Bernoulli泛函空间中截断计数算子的时间算子 The Time Operator of the TruncationOperator of the Number Operator Actingon Bernoulli Functional

doi:10.12677/PM.2023.131002

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PureMathematicsnØêÆ,2023,13(1),15-23

PublishedOnlineJanuary2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.131002

Bernoulli•¼˜m¥äOêŽfžm

Žf

x,Üw_

∗

,â¬

†

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c124F¶¹^FÏµ2023c15F¶uÙFÏµ2023c113F

Á‡

©|^gŠŽfžmŽfnØ§ ÐÚEBernoulli•¼˜m¥†þfBernoulliD(k

—ƒéXOêŽfÙäŽfžmŽf§…y²TžmŽf¿Ø•˜"

'…c

OêŽf§žmŽf§þfBernoulliD(

TheTimeOperatoroftheTruncation

OperatoroftheNumberOperatorActing

onBernoulliFunctional

TingYang,LixiaZhang

∗

,CaishiWang

†

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Dec.4

,2022;accepted:Jan.5

,2023;published:Jan.13

,2023

∗1˜Šö"

†ÏÕŠö"

©ÙÚ^:x,Üw_,â¬.Bernoulli•¼˜m¥äOêŽfžmŽf[J].nØêÆ,2023,13(1):15-23.

DOI:10.12677/pm.2023.131002

x

Abstract

Inthispaper,usingthetimeoperatortheoryofself-adjointoperator,weconstructthe

timeofthetruncationoperatorofthenumberoperatoractingonBernoullifunctional

whichiscloselyrelatedtothequantumBernoullinoiseandprovethatthethetime

operatorisnotunique.

Keywords

Numb erOperator,TimeOperator,QuantumBernoulliNoise

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

Bernoulli•¼´˜a-‡•¼[1],3êÆÚÔnNõ+•¥2•A^.½Â

3Bernoulli•¼˜mþOêŽf3ïáOrnstein-UhlenbeckŒ+¥ˆüX-‡Ú[2].§Ø

=´gŠŽf,…†þfBernoulliD(kX—ƒéX,U†þfBernoulliD(E†ƒƒ'

MarkovŒ+[3],Ny›©-‡Ôn¿Â.

“žmŽf”˜c5guþfnØµ,3²;ƒéØåÆ¥,žm´z ‡âÔ[‹I¥Uþ

;KÝCþ[4].´·‚5¿ù‡¶¡k ˜½Ø5,Ï•3Ï~þfnØ¥,žm¿Ø

U*ÿ,´˜‡þf¯‡ *ÿž,¤Ižmëê.Lé• ˜ãžmS,<‚ÊH@

•3þfnØ¥Ø•3žmŽf[5].1961c,AharonovÚBohmE˜‘HamiltonianHž

mŽf[6],2000c,AsaoArai<?ØäkšòzAŠHamiltonianHžmŽf•3

5[7].2008c,AsaoArai3©z[7]¥?ØM-òz…äkk •‘Af˜mHamiltonianH

äkžmŽf ¿©Ú7‡^‡.É©z[7]¥äklÑAŠHamiltonianžmŽfEé

u,©•ÄBernoulli•¼˜m¥äOêŽfžmŽf.©|„(Xe:31!¥,Ä

k£ÁBernoulli•¼˜mÚOêŽf.e5,0žmŽf ½Â.1n!EBernoulli

•¼˜m¥äOêŽfžmŽf,¿y²TžmŽf¿Ø•˜.

2.ý•£

!{‡0'uþfBernoulli•¼˜m,OêŽfÚžmŽf½Â,PÒÚƒ'(J,•

[SN•„©z[3][8][9]Ú[10].

DOI:10.12677/pm.2023.13100216nØêÆ

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NL«šKê8,CL«Eê8,Γ•Nk•˜8,=Γ={σ|Γ∈N,#σ<∞},Ù¥,

#(σ)•8ÜσÄê.Dom BL«ŽfB½Â•,B

∗

L«BÝŽf.

‰½˜‡VÇ˜m(Ω,F,P),Z=(Z

)

n≥0

´(Ω,F,P)þ˜Õá‘ÅCþ,÷

v^‡:

P{Z

= ε

}= θ

,P{Z

−1

}= 1−θ

,n≥0

Ù¥ε

1−θ

,0<θ

<1.F=σ(Z

,n∈N)´Z=(Z

)

n≥0

)¤σ-“ê.d©z[8]•,

Z= (Z

)

n≥0

´(Ω,F,P)þlÑžmBernoulliD(.

hL«²•ŒÈBernoulli•¼˜m,=

h= L

(Ω,F,P).(1)

½Âh·,·i•˜mhþSÈ,¿½h·,·i'u1˜‡CþÝ‚5,'u1‡Cþ‚5,k·k•

ƒA‰ê.d©z[1]Œ•Zäk·bL«5Ÿ,l{Z

|σ∈Γ}´h;KIOÄ,Ù

¥Z

∅

= 1,

j∈σ

,σ∈Γ,σ6= ∅.(2)

ùÒL²h´˜‡Ã¡‘EHilbert˜m.

éz‡šKêk≥0,3˜mhþ•3k.Žf∂

:h→h,÷vk∂

k= 1…

∂

= 1

(k)Z

σ\k

,∂

∗

= [1−1

(k)]Z

σ∪k

,σ∈Γ,(3)

Ù¥∂

∗

L«∂

ÝŽf,σ\k= σ\{k},σ∪k= σ∪{k},…1

(k)´σŠ•8ÜNf8«

5¼ê.

Žf∂

9ÙÝŽf∂

∗

¡•Š^uBernoulli•¼«ŽfÚO)Žf,…Žfx{∂

,∂

∗

}

k≥0

¡•þfBernoulliD([8].

þfBernoulliD(÷vž;K‡†'X(CAR),=

∂

= ∂

∂

,∂

∗

∂

∗

= ∂

∗

∂

∗

,∂

∗

∂

= ∂

∂

∗

(l,k≥0,k6= l),(4)

…

∂

= ∂

∗

∂

∗

= 0,∂

∂

∗

+∂

∗

∂

= I,(5)

Ù¥I´hþü Žf.

3h¥½Â‚5ŽfN

DomN= {ξ∈h|

σ∈Γ

(#σ)

|hZ

,ξi|

<∞},(6)

Nξ=

σ∈Γ

#σhZ

,ξiZ

, ξ∈DomN,(7)

Ù¥#σL«8ÜσÄê.

DOI:10.12677/pm.2023.13100217nØêÆ

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½Â1d(6)Ú(7)½ÂŽfN‰Š^3Bernoulli•¼þOêŽf.

d©z[3]Œ,é?¿ξ,η∈DomN,

∞

k=0

h∂

ξ,∂

ηi´ýéÂñ,¿…k

hξ,Nηi=

∞

k=0

h∂

ξ,∂

ηi.(8)

lé?¿ξ,η∈DomN,k

hξ,Nηi=

∞

k=0

hξ,∂

∗

∂

ηi,(9)

ùL²Œ±òOêŽfNL«•N=

∞

k=0

∂

∗

∂

e¡‰ÑžmŽf½Â,•[SNë•©z[10].

H´Hilbert˜m,H´HþgŠŽf,T´Hþé¡Žf,e•3Hf˜mD6= {0}

(Ø˜½È—),¦D⊂Dom(TH)

Dom(HT)…3Dþ÷v;K†'X(CCR)

[T,H] := TH−HT= i,(10)

K¡T•H˜‡žmŽf,½¡T•H;KÝ,Ù¥f˜mD´Žfé(T,H)CCR-•,i

´Jü Žf.

3.Ì‡(J

!Ì‡•ÄäOêŽfžmŽf9Ùƒ'(Ø.

h

=Span{Z

|σ∈Γ

},Ù¥Γ

={σ|σ⊂{0,1,2,···,n}},n•šKê.w,h

⊂h,

…{Z

|σ∈Γ

}´k•‘Hilbert˜mh

˜‡IOÄ.ŽfN

•OêŽfNäŽf,

k=0

∂

∗

∂

.(11)

½n1éu?¿šKên≥0,N

´h

þk.gŠŽf.

y²ξ∈h

,·‚k

ξk



σ∈Γ

,ξiN



σ∈Γ

k=0

(k)hZ

,ξiZ



≤

σ∈Γ



k=0

(k)|

|hZ

,ξi



≤(n+1)

|hZ

,ξi|

(n+1)

kξk

KkN

k≤n+1,lN

´k..eyŽfN

´é¡.

DOI:10.12677/pm.2023.13100218nØêÆ

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,ξi=

τ∈Γ

,ξ

τ∈Γ

,ξ

τ∈Γ

τhZ

,ξ

= h#

σZ

,ξi= hZ

σξi=

σ∈Γ

,ξiZ

σ∈Γ

σhZ

,ξiZ

σ∈Γ

,ξiN

= hZ

ξi.

nþ,N

´k.gŠŽf.

½n2éuh

þŽfN

σ(N

) = {0,1,2,···,n+1},

…ˆAŠ-ê•g•C

n+1

,···,C

n+1

,···,C

n+1

y²d(7)ªÚ(11)ªŒ•

k=0

(k)Z

= #

(σ)Z

Ù¥#

(σ)=

k=0

(k).duZ

´h

˜‡IOÄ,KdþªŒ•,#

(σ)´N

A

Š,Z

•AŠ#

(σ)éAA•þ.qN

´h

þk.‚5Ž f,lN

•kX:Ì,

σ(N

) = {0,1,2,···,n+1}.eyN

AŠ-ê.

ŠâAŠ-ê½ÂŒ•,AŠλ-êmuÙAf˜m‘ê,=m= dim

Span{Z

σ∈Γ

}

.u´,#

(σ)=0,=σ=∅,AŠ0-êm=dim

Span{Z

∅

}

n+1

=1,

#

(σ) = 1,AŠ1-êm= dim

Span{Z

|σ∈Γ

}

= C

n+1

,Ón,#

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-êm= C

n+1

,±daí,#

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n+1

,k∈{0,1,2,···,n+1}.

l,AŠ0,···,k,···,n+1-ê•g•C

n+1

,···,C

n+1

,···,C

n+1

Ï•Γ´ŒêÃ¡‘,ŒL«{Z

|σ∈Γ}•(e

)

n≥1

,¦

,...,e

}= {Z

|σ∈Γ

},n≥1

Ù¥j

= 2

n+1

,Kéz‡n≥1k

= Span{e

,···,e

},j

= 2

n+1

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,···,e

},j

= 2

n+1

´h

˜‡IOÄ.

••BOŽ,{E

}

n+1

k=0

L«N

AŠ,{e

|k∈(0,1,...,n+1),m∈(1,2,...,M

)}L

«AŠE

¤éAA•þ, Ù¥mL«AŠE

-ê, M

= max{C

n+1

|k= 0,···,n+1}.

ddŒN

= E

-e

√

m=1

,KN

= E

,w,{e

}•´h

IOX,he

i=δ

.Ù

DOI:10.12677/pm.2023.13100219nØêÆ

x

¥δ

•Dirac¼ê.e¡½Âh

þ‚5ŽfT

Xe:

ϕ:= i

n+1

k=1

(

n+1

l6=k

,ϕi

−E

,(12)

ϕk

n+1

k=0



n+1

l6=k

,ϕi

−E



.(13)

é˜‡f8D⊂H,Span{D}´dD¥k•õ‡•þ‚5|ÜÜ¤‚5f˜m,e¡-

= Span

−e

|k,l∈{0,1,···,n+1}

⊂h

.(14)

½n3Žf

:= T

´é¡.

y²•Iyé?¿ϕ∈D

,khϕ,

ϕi´¢=Œ.

hϕ,

ϕi=

ϕ,i

n+1

k=0

(

n+1

l6=k

,ϕi

−E

n+1

k=0

hϕ,e

n+1

l6=k

,ϕi

−E

hϕ,

ϕi

∗

n+1

k=0

,ϕi

n+1

l6=k

hϕ,e

−E

) =

n+1

l=0

hϕ,

n+1

l6=k

,ϕi

−E

= hϕ,

ϕi,

lhϕ,

ϕi´¢.

e¡½ny²

´N

žmŽf.

½n4é¡ŽfT

´N

˜‡žmŽf,Ù¥D

´(

)CCR-•.=D

⊂Dom(

)∩

Dom(N

)…k

[

] = i, ∀ϕ∈D

.(15)

y²duN

´h

þk.‚5Žf,w,D

´(

)CCR-•,=D

⊂Dom(

)∩

Dom(N

).eyé?¿ϕ∈D

] = i.

-ϕ

−e

n+1

p=0

(

n+1

q6=p

−E

= i

n+1

p6=k

−E

DOI:10.12677/pm.2023.13100220nØêÆ

x

u´,ÏLc[OŽ,Œ

−e

) =

n+1

p6=k

−E

−i

n+1

p6=l

−E

n+1

p6=k,l

−E

−(i

−E

n+1

p6=l,k

−E

) =

−E

−i

−E

n+1

p6=k,l

−E

−i

n+1

p6=l,k

−E

n+1

p6=k,l

(

−E

−

−E

−

−E

i(E

−E

)

n+1

p6=k,l

−E

)(E

−E

)

−E

−

−E

?˜Ú/,

[i(E

−E

)

n+1

p6=k,l

−E

)(E

−E

)

−E

−

−E

] =

i(E

−E

)

n+1

p6=k,l

−E

)(E

−E

)

−E

−

−E

,˜•¡,

= E

−E

?,

−E

) = E

−E

n+1

p6=k

−E

n+1

p6=l

−E

(

n+1

p6=k,l

−E

)−iE

(

n+1

p6=l,k

−E

) =

i(E

−E

)

n+1

p6=k,l

−E

)(E

−E

)

−E

−

−E

nÜþã'X,Œ

−

= iϕ

ù¿›Xé?¿ϕ∈D

−

= i.nþ,

´N

žmŽf.

DOI:10.12677/pm.2023.13100221nØêÆ

x

½n5ÏLé¡6Ä,?ØN

žmŽf¿Ø´•˜.

½n5T

´h

þé¡Žf,÷vD

⊂Dom(T

)∩Dom(N

),¿…é?¿ϕ∈D

ϕ= N

ϕ,=[T

] = 0,K

´N

žmŽf.

y²w,D

´(

)CCR-•.ÏL{üOŽŒ

] =(

−N

(

) =

−N

= i.

u´,

´N

žmŽf.

Ä7‘8

I[g,‰ÆÄ7‘8(No.12261080).

ë•©z

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