The Neutrosophic Logic View to Schrodinger’s Cat Paradox, Revisited

2008·0.15 MB·

by  Florentin Smarandache

Volume3 PROGRESSINPHYSICS July,2008
The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited
FlorentinSmarandache andVicChristianto
(cid:3) y
DepartmentofMathematics,UniversityofNewMexico,Gallup,NM87301,USA
(cid:3)
E-mail:[email protected]
Sciprint.org—aFreeScientificElectronicPreprintServer, http://www.sciprint.org
y
E-mail:[email protected]
The present article discusses Neutrosophic logic view to Schro¨dinger’s cat paradox.
Wearguethatthisparadox involvessomedegreeofindeterminacy(unknown)which
Neutrosophiclogiccantakeintoconsideration,whereasothermethodsincludingFuzzy
logiccannot. Tomakethispropositionclear,werevisitourpreviouspaperbyoffering
anillustrationusingmodifiedcointossingproblem,knownasParrondo’sgame.
1 Introduction where p , (cid:126) represents momentum at x direction, and ratio-
x
nalisedPlanckconstantsrespectively.
The present article discusses Neutrosophic logic view to
Byintroducingkineticenergyofthemovingparticle, T,
Schro¨dinger’s cat paradox. In this article we argue that this
andwavefunction,asfollows[9]:
paradox involves some degree of indeterminacy (unknown)
which Neutrosophic logic can take into consideration, mv2 p2 (cid:126)2
T = = x = k2; (3)
whereasothermethodsincludingFuzzylogiccannot. 2 2m 2m
In the preceding article we have discussed how Neutro- and
sophiclogicviewcanofferanalternativemethodtosolvethe (x)=exp(ikx): (4)
well-knownprobleminQuantumMechanics,i.e. theSchro¨-
Thenonehasthetime-independentSchro¨dingerequation
dinger’s cat paradox [1, 2], by introducing indeterminacy of
from[1,3,4]:
theoutcomeoftheobservation.
Inotherarticlewealsodiscusspossiblere-interpretation (cid:126) @2 (x)=T (x): (5)
of quantum measurement using Unification of Fusion Theo- (cid:0)2m@x2 (cid:1)
ries as generalization of Information Fusion [3, 4, 5], which Itisinterestingtoremarkherethatbyconventionphysi-
resultsinpropositionthatonecanexpecttoneglecttheprin- cistsassertthat“thewavefunctionissimplythemathematical
ciple of “excluded middle”; therefore Bell’s theorem can be function that describes the wave” [9]. Therefore, unlike the
considered as merely tautological. [6] This alternative view waveequationinelectromagneticfields,oneshouldnotcon-
of Quantum mechanics as Information Fusion has also been siderthatequation[5]hasanyphysicalmeaning. Bornsug-
proposed by G. Chapline [7]. Furthermore this Information gested that the square of wavefunction represents the prob-
Fusion interpretation is quite consistent with measurement ability to observe the electron at given location [9, p.56].
theoryofQuantumMechanics,wheretheactionofmeasure- Although Heisenberg rejected this interpretation, apparently
mentimpliesinformationexchange[8]. Born’sinterpretationprevailsuntiltoday.
In the first section we will discuss basic propositions of NonethelessthefoundingfathersofQuantumMechanics
Neutrosophic probability and Neutrosophic logic. Then we (Einstein,DeBroglie,Schro¨dingerhimself)weredissatisfied
discusssolutiontoSchro¨dinger’scatparadox. Insubsequent withthetheoryuntiltheendoftheirlives.Wecansummarize
sectionwediscussanillustrationusingmodifiedcointossing thesituationbyquotingasfollows[9,p.13]:
problem,anddiscussitsplausiblelinktoquantumgame.
“The interpretation of Schro¨dinger’s wave function
While it is known that derivationof Schro¨dinger’s equa-
(andofquantumtheorygenerally)remainsamatterof
tionisheuristicinthesensethatweknowtheanswertowhich
continuing concern and controversy among scientists
the algebra and logic leads, but it is interesting that Schro¨-
whoclingtophilosophicalbeliefthatthenaturalworld
dinger’sequationfollowslogicallyfromdeBroglie’sgrande
isbasicallylogicalanddeterministic.”
loi de la Nature [9, p.14]. The simplest method to derive
Furthermore,the“pragmatic”viewofBohrassertsthatfora
Schro¨dinger’sequationisbyusingsimplewaveas[9]:
givenquantummeasurement[9,p.42]:
@2
exp(ikx)= k2 exp(ikx): (1) “Asystemdoesnotpossessobjectivevaluesofitsphys-
@x2 (cid:0) (cid:1) ical properties until a measurement of one of them is
Byderivingtwicethewaveanddefining: made; the act of measurement is asserted to force the
2(cid:25)mv mv p system into an eigenstate of the quantity being mea-
k= = = x ; (2)
h (cid:126) (cid:126) sured.”
16 F.SmarandacheandV.Christianto.TheNeutrosophicLogicViewtoSchro¨dinger’sCatParadox,Revisited
July,2008 PROGRESSINPHYSICS Volume3
In1935, Einstein-Podolsky-Rosenarguedthattheaxiomatic theory, the definitive and unambiguous assignment of an el-
basisofQuantumMechanicsisincomplete,andsubsequently ementoftheset{0,1}, andsotheassignmentofaninforma-
Schro¨dinger was inspired to write his well-known cat para- tioncontentofthephotonitselfisfraughtwiththesamedif-
dox.Wewilldiscusssolutionofhiscatparadoxinsubsequent ficulties [8]. Similarly, the problem becomes more adverse
section. becausethefundamentalbasisofconventionalstatisticalthe-
oriesisthesameclassicalset{0,1}.
2 Catparadoxandimpositionofboundaryconditions For example the Schro¨dinger’s cat paradox says that the
quantumstateofaphotoncanbasicallybeinmorethanone
Asweknow,Schro¨dinger’sdeepdisagreementwiththeBorn placeinthesametimewhich, translatedtotheneutrosophic
interpretation of Quantum Mechanics is represented by his set,meansthatanelement(quantumstate)belongsanddoes
catparadox,whichessentiallyquestioningthe“statistical”in- not belong to a set (a place) in the same time; or an ele-
terpretation of the wavefunction (and by doing so, denying ment (quantum state) belongs to two different sets (two dif-
thephysicalmeaningofthewavefunction). Thecatparadox ferentplaces)inthesametime. Itisaquestionof“alternative
hasbeenwrittenelsewhere[1,2],buttheessenceseemsquite worlds”theoryverywellrepresentedbytheneutrosophicset
similartocointossingproblem: theory. InSchro¨dinger’sequationonthebehaviorofelectro-
“Given p=0.5 for each side of coin to pop up, we magnetic waves and “matter waves” in quantum theory, the
will never know the state of coin before we open our wavefunction,whichdescribesthesuperpositionofpossible
palm from it; unless we know beforehand the “state” states may be simulated by a neutrosophic function, i.e. a
of the coin (under our palm) using ESP-like phenom- functionwhosevaluesarenotuniqueforeachargumentfrom
ena. Prop. (1).” thedomainofdefinition(theverticallinetestfails,intersect-
ingthegraphinmorepoints).
The only difference here is that Schro¨dinger asserts that the
Thereforethequestioncanbesummarizedasfollows[1]:
stateofthecatishalfaliveandhalfdead,whereasinthecoin
problemabove,wecanonlysaythatwedon’tknowthestate “How to describe a particle (cid:16) in the infinite micro-
ofcoinuntilweopenourpalm;i.e. thestateofcoinisinde- universethatbelongstotwodistinctplacesP andP
1 2
terminateuntilweopenourpalm. Wewilldiscussthesolu- inthesametime? (cid:16) P and(cid:16) P isatruecon-
1 1
tionofthisprobleminsubsequentsection,butfirstofallwe tradiction,withrespe2cttoQuantum2 C:onceptdescribed
shall remark here a basic principle in Quantum Mechanics, above.”
i.e. [9,p.45]: NowwewilldiscusssomebasicpropositionsinNeutrosophic
“Quantum Concept: The first derivative of the wave- logic[1].
function (cid:9) of Schro¨dinger’s wave equation must be
single-valued everywhere. As a consequence, the 3a Non-standardrealnumberandsubsets
wavefunctionitselfmustbesingle-valuedeverywhere.”
LetT,I,Fbestandardornon-standardrealsubsets ] 0,1+[,
Theaboveassertioncorrespondstoquantumlogic,whichcan (cid:0)
(cid:18)
bedefinedasfollows[10,p.30;11]: withsupT=t sup,infT=t inf,
supI=i sup,infI=i inf,
P Q=P +Q PQ: (6)
supF=f sup,infF=f inf,
_ (cid:0)
As we will see, it is easier to resolve this cat paradox
andn sup=t sup+i sup+f sup,
by releasing the aforementioned constraint of “single-
n inf=t inf+i inf+f inf.
valuedness” of the wavefunction and its first derivative. In
fact, nonlinear fluid interpretation of Schro¨dinger’s equation Obviously,t sup,i sup,f sup(cid:54)1+;andt inf,i inf,f inf(cid:62) 0,
(cid:0)
(using the level set function) also indicates that the physical whereasn sup(cid:54)3+andn inf(cid:62) 0.ThesubsetsT,I,Farenot
(cid:0)
meaningofwavefunctionincludesthenotionofmultivalued- necessarilyintervals,butmaybeanyrealsubsets: discreteor
ness [12]. In other words, one can say that observation of continuous; single element; finite or infinite; union or inter-
spin-halfelectronatlocationxdoesnotexcludeitspossibility sectionofvarioussubsetsetc. Theymayalsooverlap. These
topopupsomewhereelse. Thiscounter-intuitiveproposition realsubsetscouldrepresenttherelativeerrorsindetermining
willbedescribedinsubsequentsection. t,i,f(inthecasewhereT,I,Farereducedtopoints).
For interpretation of this proposition, we can use modal
logic[10]. Wecanusethenotionof“world”inmodallogic,
3 NeutrosophicsolutionoftheSchro¨dingercatparadox
whichissemanticdeviceofwhattheworldmighthavebeen
In the context of physical theory of information [8], Barrett like. Then, one says that the neutrosophic truth-value of a
has noted that “there ought to be a set theoretic language statement A, NL (A)=1+ if A is “true in all possible
t
which applies directly to all quantum interactions”. This is worlds.” (syntagme first used by Leibniz) and all conjunc-
becausethe idea of a bit is itself straight out of classical set tures, that one may call “absolute truth” (in the modal logic
F.SmarandacheandV.Christianto.TheNeutrosophicLogicViewtoSchro¨dinger’sCatParadox,Revisited 17
Volume3 PROGRESSINPHYSICS July,2008
it was named necessary truth, as opposed to possible truth), normallymeansy isnotforsureinA);orz(0,1,0)belongs
whereasNL (A)=1ifAistrueinatleastoneworldatsome toA(whichmeansonedoesknowabsolutelynothingabout
t
conjuncture,wecallthis“relativetruth”becauseitisrelated z’saffiliationwithA).
toa“specific”worldandaspecificconjuncture(inthemodal More general, x((0.2–0.3), (0.40–0.45) [0.50–0.51],
logicitwasnamedpossibletruth). Becauseeach“world”is {0.2,0.24,0.28})belongstothesetA,whichm[eans:
dynamic, depending on an ensemble of parameters, we in-
— with a probability in between 20-30% particle x is in
troduce the sub-category “conjuncture” within it to reflect a
apositionofA(onecannotfindanexactapproximate
particularstateoftheworld.
becauseofvarioussourcesused);
Inaformalway,let’sconsidertheworldWasbeinggen-
— withaprobabilityof20%or24%or28%xisnotinA;
erated by the formal system FS. One says that statement A
belongs to the world W if A is a well-formed formula (wff) — the indeterminacy related to the appurtenance of x to
A is in between 40–45% or between 50–51% (limits
in W, i.e. a string of symbols from the alphabet of W that
included).
conforms to the grammar of the formal language endowing
W.Thegrammarisconceivedasasetoffunctions(formation The subsets representing the appurtenance, indeterminacy,
rules)whoseinputsaresymbolsstringsandoutputs“yes”or and falsity may overlap, and n sup=30%+51%+28%>
“no”. Aformalsystemcomprisesaformallanguage(alpha- 100%inthiscase.
bet and grammar) and a deductive apparatus (axioms and/or To summarize our proposition [1, 2], given the Schro¨-
rulesofinference). Inaformalsystemtherulesofinference dinger’scatparadoxisdefinedasastatewherethecatcanbe
aresyntacticallyandtypographicallyformalinnature, with- dead, orcanbealive, oritisundecided(i.e. wedon’tknow
outreferencetothemeaningofthestringstheymanipulate. if it is dead or alive), then herein the Neutrosophic logic,
Similarly for the Neutrosophic falsehood-value, basedonthreecomponents,truthcomponent,falsehoodcom-
NL (A)=1+ if the statement A is false in all possible ponent,indeterminacycomponent(T,I,F),worksverywell.
f
worlds,wecallit“absolutefalsehood”,whereasNL (A)=1 In Schro¨dinger’s cat problem the Neutrosophic logic offers
f
if the statement A is false in at least one world, we call it the possibility of considering the cat neither dead nor alive,
“relative falsehood”. Also, the Neutrosophic indeterminacy but undecided, while the fuzzy logic does not do this. Nor-
valueNL (A) = 1ifthestatementAisindeterminateinall mallyindeterminacy(I)issplitintouncertainty(U)andpara-
i
possibleworlds,wecallit“absoluteindeterminacy”,whereas dox(conflicting)(P).
NL (A) = 1 if the statement A is indeterminate in at least We have described Neutrosophic solution of the Schro¨-
i
oneworld,wecallit“relativeindeterminacy”. dinger’s cat paradox. Alternatively, one may hypothesize
four-valued logic to describe Schro¨dinger’s cat paradox, see
Rauscheretal. [13,14].
3b Neutrosophicprobabilitydefinition
In the subsequent section we will discuss how this Neu-
Neutrosophic probability is defined as: “Is a generalization trosophicsolutioninvolving“possibletruth”and“indetermi-
oftheclassicalprobabilityinwhichthechancethatanevent nacy” can be interpreted in terms of coin tossing problem
Aoccursist%true—wheretvariesinthesubsetT,i%in- (albeit in modified form), known as Parrondo’s game. This
determinate — where i varies in the subset I, and f% false approachseemsquiteconsistentwithnewmathematicalfor-
— where f varies in the subset F. One notes that NP(A) = mulationofgametheory[20].
(T,I,F)”.Itisalsoageneralizationoftheimpreciseprobabil-
ity,whichisaninterval-valueddistributionfunction. 4 Analternativeinterpretationusingcointossproblem
Theuniversalset,endowedwithaNeutrosophicprobabil-
itydefinedforeachofitssubset,formsaNeutrosophicprob- Apartfromtheaforementionedpuremathematics-logicalap-
abilityspace. proach to Schro¨dinger’s cat paradox, one can use a well-
knownneatlinkbetweenSchro¨dinger’sequationandFokker-
Planckequation[18]:
3c SolutionoftheSchro¨dinger’scatparadox
@2p @(cid:11) @p @p
Let’sconsideraneutrosophicsetacollectionofpossiblelo- D p (cid:11) =0: (7)
cations (positions) of particle x. And let A and B be two @z2 (cid:0) @z (cid:0) @z (cid:0) @t
neutrosophicsets. Onecansay, bylanguageabuse, thatany Aquitesimilarlinkcanbefoundbetweenrelativisticclas-
particlexneutrosophicallybelongstoanyset,duetotheper- sical field equation and non-relativistic equation, for it is
centagesoftruth/indeterminacy/falsityinvolved,whichvaries known that the time-independent Helmholtz equation and
between 0 and 1+. For example: x(0.5, 0.2, 0.3) belongs Schro¨dinger equation is formally identical [15]. From this
(cid:0)
toA(whichmeans,withaprobabilityof50%particlexisin reasoningonecanarguethatitispossibletoexplainAharo-
apositionofA,withaprobabilityof30%xisnotinA,and noveffectfrompureelectromagneticfieldtheory;andthere-
the rest is undecidable); or y(0, 0, 1) belongs to A (which fore it seems also possible to describe quantum mechan-
18 F.SmarandacheandV.Christianto.TheNeutrosophicLogicViewtoSchro¨dinger’sCatParadox,Revisited
July,2008 PROGRESSINPHYSICS Volume3
ical phenomena without postulating the decisive role of pliedexternalfield.
“observer”asBohrasserted. [16,17]. Inidiomaticform,one With respect to the aforementioned Neutrosophic solu-
can expect that quantum mechanics does not have to mean tion to Schro¨dinger’s cat paradox, one can introduce a new
that“theMoonisnottherewhennobodylooksat”. “indeterminacy”parametertorepresentconditionswherethe
With respect to the aforementioned neat link between outcome may be affected by other issues (let say, apparatus
Schro¨dinger’sequationandFokker-Planckequation, itisin- setting of Geiger counter). Therefore equation (14) can be
teresting to note here that one can introduce “finite differ- writtenas:
ence”approachtoFokker-Planckequationasfollows. First,
1
we can define local coordinates, expanded locally about a p = " (cid:17) p +
i;j i 1;j 1
point (z , t ) we can map points between a real space (z;t) 2 (cid:0) (cid:0) (cid:1) (cid:0) (cid:0)
0 0 (cid:18) (cid:19) 1
andanintegerordiscretespace(i;j). Thereforewecansam- +a p + +" (cid:17) p ; (17)
0 i;j 1 i+1;j 1
plethespaceusinglinearrelationship[19]: (cid:1) (cid:0) 2 (cid:0) (cid:1) (cid:0)
whereunlikethebiaspara(cid:18)meter( 1/2(cid:19)00),theindeterminacy
(z;t)=(z +i(cid:21);t +j(cid:28)); (8) parametercanbequitelargedepen(cid:24)dingonthesysteminques-
0 0
tion. ForinstanceintheNeutrosophicexamplegivenabove,
where (cid:21) is the sampling length and (cid:28) is the sampling time.
wecanwritethat:
UsingasetoffinitedifferenceapproximationsfortheFokker-
PlanckPDE: (cid:17) 0.2 0.3=k d (cid:0)1 =k t (cid:54)0.50: (18)
@@zp =A1 = p(z0+(cid:21);t0(cid:0)(cid:28))2(cid:0)(cid:21)p(z0(cid:0)(cid:21);t0(cid:0)(cid:28)); (9) The(cid:24)onlyp(cid:0)roblemher(cid:18)eits(cid:19)thatinor(cid:18)igdin(cid:19)alcointossing,one
cannotassertan“intermediate”outcome(wheretheoutcome
@2p
@z2 =2A2 = isneitherAnorB).Thereforeoneshallintroducemodallogic
definition of “possibility” into this model. Fortunately, we
= p(z0(cid:0)(cid:21);t0(cid:0)(cid:28))(cid:0)2p(z0;t0(cid:0)(cid:28))+p(z0+(cid:21);t0(cid:0)(cid:28)); (10) can introduce this possibility of intermediate outcome into
(cid:21)2
Parrondo’sgame,soequation(17)shallberewrittenas:
and
@p p(z ;t ) p(z ;t (cid:28))
=B1 = 0 0 (cid:0) 0 0(cid:0) : (11) p = 1 " (cid:17) p +
@t (cid:28) i;j i 1;j 1
2 (cid:0) (cid:0) (cid:1) (cid:0) (cid:0)
Wecanapplythesameproceduretoobtain:
(cid:18) (cid:19) 1
+(2(cid:17)) p + +" (cid:17) p ; (19)
i;j 1 i+1;j 1
@@(cid:11)z =A1 = (cid:11)(z0+(cid:21);t0(cid:0)(cid:28))2(cid:0)(cid:21)(cid:11)(z0(cid:0)(cid:21);t0(cid:0)(cid:28)): (12) Forinstance,(cid:1)bys(cid:0)etting(cid:18)(cid:17)2 0.2(cid:0)5,th(cid:19)en(cid:1) onege(cid:0)tsthefinite
Equations (9–12) can be substituted into equation (7) to differenceequation: (cid:24)
yieldtherequiredfinitepartialdifferentialequation[19]:
p =(0.25 ") p +(0.5) p +
i;j i 1;j 1 i;j 1
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:0)
p(z0;t0)=a 1 p(z0 (cid:21);t0 (cid:28)) a0 p(z0;t0 (cid:28))+ +(0.25+") p ; (20)
(cid:0) (cid:1) (cid:0) (cid:0) (cid:0) (cid:1) (cid:0) i+1;j 1
+a+1 p(z0+(cid:21);t0 (cid:28)): (13) whichwillyieldmoreorlessthesamere(cid:1)sultco(cid:0)mparedwith
(cid:1) (cid:0)
Thisequationcanbewrittenintermsofdiscretespaceby Neutrosophicmethoddescribedintheprecedingsection.
using[8],sowehave: For this reason, we propose to call this equation (19):
Neutrosophic-modified Parrondo’s game. A generalized ex-
p =a p +a p +a p : (14) pressionofequation[19]is:
i;j 1 i 1;j 1 0 i;j 1 +1 i+1;j 1
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0)
Equation (14) is precisely the form required for Parron-
p =(p " (cid:17)) p +(z(cid:17)) p +
i;j 0 i 1;j 1 i;j 1
do’sgame.ThemeaningofParrondo’sgamecanbedescribed (cid:0) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:0)
insimplestwayasfollows[19].Consideracointossingprob- +(p0+" (cid:17)) pi+1;j 1; (21)
(cid:0) (cid:1) (cid:0)
lemwithabiasedcoin:
wherep ,zrepresentstheprobableoutcomeinstandardcoin
0
p = 1 "; (15) tossing, and a real number, respectively. For the practical
head
2 (cid:0) meaning of (cid:17), one can think (by analogy) of this indetermi-
where"isanexternalbiasthatthegamehasto“overcome”. nacyparameterasavariablethatisinverselyproportionalto
Thisbiasistypicallyasmallnumber,forinstance1/200.Now the “thickness ratio” (d=t) of the coin in question. There-
wecanexpressequation(15)infinitedifferenceequation(14) fore using equation (18), by assuming k=0.2, coin thick-
asfollows: ness=1.0mm, and coin diameter d=50mm, then we get
pi;j = 12 (cid:0)" (cid:1)pi(cid:0)1;j(cid:0)1+0(cid:1)pi;j(cid:0)1+ 21 +" (cid:1)pi+1;j(cid:0)1: (16) dif=wt=eu5s0e,aorth(cid:17)ic=k0c.o2i(n5(0f)o(cid:0)r1in=st0a.n0c0e4b,ywhgilcuhinigs1n0eg0licgoiibnlse.aBltou-t
Furt(cid:16)hermor(cid:17)e, thebiasparameterc(cid:16)anber(cid:17)elated to an ap- gether),thenbyassumingk=0.2,cointhickness=100mm,
F.SmarandacheandV.Christianto.TheNeutrosophicLogicViewtoSchro¨dinger’sCatParadox,Revisited 19
Volume3 PROGRESSINPHYSICS July,2008
and coin diameter d=50mm, we get d=t=0.5, or 3. SmarandacheF.andChristiantoV.Anoteongeometricandin-
(cid:17)=0.2(0.5) 1=0.4,whichindicatesthatchancetogetout- formationfusioninterpretationofBell’stheoremandquantum
(cid:0)
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