Research Article An Inconsistency in the Spectrum …downloads.hindawi.com/journals/ahep/2015/360356.pdfResearch Article An Inconsistency in the Spectrum of Bosonic Open 2-Brane M.AbdulWasay

Research ArticleAn Inconsistency in the Spectrum of Bosonic Open 2-Brane

M Abdul Wasay12 and Ding-fang Zeng1

1 Institute of Theoretical Physics Beijing University of Technology Beijing 100124 China2Department of Physics University of Agriculture Faisalabad 38040 Pakistan

Correspondence should be addressed to M Abdul Wasay wasay 31hotmailcom

Received 14 July 2015 Revised 3 October 2015 Accepted 7 October 2015

Academic Editor Anastasios Petkou

Copyright copy 2015 M A Wasay and D-f Zeng This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

We show that the spectrum of a bosonic open 2-brane does not contain any massless states to take the role of gravitons Moreoverthe spectrum of this open 2-brane only contains half integer mass squared values

1 Introduction

Besides the impressive progress of string theories in thepast few decades which compelled physicists to believe thata consistent theory of quantum gravity based on stringphilosophy would be built soon has slowed down Theearly success of string theory also sparked the motivationto study higher dimensional extended objects Indeed ifone can give up 0-dimensional particles in favor of stringswhich are one-dimensional objects then why not strings infavor of two-dimensional membranes The basic idea thatelementary particles could be interpreted as vibrating modesof a membrane originally came in 1962 by Dirac [1]

Since the task of achieving a consistent theory of quan-tum gravity based on string philosophy became more andmore subtle physicists tried to find alternative candidates toconstruct a theory of fundamental interactions one of thesecandidates was supermembrane theory [2ndash4] However itwas soon realized that the supermembrane has an unstableground state [5]

Compared to the works on string quantization workson membrane quantization are very few precisely becauseof the difficulty the Nambu action meets when one tries togeneralize it to 2+ 1 dimensions or general 119901+1 dimensionsHowever it is possible to quantize a bosonic membranetheory in a flat background [6 7]

We will analyze the mass squared values in the spectrumof the bosonic open 2-brane The bosonic membranes could

be related to the bosonic string theory via dimensionalreduction [8] The critical spacetime dimension in analogywith bosonic string theory is taken to be 119863 = 26 Wewill focus on the spectrum of states after reviewing theirquantization

The organization of this paper is as follows Section 2 is abrief review of the quantization scheme in a flat background[6] Section 3 deals with the spectrum of this open 2-braneand Section 4 is summary and conclusion

2 A Brief Review

The quantization of bosonic open branes was studied in [6 79] The open 2-brane dynamics is captured by the Nambu-Goto action which physically describes the world-volumeswept out by the brane However for quantization purposes itis not suitable because of the square root However one mayfollow [10] and write a Polyakov-type action for the generalbosonic open 2-brane given by

119878 = minus1

41205871205721015840int1198893120590radicminusℎ [ℎ

120572120573120597120572119883120583120597120573119883120583+ 119877 minus 2Λ] (1)

where 119877 and Λ give the contribution from the cosmologicalconstant term A 2-brane sweeps out a 3-dimensional world-volume which is parameterized by 120591 1205901 1205902

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 360356 4 pageshttpdxdoiorg1011552015360356

2 Advances in High Energy Physics

The energy-momentum tensor is given by the variation ofthe action (1) with respect to the world-volume metric ℎ120572120573as

119879120572120573

=minus21205871205721015840

radicminusℎ

120575119878

120575ℎ120572120573

= 120597120572119883120583120597120573119883120583minus1

2ℎ120572120573ℎ120574120575120597120574119883120583120597120575119883120583+1

2ℎ120572120573

= 0

(2)

Under the flat metric condition ℎ120572120573

= 120578120572120573 the compo-

nents of this energy-momentum tensor can be written as

119879120572120573

= 120597120572119883120583120597120573119883120583minus1

2

sdot ℎ120572120573

[minus120597120591119883120583120597120591119883120583+ 12059712059011198831205831205971205901119883120583+ 12059712059021198831205831205971205902119883120583]

+1

2ℎ120572120573

(3)

with

11987900

=1

2[120597120591119883120583120597120591119883120583+ 12059712059011198831205831205971205901119883120583+ 12059712059021198831205831205971205902119883120583minus 1]

= 0

11987911

=1

2[120597120591119883120583120597120591119883120583+ 12059712059011198831205831205971205901119883120583minus 12059712059021198831205831205971205902119883120583+ 1]

= 0

11987922

=1

2[120597120591119883120583120597120591119883120583minus 12059712059011198831205831205971205901119883120583+ 12059712059021198831205831205971205902119883120583+ 1]

= 0

(4)

and the Euler-Lagrange equation for the119883120583 fields is given by

(1205972

120591minus 1205972

1205901

minus 1205972

1205902

)119883120583(120591 1205901 1205902) = 0 (5)

By imposing Neumann boundary condition

1205971205901119883120583(120591 0 120590

2) = 1205971205901119883120583(120591 120587 120590

2) = 0

1205971205902119883120583(120591 1205901 0) = 120597

1205902119883120583(120591 1205901 120587) = 0

(6)

one can get the following modes expansion for the119883120583 fields

119883120583(120591 1205901 1205902) =

119909120583

radic120587+2119901120583

radic120587120591

+ 119894radic2

+infin

sum

119898119899=0

(1198992+ 1198982)minus14

sdot (119883120583

119899119898119890119894120591radic1198992+1198982

minus 119883dagger120583

119899119898119890minus119894120591radic119899

2+1198982

) cos 1198991205901 cos1198981205902

(7)

and the canonical momentum as

119875120583

(120590) =119901120583

120587radic120587+1

120587

radic2

+infin

sum

119898119899=0

(1198992+ 1198982)14

sdot (119875120583dagger

119899119898119890119894120591radic1198992+1198982

+ 119875120583

119899119898119890minus119894120591radic119899

2+1198982

) cos 1198991205901

sdot cos1198981205902

(8)

According to the standard commutation relation

[119883120583 119875

]] = 120578120583]120575 (1205901minus 12059010158401) 120575 (120590

2minus 12059010158402) (9)

the following commutation relations between the creationannihilation operators can be obtained

[119883120583

119899119898 119875

]11989910158401198981015840]

=1

120587120578120583][12057511989911989910158401205751198981198981015840 minus

1

2120575minus11989911989910158401205751198981198981015840 minus

1

21205751198991198991015840120575minus119898119898

1015840]

(10)

Using these creationannihilation operators theHamiltonianof the system is expressed as

41205871205721015840119867 = int

120587

0

1198891205901int

120587

0

1198891205902(119875120583120583minusL)

= 212057210158401205872120578120583]

infin

sum

119899=1

119899 [119883120583dagger

1198990119883

]1198990+ 119883120583

1198990119883

]dagger1198990]

+ 212057210158401205872120578120583]

infin

sum

119898=1

119898[119883120583dagger

0119898119883

]0119898

+ 119883120583

0119898119883

]dagger0119898]

+ 12057210158401205872120578120583]

infin

sum

119899119898=1

(1198992+ 1198982)12

[119883120583dagger

119899119898119883

]119899119898

+ 119883120583

119899119898119883

]dagger119899119898]

+ 4120587120572101584021199012

(11)

Under the tensorial assumption

119883120583

119899119898= radic

2

120587120601120583dagger

119899otimes 120601120583dagger

119898

119883120583dagger

119899119898= radic

2

120587120601120583

119899otimes 120601120583

119898

(12)

the Hamiltonian (11) is translated into

119867 = 1198731+ 1198732+ 11987312+ 119886 + 119887 minus 120572

10158401198722 (13)

Advances in High Energy Physics 3

where

1198731= 120578120583]

infin

sum

119899=1

119899 120601120583dagger

119899otimes 120601120583dagger

0 120601120583

119899otimes 120601120583

0 (14)

1198732= 120578120583]

infin

sum

119898=1

119898120601120583dagger

0otimes 120601120583dagger

119898 120601120583

0otimes 120601120583

119898 (15)

11987312

= 120578120583]

infin

sum

119899119898=1

(1198992+ 1198982)12

120601120583dagger

119899otimes 120601120583dagger

119898 120601120583

119899otimes 120601120583

119898 (16)

119886 = 120578120583

120583

infin

sum

119899=1

119899 (17)

119887 =1

2120578120583

120583

infin

sum

119899119898=1

radic1198992+ 1198982 (18)

3 Spectrum

We will employ the Zeta function regularization scheme toregularize the infinite divergent summation in (17) and (18)The contributions arising from the infinite sum (18) have notbeen considered carefully in earlier works and have usuallybeen taken as some background field effects

In fact there is another scheme [11] the so-called EpsteinZeta functions to regularize the infinite sums of type (18)

infin

sum

119899119898=1

radic(119899

1198881

)

2

+ (119898

1198882

)

2

=1

24(1

1198881

+1

1198882

) minus120577 (3)

81205872(1198881

1198882

2

+1198882

1198882

1

)

minus12058732

2radic11988811198882

[exp(minus212058711988811198882

) (1 + O (10minus3))]

(19)

with 120577(3) asymp 1202 and 1198881le 1198882 The infinite sums (17) and

(18) appearing in the quantization of the open 2-brane can bewritten as

119886 = 120578120583

120583

infin

sum

119899=1

119899 = (119863 minus 2)

infin

sum

119899=1

119899 = (119863 minus 2) 120577 (minus1)

119887 =1

2120578120583

120583

infin

sum

119899119898=1

radic1198992+ 1198982=119863 minus 2

2

infin

sum

119899119898=1

radic1198992+ 1198982

(20)

Numerically 119886 = minus2 while 119887 asymp 12 given that the spacetimedimension119863 = 26

According to the above calculation and the fact that 11987900

=

0 = 119867 we can rewrite (13) the mass formula of the open 2-brane spectrum as

12057210158401198722= 1198731+ 1198732+ 11987312minus3

2 (21)

By definition the ground state of the system is annihilated bythe operators 120601120583

119899⩾0and 120601120583119898⩾0

with the condition that 119899sdot119898 = 0

120601119899otimes 120601119898|0 0⟩ = 0 (22)

A general state |120593 120594⟩ in the Fock space can be obtainedby application of the creation operators 120601120583dagger

119899and 120601

120583dagger

119898on the

ground state

1003816100381610038161003816120593 120594⟩ = Π119899119898

120601120583dagger

119899119894

otimes 120601120583dagger

119898119894

|0 0⟩ (23)

withsum119894119899119894= 120593 andsum

119894119898119894= 120594The states |120593 120594⟩ and |120594 120593⟩ have

the same mass squaredAt the lowest mass level the number operators 119873

1 1198732

and11987312are zero so

12057210158401198722= minus

3

2 (24)

Thefirst excited level contains two kinds of states correspond-ing to 119899 = 1119898 = 0 and 119899 = 0119898 = 1 that is

|1 0⟩ = 119896120583(120601120583dagger


0) |0 0⟩

|0 1⟩ = 119897120583(120601120583dagger


1) |0 0⟩

(25)

whose mass-square operator reads

12057210158401198722

119896= 1198731+ 1198732+ 11987312minus3

2= 1 + 0 + 0 minus

3

2= minus

1

2

12057210158401198722

119897= 1198731+ 1198732+ 11987312minus3

2= 0 + 1 + 0 minus

3

2= minus

1

2

(26)

The ground and first excited states are both tachyonicHowever the value of mass squared operator in [6] is aninteger while we show that it is half integer valued Theground states are scalar states We have two different kindsof vector states at the first excited state level corresponding to|1 0⟩ and |0 1⟩

At the second excited level there are four kinds of tensorstates featured by

119899 = 0

119898 = 2

119899 = 2

119898 = 0

1198991= 1198992= 1

1198981= 1198982= 0

1198991= 1198992= 0

1198981= 1198982= 1

(27)

with the creation operators arranging at this level as follows

|2 0⟩1= 119896120583] (120601120583dagger


0) (120601

]dagger1otimes 120601

]dagger0) |0 0⟩

|2 0⟩2= 119896120583(120601


]dagger0) |0 0⟩

|0 2⟩1= 119897120583] (120601120583dagger


1) (120601


]dagger1) |0 0⟩

|0 2⟩2= 119897120583(120601


]dagger2) |0 0⟩

(28)


For these states the number operators are 1198731= 2 119873

2=

11987312

= 0 or1198732= 2119873

1= 11987312

= 0 Themass squared operatorwill become

12057210158401198722= 1198731+ 1198732+ 11987312minus3

2

= 0 + 2 + 0 minus3

2or 2 + 0 + 0 minus

3

2=1

2

(29)

We can see that the spectrum of the bosonic 2-brane doesnot contain any massless states to play the role of gravitonsMoreover the mass squared operator for the open 2-braneonly gives half integer mass squared values

4 Summary and Conclusion

In this paper firstly we review the quantization of open 2-brane starting from the Polyakov action We then investigatethe spectrum of the open 2-brane by taking into account thecontributions of both the infinite sums (17) and (18) On thebasis of this we can draw the conclusion that there are nogravitons present in the bosonic open 2-brane at the masslesslevel or there are no massless states in the bosonic open 2-brane spectrum This further implies that the bosonic open2-brane theory is not a theory of quantum gravity This isreminiscent of the fact that in the string case [12] simpleopen strings do not form self-complete system of quantumgravity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theauthors thank Jian-FengWu for suggesting to them litera-tures on the special zeta functions [11]The work is supportedby the Open Project Program of State Key Laboratory ofTheoretical Physics Institute of Theoretical Physics ChineseAcademy of Sciences China M Abdul Wasay would alsolike to thank Yasir Jamil for providing the facilities at PhysicsDepartment University of Agriculture Faisalabad wherepart of this work was carried out

References

[1] P A M Dirac ldquoAn extensible model of the electronrdquo Proceed-ings of the Royal Society of London Series A Mathematical andPhysical Sciences vol 268 no 1332 pp 57ndash67 1962

[2] J Hughes J Liu and J Polchinski ldquoSupermembranesrdquo PhysicsLetters B vol 180 no 4 pp 370ndash374 1986

[3] E Bergshoeff E Sezgin and P K Townsend ldquoSupermembranesand eleven-dimensional supergravityrdquoPhysics Letters B vol 189no 1-2 pp 75ndash78 1987

[4] E Bergshoeff E Sezgin and P K Townsend ldquoProperties of theeleven-dimensional supermembrane theoryrdquoAnnals of Physicsvol 185 no 2 pp 330ndash368 1988

[5] B deWit M Luscher and H Nicolai ldquoThe supermem-brane isunstablerdquo Nuclear Physics B vol 320 no 1 pp 135ndash159 1989

[6] Y-C Huang and C-X Yu ldquoQuantization and spectrum of anopen 2-branerdquo Physical Review D vol 75 no 4 Article ID044011 2007

[7] C-X Yu and Y-C Huang ldquoDirac quantization of open p-branerdquo Physics Letters B vol 647 no 1 pp 49ndash55 2007

[8] G T Horowitz and L Susskind ldquoBosonic M theoryrdquo Journal ofMathematical Physics vol 42 no 7 pp 3152ndash3160 2001

[9] C-X Yu C Huang P Zhang and Y-C Huang ldquoFirst andsecond quantization theories of open p-brane and their spectrardquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 697 no 4 pp 378ndash384 2011

[10] R Banerjee P Mukherjee and A Saha ldquoBosonic P-brane andarnowitt-deser-misner decompositionrdquo Physical Review D vol72 no 6 Article ID 066015 2005

[11] E Elizalde Ten Physical Applications of Spectral Zeta FunctionsSpringer Berlin Germany 2nd edition 2012

[12] J Polchinski String Theory vol 1 Cambridge University PressCambridge UK 2001

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The energy-momentum tensor is given by the variation ofthe action (1) with respect to the world-volume metric ℎ120572120573as

119879120572120573

=minus21205871205721015840

radicminusℎ

120575119878

120575ℎ120572120573

= 120597120572119883120583120597120573119883120583minus1

2ℎ120572120573ℎ120574120575120597120574119883120583120597120575119883120583+1

2ℎ120572120573

= 0

(2)

Under the flat metric condition ℎ120572120573

= 120578120572120573 the compo-

nents of this energy-momentum tensor can be written as

119879120572120573

= 120597120572119883120583120597120573119883120583minus1

2

sdot ℎ120572120573

[minus120597120591119883120583120597120591119883120583+ 12059712059011198831205831205971205901119883120583+ 12059712059021198831205831205971205902119883120583]

+1

2ℎ120572120573

(3)

with

11987900

=1

2[120597120591119883120583120597120591119883120583+ 12059712059011198831205831205971205901119883120583+ 12059712059021198831205831205971205902119883120583minus 1]

= 0

11987911

=1

2[120597120591119883120583120597120591119883120583+ 12059712059011198831205831205971205901119883120583minus 12059712059021198831205831205971205902119883120583+ 1]

= 0

11987922

=1

2[120597120591119883120583120597120591119883120583minus 12059712059011198831205831205971205901119883120583+ 12059712059021198831205831205971205902119883120583+ 1]

= 0

(4)

and the Euler-Lagrange equation for the119883120583 fields is given by

(1205972

120591minus 1205972

1205901

minus 1205972

1205902

)119883120583(120591 1205901 1205902) = 0 (5)

By imposing Neumann boundary condition

1205971205901119883120583(120591 0 120590

2) = 1205971205901119883120583(120591 120587 120590

2) = 0

1205971205902119883120583(120591 1205901 0) = 120597

1205902119883120583(120591 1205901 120587) = 0

(6)

one can get the following modes expansion for the119883120583 fields

119883120583(120591 1205901 1205902) =

119909120583

radic120587+2119901120583

radic120587120591

+ 119894radic2

+infin

sum

119898119899=0

(1198992+ 1198982)minus14

sdot (119883120583

119899119898119890119894120591radic1198992+1198982

minus 119883dagger120583

119899119898119890minus119894120591radic119899

2+1198982

) cos 1198991205901 cos1198981205902

(7)

and the canonical momentum as

119875120583

(120590) =119901120583

120587radic120587+1

120587

radic2

+infin

sum

119898119899=0

(1198992+ 1198982)14

sdot (119875120583dagger

119899119898119890119894120591radic1198992+1198982

+ 119875120583

119899119898119890minus119894120591radic119899

2+1198982

) cos 1198991205901

sdot cos1198981205902

(8)

According to the standard commutation relation

[119883120583 119875

]] = 120578120583]120575 (1205901minus 12059010158401) 120575 (120590

2minus 12059010158402) (9)

the following commutation relations between the creationannihilation operators can be obtained

[119883120583

119899119898 119875

]11989910158401198981015840]

=1

120587120578120583][12057511989911989910158401205751198981198981015840 minus

1

2120575minus11989911989910158401205751198981198981015840 minus

1

21205751198991198991015840120575minus119898119898

1015840]

(10)

Using these creationannihilation operators theHamiltonianof the system is expressed as

41205871205721015840119867 = int

120587

0

1198891205901int

120587

0

1198891205902(119875120583120583minusL)

= 212057210158401205872120578120583]

infin

sum

119899=1

119899 [119883120583dagger

1198990119883

]1198990+ 119883120583

1198990119883

]dagger1198990]

+ 212057210158401205872120578120583]

infin

sum

119898=1

119898[119883120583dagger

0119898119883

]0119898

+ 119883120583

0119898119883

]dagger0119898]

+ 12057210158401205872120578120583]

infin

sum

119899119898=1

(1198992+ 1198982)12

[119883120583dagger

119899119898119883

]119899119898

+ 119883120583

119899119898119883

]dagger119899119898]

+ 4120587120572101584021199012

(11)

Under the tensorial assumption

119883120583

119899119898= radic

2

120587120601120583dagger

119899otimes 120601120583dagger

119898

119883120583dagger

119899119898= radic

2

120587120601120583

119899otimes 120601120583

119898

(12)

the Hamiltonian (11) is translated into

119867 = 1198731+ 1198732+ 11987312+ 119886 + 119887 minus 120572

10158401198722 (13)


where

1198731= 120578120583]

infin

sum

119899=1

119899 120601120583dagger

119899otimes 120601120583dagger

0 120601120583

119899otimes 120601120583

0 (14)

1198732= 120578120583]

infin

sum

119898=1

119898120601120583dagger


119898 120601120583

0otimes 120601120583

119898 (15)

11987312

= 120578120583]

infin

sum

119899119898=1

(1198992+ 1198982)12

120601120583dagger

119899otimes 120601120583dagger

119898 120601120583

119899otimes 120601120583

119898 (16)

119886 = 120578120583

120583

infin

sum

119899=1

119899 (17)

119887 =1

2120578120583

120583

infin

sum

119899119898=1

radic1198992+ 1198982 (18)

3 Spectrum



infin

sum

119899119898=1

radic(119899

1198881

)

2

+ (119898

1198882

)

2

=1

24(1

1198881

+1

1198882

) minus120577 (3)

81205872(1198881

1198882

2

+1198882

1198882

1

)

minus12058732

2radic11988811198882

[exp(minus212058711988811198882

) (1 + O (10minus3))]

(19)



119886 = 120578120583

120583

infin

sum

119899=1

119899 = (119863 minus 2)

infin

sum

119899=1

119899 = (119863 minus 2) 120577 (minus1)

119887 =1

2120578120583

120583

infin

sum

119899119898=1

radic1198992+ 1198982=119863 minus 2

2

infin

sum

119899119898=1

radic1198992+ 1198982

(20)



=


12057210158401198722= 1198731+ 1198732+ 11987312minus3

2 (21)


119899⩾0and 120601120583119898⩾0


120601119899otimes 120601119898|0 0⟩ = 0 (22)


119899and 120601

120583dagger

119898on the

ground state

1003816100381610038161003816120593 120594⟩ = Π119899119898

120601120583dagger

119899119894


119898119894

|0 0⟩ (23)

withsum119894119899119894= 120593 andsum



1 1198732


12057210158401198722= minus

3

2 (24)


|1 0⟩ = 119896120583(120601120583dagger


0) |0 0⟩

|0 1⟩ = 119897120583(120601120583dagger


1) |0 0⟩

(25)


12057210158401198722

119896= 1198731+ 1198732+ 11987312minus3

2= 1 + 0 + 0 minus

3

2= minus

1

2

12057210158401198722

119897= 1198731+ 1198732+ 11987312minus3

2= 0 + 1 + 0 minus

3

2= minus

1

2

(26)



119899 = 0

119898 = 2

119899 = 2

119898 = 0

1198991= 1198992= 1

1198981= 1198982= 0

1198991= 1198992= 0

1198981= 1198982= 1

(27)


|2 0⟩1= 119896120583] (120601120583dagger


0) (120601


]dagger0) |0 0⟩

|2 0⟩2= 119896120583(120601


]dagger0) |0 0⟩

|0 2⟩1= 119897120583] (120601120583dagger


1) (120601


]dagger1) |0 0⟩

|0 2⟩2= 119897120583(120601


]dagger2) |0 0⟩

(28)



2=

11987312

= 0 or1198732= 2119873

1= 11987312


12057210158401198722= 1198731+ 1198732+ 11987312minus3

2

= 0 + 2 + 0 minus3

2or 2 + 0 + 0 minus

3

2=1

2

(29)






Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where

1198731= 120578120583]

infin

sum

119899=1

119899 120601120583dagger

119899otimes 120601120583dagger

0 120601120583

119899otimes 120601120583

0 (14)

1198732= 120578120583]

infin

sum

119898=1

119898120601120583dagger


119898 120601120583

0otimes 120601120583

119898 (15)

11987312

= 120578120583]

infin

sum

119899119898=1

(1198992+ 1198982)12

120601120583dagger

119899otimes 120601120583dagger

119898 120601120583

119899otimes 120601120583

119898 (16)

119886 = 120578120583

120583

infin

sum

119899=1

119899 (17)

119887 =1

2120578120583

120583

infin

sum

119899119898=1

radic1198992+ 1198982 (18)

3 Spectrum



infin

sum

119899119898=1

radic(119899

1198881

)

2

+ (119898

1198882

)

2

=1

24(1

1198881

+1

1198882

) minus120577 (3)

81205872(1198881

1198882

2

+1198882

1198882

1

)

minus12058732

2radic11988811198882

[exp(minus212058711988811198882

) (1 + O (10minus3))]

(19)



119886 = 120578120583

120583

infin

sum

119899=1

119899 = (119863 minus 2)

infin

sum

119899=1

119899 = (119863 minus 2) 120577 (minus1)

119887 =1

2120578120583

120583

infin

sum

119899119898=1

radic1198992+ 1198982=119863 minus 2

2

infin

sum

119899119898=1

radic1198992+ 1198982

(20)



=


12057210158401198722= 1198731+ 1198732+ 11987312minus3

2 (21)


119899⩾0and 120601120583119898⩾0


120601119899otimes 120601119898|0 0⟩ = 0 (22)


119899and 120601

120583dagger

119898on the

ground state

1003816100381610038161003816120593 120594⟩ = Π119899119898

120601120583dagger

119899119894


119898119894

|0 0⟩ (23)

withsum119894119899119894= 120593 andsum



1 1198732


12057210158401198722= minus

3

2 (24)


|1 0⟩ = 119896120583(120601120583dagger


0) |0 0⟩

|0 1⟩ = 119897120583(120601120583dagger


1) |0 0⟩

(25)


12057210158401198722

119896= 1198731+ 1198732+ 11987312minus3

2= 1 + 0 + 0 minus

3

2= minus

1

2

12057210158401198722

119897= 1198731+ 1198732+ 11987312minus3

2= 0 + 1 + 0 minus

3

2= minus

1

2

(26)



119899 = 0

119898 = 2

119899 = 2

119898 = 0

1198991= 1198992= 1

1198981= 1198982= 0

1198991= 1198992= 0

1198981= 1198982= 1

(27)


|2 0⟩1= 119896120583] (120601120583dagger


0) (120601


]dagger0) |0 0⟩

|2 0⟩2= 119896120583(120601


]dagger0) |0 0⟩

|0 2⟩1= 119897120583] (120601120583dagger


1) (120601


]dagger1) |0 0⟩

|0 2⟩2= 119897120583(120601


]dagger2) |0 0⟩

(28)



2=

11987312

= 0 or1198732= 2119873

1= 11987312


12057210158401198722= 1198731+ 1198732+ 11987312minus3

2

= 0 + 2 + 0 minus3

2or 2 + 0 + 0 minus

3

2=1

2

(29)






Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2=

11987312

= 0 or1198732= 2119873

1= 11987312


12057210158401198722= 1198731+ 1198732+ 11987312minus3

2

= 0 + 2 + 0 minus3

2or 2 + 0 + 0 minus

3

2=1

2

(29)






Acknowledgments


References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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