Table Of ContentJanuary,2009 PROGRESSINPHYSICS Volume1
On PT-Symmetric Periodic Potential, Quark Confinement,
and Other Impossible Pursuits
VicChristianto andFlorentinSmarandache
(cid:3) y
Sciprint.org—aFreeScientificElectronicPreprintServer, http://www.sciprint.org
(cid:3)
E-mail:[email protected]
DepartmentofMathematics,UniversityofNewMexico,Gallup,NM87301,USA
y
E-mail:[email protected]
As we know, it has been quite common nowadays for particle physicists to think of
siximpossiblethingsbeforebreakfast,justlikewhattheircosmologyfellowsusedto
do. In the present paper, we discuss a number of those impossible things, including
PT-symmetric periodic potential, its link with condensed matter nuclear science, and
possible neat link with Quark confinement theory. In recent years, the PT-symmetry
anditsrelatedperiodicpotentialhavegainedconsiderableinterestsamongphysicists.
Webeginwithareviewofsomeresultsfromaprecedingpaperdiscussingderivationof
PT-symmetricperiodicpotentialfrombiquaternionKlein-Gordonequationandproceed
further with the remaining issues. Further observation is of course recommended in
ordertorefuteorverifythisproposition.
1 Introduction 2 PT-symmetricperiodicpotential
As we know, it has been quite common nowadays for parti- It has been argued elsewhere that it is plausible to derive a
clephysiciststothinkofsiximpossiblethingsbeforebreak- newPT-symmetricQuantumMechanics(PT-QM;sometimes
fast [1], just like what their cosmology fellows used to do. it is called pseudo-Hermitian Quantum Mechanics [3,9])
In the present paper, we discuss a number of those impossi- whichischaracterizedbyaPT-symmetricpotential[2]
blethings,includingPT-symmetricperiodicpotential,itslink
V(x)=V( x): (1)
withcondensedmatternuclearscience,andpossibleneatlink
(cid:0)
withQuarkConfinementtheory. One particular example of such PT-symmetric potential
Inthisregards,itisworthtoremarkherethattherewere canbefoundinsinusoidal-formpotential
some attempts in literature to generalise the notion of sym-
V =sin’: (2)
metries in Quantum Mechanics, for instance by introducing
PT-symmetricharmonicoscillatorcanbewrittenaccord-
CPTsymmetry,chiralsymmetryetc. Inrecentyears,thePT-
ingly [3]. Znojil has argued too [2] that condition (1) will
symmetryanditsrelatedperiodicpotentialhavegainedcon-
yieldHulthenpotential
siderableinterestsamongphysicists[2,3]. Itisexpectedthat
thediscussionspresentedherewouldshedsomelightonthese A B
V((cid:24))= + : (3)
issues. (1 e2i(cid:24))2 (1 e2i(cid:24))
Webeginwithareviewofresultsfromourprecedingpa- (cid:0) (cid:0)
Interestingly,asimilarperiodicpotentialhasbeenknown
persdiscussingderivationofPT-symmetricperiodicpotential
for quite a long time as Posch-Teller potential [9], although
from biquaternion Klein-Gordon equation [4–6]. Thereafter
itisnotalwaysrelatedtoPT-Symmetryconsiderations. The
wediscusshowthiscanberelatedwithbothGribov’stheory
Posch-Tellersystemhasauniquepotentialintheform[9]
ofQuarkConfinement,andalsowithEQPET/TSCmodelfor
condensed matter nuclear science (aka low-energy reaction U(x)= (cid:21)cosh 2x: (4)
(cid:0)
(cid:0)
or “cold fusion”) [7]. We also highlight its plausible impli- It appears worth to note here that Posch-Teller periodic
cationtothecalculationofGamowintegralforthe(periodic) potential can be derived from conformal D’Alembert equa-
non-Coulombpotential. tions[10,p.27]. ItisalsoknownasthesecondPosch-Teller
In other words, we would like to discuss in this paper, potential
whether there is PT symmetric potential which can be ob- (cid:22)((cid:22) 1) ‘(‘+1)
served in Nature, in particular in the context of condensed V(cid:22)((cid:24))= sinh(cid:0)2(cid:24) + cosh2(cid:24) : (5)
matter nuclear science (CMNS) and Quark confinement ThenextSectionwilldiscussbiquaternionKlein-Gordon
theory. equation [4,5] and how its radial version will yield a sinu-
Nonetheless,furtherobservationisofcourserecommend- soidal form potential which appears to be related to equa-
edinordertorefuteorverifythisproposition. tion(2).
V.ChristiantoandF.Smarandache.OnPT-SymmetricPeriodicPotential,QuarkConfinement,andOtherImpossiblePursuits 63
Volume1 PROGRESSINPHYSICS January,2009
3 Solution of radial biquaternion Klein-Gordon equa- Using Maxima computer package we find solution of
tionandanewsinusoidalformpotential equation(15)asanewpotentialtakingtheformofsinusoidal
potential
In our preceding paper [4], we argue that it is possible to
m r m r
write biquaternionic extension of Klein-Gordon equation as y =k1sin j j +k2cos j j ; (16)
follows p i 1 p i 1
(cid:18) (cid:0) (cid:0) (cid:19) (cid:18) (cid:0) (cid:0) (cid:19)
@2 @2 wherek1 andk2 areparameterstobedetermined. Itappears
2 +i 2 ’(x;t)= veryinterestingtoremarkhere,whenk issetto0,thenequa-
@t2 (cid:0)r @t2 (cid:0)r tion(16)canbewrittenintheformofe2quation(2)
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19)=(cid:21) m2’(x;t); (6)
(cid:0) V =k sin’; (17)
1
orthisequationcanberewrittenas
byusingdefinition
(cid:22) +m2 ’(x;t)=0 (7) m r
}} ’=sin j j : (18)
providedweusethisdefinition p i 1
(cid:0) (cid:1) (cid:18) (cid:0) (cid:0) (cid:19)
In retrospect, the same procedure which has been tradi-
= q+i q = i @ +e @ +e @ +e @ + tionallyusedtoderivetheYukawapotential, byusingradial
1 2 3
} r r (cid:0) @t @x @y @z biquaternionKlein-Gordon potential, yields a PT-symmetric
@ (cid:18)@ @ @ (cid:19) periodicpotentialwhichtakestheformofequation(1).
+i i +e +e +e ; (8)
1 2 3
(cid:0) @T @X @Y @Z
(cid:18) (cid:19) 4 PlausiblelinkwithGribov’stheoryofQuarkConfine-
where e , e , e are quaternion imaginary units obeying
1 2 3 ment
(withordinaryquaternionsymbolse =i,e =j,e =k):
1 2 3
i2 =j2 =k2 = 1; ij = ji=k; (9) Interestingly, and quite oddly enough, we find the solution
(cid:0) (cid:0) (17)mayhavedeeplinkwithGribov’stheoryofQuarkcon-
jk= kj =i; ki= ik=j; (10) finement[8,11]. InhisThirdOrsayLectureshedescribeda
(cid:0) (cid:0)
andquaternionNablaoperatorisdefinedas[4] periodicpotentialintheform[8,p.12]
@ @ @ @ (cid:127) 3sin =0: (19)
q = i +e1 +e2 +e3 : (11) (cid:0)
r (cid:0) @t @x @y @z ByusingMaximapackage,thesolutionofequation(19)
Note that equation (11) already included partial time- isgivenby
differentiation.
1 dy
Thereafter one can expect to find solution of radial bi- x1 =k2 pk1(cid:0)cos(y)
quaternionKlein-GordonEquation[5,6]. (cid:0) p6 ; (20)
First,thestandardKlein-Gordonequationreads R 1 dy 9
x2 =k2+ pk1p(cid:0)c6os(y) >>=
@@t22 (cid:0)r2 ’(x;t)=(cid:0)m2’(x;t): (12) wnohnilleineGarriboosvciallragtuioens twhiatthRadcatumaplliyngth,etheequeaqtui>>;oatniosnha(l1l9b)einlidkie-
Atthis(cid:18)pointwecan(cid:19)introducepolarcoordinatebyusing catesclosesimilaritywithequation(2).
thefollowingtransformation ThereforeonemaythinkthatPT-symmetricperiodicpo-
tential in the form of (2) and also (17) may have neat link
1 @ @ ‘2 with the Quark Confinement processes, at least in the con-
= r2 : (13)
r r2@r @r (cid:0) r2 text of Gribov’s theory. Nonetheless, further observation is
(cid:18) (cid:19) ofcourserecommendedinordertorefuteorverifythispro-
Therefore by introducing this transformation (13) into
position.
(12)onegets(setting‘=0)
1 @ @ 5 Implication to condensed matter nuclear science.
r2 +m2 ’(x;t)=0: (14)
r2@r @r ComparingtoEQPET/TSCmodel. Gamowintegral
(cid:18) (cid:18) (cid:19) (cid:19)
By using the same method, and then one gets radial ex-
Inaccordancewitharecentpaper[6],weinterpretandcom-
pression of BQKGE (6) for 1-dimensional condition as fol-
pare this result from the viewpoint of EQPET/TSC model
lows[5,6]
whichhasbeensuggestedbyProf.Takahashiinordertoex-
plain some phenomena related to Condensed matter nuclear
1 @ @ 1 @ @
r2 i r2 +m2 ’(x;t)=0: (15) Science(CMNS).
r2@r @r (cid:0) r2@r @r
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
64 V.ChristiantoandF.Smarandache.OnPT-SymmetricPeriodicPotential,QuarkConfinement,andOtherImpossiblePursuits
January,2009 PROGRESSINPHYSICS Volume1
Takahashi [7] has discussed key experimental results 6 Concludingremarks
in condensed matter nuclear effects in the light of his
EQPET/TSCmodel. Weargueherethathispotentialmodel Inrecentyears,thePT-symmetryanditsrelatedperiodicpo-
withinversebarrierreversal(STTBA)maybecomparableto tentialhavegainedconsiderableinterestsamongphysicists.
theperiodicpotentialdescribedabove(17). Inthepresentpaper, ithasbeenshownthatonecanfind
In[7]Takahashireportedsomefindingsfromcondensed a new type of PT-symmetric periodic potential from solu-
matter nuclear experiments, including intense production of tion of the radial biquaternion Klein-Gordon Equation. We
helium-4,4Heatoms,byelectrolysisandlaserirradiationex- alsohavediscusseditsplausiblelinkwithGribov’stheoryof
periments. Furthermore he [7] analyzed those experimental Quark Confinement and also with Takahashi’s EQPET/TSC
results using EQPET (Electronic Quasi-Particle Expansion model for condensed matter nuclear science. All of which
Theory). Formation of TSC (tetrahedral symmetric conden- seems to suggest that the Gribov’s Quark Confinement the-
sate) were modeled with numerical estimations by STTBA orymayindicatesimilarity,orperhapsahiddenlink,withthe
(Sudden Tall Thin Barrier Approximation). This STTBA CondensedMatterNuclearScience(CMNS).Itcouldalsobe
modelincludesstronginteractionwithnegativepotentialnear expectedthatthoroughunderstandingoftheprocessesbehind
thecenter. CMNSmayalsorequirerevisionoftheGamowfactortotake
One can think that apparently to understand the physics intoconsiderationtheclusterdeuteriuminteractionsandalso
behind Quark Confinement, it requires fusion of different PT-symmetricperiodicpotentialasdiscussedherein.
fields in physics, perhaps just like what Langland program Furthertheoreticalandexperimentsarethereforerecom-
wantstofusedifferentbranchesinmathematics. mended to verify or refute the proposed new PT symmetric
Interestingly, Takahashi also described the Gamow inte- potentialinNature.
gralofhisSTTBAmodelasfollows[7]
SubmittedonNovember14,2008/AcceptedonNovember20,2008
b
(cid:0) =0:218 (cid:22)1=2 (V E )1=2dr: (21) References
n b d
(cid:0)
(cid:16) (cid:17)rZ0 1. http://www-groups.dcs.st-and.ac.uk/ history/Quotations/
Usingb=5:6fmandr =5fm,heobtained[7] Dodgson.html (cid:24)
2. ZnojilM.arXiv:math-ph/0002017.
P4D =0:77; (22) 3. ZnojilM.arXiv:math-ph/0104012,math-ph/0501058.
and
4. YefremovA.F., SmarandacheF., andChristiantoV. Progress
VB =0:257MeV; (23) inPhysics,2007,v.3,42;alsoin: HadronModelsandRelated
NewEnergyIssues,InfoLearnQuest,USA,2008.
whichgavesignificantunderestimatefor4Dfusionratewhen
rigidconstraintofmotionin3Dspaceattained. Nonetheless 5. ChristiantoV.andSmarandacheF. ProgressinPhysics,2008,
v.1,40.
byintroducingdifferentvaluesfor(cid:21) theestimateresultcan
4D
be improved. Therefore we may conclude that Takahashi’s 6. ChristiantoV.EJTP,2006,v.3,no.12.
STTBApotentialoffersagoodapproximation(justwhatthe 7. TakahashiA.In: SienaWorkshoponAnomaliesinMetal-D/H
nameimplies,STTBA)ofthefusionrateincondensedmatter Systems, Siena, May 2005; also in: J. Condensed Mat. Nucl.
nuclearexperiments. Sci.,2007,v.1,129.
Itshallbenoted,however,thathisSTTBAlackssufficient 8. GribovV.N.arXiv:hep-ph/9905285.
theoretical basis, therefore one can expect that a sinusoidal 9. CorreaF.,JakubskyV.,PlyushckayM.arXiv:0809.2854.
periodic potential such as equation (17) may offer better re-
10. DeOliveiraE.C.anddaRochaR.EJTP,2008,v.5,no.18,27.
sult.
11. GribovV.N.arXiv:hep-ph/9512352.
All of these seem to suggest that the cluster deuterium
12. Fernandez-GarciaN.andRosas-OrtizO.arXiv:0810.5597.
mayyieldadifferentinversebarrierreversalwhichcannotbe
predictedusingtheD-Dprocessasinstandardfusiontheory. 13. ChugunovA.I.,DeWittH.E.,andYakovlevD.G.arXiv: astro-
Inotherwords,thestandardproceduretoderiveGamowfac- ph/0707.3500.
tor should also be revised [12]. Nonetheless, it would need
furtherresearchtodeterminethepreciseGamowenergyand
Gamowfactorfortheclusterdeuteriumwiththeperiodicpo-
tentialdefinedbyequation(17);seeforinstance[13].
In turn, one can expect that Takahashi’s EQPET/TSC
modelalongwiththeproposedPT-symmetricperiodicpoten-
tial (17) may offer new clues to understand both the CMNS
processesandalsothephysicsbehindQuarkconfinement.
V.ChristiantoandF.Smarandache.OnPT-SymmetricPeriodicPotential,QuarkConfinement,andOtherImpossiblePursuits 65
