ s♣ rt♦♥s ♥ ♠♦♠♥t♠ rt♦♥s ♦r ♥ q♥t

③t♦♥ ♦ ♦t ♦♥rt♦♥ ♥ ♠♦♠♥t♠ s♣

♦♥♦♠♠ttt② s r♥t② ♥ ♥trst t♦ str♥ t♦r② s t ②♥♠s

♦ str♥s ♥ sr ② t♦r② ♥ ♥♦♥♦♠♠tt s♣ s ♦r ①♠

♣ s ❬ ❪ ♥ st♥r s t ♦♥r② ♥t ♥♠r ♦ ♣rts ♠t

♦ ♦tr ♥♠♥t t♦rs ♦♥♦♠♠ttt② ♠t ♣♣r s s♠ t t t

q♥t♠ ♠♥ s ♦ ts rt ♠♦♥t ♦ ♦r s ♥ ♦t t♦

♦♥♦♠♠ttt② ♥♥ t♦♣s s s ♦♥♦♠♠tt ♦♠tr② s ♦r ①♠♣

❬❪ t ♣♣r♥ ♦ ♦♥♦♠♠ttt② ♥ ♣rts t ② ♠♥t s

♦♥♦♠♠ttt② ♥ ♥t♠ ♦r② s ♦r ①♠♣ ❬❪ ♣rts ♥

st ♣♦t♥ts t ♥ Ps ♣ s s ♣rt ♥ ♥tr ♣♦t♥t s

♦r ①♠♣ ❬❪ t rtt♦♥ ♥t♠ ❲ s ♦r ①♠♣ ❬ ❪

t r♠♦♥ st♦r s ♦r ①♠♣ ❬❪ ♥ t ②r♦♥ t♦♠ ♣♣t♦♥s t♦

♦s♠♦♦② s♦ ♥ ♦♥sr s ♦r ①♠♣ s ❬ ❪ s s ♦tr ♦rs

r ♣rt ♦ ts ♦rs s ♥ ♦t t♦ rt♥ ♥ tr♥t ♦r♠t♦♥ ♦

s ♦♥ t ❲②❲♥r ♦r♠t♦♥ ♦ q♥t♠ ♠♥ss ♦r ①♠♣ s

❬ ❪ ♥ s ♥ t trt♠♥t ♦ ts ♥ ♥rt♥t② rt♦♥s s s

❬ ❪

♥ ts ♦r ♦s ♦♥ s♥ ♦♥♦♠♠ttt② s♥ ❲②❲♥r ♦r♠

t♦♥ s ♥② ♥♥ ♥ ♦♦ ♦r♠s s s t ♦♦♥♥ ♦r♠

♣tr

❲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 ♥ ts st♦♥

r② sr t ❲②❲♥r ❲❲ ♦r♠t♦♥ s ♣tr ♥ 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 rt t♦ t ♥t♦♥ ②

f(qi, pi) =

ˆ

ψ∗(−→q −

−→y

2

)ψ

(−→q +

−→y

2

)e−

ipiyiℏ ddy,

r qi r ♣♦st♦♥s pi r t ♠♦♠♥t ℏ s t r P♥ ♦♥st♥t ♥ d s t

♥♠r ♦ s♣ ♠♥s♦♥s

♥♦tr ♠♣♦rt♥t t♦♦ s t ❲♥r❲② tr♥s♦r♠t♦♥ s ❬❪ ♠♣s

♦♣rt♦rs ♥ rt s♣ t♦ ♥t♦♥s ♥ ♣s s♣ ♥ rs ♥ ♥ ♦♣rt♦r

A t ❲♥r tr♥s♦r♠ s ♥ s

W(A)(qi, pi) =

ˆ

ddy

⟨−→q −

−→y

2|A|−→q +

−→y

2

⟩e−

ipiyiℏ .

♦t tt t ❲♥r tr♥s♦r♠ s t ♣r♦♣rts

♣tr ❲②❲♥r ♦r♠s♠ ♦ ♥t♠ ♥s

W (qi) = qi,

W (pi) = pi,

W (Id) = 1.

❲ ♥ ♥♦ s tt ♥ t

f(qi, pi) =W (ρ) =W (|ψ〉 〈ψ|) ,

r ρ = |ψ〉 〈ψ| s t ♥st② ♠tr① ss♦t t stt |ψ〉

s ♠♣♣♥ s ♦♥t♦♦♥ ♥ ♠ts ♥ ♥rs t ❲② tr♥s♦r♠

W−1 (g) =

ˆ

d2dk

(2π)2d

ˆ

d2dz g · eikizi

e−ikizi

,

r s z = (qi, pi) ♦r t ♦♦r♥ts ♥ t ♣s s♣ ♥ z = (qi, pi) r t

♣♦st♦♥ ♥ ♠♦♠♥t ♦♣rt♦rs ♥ rt s♣ ♥ 2d s t ♠♥s♦♥ ♦ t ♣s s♣

s ♠♥s tt ♦r ♥② ♦♣rt♦r A s ❬❪

A =W−1(W(A))

=

ˆ

d2dk

(2π)2d

ˆ

d2dzW(A)· eikiz

i

e−ikizi

.

♦t tt W(A)s ♥t♦♥ ♦ z = (qi, pi) r ♥s ♦ z = (qi, pi) ♥

ts t t tt eikizi

♦r♠s ♦♠♣t ss ♦r ♦♣rt♦rs ♦s ts tr♥s♦r♠t♦♥s t♦

♦♥t♦♦♥

♦② ♣r♦t

st ♠t♠t ♦t t♦ ♥tr♦ s t s♦ ♦② ♦r str ♣r♦t

♥ s♦ tt

W(A)⋆ W

(B)=W

(AB).

♥ ♥r t ♥ ♣r♦♥ tt ♦② ♣r♦t s t ♦r♠ s s ❬ ❪

❲②❲♥r ♦r♠t♦♥ ♦ ♥t♠ ♥s

A ⋆ B = Aeiℏ2

←−∂ziΩij

−→∂zjB

= AB +∞∑

n=1

1

n!

(iℏ

2

)nA(←−∂ziΩij

−→∂zj

)nB

= AB +

∞∑

n=1

(iℏ)n

n!2n

(∂(n)zα1

...zαnA)(

∂(n)zβ1...zβn

B)Ωα1β1

. . .Ωαnβn,

r t rr♦s ♠♥ t rts r ♣♣ t♦ t t ♦r rt ♥

Ω =

(0 Idd×d

−Idd×d 0

)

s t ♠tr① ♦ ♦♠♠tt♦♥ rt♦♥s ♦r z = (qi, pi)

[zi, zj ] = iℏΩij ,

♦r♥ t♦ t s♥r❲② r q

♦t tt ② ♥t♦♥ Ωij s ♥ts②♠♠tr s [zi, zj ] = − [zj , zi]

♥ t♦♥ t♥ t ❲② tr♥s♦r♠ ♦ q s tt

W−1(W(A)⋆ W

(B))

= W−1(W(AB))

= AB.

❲②❲♥r ♦r♠t♦♥ ♦ ♥t♠ ♥s

♥ t ❲❲ ♦r♠t♦♥ ♦ ①♣tt♦♥ s ♦ ♦♣rt♦rs ♥ t s♥

t ❲♥r❲② tr♥s♦r♠t♦♥ ♥ t ♦② qt♦♥❬ ❪ ♦♥ts ♦r t ②♥♠

♦t♦♥ st s t rö♥r qt♦♥ ♦r t st♥r ♦r♠t♦♥ ♦

❯s♥ t ♥t♦♥ ♦ t ❲② tr♥s♦r♠ ♦♥ ♥ s♦ tt t ①♣tt♦♥ s ♦

♦♣rt♦rs r ♥ ②

⟨G⟩=

ˆ

f(z)g(z) d2dz.

♠r② ♦♥ ♥ ♣r♦ tt t ♦t♦♥ ♦ t s②st♠ s sr ② t ♦②

qt♦♥

♣tr ❲②❲♥r ♦r♠s♠ ♦ ♥t♠ ♥s

∂f

∂t=H ⋆ f − f ⋆ H

iℏ:=

1

iℏH, f⋆ ,

r H = W(H)s t ♣s s♣ rs♦♥ ♦ t ♠t♦♥♥ ♥ r ♥tr♦

t ♦② rts ♥ s

A,B⋆ = A ⋆ B −B ⋆ A.

rtr♠♦r ♦♥ ♥ s♦ ♣r♦ tt ♦r stt♦♥r② s②st♠s ts ♥ ①♣rss s s

❬❪

H (z) ⋆ f (z) = E f (z) .

♥ Pr♦♣rts

r ♦ ♦♣rt♦rs

s ♣r♦♣rt② s t rt♦♥ t♥ t tr ♦ ♥ ♦♣rt♦r ♥ ts ❲♥r tr♥s

♦r♠ ♣r♦♦ s strt♦rr

♦r♠ ♦r ♥② ♦♣rt♦r A

Tr(A)=

ˆ

d2dzW(A).

Pr♦♦ ❯s♥ t ♥t♦♥ ♦ t ❲② tr♥s♦r♠

Tr(A)

= Tr(W−1

[W(A)])

= Tr

(ˆ

d2dk

(2π)2d

ˆ

d2dzW(A)eikiz

i

e−ikizi

)

=

ˆ

d2dzW(A)Tr

(ˆ

d2dk

(2π)2deikiz

i

e−ikizi

),

♥ tt ❲♥r tr♥s♦r♠ s ♥♦t ♥ ♦♣rt♦r ♥ ts s ♥♦t t ② tr ♥

② ♣r♦♣rt② ♦ t tr ♥ ♦ t ♥trt♦♥ ♥ k

♥ Pr♦♣rts

Tr

(ˆ

d2dk

(2π)2deikiz

i

e−ikizi

)=

ˆ

d2dk

(2π)2deikiz

i

e−ikizi

=

ˆ

d2dk

(2π)2d

= 1,

r t t tr ♥ t ♥ss ♦ z ♥ t rst st♣ ♥ ts

Tr(A)

=

ˆ

d2dzW(A)Tr

(ˆ

d2dk

(2π)2deikiz

i

e−ikizi

)

=

ˆ

d2dzW(A).

♥r♥ ♥r ♣s s♣ ♥trt♦♥

♥ ♦ t ♠♦st ♠♣♦rt♥t ♣r♦♣rts tt s s t ♥r♥ ♥r ♣s

s♣ ♥trt♦♥ ♦ t ♦② Pr♦t

♦r♠ ♦r ♥② Ωij tt ♥s ♦② Pr♦t ⋆ ♥ ♦r ♥② t♦ ♥t♦♥s A,B

♥ ♥ Ps♣ t ♦♦♥ s tr

ˆ

d2dz A ⋆ B =

ˆ

d2dz AB.

Pr♦♦ rst ♦♥ ①♣♥s t ♦② Pr♦t ♥ ts srs ①♣♥s♦♥

ˆ

d2dz A ⋆ B =

ˆ

d2dz AB +

∞∑

n=1

1

n!

(iℏ

2

)n ˆd2dz

(A(←−∂ziΩij

−→∂zj

)nB).

♦ ♦♥ s ♦♥② t♦ ♣r♦ tt t ♥trs ♥ t s♦♥ tr♠ ② ③r♦ s s

♦♥ ② ①♣♥♥(←−∂ziΩij

−→∂zj

)n♥t♦ ♥ tr♠s q ♥ t♥ tr♥s♦r♠♥

t ♥tr♥ ♥t♦ t♦t rt ♠♥s tr♠ tt s s②♠♠tr ♥ t ♥ ♦

♣rtr ♣r ♦ ♥①s i, j ♦r tr s ♥ ss♦t Ωij ♠tr① ♠♥t ♥ s♥

t Ω ♠tr① s ♥ts②♠♠tr tt tr♠ ②s ③r♦ ♥♦t tt ♦♥ ss♠s tt A B ♥

tr rts ♥s t ♥♥t②

(∂(n)zα1

...zαnA)(

∂(n)zβ1...zβn

B)Ωα1β1

. . .Ωαnβn=

♣tr ❲②❲♥r ♦r♠s♠ ♦ ♥t♠ ♥s

∂zα1

((∂(n−1)zα2

...zαnA)(

∂(n)zβ1...zβn

B))

Ωα1β1. . .Ωαnβn

−

−(∂(n)zα1

...zαnA)(

∂(n+1)zα1

zβ1...zβn

B)Ωα1β1

. . .Ωαnβn.

♥ t rst tr♠ s t♦t rt

ˆ

d2dz ∂zα1

((∂(n−1)zα2

...zαnA)(

∂(n)zβ1...zβn

B))

Ωα1β1. . .Ωαnβn

= 0.

s ∂zα1∂zβ1

= ∂zβ1∂zα1

♥ Ωzα1zβ1

s ♥ts②♠♠tr ♥ t ①♥ ♦ ♥①s

α1 ↔ β1

(∂(n)zα1

...zαnA)(

∂(n+1)zα1

zβ1...zβn

B)Ωα1β1

. . .Ωαnβn= 0,

♥ tr♦r

∞∑

n=1

1

n!

(iℏ

2

)n ˆd2dz

(A(←−∂ziΩij

−→∂zj

)nB)= 0,

♠♥s

ˆ

d2dz A ⋆ B =

ˆ

d2dz AB.

♣tr

❲②❲♥r ♦r♠s♠ ♦ ♦♥

♦♠♠tt ♥t♠ ♥s

♦♥♦♠♠tt ❲♥r r♥s♦r♠

♥ ♦rr t♦ sr ♦♥♦♠♠tt ♥t♠ ♥s ♥ Ps ♣

♦♥ ♥s ♣r♦♣r ② t♦ ♠♣ ♦♣rt♦rs ♥ rt ♣ ♥t♦ ♥t♦♥s ♥ R2d

s ♠♥s ♥♥ ♦♥t♦♦♥ ♥r ♠♣ V s♦ tt

V (Id) = 1

V (q) = q

V (p) = p

V(AB)= V

(A)⋆NC V

(B)

♦r ♣s s♣ t ♦r♠ ♦♠♠tt♦♥ rt♦♥s ♦ t s s♥r❲② r

s q

[qi, qj ] = iθij , [pi, pj ] = iηij , [qi, pj ] = iℏ′δij ,

r θ ♥ η r r ♥ts②♠♠tr ♠trs

♦r ts ♠♣ s ♥♦t ♥q ♥ ts tr r sr ♣r♦♣♦ss ♦r ts ♠♣ ♥

❬❪ ts ss s sss ♥ r♥t ♠♣s r ♦♠♣r t s r tt st

♠♣ s t ♦♦♥

♣tr ❲②❲♥r ♦r♠s♠ ♦ ♦♥ ♦♠♠tt ♥t♠ ♥s

WNC

(A)(z) = h−d

ˆ

ddy ddx e−iΠ(z)·yδ (x−R(z))

⟨x+

ℏ

2y|A|x−

ℏ

2y

⟩

R

,

r x, y r ♣♦st♦♥s R(z),Π(z) r ♥♦♥② ♦♥t rs tt r rt t♦ z

r♦① tr♥s♦r♠t♦♥ t♦ sr ♥ t ♥①t st♦♥ s ❬❪ ♦r t

①♣♥t♦♥ t s s♦ r tt ts ♠♣ s ①t② t s♠ ♣r♦♣rts s ♥ st♦♥

WNC

(AB)=WNC

(A)⋆NC WNC

(B),

Tr(A)=

ˆ

d2dzWNC

(A),

ˆ

d2dz A ⋆NC B =

ˆ

d2dz AB,

t

A ⋆NC B = Aeiℏ′

2

←−∂ziΩ

NCij

−→∂zjB,

r t Ω ♠tr① s ♥♦

ΩNC =

(1ℏΘ Idd×d

−Idd×d1ℏN

),

♥ r

Θ = (θij) , N = (ηij)

r t ♠trs ♦ t ♦♠♠tt♦♥ rt♦♥s ♠♥ts

s ♠♣s ♦s ♦r t♦ sr ♥ ♣ss♣ st s ♥ ♥♦ ♥s

② t♦ ♦♥♥t rs ♥ ♥ rs ♥

r♦① tr♥s♦r♠t♦♥

r♦① tr♥s♦r♠t♦♥ ♦r r❲tt♥ ♠♣ s ♥♦♥♥♦♥ ♥r tr♥s

♦r♠t♦♥ t♥ t t♦ sts ♦ ♣s s♣ rs t r♥t ♦♠♠tt♦♥ r

t♦♥s s② t♥ ♦♠♠tt rs tt ♦② t s♥r❲② r ♥

♥♦♥♦♠♠tt rs tt ♦② t r s♦♥ ♦ ♥ tt r♣t ♦r ♦♥

♥♥

r♦① tr♥s♦r♠t♦♥

[qi, qj ] = iθij ,

[pi, pj ] = iηij ,

[qi, pj ] = iℏ′δij .

♠ ♦ ts st♦♥ s t♦ ♦r ♦t ts tr♥s♦r♠t♦♥ sts ts ♣r♦♣rts ♥ ♦rr

t♦ s♣② ♥ t♥ rs ♥ t t ♦♥s ♥

♥ ♦rr t♦ rt ♦♠♠tt ♥ ♥♦♥♦♠♠tt rs ♦♥sr ♥r tr♥s

♦r♠t♦♥ t♦ ♥ t rs z = (qi, pi) tt ♦② t ♦ ♦♠♠tt♦♥ rt♦♥s

♥t♦ st♥r ♦♠♠tt rs zC =(qCi , p

Ci

)tt sts② t s♥r❲② ♦♠

♠tt♦♥ rt♦♥s

qi = AijqCj +Bijp

Cj ,

pi = CijqCj +Dijp

Cj .

♦r♠ s ❬❪ ♠trs A B C ♥ D ♦② t rt♦♥s♣s

ADT −BCT =ℏ′

ℏIdd×d,

ABT −BAT =1

ℏΘ,

CDT −DCT =1

ℏN,

r

Θ = (θij) , N = (ηij)

r t ♠trs ♦ t ♦♠♠tt♦♥ rt♦♥s ♠♥ts ♦r ♣♦st♦♥ ♥ ♠♦♠♥t rs♣

t② ♦r t ♥♦♥♦♠♠tt rs

♣tr ❲②❲♥r ♦r♠s♠ ♦ ♦♥ ♦♠♠tt ♥t♠ ♥s

Pr♦♦ ♦ q

iθij = [qi, qj ]

= AikAjl[qCk , q

Cl

]+AikBjl

[qCk , p

Cl

]

−BikAjl[qCl , p

Ck

]+BikBjl

[pCk , p

Cl

]

= iℏ (AikBjlδkl −BikAjlδkl)

= iℏ(AikB

Tkj −BikA

Tkj

)

= iℏ(ABT −BAT

)ij.

Pr♦♦ ♦ q

iηij = [pi, pj ]

= CikCjl[qCk , q

Cl

]+ CikDjl

[qCk , p

Cl

]

−DikCjl[qCl , p

Ck

]+DikDjl

[pCk , p

Cl

]

= iℏ (CikDjlδkl −DikCjlδkl)

= iℏ(CikD

Tkj −DikC

Tkj

)

= iℏ(CDT −DCT

)ij.

Pr♦♦ ♦ q

iℏ′δij = [qi, pj ]

= AikCjl[qCk , q

Cl

]+AikDjl

[qCk , p

Cl

]

−BikCjl[qCl , p

Ck

]+BikDjl

[pCk , p

Cl

]

= iℏ (AikDjlδkl −BikCjlδkl)

= iℏ(AikD

Tkj −BikC

Tkj

)

= iℏ(ADT −BCT

)ij.

♦t ♦r tt ts rt♦♥s r ♥♦t ♥♦ t♦ ② tr♠♥t t ♠trs Θ

♥ N s tr r 4d2 ♣r♠trs ♥ d2 (3d− 1) ♥♣♥♥t qt♦♥s ♥ ts tr

r d2 (5d+ 1) r ♣r♠trs

♥ ♥ ♦r ♦♥sr s♠♣t♦♥s ♦r ♥st♥ t♦t ♦ss ♦ ♥rt②

♥ t A ♥ D t♦ t ♥tt② ♠tr① ❲ t♥

r♦① tr♥s♦r♠t♦♥

−BCT =ℏ′

ℏIdd×d,

BT −B =1

ℏΘ,

C − CT =1

ℏN.

♠♦st ♦♠♠♦♥② ♦♥sr s s Θ = θǫij ♥ N = ηǫij r ǫij = ǫijk t

k 6= i, j s ♥ts②♠♠tr ♥ i, j ♥ ♥ t

B =Θ

2ℏ, C = −

N

2ℏ.

s t♥ ♥ ♣ t t ♦♦♥ r♦① tr♥s♦r♠t♦♥

qi = qCi +θ

2ℏǫijp

Cj ,

pi = pCi −η

2ℏǫijq

Cj .

♥② ♥♦t tt ℏ′ qs ♥ ♥ ℏ r ts rt ②

ℏ′ = ℏ

(1 +

θη

4ℏ2

).

♣tr

♦♦ ♦r♠s ♥ ♥t♠

♥s

♥ ts ♣tr ♣r♦ ♥ ♦r ♦ ♦♦ ♦r♠s ♥ rst strt ②

♣r♦♥ t ♦♦♥♥ ♦r♠ ♥ t ♦♥t①t ♦ ♥ t♥ ♣r♦ ♥r③t♦♥

♦r ♦♦ ♦r♠s tt ♥ t ♦♦♥♥ ♥ ♦t♥ ♦r♠ s s♣ ss

♦♦♥♥ ♥ ♥t♠ ♥s

♦♥♣t ♦ ♦♥♥ s s♠♣ ♦♥ s t♦ t ♥r s②st♠ ♥ ♥ ♠♣t②

♦♥ ♥ ♦ t♠ s♦ tt ♦♥ ♥s ♣ t t♦ ♦♣s ♦ t ♦r♥ s②st♠ ♦r s

s ♥①t ss♠♥ ♦♥♥ ♦r ♥r stt rstrts t stts tt ♥ ♦♥

♥ ts ♦♥♥ ♥♥♦t ♦♥ ♥ ♥rt②

♦r♠ ♦♦♥♥ ♥ t |ψ〉 ♥r q♥t♠ stt ♥ t |0〉 ♥

♠♣t② stt ♥ t s ♥♦t ♣♦ss t♦ ♦ ts t♦ stts ♥t♦ t♦ ♦♣s ♦ |ψ〉 ♦r ♥②

q♥t♠ stt |ψ〉

Pr♦♦ ss ♣r♦♦ ♦ t t♦r♠ s ♦♥ ② rt♦ sr♠

ss♠ ♦♥♥ s ♣♦ss ♥ tr s ♥ ♠t♦♥♥ H t ♥ ss♦t ♥tr②

t♠ ♦t♦♥ ♦♣rt♦r U = eiℏ

´

Hdt s♦ tt

U |ψ〉A |0〉B = |ψ〉A |ψ〉B ,

♦r ♥② ♥ ♥♥♦♥ stt |ψ〉 ♥ ♥ ♠♣t② stt |0〉

♣tr ♦♦ ♦r♠s ♥ ♥t♠ ♥s

♥ |φ〉 s ♥♦tr stt

U |φ〉A |0〉B = |φ〉A |φ〉B ,

♦♥ s

〈φ|ψ〉 = 〈φ|ψ〉A 〈0|0〉B

= (〈φ|A 〈0|B) (|ψ〉A |0〉B)

= (〈φ|A 〈0|B) U†U (|ψ〉A |0〉B)

= (〈φ|A 〈φ|B) (|ψ〉A |ψ〉B)

= 〈φ|ψ〉2,

r t s s tt U†U = Id ♥ 〈0|0〉 = 1

s 〈φ|ψ〉 = 0 ♦r 〈φ|ψ〉 = 1 ♥♥♦t tr ♦r stts|φ〉 ♥ |ψ〉

♦♥trt♦♥ rss r♦♠ t t tt ss♠ tt ♦♥♥ s ♣♦ss ♦r ♥② ♥

stt ♥ tr ♥ ♥♦ ♦♥♥

♦t tt s♠r ♣r♦♦ ♥ ♣r♦r♠ ♥ rrs ♦rr tr s ♥♦ t♠ ♦t♦♥

♦♣rt♦r U ′ s♦ tt

U ′ |ψ〉A |ψ〉B = |ψ〉A |0〉B

♦r ♥② ♥r q♥t♠ stt |ψ〉 ts ♥ t ♦t♥ ♦r♠ s ♥

s t♠rrs ♦ t ♦♦♥♥ ♦r♠ s ② ♥t♦♥ ♦ t♠ ♦t♦♥

♦♣rt♦r

U (t0, t1) U (t1, t0) = Id,

♥ ts

U (t0, t1) |ψ〉A |0〉B = |ψ〉A |ψ〉B ⇐⇒ U (t1, t0) |ψ〉A |ψ〉B = |ψ〉A |0〉B .

♦♦ ♥r③t♦♥ ♥ ♥t♠ ♥s

♥ ts st♦♥ ♥tr♦ t♦r♠ str♦♥r t♥ t ♦♦♥♥ ♥ ♦t♥

♦r♠s ♥ ♥s ♦t s s♣ ss ② rst ♦ t t♦r♠ s tt

♦♦ ♥r③t♦♥ ♥ ♥t♠ ♥s

♦♥ ♥♥♦t t ♦♥ ♦r ♠♦r ♦♣s ♦ q♥t♠ s②st♠s ♥ ♣rt② s♣r♣♦s t♠ t

① stt s ♣r♦♦ s s♠r t♦ ♦♥ ♥ ♥ ❬❪

♦r♠ r s ♥♦ ♠t♦♥♥ H t ♥ ss♦t t♠ ♦t♦♥ ♦♣rt♦r U s♦

tt ♦r ① stt |φ〉 ♥ ♦r ♥② stt |ψ〉 t ♦♦♥ s tr

U |ψ〉⊗k|0〉⊗N−k

= |ϕ〉⊗n|0〉⊗N−n

,

r |ϕ〉 = α |ψ〉+ β |φ〉 t |α|2+ |β|

2= 1 ♥ r s t ♥♦tt♦♥

|ψ〉⊗k

= |ψ〉 ⊗ . . .⊗ |ψ〉k times

.

Pr♦♦ rst ♥♦t tt ♥ β = 0 tr s ♥♦ s♣r♣♦st♦♥ t ♥♦tr stt

k < n t ♦♦♥♥ ♦r♠ ♥ k > n t ♦t♥ ♦r♠

s ♦♥② ♥ t♦ ♣r♦ t s 0 < β < 1 ♦ s t♦ s♦ tt ts ♠♣s

♦♥trt♦♥ t rt♦ sr♠ ♠t♦

♣♣♦s tr s ♥ ♠t♦♥♥ H s♦ tt

U |ψ〉⊗k|0〉⊗N−k

= |ϕ〉⊗n|0〉⊗N−n

♦r ♥② stt |ψ〉 ♥ t |ϕ〉 = α |ψ〉+ β |φ〉

♥ ♥st s t stt eiθ |ψ〉 ♦

Ueikθ |ψ〉⊗k|0〉⊗N−k

= |ϕ′〉⊗n|0〉⊗N−n

,

r |ϕ′〉 = αeiθ |ψ〉+ β |φ〉

♦r s |ψ〉 ∝ eiθ |ψ〉 ss♠♥ tt stts r ♥♦r♠③ t t r♠t

♦♥t ♦ q ♥ ♠t♣② t ② q t♦ ♦t♥

eikθ = 〈ϕ|ϕ′〉n.

♥ ② ♥t♦♥ ♦ |ϕ〉 ♥ |ϕ′〉

〈ϕ|ϕ′〉 = eiθ |α|2+ |β|

2,

♥ ts s |α|2♥ |β|

2r r ♥♠rs q ♥ ♦♥② tr β = 0

s ♦♥trt♦♥ s ♦♥s t ♣r♦♦

♣tr

r♥ ♦♥♦♠♠tt

♥t♠ ♥s ♥ ♥t♠

♥s

♦r ♣♣r♦♥ t ♣r♦s t♦r♠s ♥ ♦♥ ♥s ② t♦ rt stts ♥

t stts ♥ s ♣tr ♥t♥s t♦ s♦ tt t tr♥s♦r♠t♦♥

A −→W−1(D WNC

[A])

♦s ♦r sr♥ stts ♥ tr♦ stts ♥ ♠♦ ♠t♦♥♥

♦r ♠♣ WNC ♥ r D s t r♦① tr♥s♦r♠t♦♥ ♦r rs ♥ ♣ss♣

sr ♥ ♣tr

r♥s♦r♠♥ ♣rt♦rs ♥ t♦ ♣rt♦rs ♥

s s ♥ t ♣r♦s ♣trs t ❲♥r r♥s♦r♠ ♠♣s ♦♣rt♦rs t♦ ♣s

s♣ ♥t♦♥s ♥ ts ♥rs t ❲② tr♥s♦r♠ ♠♣s ts ♣ss♣ ♥t♦♥s ♥

t♦ ♦♣rt♦rs ♥

W : HC −→ C[R

2d],

♣tr r♥ ♦♥♦♠♠tt ♥t♠ ♥s ♥ ♥t♠ ♥s

W−1 : C[R

2d]−→ HC .

♠r② ♥② ❲♥r r♥s♦r♠ ♠♣s ♦♣rt♦rs t♦ ♣ss♣ ♥t♦♥s

WNC : HNC −→ C[R

2d].

♥ ♦rr t♦ st② stts ♥ ♥♦ ♥t♥ t♦ s qs ♥ t♦ ♠♣

♦♣rt♦rs ♥ t♦ s♦♠ ♦tr ♦♣rt♦rs ♥ ② s♥ t t tt WNC ②s

♥t♦♥s ♥ W−1 s ♥t♦♥s s r♠♥ts s ♠♥s ♦r ♥ ♦♣rt♦r A ♦♥ ♥

♥ ♥ ♦♣rt♦r A ♥ s

A =W−1(WNC

[A])

♥ ♦t tt ts s s♠♣② ♠t♠t ♦t ♥ ♦s ♥♦t ♥ssr② t

s♠ ♣②s ♠♥♥ s A

♦r ♥ ♦♥ ♦♥srs ♣r♦t ♦ t♦ ♦♣rt♦rs A ♥ B r♠♠r♥ tt

s qs ♥

WNC

[AB]=WNC

[A]⋆NC WNC

[B]

W[AB]=W

[A]⋆C W

[B],

t

W−1(WNC

[AB])

= W−1(WNC

[A]⋆NC WNC

[B])

6= W−1(WNC

[A])W−1

(WNC

[B]),

s♥ W−1 s t ♣r♦♣rt②

W−1 (f ⋆C g) =W−1 (f)W−1 (g)

♦r ♥t♦♥s f ♥ g

r♦r ♥ ♥r tr♥s♦r♠t♦♥ ♦ rs T

z′i = (T z)i = Sijzj

♥ ♦ rs T

♥

T f (z) = f (z′) = f (Sijzj) ,

s♦ tt

W−1(T

(WNC

[A]⋆NC WNC

[B]))

=

=W−1((T WNC

[A])⋆C

(T WNC

[B]))

,

♥ ♥

O =W−1(T WNC

[O])

♦r ♥② ♦♣rt♦r O ♥ ♥ t

AB =W−1(T WNC

[AB]).

♣r♦♠ ♥♦ s t♦ t ts tr♥s♦r♠t♦♥ T

♥ ♦ rs T

s ♦ q ♥ t♦ ♥ ♥r tr♥s♦r♠t♦♥ ♦ rs T s♦ tt ♦r

♥② t♦ ♣ss♣ ♥t♦♥s f ♥ g

T (f (z) ⋆NC g (z)) = (T f (z)) ⋆C (T g (z)) .

①♣♥♥ t ♦② ♣r♦ts s qs t

T (fe

iℏ′

2

←−∂ziΩ

NCij

−→∂zj g

)= (T f) e

iℏ2

←−∂ziΩ

Cij

−→∂zj (T g) ,

r

ΩC =

(0 Idd×d

−Idd×d 0

), ΩNC =

(1ℏΘ Idd×d

−Idd×d1ℏN

),

s ♥ ♣r♦s②

ss♠♥ T s ♥r tr♥♦r♠t♦♥ s♠r t♦ t s ♦♥ ♦r t r♦① tr♥s

♦r♠t♦♥ ♥ rt

zi −→ z′i = (T z)i = Sijzj ,

t i, j = 1, ..., 2d ♦r s♣rt♥ ♣♦st♦♥s r♦♠ ♠♦♠♥t

♣tr r♥ ♦♥♦♠♠tt ♥t♠ ♥s ♥ ♥t♠ ♥s

qiT−→ q′i = αijqj + βijpj ,

piT−→ p′i = γijqj + ζijpj ,

t i, j = 1, ..., 2d r ♥

S = (Sij) =

(α β

γ ζ

),

r α, β, γ, ζ r t ♠trs t ♦♥ts αij , βij , γij , ζij rs♣t②

s ♥ ①♣♥ t ♦② ♣r♦t q s

f ⋆ g = fg +

∞∑

n=1

(iℏ)n

n!2n

(∂(n)zα1

...zαnf)(

∂(n)zβ1...zβn

g)Ωα1β1

. . .Ωαnβn,

♥ t♦ t tr♠s ∂zif (T z)

∂zif (T z) = ∂zif (z′)

=∂z′j

∂zi

∂

∂z′jf (z′)

= Sji∂z′jf (z′)

r♦r

(∂zi (T f))(∂zj (T g)

)ΩCij =

(∂z′

kf (z′)

)SkiSlj

(∂z′

lg (z′)

)ΩCij

=(∂z′

kf (z′)

)(∂z′

lg (z′)

)SkiΩ

CijS

Tjl

s ①♣♥♥ q s♥ q ②s s♠♣②

ℏSkiΩCijS

Tjl = ℏ

′ΩNCkl ,

♦r ♥ ♠tr① ♦r♠

ℏSΩCST = ℏ′ΩNC .

♦t tt ℏ ♥ ℏ′ ♣♣r s♥ t r♠♥t ♦ t ①♣♦♥♥t ♦ t ♦♠♠tt

♦② ♣r♦t siℏ

2

←−∂ziΩ

NCij

−→∂zj ,

♥st② tr① ♥ ♥

♥ t ♥♦♥♦♠♠tt ♥♦♦s s

iℏ′

2

←−∂ziΩ

NCij

−→∂zj .

♣♥ qs ♥ ♥t♦ q t

(1ℏΘ ℏ

′

ℏIdd×d

ℏ′

ℏIdd×d

1ℏN

)=

(α β

γ ζ

)(0 Idd×d

−Idd×d 0

)(αT γT

βT ζT

)

=

(α β

γ ζ

)(βT ζT

−αT −γT

)

=

(αβT − βαT αζT − βγT

γβT − ζαT γζT − ζγT

).

t ts r ①t② t qt♦♥s ♦t♥ ♥ ♦♣♥ t r♦① tr♥s♦r♠

♥ ♣tr qs ♥ r♦r t rqr tr♥s♦r♠t♦♥ s

r♦① tr♥s♦r♠t♦♥ ♥ tr♦r t tr♥s♦r♠t♦♥

W−1 (D WNC [∗])

rts ♦♣rt♦rs ♥ t ♦♣rt♦rs ♥ ♥ rs♣ts q

♥st② tr① ♥ ♥

♦ s♥ ♥t t♦ st② stts ♥ ♥ s♥ t ♥st② ♠tr① ρψ ss♦

t t stt |ψ〉 s ♥ ♦♣rt♦r ρψ = |ψ〉 〈ψ| ♥ s t tr♥s♦r♠t♦♥

t♦ rt ts ♥st② ♠tr① ♥ t♦ ♥ ♦♣rt♦r M ♥

M =W−1(D WNC

[ρNCψ

]).

♦t ♥♦ s t♦ ♣r♦ tt ts ♦♣rt♦r s s ♥st② ♠tr① M = ρ =

|ψ′〉 〈ψ′|

Pr♦♣rts

⟨A⟩= Tr

(MA)♦r ♥② ♦♣rt♦r A ♥ r A =W−1

(D WNC

[A]).

Tr(M)= 1 ♥♦r♠③t♦♥

♣tr r♥ ♦♥♦♠♠tt ♥t♠ ♥s ♥ ♥t♠ ♥s

M† = M r♠t②

Tr(M2)= 1 ♣rt②

Pr♦♦ ♦

Tr(MA)=

ˆ

d2DzW(MA)

s ♦ q s ♦ t ♣r♦♣rt② ♦ W

ˆ

d2dzW(MA)

=

ˆ

d2dzW(M)⋆C W

(A)

=

ˆ

d2dz(D WNC

[ρNCψ

])⋆C

(D WNC

[A]),

r s t ♥t♦♥ ♦ M ♥ A ♥ tt W(W−1 [∗]

)= Id ♥

Tr(MA)

=

ˆ

d2dz(D WNC

[ρNCψ

])⋆C

(D WNC

[A])

=

ˆ

d2dz D (WNC

[ρNCψ

]⋆NC WNC

[A]),

s s ♥ t ♣r♦s st♦♥ ♥ D s ♥ ♦ rs ♥ ♥♦r♣♦rt t

♥t♦ t ♥trt♦♥ ♥ ts

Tr(MA)

=

ˆ

d2dzWNC

[ρNCψ

]⋆NC WNC

[A]

=

ˆ

d2dzWNC

[ρNCψ A

]

= Tr(ρNCψ A

)=⟨A⟩=⟨A⟩,

r s qs ♥

♦t tt 〈A〉 =⟨A⟩

s t rst ♦ ♠sr♠♥t ♥♥♦t ♣♥ ♦♥ t ②

♦♥ ♦♦ss t♦ r♣rs♥t t ♦♣rt♦rs

Pr♦♦ ♦ ♠r② t♦

♥st② tr① ♥ ♥

Tr(M)

=

ˆ

d2dzW(M)

=

ˆ

d2dz(D WNC

[ρNCψ

]),

② ♥t♦♥ ♦ M ♥ s♥ q ♥ s♥ D s ♥ ♦ rs ♥

♥♦r♣♦rt t ♥t♦ t ♥trt♦♥ ♥ ts

Tr(M)

=

ˆ

d2dzWNC

[ρNCψ

]

= Tr(ρNCψ

)= 1.

Pr♦♦ ♦ ② ♥t♦♥ ♦ M

M† =[W−1

(D WNC

[ρNCψ

])]†

=

[ˆ

d2dk

(2π)2d

ˆ

d2dz(D WNC

[ρNCψ

])∗· eikiz

i

e−ikizi

]†

=

ˆ

d2dk

(2π)2d

ˆ

d2dz(D WNC

[ρNCψ

])∗· e−ikiz

i

eikizi

,

r s t ①♣t ♦r♠ ♦r t ❲② tr♥s♦r♠ s q ♥ t s♦♥ ♥

♦t tt s♦ s ♥ t st ♥ t t tt ♦♥② eikizi

s ♥ ♦♣rt♦r ♥ tt

♣♦st♦♥ ♥ ♠♦♠♥t ♦♣rt♦rs ♥ r r♠t♥ ♥♥ t ♥tr rs

t tr♥s♦r♠t♦♥ zi → −zi t

M† =

ˆ

d2dk

(2π)2d

ˆ

d2dz(D WNC

[ρNCψ

])∗· e−ikiz

i

eikizi

= (−1)2dˆ

d2dk

(2π)2d

ˆ

d2dz(D WNC

[ρNCψ

])∗· eikiz

i

e−ikizi

= W−1((D WNC

[ρNCψ

])∗).

s t r♦① tr♥s♦r♠t♦♥ s r

(D WNC

[ρNCψ

])∗= D

(WNC

[ρNCψ

])∗.

♣tr r♥ ♦♥♦♠♠tt ♥t♠ ♥s ♥ ♥t♠ ♥s

s ρNCψ s r♠t♥

WNC

[ρNCψ

]= WNC

[ρNC,†ψ

]

=(WNC

[ρNCψ

])∗,

♥ ts

M† = W−1((D WNC

[ρNCψ

])∗)

= W−1(D WNC

[ρNCψ

])

= M.

Pr♦♦ ♦ s ♦r

Tr(M2)

=

ˆ

d2dzW(M2)

=

ˆ

d2dz(D WNC

[ρNCψ

])⋆C(D WNC

[ρNCψ

])

=

ˆ

d2dz D (WNC

[ρNCψ

]⋆NC WNC

[ρNCψ

])

=

ˆ

d2dzWNC

[ρNCψ

]⋆NC WNC

[ρNCψ

]

=

ˆ

d2dzWNC

[ρNCψ ρNCψ

]

= Tr(ρNCψ ρNCψ

)= 1

♦t tt ts s ①t② Pr♦♣rt② t A = M

♥ M ♦②s t ♣r♦♣rts ♦ t s s ♥st② ♠tr① ss♦t t s♦♠

stt |ψ′〉 ♥

M = ρ = |ψ′〉 〈ψ′| .

rtr♠♦r s ♦ ♣r♦♣rt②

E = Tr(ρNCψ HNC

)= Tr

(ρHC

),

r♥s♦r♠t♦♥ ♦ f(QNC , PNC

)♦♣rt♦rs

r HC =W−1(D WNC

[HNC

])s ♦rrs♣♦♥♥ t ♠t♦♥♥ ♥

❲ s♦ ♦t♥

|〈ψNC |φNC〉|2= Tr

(ρNCψ ρNCφ

)= Tr (ρψρφ) = |〈ψ|φ〉|

2,

♥ ts t ♦rt♦♦♥t② ♦ stts s ♣rsr

♥ ♦t♥ ♦♥t♦♦♥ ♦rrs♣♦♥♥ t♥ stts ♥ ♥ stts ♥

r♥s♦r♠t♦♥ ♦ f(QNC , PNC

)♦♣rt♦rs

♥ ts st♦♥ s♦ t tr♥s♦r♠t♦♥ ♦ f(QNC , PNC

)♦♣rt♦rs ② q

♦r t WNC ♠♣ ♥ ❬❪

♦♥sr t ❲♥r r♥s♦r♠ ♥ q

WNC

(A)= ℏ

−d

ˆ

ddx ddy e−iPC(z)·yδ

(x−QC(z)

)⟨x+

ℏ

2y|A|x−

ℏ

2y

⟩

QC

,

t t r♦① tr♥s♦r♠t♦♥ s qs ♥

Qi = QCi +θij

2ℏPCj ,

Pi = PCi −ηij

2ℏQCj ,

r QCi ♥ PCi r ♦♠♠tt ♣♦st♦♥ ♥ ♠♦♠♥t rs ♥ Qi ♥ Pi r t

♥♦♥♦♠♠tt rs

♦r θ, η ≪ ℏ ℏ′ ≈ ℏ ♥ ts tr♥s♦r♠t♦♥ s s② ♥rt

QCi = Qi −θij

2ℏPj

PCi = Pi +ηij

2ℏQj .

s

WNC

(Qi

)= h−d

ˆ

ddx ddy e−iPC ·yδ

(x−QC

)⟨x+

ℏ

2y|Qi|x−

ℏ

2y

⟩

QC

♣tr r♥ ♦♥♦♠♠tt ♥t♠ ♥s ♥ ♥t♠ ♥s

= h−dˆ

ddx ddy e−iPC ·yδ

(x−QC

)⟨x+

ℏ

2y|QCi +

θij

2ℏPCj |x−

ℏ

2y

⟩

QC

= h−dˆ

ddx ddy e−iPC ·yδ

(x−QC

)(xiδ

(ℏ

2y

)+θij

2ℏ

d

dQCjδ

(ℏ

2y

)).

♥trt♥ ② ♣rts ♦♥ t s♦♥ tr♠ ♥ ♥ ♦♥r② tr♠ ♥s t

WNC

(Qi

)= ℏ

−d

ˆ

ddx ddy e−iPC ·yδ

(x−QC(z)

)(xiδ

(ℏ

2y

)+θij

2ℏδ

(ℏ

2y

)PCj

)

=

ˆ

ddx δ(x−QC(z)

)(xi +

θij

2ℏPCj

)

=

(QCi +

θij

2ℏPCj

)= Qi,

s ①♣t r♦♠ ❲♥r tr♥s♦r♠

♥ s♥ q t tr♥s♦r♠t♦♥ ♦♠s

W−1(D WNC

[Qi

])=W−1 (D Qi) =W−1

(QCi +

θij

2ℏPCj

)=

=

ˆ

d2dk d2dk

(2π)2d

ˆ

ddQC ddPC(QCi +

θij

2ℏPCj

)· eikl

QCi+ik′l

PCi e−iklQ

Ci −ik

′

lPCi .

♥trt♦♥ ♦ tr♠ s ♦♥ ② s♠♣ ♥trt♦♥ ② ♣rts ♦ t②♣´

dx xe−x

♦r t rst tr♠

ˆ

d2dk d2dk

(2π)2d

ˆ

ddQC ddPC QCi · eikl

QCi+ik′l

PCi e−iklQ

Ci −ik

′

lPCi =

=

ˆ

d2dk d2dk

(2π)2d

eiklQC

i+ik′l

PCi

ˆ

ddPC ki(−xe−x − e−x

)|∞−∞e

−ik′lPCi =

=

ˆ

d2dk d2dk

(2π)2d

kieikl

QCi+ik′l

PCi

ˆ

ddPC e−ik′

lPCi =

=

ˆ

d2dk d2dk

(2π)2d

kieikl

QCi+ik′l

PCi = QCi .

♠r② ♦r t s♦♥ tr♠

r♥s♦r♠t♦♥ ♦ f(QNC , PNC

)♦♣rt♦rs

ˆ

d2dk d2dk

(2π)2d

ˆ

ddQC ddPC PCi · eikl

QCi+ik′l

PCi e−iklQ

Ci −ik

′

lPCi = PCi .

s

W−1(D WNC

[Qi

])= QCi +

θij

2ℏPCj .

♦t tt ts ♦♣rt♦rs ♦② t s♠r r ② ♥t♦♥ s ❬❪

s♠ ♣r♦ss ♥ r♣t ♦r W−1(D WNC

[Pi

]) ②♥

WNC

(Pi

)= ℏ

−d

ˆ

ddx ddy e−iPC ·yδ

(x−QC

)⟨x+

ℏ

2y|Pi|x−

ℏ

2y

⟩

QC

= Pi,

♥ ts

W−1(D WNC

[Pi

])=W−1 (D Pi) =

=W−1(PCi −

ηij

2ℏQCj

)= PCi −

ηij

2ℏQCi .

♦t tt ts s s♠♣② t r♦① tr♥s♦r♠t♦♥ ♦ t ♦♣rt♦rs s t s

①♣t ♦r ♦r ♥r ♥t♦♥ f(Q, P

) ts ♠t ♥♦t tr

♥ t s

f(Q, P

)=

∑

n,m,i,j

αnmQni P

mj ,

WNC

(f(Q, P

))=

=∑

n,m,i,j

αnmijh−d

ˆ

ddx ddy e−iPC ·yδ

(x−QC

)⟨x+

ℏ

2y|Qni P

mj |x−

ℏ

2y

⟩

QC

=∑

n,m,i,j

αnmijQi ⋆NC . . . ⋆NC Qin times

⋆NC Pj ⋆NC . . . ⋆NC Pjm times

♣tr r♥ ♦♥♦♠♠tt ♥t♠ ♥s ♥ ♥t♠ ♥s

=∑

n,m,i,j

αnmijQni ⋆NC P

mj .

♦r s ♦ q

D WNC

(f(Q, P

))=

∑

n,m,i,j

αnmij

(QCi +

θik

2ℏPCk

)n⋆C

(PCj −

ηjl

2ℏQCl

)m,

♥ ts

W−1(D WNC

[f(Q, P

)])=

∑

n,m,i,j

αnmij

(QCi +

θik

2ℏPCk

)n (PCj −

ηjl

2ℏQCl

)m

= f

(QCi +

θik

2ℏPCk , P

Cj −

ηjl

2ℏQCl

).

r s q

♦t tt ts s s♠♣② t r♦① tr♥s♦r♠t♦♥ ♦ t ♦♣rt♦rs s ♥sr♣rs

♥ s♥ ts ♣rtr ❲♥r r♥s♦r♠ s t t♦ ♦② t ♠♣ s ❬❪

A(ξ)

D−→ A′ (z) = A

(ξ (z)

)

Wξ ↓ ↓W ξz

A (ξ) −→D

A′ (z) = A (ξ (z))

r A ♥♦t ♦♣rt♦rs ♥ A ♥♦ts ♣ss♣ ♥t♦♥s ♥ ξ ♥ z ♥♦t ♣♦st♦♥s

♥ ♠♦♠♥t rt ② r♦① tr♥s♦r♠ s ♣r♦♣rt② s t ♦ q

❲ ts ♠t s♠ r♥♥t ♦♥ ♥st ♦ ♦r♥ t ♦♣rt♦rs ♥

♦rs ♦♥② t ♥t♦♥s ♥ ♣ss♣ ♦r ①♠♣ s♥ ♦②s qt♦♥ q

♦♥ ♥s ♦♥② ♦ t tr♥s♦r♠t♦♥

F =W−1 (D f(z)) ,

♦r ♥ ♣ss♣ ♥t♦♥ f (z) t♦ ♦t♥ ♦rrs♣♦♥♥ ♦♣rt♦r ♥

rtr♠♦r ts ♦s s t♦ ♥rst♥ t t♥ ♦ ♦♣rt♦rs ♦♥ stt ♥ ♣rtr

s♥ ♥ ♣♦st♦♥ ♥ ♠♦♠♥t ♦♣rt♦rs ♦ ♥♦t ♦♠♠t t② ♦ ♥♦t ♦r♠ ♦♠

♣t t ♦ ♦♠♠t♥ srs ♥ ts ♦♥ ♥♥♦t ♥ ♣♦st♦♥ ss ♦r

♠♦♠♥t♠ ss ♦r ② s♥ ♦♣rt♦rs ♥ ts ♥ ♦♥ rt② s♥ ♥

ts ♦♣rt♦rs ♦ ♦r♠ s t ♠♦tt♦♥ ♦r t ♣r♦s st♦♥

♣tr

♦♦ ♦r♠s ♥

♦♥♦♠♠tt ♥t♠

♥s

♥ sts tr♥s♦r♠t♦♥ t♥ ♦♣rt♦rs ♥ stts ♥ ♥

♦♥ ♥ ♣r♦ tt ♦♦ ♦r♠s ♥ s♦ ♦ ♥ s t♦ rt

t ♣r♦♦ ♦r s t tr♥s♦r♠t♦♥ t♦ st q♥t t♦r♠s ♥

♦♦♥♥ ♥ ♦♥♦♠♠tt ♥t♠ ♥s

♦t♥ ♦ t ♣r♦♦ s s♠♣ ❯s t s♠ ss♠♣t♦♥s s ♥ s t rt♦♥

t♥ ♥ ♥ t t tt t ♦rrs♣♦♥♥ ♦♣rt♦r t♦ UNC s s♦ ♥tr②

t♦ s♦ tt ♦♥♥ r ♣♦ss 〈φ|ψ〉NC = 〈φ|ψ〉2NC ♦r ♥② stts |φ〉NC ♥ |ψ〉NC

♦r♠ ♦♦♥♥ ♥ t |ψ〉NC ♥r ♥♦♥♦♠♠tt q♥t♠

stt ♥ |0〉NC ♥ ♠♣t② stt ♥ t s ♥♦t ♣♦ss ♦ t♦ ts t♦ stts ♥t♦

t♦ ♦♣s ♦ |ψ〉NC ♦r ♥② q♥t♠ stt |ψ〉NC

Pr♦♦ ss♠ tt ♦♥♥ s ♣♦ss ♥ ♥ tr s ♠t♦♥♥ HNC s♦

tt

UNC |ψ〉A,NC |0〉B,NC = |ψ〉A,NC |ψ〉B,NC

♣tr ♦♦ ♦r♠s ♥ ♦♥♦♠♠tt ♥t♠ ♥s

♦r ♥② ♥ stt |ψ〉NC ♥ ♥ ♠♣t② stt |0〉NC ♥ r UNC s t t♠ ♦t♦♥

♦♣rt♦r ss♦t t HNC

♥ |φ〉NC s ♥♦tr stt

UNC |φ〉A,NC |0〉B,NC = |φ〉A,NC |φ〉B,NC ,

♦♥ s

〈φ|ψ〉NC =(Tr(ρNCφ ρNCψ

)) 1

2

= (Tr (ρφ′ρψ′))1

2

= eiα 〈φ′|ψ′〉 ,

r α s r ♥♠r ♥ r s q

♦ ♣r♦ tt UNC s ♥tr② V ♥ s q

V =W−1(D WNC

[UNC

]),

s s♦ ♥tr② ♥ rs

Pr♦♦ ♥

V =W−1(D WNC

[U†NC

]),

V †V = W−1(D WNC

[U†NC

])·W−1

(D WNC

[UNC

])

= W−1((D WNC

[U†NC

])⋆C

(D WNC

[UNC

])),

② q ❯s♥ q ♦♥ s♦ s

V †V = W−1(D

(WNC

[U†NC

]⋆NC WNC

[UNC

]))

= W−1(D

(WNC

[U†NCUNC

])),

r s q ♥ t st st♣

♥ ♦t WNC ♥ W ♠♣ t ♥tt② t♦

♦♦ ♥r③t♦♥ ♥ ♦♥♦♠♠tt ♥t♠ ♥s

V †V = Idd×d ⇔ U†NCUNC = Idd×d.

♥ tt V s s♦ t t♠ ♦t♦♥ ♦♣rt♦r ss♦t t♦ ♠t♦♥♥ H ♥

s

H =W−1(D WNC

[HNC

]),

♦♥ s

V |ψ′〉A |0〉B = |ψ′〉A |ψ′〉B ,

V |φ′〉A |0〉B = |φ′〉A |φ′〉B ,

♥ tr♦r

〈φ|ψ〉NC = eiα 〈φ′|ψ′〉

= eiα 〈φ′|ψ′〉A 〈0|0〉B

= eiα (〈φ′|A 〈0|B) V†V (|ψ′〉A |0〉B)

= eiα 〈φ′|ψ′〉2

= e−iα 〈φ|ψ〉2NC ,

r♦♠ q ♥ ts 〈φ|ψ〉NC = 0 ♦r 〈φ|ψ〉NC = eiα ♥♥♦t tr ♦r stts

|φ〉NC ♥ |ψ〉NC ♦♥trt♦♥ rss r♦♠ t ss♠♣t♦♥ tt ♦♥♥ s ♣♦ss

♦r ♥② ♥ stt ♥ s ♥ tr ♥ ♥♦ ♦♥♥ ♥

s ♥ t ♣r♦♦ ♥ ♣r♦r♠ ♥ rrs ♦rr ♥ ts t ♦t♥

♦r♠ s s♦ ♥

♦♦ ♥r③t♦♥ ♥ ♦♥♦♠♠tt ♥t♠

♥s

♣r♦♦ s s♠r t♦ t ♦♥ ♥ ss r tr r ♥♦ s♣r♣♦st♦♥ r

t ♦♦♥♥ ♥ ♦t♥ ♦r♠s ♥ s♥ t ♥rt② ♦ t ❲♥r❲②

tr♥s♦r♠t♦♥ ♥ ts ♣rsrt♦♥ ♦ ♥trt② ♥ ♣r♦ t ♥r s

♣tr ♦♦ ♦r♠s ♥ ♦♥♦♠♠tt ♥t♠ ♥s

♦r♠ r s ♥♦ ♠t♦♥♥ HNC t ♥ ss♦t t♠ ♦t♦♥ ♦♣rt♦r UNC

s♦ tt ♦r ① stt |φ〉NC ♥ ♦r ♥② stt |ψ〉NC t ♦♦♥ s tr

UNC |ψ〉⊗kNC |0〉

⊗N−kNC = |ϕ〉

⊗nNC |0〉

⊗N−nNC ,

r |ϕ〉NC = α |ψ〉NC + β |φ〉NC t |α|2+ |β|

2= 1 ♥ r s t ♥♦tt♦♥

|ψ〉⊗k

= |ψ〉 ⊗ . . .⊗ |ψ〉k times

.

Pr♦♦ rst ♥♦t tt s ♥ ♥ β = 0 tr s ♥♦ s♣r♣♦st♦♥ t ♥♦tr

stt k < n t ♦♦♥♥ ♦r♠ ♥ k > n t ♦t♥

♦r♠ s ♦♥② ♥ t♦ ♣r♦ t s 0 < β < 1

♣♣♦s tr s ♠t♦♥♥ HNC s♦ tt

UNC |ψ〉⊗kNC |0〉

⊗N−kNC = |ϕ〉

⊗nNC |0〉

⊗N−nNC

♦r stt |ψ〉NC ♥ t |ϕ〉NC = α |ψ〉NC + β |φ〉NC

♥ ♥st s t stt eiθ |ψ〉 t♥

UNCeikθ |ψ〉

⊗kNC |0〉

⊗N−kNC = |ϕ′〉

⊗n

NC |0〉⊗N−nNC ,

r |ϕ′〉NC = αeiθ |ψ〉NC + β |φ〉NC

♦r s |ψ〉NC ∝ eiθ |ψ〉NC ss♠♥ tt t stts r ♥♦r♠③ t♥

t r♠t♥ ♦♥t ♦ q ♥ ♠t♣②♥ t t♦ q ②s

eikθ = 〈ϕ|ϕ′〉n

NC .

❯s♥ t rt♦♥ t♥ ♥ s ♦r

eiθkn = 〈ϕ|ϕ′〉NC

=(Tr(ρNCϕ ρNCϕ′

)) 1

2

= (Tr (ρϕρϕ′))1

2

= eiα 〈ϕ|ϕ′〉 .

r ♥

ρϕ =W−1(D WNC

[ρNCϕ

]).

s r♦♠ t ♥rt② ♦ t ❲② ♥ ❲♥r tr♥s♦r♠

♦♦ ♥r③t♦♥ ♥ ♦♥♦♠♠tt ♥t♠ ♥s

|ϕ〉 = α |ψ〉+ β |φ〉

|ϕ′〉 = αeiθ |ψ〉+ β |φ〉 ,

r |ψ〉 ♥ |φ〉 r t stt r♣rs♥tts ♦

ρψ =W−1(D WNC

[ρNCψ

]),

ρφ =W−1(D WNC

[ρNCφ

]).

s ♠st

〈ϕC |ϕ′C〉 ∝ e

iθ |α|2+ |β|

2,

♥ s♥ |α|2♥ |β|

2r r ♥♠rs q ♥ ♦♥② tr β = 0 s

♦♥trt♦♥ t♦ t ②♣♦tss 0 < β < 1 s ♦♥s t ♣r♦♦ ♦r

♣tr

♦♥s♦♥s

♥ ts ♦r ♦♦ ♦r♠s ♥ t ♦♥t①t ♦ r rss tr r

r ♦ t ❲♥r❲② ♦r♠s♠ ♦ ♥ t ♦♦♥♥ ♦r♠ ♥ s♦

♥r③t♦♥ ♦ t ♦♦ ♦r♠s s ♥ sss ♥ t ♦♥t①t ♦

② sts♥ rt♦♥ t♥ ♥ t s ♣r♦♥ tt ts t♦r♠s

st ♦ ♥ ♦♥ ♦♥srs ♣s s♣ t ♦r♠ ♦♠♠tt♦♥ rt♦♥s s s

♠♦st② t♦ t t tt ♥trt② ♥ ♥rt② ② trs ♥ r ♣rsr ♥

t ❲♥r❲② ♦r♠s♠ ♥ tr♦r ♠♦st ♦ t rsts r r♦♠ ♥rt②

s♦ st tr ♥

rtr♠♦r t t tt ♣ss♣ ♥trt♦♥s r ♥r♥t ♥r r♥t ♦②

♣r♦ts s♦ ♦♥trt ♦r ts rsts t♦ ♦ rrss ♦ t ♦♠♠tt♦♥ rt♦♥s

t♥ ♣s s♣ rs ♦♥sr ♥ s ② t♦r ♥ t t♦♥ ♦ ①♣tt♦♥

s s t② ♥ t ♥♣♥♥t② ♦ t ② ♦♥ ♦♦ss t♦ r♣rs♥t t

♦♣rt♦rs

♦r♣②

❬❪ s ♦♠♠♥t♦♥ ② ♣r s P②s tt ♣♣

❬❪ ❲ ❲♦♦ttrs ♥ ❲ ❩r s♥ q♥t♠ ♥♥♦t ♦♥

tr ♣♣

❬❪ Pt ♥ r♥st♥ ♠♣♦sst② ♦ t♥ ♥ ♥♥♦♥ q♥t♠ stt

r❳ ♣r♣r♥t q♥t♣

❬❪ r ♥ ❲tt♥ tr♥ t♦r② ♥ ♥♦♥♦♠♠tt ♦♠tr② ♥r②

P②s ♣

❬❪ ♥②r ♥t③ s♣t♠ P②s ♣

❬❪ ♦♥♥s ♥ r♦ ♦♥♦♠♠tt ♦♠tr② q♥t♠ s ♥ ♠♦ts

♦ ♠r♥ t♠t ♦

❬❪ ③♦ ♥t♠ t♦r② ♦♥ ♥♦♥♦♠♠tt s♣s P②s ♣

♣♣

❬❪ ♠♦ ♦ ♥ ♦s ♦♥♦♠♠tt q♥t♠ ♠♥s P②s

♣

❬❪ rt♦♠ ♦s rã♦ P st♦r♥ ♥ ❩♣♣ ♦♥♦♠♠tt

rtt♦♥ q♥t♠ P②s ♣

❬❪ rt♦♠ ♦s rã♦ P st♦r♥ ♥ ❩♣♣ ♥ ♦ rs

♥ t rt♦♥ t♥ ♥♦♥♦♠♠tt ♣r♠trs ♥ ♥♦♥♦♠♠tt q♥t♠ ♠

♥s ♦r♥ P②s tt ♣♣

❬❪ rt♦♠ ♥ P s♣ts ♦ ♣ss♣ ♥♦♥♦♠♠tt q♥t♠ ♠♥

s P②s tt ♣♣

♦r♣②

❬❪ ❲♥ ♥ t ❲♥r ♥t♦♥s ♦r r♠♦♥ ♦st♦r ♥ ♥♦♥♦♠♠

tt ♣s s♣ r❳ ♣r♣r♥t r❳

❬❪ st♦s rt♦♠ s ♥ Prt Pss♣ ♥♦♥♦♠♠tt

q♥t♠ ♦s♠♦♦② P②s ♣

❬❪ st♦s rt♦♠ s ♥ Prt ❲②♥r ♦r♠t♦♥ ♦

♥♦♥♦♠♠tt q♥t♠ ♠♥s t P②s ♣

❬❪ st♦s s ♥ Prt ❲♥r ♠srs ♥ ♥♦♥♦♠♠tt q♥t♠ ♠

♥s ♦♠♠ t P②s ♣♣

❬❪ st♦s r♥r♥ rt♦♠ s ♥ Prt ♥t♥♠♥t

t♦ ♥♦♥♦♠♠ttt② ♥ ♣s s♣ P②s ♣

❬❪ st♦s r♥r♥ rt♦♠ s ♥ Prt Pss♣

♥♦♥♦♠♠tt ♦r♠t♦♥ ♦ ♦③s ♥rt♥t② ♣r♥♣ P②s

♣

❬❪ st♦s r♥r♥ rt♦♠ s ♥ Prt Pss♣

♥♦♥♦♠♠tt ①t♥s♦♥ ♦ t r♦rts♦♥srö♥r ♦r♠t♦♥ ♦ ♦③s ♥r

t♥t② ♣r♥♣ P②s ♣

❬❪ st♦s r♥r♥ rt♦♠ s ♥ Prt ♦♣r

t♦r ♥ ss♥ sq③ stts ♥ ♥♦♥♦♠♠tt q♥t♠ ♠♥s P②s

♣

❬❪ ❲② ♦r② ♦ r♦♣s ♥ ♥t♠ ♥s ♦ ♦r Pt♦♥s

♥

❬❪ ♦② ♥t♠ ♠♥s s sttst t♦r② ♥ t♠t Pr♦♥s

♦ t ♠r P♦s♦♣ ♦t② ♣♣ ♠r ❯♥rst② Prss

❬❪ ❩♦s r ♥ rtrt ♥t♠ ♠♥s ♥ ♣s s♣ ♥ ♦r

t st ♣♣rs ♦ ❲♦r ♥t

❬❪ s ♥ Prt ♥r③ ②♥r ♠♣ ♥ ② q♥t♠ ♠♥s

♦r♥ ♦ t♠t P②ss ♣♣

❬❪ ❳ ♦ ♥ ❳ ❲♥ ❯♥ q♥t♠ ♥♦♦ t♦r♠s ♥

tr♥s♦r♠♥ ♦ q♥t♠ stts ♥ rstrt st r❳ ♣r♣r♥t r❳

♦r♣②

❬❪ r♥r♥ st♦s rt♦♠ s ♥ Prt ♥t♥♠♥t

♥ s♣rt② ♥ t ♥♦♥♦♠♠tt ♣ss♣ s♥r♦ ♥ ♦r♥ ♦ P②ss

♦♥r♥ rs ♣ P Ps♥

❬❪ ♥ r ♥ r♥ ②r♦♥ t♦♠ s♣tr♠ ♥ t

♠ st ♥ ♥♦♥♦♠♠tt q P②s tt ♣

❬❪ ♦♥♥♥♦ ♥ ♦ë ♥t♠ ♠♥s ♥t♠ ♥s

❲②

❬❪ ♥ ♥ ♠♦♥ ②r♦♥ t♦♠ s♣tr♠ ♥ ♥♦♥♦♠♠tt ♣s s♣

♥s P②s tt ♣

❬❪ ❲♥ ♥ ♥ ♣rs♥tt♦♥ ♦ ♥♦♥♦♠♠tt ♣s s♣ ♦r♥

P②s tt ♣♣

❬❪ ♦s♥♠ ♥ ❱rr qt♦♥ ♥ ♥r strt♦♥s ♥ ♥♦♥

♦♠♠tt s♥r rs ♥r tt② ♥ rtt♦♥ ♣♣