Experimental Constraints on the Neutrino Oscillations and a Simple Model of Three Flavour M

a r X i v :h e p -t h /9705178v 1 23 M a y 1997Generalized Chiral QED 2:Anomaly and Exotic StatisticsFuad M.SaradzhevInstitute of Physics,Academy of Sciences of Azerbaijan,Huseyn Javid pr.33,370143Baku,AZERBAIJANABSTRACTWe study the influence of the anomaly on the physical quantum picture of the generalized chiral Schwinger model defined on S 1.We show that the anomaly i)results in the background linearly rising electric field and ii)makes the spectrum of the physical Hamiltonian nonrelativistic without a massive boson.The physical matter fields acquire exotic statistics.We construct explicitly the algebra of the Poincare generators and show that it differs from the Poincare one.We exhibit the role of the vacuum Berry phase in the failure of the Poincare algebra to close.We prove that,in spite of the background electric field,such phenomenon as the total screening of external charges characteristic for the standard Schwinger model takes place in the generalized chiral Schwinger model,too.PACS numbers:03.70+k ,11.10.Mn.1IntroductionThe two-dimensional QED with massless fermions,i.e.the Schwinger model(SM),demonstrates such phenomena as the dynamical mass generation and the total screening of the charge[1].Although the Lagrangian of the SM contains only masslessfields,a massive bosonfield emerges out of the interplay of the dynamics that govern the originalfields.This mass generation is due to the complete compensation of any external charge inserted into the vacuum.In the chiral Schwinger model(CSM)[2,3]the right and left chiral components of the fermionic field have different charges.The left-right asymmetric matter content leads to an anomaly.At the quantum level,the local gauge symmetry is not realized by a unitary action of the gauge symmetry group on Hilbert space.The Hilbert space furnishes a projective representation of the symmetry group[4,5,6].In this paper,we aim to study the influence of the anomaly on the physical quantum picture of the CSM.Do the dynamical mass generation and the total screening of charges take place also in the CSM?Are there any new physical effects caused just by the left-right asymmetry?These are the questions which we want to answer.To get the physical quantum picture of the CSM we needfirst to construct a self-consistent quantum theory of the model and then solve all the quantum constraints.In the quantization procedure,the anomaly manifests itself through a special Schwinger term in the commutator algebra of the Gauss law generators.This term changes the nature of the Gauss law constraint:instead of beingfirst-class constraint,it turns into second-class one.As a consequence,the physical quantum states cannot be defined as annihilated by the Gauss law generator.There are different approaches to overcome this problem and to consistently quantize the CSM. The fact that the second class constraint appears only after quantization means that the number of degrees of freedom of the quantum theory is larger than that of the classical theory.To keep the Gauss law constraintfirst-class,Faddeev and Shatashvili proposed adding an auxiliaryfield in such a way that the dynamical content of the model does not change[7].At the same time,after quantization it is the auxiliaryfield that furnishes the additional”irrelevant”quantum degrees of freedom.The auxiliaryfield is described by the Wess-Zumino term.When this term is added to the Lagrangian of the original model,a new,anomaly-free model is obtained.Subsequent canonical quantization of the new model is achieved by the Dirac procedure.For the CSM,the correspondig WZ-term is not defined uniquely.It contains the so called Jackiw-Rajaraman parameter a>1.This parameter reflects an ambiguity in the bosonization procedure and in the construction of the WZ-term.The spectrum of the new,anomaly-free model turns out to be relativistic and contains a relativistic boson.However,the mass of the boson also depends on the Jackiw-Rajaraman parameter[2,3].This mass corresponds therefore to the”irrelevant”quantum degrees of freedom.The quantum theory with such a parameter in the spectrum is not physical,i.e. thatfinal version of the quantum theory which we would like to get.The latter should not contain any nonphysical parameters,otherwise one can not say anything about a physical quantum picture.In another approach also formulated by Faddeev[8],the auxiliaryfield is not added,so the quantum Gauss law constraint remains second-class.The standard Gauss law is assumed to be regained as a statement valid in matrix elements between some states of the total Hilbert space,and it is the states that are called physical.The theory is regularized in such a way that the quantum Hamiltonian commutes with the nonmodified,i.e.second-class quantum Gauss law constraint.The spectrum turns out to be non-relativistic[9,10].Here,we follow the approach given in our previous work[11].The pecularity of the CSM is that its anomalous behaviour is trivial in the sense that the second class constraint which appears afterquantization can be turned intofirst class by a simple redefinition of the canonical variables.This allows us to formulate a modified Gauss law to constrain physical states.The physical states are gauge-invariant up to a phase,the phase being1-cocycle of the gauge symmetry group algebra.In [12,13,14],the modification of the Gauss law constraint is obtained by making use of the adiabatic approach.Contrary to[11]where the CSM is defined on R1,we suppose here that space is a circle of lengthL,−L2,so space-time manifold is a cylinder S1×R1.The gaugefield then acquires aglobal physical degree of freedom represented by the non-integrable phase of the Wilson integral on S1.We show that this brings in the physical quantum picture new features of principle.Another way of making two-dimensional gaugefield dynamics nontrivial is byfixing the spatial asymptotics of the gaugefield[15,16].If we assume that the gaugefield defined on R1diminishes rather rapidly at spatial infinities,then it again acquires a global physical degree of freedom.We will see that the physical quantum picture for the model defined on S1is equivalent to that obtained in[15,16].We consider the general version of the CSM with a U(1)gaugefield coupled with different charges to both chiral components of a fermionicfield.We show that the charges are not arbitrary,but satisfy a quantization condition.The SM where these charges are equal is a special case of the generalized CSM.This will allow us at each step of our consideration to see the distinction between the two models.We work in the temporal gauge A0=0in the framework of the canonical quantization scheme and the Dirac’s quantization method for the constrained systems[17].We use the system of units where c=1.In Section2,we quantize our model in two steps.First,the matterfields are quantized, while A1is handled as a classical backgroundfield.The gaugefield A1is quantized afterwords,using the functional Schrodinger representation.We derive the anomalous commutators with nonvanishing Schwinger terms which indicate that our model is anomalous.In Section3,we show that the Schwinger term in the commutator of the Gauss law generators is removed by a redefinition of these generators and formulate the modified quantum Gauss law constraint.We prove that this constraint can be also obtained by using the adiabatic approximation and the notion of quantum holonomy.In Section4,we construct the physical quantum Hamiltonian consistent with the modified quan-tum Gauss law constraint,i.e.invariant under the modified gauge transformations both topologically trivial and non-trivial.We introduce the modified topologically non-trivial gauge transformation op-erator and defineθ–states which are its eigenstates.We consider in detail the case of the SM and demonstrate its equivalence to the freefield theory of a massive scalarfield.For the generalized CSM,we define the exotic statistics matterfield and reformulate the quantum theory in terms of thisfield.In Section5,we construct two other Poincare generators,i.e.the momentum and the boost.We act in the same way as before with the Hamiltonian,namely we define the physical generators as those which are invariant under both topologically trivial and non-trivial gauge transformations.We show that the algebra of the constructed generators is not a Poincare one and that the failure of the Poincare algebra to close is connected to the nonvanishing vacuum Berry curvature.In Section6,we study the charge screening.We introduce external charges and calculate(i)the energy of the ground state of the physical Hamiltonian with the external charges and(ii)the current density induced by these charges.Section7contains our conclusions and discussion.2Quantization Procedure2.1Classical TheoryThe Lagrangian density of the generalized CSM isL=−10,1,γ0=σ1,γ1=−iσ2,γ0γ1=γ5=σ3,σi(i=2(1±γ5)ψ.In the temporal gauge A0=0,the Hamiltonian density isH=H EM+H F,(2) where H EM=12)=A1(L2)=ψ±(L¯he±λ}ψ±,generated byG=∂1E+e+j++e−j−,λbeing a gauge function,as well as under global gauge transformations of the right-handed and left-handed Diracfields which are generated byQ±=e± L/2−L/2dxj±(x).Due to the gauge invariance,the Hamiltonian density is not uniquely determined.On the con-strained submanifold G≈0of the full phase space,the Hamiltonian density˜H=H+v H·G,(4) where v H is an arbitrary Lagrange multiplier which can be any function of thefield variables and their momenta,reduces to the Hamiltonian density H.In this sense,our theory cannot distinguish between H and˜H,and so both Hamiltonian densities are physically equivalent to each other.For arbitrary e+,e−the gauge transformations do not respect the boundary conditions 3.The gauge transformations compatible with the boundary conditions must be either of the formλ(L2)+¯h2πe+=N,N∈Z,(6)or of the formλ(L2)+¯h2πe−=N∈Z.(7)Eqs.6or7imply the charge quantization condition for our system.Without loss of generality, we choose the condition 6.For N=1,e−=e+and we have the standard Schwinger model.For N=0,we get the model in which only the right-handed component of the Diracfield is coupled to the gaugefield.From Eq.5we see that the gauge transformations under consideration are divided into topo-logical classes characterized by the integer n.Ifλ(L2),then the gauge transformation istopologically trivial and belongs to the n=0class.If n=0it is nontrivial and has winding number n.Given Eq.5,the nonintegrable phaseΓ(A)=exp{i¯he+L b(t)}.In contrast toΓ(A),the line integralb(t)=1e+Ln.By a non-trivial gauge transformation of the formg n=exp{i2πe+L].The configurations b=0and b=¯h2πe+L.2.2Quantization and AnomalyThe eigenfunctions and the eigenvalues of thefirst quantized fermionic Hamiltonians ared± x|n;± =±εn,± x|n;± ,wherex|n;± =1L exp{i¯hεn,±·x},εn,±=2π2π).We see that the spectrum of the eigenvalues depends on b.For e+b Le+L,the energies ofεn,+decrease by¯h2πL N.Some of energy levels change sign.However,the spectrum atthe configurations b=0and b=¯h2π2π¯h (and e−b L2π¯h]and{e±b L2π¯h],a†n|vac;A;+ =0for n≤[e+b Landb n |vac;A ;− =0for n ≤[e −b L2π¯h ].(11)Excited states are constructed by operating creation operators on the Fock vacuum.In the ζ–function regularization scheme,we define the action of the functional derivative on first quantized fermionic kets and bras byδδA 1(x )|n ;± ·|λεm,±|−s/2,n ;±|←δδA 1(x )|m ;± m ;±|·|λεm,±|−s/2.From 8we get the action ofδδA 1(x )a n =−lim s →0m ∈Zn ;+|δδA 1(x )a †n=lims →0m ∈Zm ;+|δδA 1(x )on b n ,b †n can be written analogously.Next we define the quantum fermionic currents and fermionic parts of the second-quantized Hamiltonian asˆj s ±(x )=12L /2−L /2dx (ψ†s ±d ±ψs ±−ψs ±d ⋆±ψ†s±).Substituting 8into these expressions,we obtainˆj s ±(x )=n ∈Z1Lnx }ρs ±(n ),whereρs +(n )≡k ∈Z12[b †k ,b k +n ]−·|λεk,−|−s/2|λεk +n,−|−s/2are momentum space charge density (or current)operators,andˆH s ±(x )=n ∈Z1Lnx }H s±(n ),H s ±(n )≡H s 0,±(n )∓e ±bρs±(n ),(12)whereH s0,+(n)≡¯hπ2[a†k,a k+n]−·|λεk,+|−s/2|λεk+n,+|−s/2,H s0,−(n)≡¯hπ2[b k+n,b†k]−·|λεk,−|−s/2|λεk+n,−|−s/2. The charges corresponding to the currentsˆj s±(x)areˆQ s±=e± L/2−L/2dxˆj s±(x)=e±ρs±(0).With Eqs.10and11,we have for the vacuum expectation values:vac;A;±|ˆj±(x)|vac;A;± =−12(ξ++ξ−),whereη±≡±lim s→01λ k∈Z|λεk,±|−s+1.Taking the sums,we obtainη±=±22π¯h}−1L(({e±b L2)2−12η±,ˆQ ±=e±:ρ±(0):−L2ξ±,where double dots indicate normal ordering with respect to|vac,A ,ˆH 0,+=¯h2π2π¯h]ka†k a k|λεk,+|−s− k≤[e+b LLlims→0{k>[e−b L2π¯h]kb†k b k|λεk,−|−s}and:ρ+(0):=lims→0{ k>[e+b L2π¯h]a k a†k|λεk,+|−s},:ρ−(0):=lims→0{ k≤[e−b L2π¯h]b k b†k|λεk,−|−s}.The operators:ˆj±(x):and:ˆH±:are well defined when acting onfinitely excited states which have only afinite number of excitations relative to the Fock vacuum.To construct the quantum electromagnetic Hamiltonian,we quantize the gaugefield using the functional Schrodinger representation.In this representation,when the vacuum and excited fermionic Fock states are functionals of A1,the gaugefield operators are represented asˆA1(x)→A1(x),ˆE(x)→−i¯hδL pxαp.Since A1(x)is a real function,αp satisfiesαp=α⋆−p.The Fourier expansion for the canonical momentum conjugate to A1(x)is thenˆE(x)=1L¯h p∈Z p=0e−i2πdαp, whereˆπb≡−i¯h dL exp{i2πL ¯h2ddb−1dα−p+qd2Lˆπ2b−1dαqd2(ξ++ξ−).If we multiply two operators that arefinite linear combinations of the fermionic creation and annihilation operators,theζ–function regulated operator product agrees with the naive product. However,if the operators involve infinite summations their naive product is not generally well defined. We then define the operator product by mutiplying the regulated operators with s large and positive and analytically continue the result to s=0.In this way we obtain the following relations[ρ±(m),ρ±(n)]−=±mδm,−n,(15) [H0,±(n),H0,±(m)]−=±¯h2π[ˆH0,±,ρ±(m)]−=∓¯h2πdbρ±(m)=0,d2π¯hδp,±m,d2π¯hδp,±m,(p>0).(16) The quantum Gauss operator isˆG=ˆG0+2πLpx−ˆG−(p)e−i2πLe+ρN(0),ˆG ±(p)≡¯h pd2πρN(±p)andρN=ρ++Nρ−is momentum space total charge density operator.Using15and16,we easily get thatρ+(±p)(andρ−(±p))are gauge invariant.For example, forρ+(±p)we have:[ˆG+(p),ρ+(±q)]−=0,[ˆG−(p),ρ+(±q)]−=0,(p>0,q>0).The operatorsˆG±(p)don’t commute with themselves,[ˆG+(p),ˆG−(q)]−=(1−N2)e2+L24π2d3Quantum Constraints3.1Quantum SymmetryIn non-anomalous gauge theories,Gauss law is considered to be valid for physical states only.This identifies physical states as those which are gauge-invariant.The problem with the anomalous be-haviour of the generalized CSM,in terms of states in Hilbert space,is apparent:owing to theSchwinger terms we cannot require that states be annihilated by the Gauss law generators ˆG±(p ).Let us represent the action of the topologically trivial gauge transformations by the operatorsU 0(τ)=exp {i¯hp>0(ˆG+τ++ˆG −τ−)}(17)with τ0,τ±(p )smooth,thenU −10(τ)α±p U 0(τ)=α±−ipτ∓(p ),U −1(τ)d dα±p∓i 2π)2τ±(p ),(p >0).The composition law for the operators U 0isU 0(τ(1))U 0(τ(2))=exp {2πiω2(τ(1),τ(2))}U 0(τ(1)+τ(2)),whereω2(τ(1),τ(2))≡−i2π¯h )2p>0p (τ(1)−τ(2)+−τ(1)+τ(2)−)is a 2-cocycle of the gauge group algebra.Thus for N =±1we are dealing with a projectiverepresentation.The 2-cocycle ω2(τ(1),τ(2))is trivial,since it can be removed by a simple redefinition of U 0(τ).Indeed,the modified operators˜U0(τ)=exp {i 2πα1(γ;τ)}·U 0(τ),(18)whereα1(γ,τ)≡−12π¯h )2p>0(α−p τ−−αp τ+)is a 1-cocycle,satisfy the ordinary composition law˜U0(τ(1))˜U 0(τ(2))=˜U 0(τ(1)+τ(2)),i.e.the action of the topologically trivial gauge transformations represented by 18is unitary.The modified Gauss law generators corresponding to 18areˆ˜G±(p )=ˆG ±(p )±18π2α±p .(19)The generators ˆ˜G±(p )commute:[ˆ˜G+(p ),ˆ˜G −(q )]−=0.This means that Gauss law can be maintained at the quantum level for N=±1,too.We define physical states as those which are annihilated byˆ˜G±(p)[11]:ˆ˜G(p)|phys;A =0.(20)±The zero componentˆG0is a sum of quantum generators of the global gauge transformations of the right-handed and left-handed fermionicfields,so the other quantum constraints are:ρ±(0):|phys;A =0.(21) It follows from20that the physical states|phys;A respond to a gauge transformation from the zero topological class with a phase:U0(τ)|phys;A =exp{−i2πα1(γ;τ)}|phys;A .(22) Only for models without anomaly,i.e.for N=±1,this equation translates into the statement that physical states are gauge invariant.Equation22expresses in an exact form the nature of anomaly in the CSM.At the quantum level the gauge invariance is not broken,but realized projectively.The1-cocycleα1occuring in the projective representation contributes to the commutator of the Gauss law generators by a Schwinger term and produces therefore the anomaly.3.2Adiabatic ApproachLet us show now that we can come to the quantum constraints20and21in a different way,using the adiabatic approximation[23,24].In the adiabatic approach,the dynamical variables are divided into two sets,one which we call fast variables and the other which we call slow variables.In our case, we treat the fermions as fast variables and the gaugefields as slow variables.Let A1be a manifold of all static gaugefield configurations A1(x).On A1a time-dependent gaugefield A1(x,t)corresponds to a path and a periodic gaugefield to a closed loop.We consider the fermionic part of the second-quantized Hamiltonian:ˆH F:which depends on t through the background gaugefield A1and so changes very slowly with time.We consider next the periodic gaugefield A1(x,t)(0≤t<T).After a time T the periodicfield A1(x,t)returns to its original value:A1(x,0)=A1(x,T),so that:ˆH F:(0)=:ˆH F:(T).At each instant t we define eigenstates for:ˆH F:(t)by:ˆH F:(t)|F,A(t) =εF(t)|F,A(t) .The state|F=0,A(t) ≡|vac,A(t) is a ground state of:ˆH F:(t),:ˆH F:(t)|vac,A(t) =0.The Fock states|F,A(t) depend on t only through their implicit dependence on A1.They are assumed to be periodic in time,|F,A(T) =|F,A(0) ,orthonormalized,F′,A(t)|F,A(t) =δF,F′,and nondegenerate.The time evolution of the wave function of our system(fermions in a background gaugefield)is clearly governed by the Schrodinger equation:∂ψ(t)i¯h¯h T0dt·εF(t),whileT0dt L/2−L/2dx˙A1(x,t) F,A(t)|iδγBerryF≡δA1(x,t)|F,A(t) ,(24) then= T0dt L/2−L/2dx˙A1(x,t)A F(x,t).γBerryFWe see that upon parallel transport around a closed loop on A1the Fock state|F,A(t) acquiresan additional phase which is integrated exponential of A F(x,t).Whereas the dynamical phaseγdynF provides information about the duration of the evolution,the Berry’s phase reflects the nontrivial holonomy of the Fock states on A1.However,a direct computation of the diagonal matrix elements ofδδδA1(x,t)A F(y,t)−2π2¯h2 n>01L n(x−y))=(1−N2)e2+2ǫ(x−y)−1The corresponding U(1)connection is easily deduced asA F=0(x,t)=−12 T0dt L/2−L/2dx L/2−L/2dy˙A1(x,t)F F=0(x,y,t)A1(y,t).In terms of the Fourier components,the connection A F=0is rewritten as vac,A(t)|ddα±p(t)|vac,A(t) ≡A±(p,t)=±(1−N2)e2+L2pα∓p,so the nonvanishing curvature isF+−(p)≡d dαpA−=(1−N2)e2+L2p.A parallel transportation of the vacuum|vac,A(t) around a closed loop in(αp,α−p)–space(p>0) yields back the same vacuum state multiplied by the phaseγBerry F=0=(1−N2)e2+L2piαp˙α−p.This phase is associated with the projective representation of the gauge group.For N=±1,when the representation is unitary,the curvature F+−and the Berry phase vanish.As mentioned in the beginning of this Section,the projective representation is trivial and the2-cocycle in the composition law of the gauge transformation operators can be removed by a redefinition of these operators.Analogously,if we redefine the momentum operators asddα±p≡d8π2¯h21 dα±p|vac,A(t) =0,˜F+−=˜ddαp˜A−=0.However,the nonvanishing curvature F+−(p)shows itself in the algebra of the modified momentum operators which are noncommuting:[˜ddα−q]−=F+−(p)δp,q.Following27,we modify the Gauss law generators asˆG ±(p)−→ˆ˜G±(p)=¯h p˜d2πρN(±p)that coincides with19.The modified Gauss law generators have vanishing vacuum expectation values,vac,A(t)|ˆ˜G±(p,t)|vac,A(t) =0.This justifies the definition20.For the zero componentˆG0,the vacuum expectation valuevac,A(t)|ˆG0|vac,A(t) =−12(e+η++e−η−)=1The quantum theory consistently describing the dynamics of the CSM should be definitely compatible with20.The corresponding quantum Hamiltonian is then defined by the conditions[ˆ˜H,ˆ˜G±(p)]−=0(p>0)(29)which specify thatˆ˜H must be invariant under the modified topologically trivial gauge transformations generated byˆ˜G±(p).We have in29a system of non-homogeneous equations in the Lagrange multipliersˆv H,±which become operators at the quantum level.The solution of these equations isˆv H,±(p)=¯hp2{pd4π¯h)2α∓p}.Substituting this expression forˆv H,±(p)into the quantum counterpart of28,on the physical states |phys;A we obtain1L2¯h2 p>0(d dα−p−1dαp,˜ddα±by˜d2L ˆπ2b−1dαp,˜d2Lˆπ2b+V(ρN;ρN),whereV(ρN;ρN)≡e2+Lp2ρN(−p)ρN(p)is the energy of the Coulomb current-current interaction.In order to make the dependence on N for the Hamiltonian more obvious,let us representρN asρN=12(1−N)σ,whereρ≡ρ1=ρ++ρ−,σ≡ρ−1=ρ+−ρ−,and[ρ(p),ρ(q)]−=[σ(p),σ(q)]−=0,[σ(p),ρ(q)]−=2pδp,−q.Then the Coulomb interaction energy takes the formV(ρN;ρN)=14(1−N)2V(σ;σ)+12Lˆπ2b+V(ρ;ρ).For N=−1,the momentum space electric charge density operator isσ(p)andˆ˜H EM =12π¯h:[e+b L2π¯h]+n,ˆψ+→exp{i2πn2π¯h ]→[e−b LLx}ˆψ−.The action of the topologically nontrivial gauge transformations on the states can be represented by the operatorsU n=exp{−i2π¯h ]−2πd[e+b L nρN(n)and U0is given by17.To identify the gauge transformation as belonging to the n th topological class we use the index n in31.The case n=0corresponds to the topologically trivial gauge transformations.The topologically nontrivial gauge transformation operators satisfy the same composition law as the topologically trivial ones.The modified operators are˜U n =exp{−i¯hˆTb})n|phys;A .Among all states|phys;A one may identify the eigenstates of the operators of the physical variables.The action of the topologically nontrivial gauge transformations on such states may, generally speaking,change only the phase of these states by a C–number,since with any gauge transformations both topologically trivial and nontrivial,the operators of the physical variables and the observables cannot be ing|phys;θ to designate these physical states,we haveexp{∓i¯h ˆTb})n|phys;A(so calledθ–states[26,27]),where|phys;A is an arbitrary physical state from20.In one dimension the parameterθis related to a constant background electricfield.To show this, let us introduce states which are invariant even against the topologically nontrivial gauge transfor-mations.Recalling that[e+b L2π¯h]θ}|phys;θ .(32)The new states|phys continue to be annihilated byˆ˜G±(p),and are also invariant under the topo-logically nontrivial gauge transformations.The electromagnetic part of the Hamiltonian transforms asˆHEM→exp{i[e+b L2π¯h]θ}=12L¯h2 p>0[˜d dα−p]+,i.e.in the new Hamiltonian the momentumˆπb is supplemented by the electricfield strength Eθ≡e+The condition34can be then rewritten as a system of linear equations in(β0,β±).We can easilyfind a solution of these equations,which gives us(β0,β±)as functions of[e+b L2π¯h}.However,these constants are irrelevant for our consideration and we neglect them.Finding(β0,β±)from34and substituting them into the expression33,on the physical states we obtainˆ˜H|phys;A =ˆHphys|phys;AwhereˆH phys =ˆH physF+ˆH physEM,ˆH physF=ˆH0,++ˆH0,−−1L¯h(1+N2)([e+b LL ¯h[e+b L2L ˆπ2b+V(ρN;ρN)+e2+L2π¯h] p∈Z p=0(−1)p 24(1−N2)2([e+b LL¯h p>0|λεp,±|−sρs±(−p)ρs±(p).Eqs.35and36give us a physical Hamiltonian invariant under both topologically trivial andnontrivial gauge transformations,ˆH physF andˆH physEMbeing invariant separately.The last two terms in35make invariant the free fermionic part of the Hamiltonian,while the ones in36the electromagnetic part.For N=±1,the last two terms in36vanish.These terms are therefore caused by the anomaly and represent new types of interaction which are absent in the nonanomalous models.The new interactions admit the following interpretation.Let us combine the last term in36with the kinetic part of the electromagnetic Hamiltonian,then124(1−N2)2([e+b L2L2 L/2−L/2dx(ˆπb−L E(x))2,i.e.the momentumˆπb is supplemented by the linearly rising electricfield strengthE(x)≡−e+2π¯h].As in four-dimensional models of a relativistic particle moving in an externalfield,we may define a generalized momentum operator in the formˆ˜πb(x)≡ˆπb−L E(x).The commutation relations for ˆ˜πb are[ˆ˜πb (x ),ˆ˜πb (y )]−=i (1−N 2)e 2+LL(1−N 2)[e +b L2L 2ˆ˜π2b→14π2(1−N 2)[e +b Lp 2ρN (p )=−e 2+L2p 2ρbgrd ·ρN (p ).It is just the background linearly rising electric field that couples b to the fermionic physical degreesof freedom in the Coulomb interaction.As a consequence,the eigenstates of the physical Hamiltonian are not a direct product of the purely fermionic Fock states and wave functionals of b .This is a common feature of gauge theories with anomaly.That the Hilbert space in such theories is not a tensor product of the Hilbert space for a gauge field and the fixed Hilbert space for fermions was shown in [6],[7].The background charge interpretation is related to the definition of the Fock vacuum.The definition given in Eqs.10-11depends on [e +b L2π¯h]is fixed.The values of the gauge field in regions of different [e +b L2π¯h]changes,then there is a nontrivial spectral flow,i.e.some of energy levels of the first quantized fermionic Hamiltonians cross zero and change sign.This means that the definition of the Fock vacuum changes.The charge operators ˆQ ±also change.Let :ˆQ (0)±:be charge operators defined in the region where[e +b L 2π¯h]the charge operators become :ˆQ (0)±:∓e ±[e ±b L。

最新物理实验报告(英文)Abstract:This report presents the findings of a recent physics experiment conducted to investigate the effects of quantum entanglement on particle behavior at the subatomic level. Utilizing a sophisticated setup involving photon detectors and a vacuum chamber, the experiment aimed to quantify the degree of correlation between entangled particles and to test the limits of nonlocal communication.Introduction:Quantum entanglement is a phenomenon that lies at the heart of quantum physics, where the quantum states of two or more particles become interlinked such that the state of one particle instantaneously influences the state of the other, regardless of the distance separating them. This experiment was designed to further our understanding of this phenomenon and its implications for the fundamental principles of physics.Methods:The experiment was carried out in a controlled environment to minimize external interference. A pair of photons was generated and entangled using a nonlinear crystal. The photons were then separated and sent to two distinct detection stations. The detection process was synchronized, and the data collected included the time, position, and polarization state of each photon.Results:The results indicated a high degree of correlation between the entangled photons. Despite being separated by a significant distance, the photons exhibited a consistent pattern in their polarization states, suggesting a strong entanglement effect. The data also showed that the collapse of the quantum state upon measurement occurred simultaneously for both particles, supporting the theory of nonlocality.Discussion:The findings of this experiment contribute to the ongoing debate about the nature of quantum entanglement and its potential applications. The consistent correlations observed between the entangled particles provide strong evidence for the nonlocal properties of quantum mechanics. This has implications for the development of quantum computing and secure communication technologies.Conclusion:The experiment has successfully demonstrated the robustness of quantum entanglement and its potential for practical applications. Further research is needed to explore the broader implications of these findings and to refine the experimental techniques for probing the quantum realm.References:[1] Einstein, A., Podolsky, B., & Rosen, N. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47(8), 777-780.[2] Bell, J. S. (1964). On the Einstein Podolsky RosenParadox. Physics, 1(3), 195-200.[3] Aspect, A., Grangier, P., & Roger, G. (1982). Experimental Tests of Realistic Local Theories via Bell's Theorem. Physical Review Letters, 49(2), 91-94.。

实验伦理英文作文Experiment ethics is a crucial aspect of scientific research. It involves the consideration of ethical principles and guidelines when conducting experiments involving human subjects or animals. The aim is to ensure the well-being and protection of the participants, as well as the integrity and validity of the research findings.When it comes to human subjects, informed consent is of utmost importance. Researchers must clearly explain the purpose, procedures, and potential risks of the experiment to the participants. They should also ensure that the participants have the right to withdraw from the study at any time without facing any negative consequences. This ensures that the participants are fully aware of what they are getting into and have the autonomy to make informed decisions.Another important aspect of experiment ethics is the protection of vulnerable populations. This includeschildren, pregnant women, prisoners, and individuals with mental disabilities. Extra precautions must be taken to ensure their safety and well-being during the experiment. Researchers should also consider the potential impact of the experiment on these vulnerable populations and weigh the benefits against the risks before proceeding.Animal experimentation is another area where experiment ethics come into play. While it is often necessary for scientific progress, it is important to minimize the suffering and pain inflicted on animals. Researchers should use alternatives to animal testing whenever possible and follow strict guidelines for the humane treatment of animals. This includes providing proper housing, nutrition, and veterinary care, as well as minimizing the number of animals used and the duration of their involvement in the experiment.Confidentiality and privacy are also important considerations in experiment ethics. Researchers must ensure that the personal information and data collected from participants are kept confidential and used only forthe purpose of the study. This helps to build trust with the participants and protects their privacy rights.In addition to these ethical considerations, researchers should also be aware of potential conflicts of interest. They should disclose any financial or personal relationships that may influence the outcome of the experiment. This helps to maintain the integrity and objectivity of the research findings.Overall, experiment ethics plays a vital role in ensuring the responsible and ethical conduct of scientific research. By considering the well-being and rights of participants, as well as the integrity of the research process, researchers can contribute to the advancement of knowledge while upholding ethical standards.。

a rXiv:q uant-ph/02593v218N ov22Quantum Principles and Mathematical Computability Tien D Kieu ∗,Centre for Atom Optics and Ultrafast Spectroscopy,Swinburne University of Technology,Hawthorn 3122,Australia February 1,2008Abstract Taking the view that computation is after all physical,we argue that physics,particularly quantum physics,could help extend the notion of computability.Here,we list the important and unique features of quantum mechanics and then outline a quantum mechanical “algo-rithm”for one of the insoluble problems of mathematics,the Hilbert’s tenth and equivalently the Turing halting problem.The key element of this algorithm is the computability and mea-surability of both the values of physical observables and of the quantum-mechanical probability distributions for these values.The fact is that quantum computers can prove theorems by methods that neither a human brain nor any other Turing-computational arbiter will ever be able to reproduce.What if a quantum algorithm delivered a theorem that it was infeasible to prove classically.No such algorithm is yet known,but nor is anything known to rule out such a possibility,and this raises a question of principle:should we still accept such a theorem as undoubtedly proved?We believe that the rational answer ot this question is yes,for our confidence in quantum proofs rests upon the same foundation as our confidence in classical proofs:our acceptance of the physical laws underlying the computing operations.D.Deustch,A.Ekert and R.Lupacchini [1]1IntroductionSupported by the convergence of many seemingly different models of computation put forward independently by Turing,Post,Markov and others [2],a thesis on the notion of computatbility has been formed and gained much credibility.The Church-Turing thesis which can be phrased asEvery function which would naturally be regarded as computable can be computed by a universal Turing machine.This is neither a theorem nor a conjecture,for it is not and cannot even be hoped to be proven.The thesis simply asserts some correspondence between certain informal concept,that of computability in this case,with certain logically well-defined (i.e.mathematical)object,namely,the universal Turing machine.The thesis thus imposes an upper limit of what any computing machine can be designed to do.Can this computability notion be enlarged?In principles,there is no reason why not.Proposals to overcome the Turing-machine limit range from the models of mathematical principles such as continous valued neural networks [3],DNA computing [4]to those of physical nature based on general arguments [5],relativity principles,and quantum mechanical principles [6,7,8].We summarise a quantum computing model in this paper.But first we present the quantum principles in the next section.A BFigure1:Plane-wave electrons passing through the two slits from the top to arrive at the screen at the bottom:(A)The intensity pattern on the screen shows no interference(continuous curve)when there is some mechanism to detect which slit the electrons have passed through(dashed curves are the intensities obtained when the other slit is closed);(B)Interference is clearly exhibited in the intensity pattern on the screen when no record is kept of which slit the electrons have passed through.2Quantum principlesSo,what are the extra-logical features of Quantum Mechanics that would enable an enlargement of computability?Following Feynman,we will employ the gedanken“simple”two-slit experiment, Figure1,to point out all that can and cannot be known about,but will be manifest in the weird reality of quantum physics.This thought experiment is about a plane wave of electrons–all of the electrons in which have a single,well-defined value for the momentum–passing through two slits one by one before arriving at a detection screen where each electron can be recorded at a definite position on the screen and at a definite moment in(laboratory)time.2.1Intrinsic randomnessOne important property of Quantum Mechanics is the randomness in the outcome of a quantum measurement.Even if we prepare the initial quantum states to be exactly the same in principle, (say,the plane-wave state for the electrons)we can still have different and random outcomes in subsequent measurements(likefinding out which slit of the two that an electron so prepared would go through,or where the electron would land on thefinal screen).Such randomness is a fact of life in the quantum reality of our universe.To reflect that intrinsic and inevitable randomness,the best that Quantum Mechanics,as a physical theory of nature,can do is to list,given the initial conditions,the possible values for measured quantities and the probability distributions for those values.Both the values and the probability distributions are computable in the sense that they can be evaluated algorithmically to any desirable accuracy[9]1.This definition of computability of a number when it can be evaluated to any degree of accuracy is sufficient to interpret the number and to establish its relationship with other numbers.On the other hand,not only the values registered in the measurement of some measurable but also the associated probability distributions are measurable in the sense that they can be obtained to any desirable accuracy by the act of physical measurements[9].Normally,the values for measurable are quantised so they can be obtained exactly;the probability distributions are real numbers but can be obtained to any given accuracy by repeating the measurements again and again(each time from the same initial quantum state)until the desired statistics can be reached.That is how the computable numbers from Quantum Mechanics can be judged against the measurable numbers obtained from physical experiments.Thus far,there is no evidence ofany discrepancy between theory and experiments.2.2Implied infinityRandomness is,by mathematical definition,incompressible and irreducible.In Algorithmic In-formation Theory,Chaitin[11]defines randomness by program-size complexity:a binary string is considered random when the size of the shortest program that generates that string is not “smaller”than the size of the string itself.We refer the readers to the original literature for more technically precise definitions for the cases offinite and infinite strings.Another way to see that randomness does entail infinity is given by an interesting argument by Stannett[5]based on K¨o nig’s Lemma which states that anyfinitely-branching tree which contains infinitely many terminal nodes must also contain an infinite path.It is pointed out that a classical algorithm which could generate a truly random binary sequence must contain infinitely many terminal nodes(c.f.the program-size definition for randomness).If this algorithm is recursive then it is possible that it may run forever without halting(that is,along the infinite path enabled by the K¨o nig’s Lemma)in the generation of some single digit of the sequence.As such,the algorithm does not really exist,for it cannot really generate a sequence if it is stuck indefinitely at the generation of some intermediate bit somewhere in the sequence.In sharp contrast,we can exploit Nature to generate an infinite binary sequence which is random just by,say,repeatedly detecting which of the two slits(hence the binary valuedness)the plane-wave electron goes through one by one2.Thus,paradoxically,the quantum reality of Nature somehow allows us to compress the infinitely incompressible randomness into the apparentlyfinite act of preparing the same quantum state over and over again for subsequent measurements!3Infiniteness implied by randomness is not,however,the only implied infinity that is embraced by quantum reality.Quantum Mechanics suggests that an electron would explore an infinite number of paths in going from one point to another(say,from one slit to a point on the screen, in which case an infinity is somehow“contained”in thefinite distance between the slit and the screen!).This infinite multiplicity of the paths taken forms the basis for Feynmann path integral formulation of Quantum Mechanics.An infinity within thefinite would normally entail inconsistency–or so one would deduce from mathematical logic.Amazingly,quantum reality manages to maintain the required consistency by changing the outcome of the measurement as soon as we try to detect/confirm the infinity by identifying the paths taken in between thefinite separation.This can be illustrated by and is in fact born out in the entirely different pattern(of no interference,see(A)in Fig.1.)which will be recorded on the screen if we try to detect which of the two slits the electrons have gone through on their way there.The quantum mechanically implied infinity is both needed for and consistent with thefinitely measured!2.3Quantum logicThe above peculiar properties can be captured and manisfest in the peculiarity of Quantum Logic. In contrast to the classical logic of propositional calculus,quantum propositions A,B and C (each has a value of being either TRUE or FALSE)do not in general satisfy the distributivity or modularity property[12];that is,A∧(B∨C)=(A∧B)∨(A∧C).(1) An heuristic example of this inequality can also be found in the gedanken two-slit experiment if we label the propositions as•A:the detection of an electron at a position x on thefinal screen;•B:the detection of an electron passing through one particular slit(say,the left one in Fig.1)on its way to thefinal screen;•C:the detection of an electron passing through the other slit on its way to thefinal screen.With these choices and with the assumption that the electron intensity is low enough such that on average the electrons arrive at the screen one by one(i.e.no coincidence in the detection), the lhs of(1)represents the detection of an electron at a position x on the screen without ever knowing which slit it has gone through to get there.On the other hand,the rhs of(1)represents the detection of an electron at a position x on the screen when it is known which slit of the two it definitely has passed through on its way there.The former case experimentally gives rise to an inteference pattern on the screen,built up by the electrons one by one as in(B)of Fig.1.The latter gives rise to a non-interference pattern as in(A)of Fig.1,and thus is distinguishable from the former.In particular,we canfind a position x on the screen where no electron is ever detected if we have interference(that is,at the node of interference pattern).For this position x the truth value of the combined proposition on the lhs of(1)is thus FALSE;while that of the rhs is TRUE. We then have the inequality in(1).3A quantum algorithmWe will consider the equivalent Hilbert’s tenth problem[13,14].This problem,appropriately rephrased,asks for a general algorithm to determine if any given Diophantine equation has a (non-negative)integer solution or not.A Diophantine equation involves polynomial equation of many unknowns and integer coefficients.If we canfind a general algorithm asked for by the Hilbert’s tenth problem then we will have a general algorithm for the well-known Turing halting problem;that is,we will be able to tell if any given Turing program will halt or not upon starting with some input.Classically,there is no such algorithm by the Cantor’s diagonal arguments.For this particular Hilbert’s tenth problem,we can see that its noncomputability originates from the lack of a general method to verify a negative statement concerning solution of a Diophantine equation.By direct subsitution into the Diophantine polynomial,it is straightforward to verify whether a set of integers is indeed a zero of the polynomial as it is claimed or not.But substitution can not be used to verify in general a negative statement that a Diophantine polynomial has no zero as it would require the infinite task of substituting all integers!For a particular equation,such as the Diophantine equation of the Fermat’s last theorem,one may be able tofind a specific way to confirm that the equation has no solution.But that specific way is peculiar and only applicable to the equation in consideration,or some related equations,and not to any Diophantine equations in general.Nevertheless,a quantum algorithm has been proposed recently[6,7]for the Hilbert’s tenth problem.We will summarise the main points of the algorithm below and only wish to mention here that we consider quantum algorithms are as good as any algorithms in the sense that they can be implementable in the physical world,occupyingfinite time duration andfinite spatial extent and consumingfinite physical resources.3.1OutlinesOur strategy is that we do not look for the zeroes of the Diophantine polynomial in question, which may not exist,but instead search within the domain of non-negative integers for the absolute minimum of the square of the polynomial,which always exists and isfinite.While it is equally hard tofind either the zeroes or the absolute minimum in classical computation,we have converted the problem to the realisation of the ground state of a quantum Hamiltonian and there is no known quantum principle against such an act.In fact,there is no known physical principles against it.Let us consider the three laws of thermodynamics concerning energy conservation,entropy of closed systems and the unattainability of absolute zero temperature.The energy involved in our algorithm isfinite,being the ground state energy of some Hamiltonian.The entropy increase whichultimately connects to decoherence effects is a technical problem for all quantum computation in general.It may appear that even the quantum process can only explore afinite domain in afinite time and is thus no better than a classical machine in terms of computability.But there is a crucial difference.In a classical search,even if the global minimum is encountered,it cannot generally be proved that it is the global minimum(unless it is a zero of the Diophantine equation).Armed only with classical logic,we would still have to compare it with all other numbers from the infinite domain yet to come,but we obviously can never complete this comparison infinite time–thus, the noncomputability.In the quantum case,the global minimum is encoded in the energy of the ground state of a suitable Hamiltonian.Then,by energetic tagging,the global minimum can be found infinite time and confirmed,if it is the ground state that is obtained at the end of the computation.It is the physical principles that can be utilised to identify and/or verify the ground state.These principles are over and above the mathematics which govern the logic of a classical machine and help differentiate the quantum from the classical.Quantum mechanics could“explore”an infinite domain,but only in the sense that it can select,among an infinite number of states,one single state(or a subspace in case of degeneracy)to be identified as the ground state of some given Hamiltonian(which is bounded from below).Our proposal is in contrast to the claim in[15]that quantum Turing machines compute exactly the same class of functions as do Turing machines,albeit perhaps more efficiently.The quantum Turing machine approach considered there is a direct generalisation of that of the classical Turing machines but with qubits and some universal set of one-qubit and two-qubit unitary gates to build up,step by step,dimensionally larger,but still dimensionallyfinite unitary operations. This universal set is chosen on its ability to evaluate any desirable classical logic function.Our approach,on the other hand,is from the start based on infinite-dimension Hamiltonians and also based on the special properties and unique status of their ground states.The unitary operations are then followed as the Schr¨o dinger time evolutions.3.2Verification of the ground stateThe quantum algorithm is based on the key ingredients of:•The exactitude,to the level required,of the theory of Quantum Mechanics in describing and predicting physical processes.•Our ability to physically implement certain Hamiltonians having infinite numbers of energy levels;•Our ability to physically obtain and verify some state as the desirable ground state;If any of these is not achievable or approximable with controllable accuracy,the quantum algorithm simply fails and further modifications may or may not work.Without any known physical principles outlawing these key assumptions,we sketch here an approach to obtain and verify the desirable ground state of the Hamiltonian corresponding to the Diophantine polynomial in consideration.It is in general easier to implement a hamiltonian H P than to obtain its ground state|g .We thus should start the computation in yet a different and readily obtainable initial state|g I ,which is the ground state of some other hamiltonian,H I,then deform this hamiltonian H I adiabatically in time into the hamiltonian whose ground state is the desired one,through a time-dependent process represented by an interpolating Hamiltonian H(s)=(1−s)H I+sH P,for s changes from 0to1.The theorem of quantum adiabtic processes ensures that we can get arbitrarily close to the ground state|g of H P.Figures2and3below give an heuristic illustration of the quantum adiabatic theorem,which can be exploited for optimisation problems.In order to solve the Hilbert’s tenth problem we need on the one hand such time-dependent physical(adiabatic)processes to arrive at a candidate state.On the other hand,the theory ofFigure2:An example of a landscape with a small basin attraction for the global minimum. Finding the quantum mechanical ground state of such landscape is quite difficult.00.10.20.30.40.50.60.70.80.9 1.Figure3:Exploiting Quantum Adiabatic Theorem,we start with a simple landscape at the reduced time s=0then adiabatically change it to the landscape in Figure2at s=1.As time evolves,the readily available initial ground state quantum mechanically tunnels into the sought-after ground state of the landscape of Figure2.Quantum Mechanics can be used to verify whether this candidate is the ground state through the usual statistical predictions from the Schr¨o dinger equation with a truncated number of energy states of the time-dependent Hamiltonian H(s).This way,we can overcome the problem of which states are to be included in the truncated basis for a numerical study of Quantum Mechanics. This also reconciles with the Cantor diagonal arguments which state that the problem could not be solved entirely in the framework of classical computation.The key factor in the ground state verification is the probability distributions,which are both computable from a numerical study of Quantum Mechanics(that is,with the control in the calculation to reach any desirable accuracy)and measurable in practice(i.e.,by repeating the physical processes to obtain the statistics to any desirable accuracy).By matching the calculated with the measured,both of which depend on the evolution time which we can vary,we then can unambiguously identify the ground state of thefinal Hamiltonian.The information about the existence of solution,or lack of it,for the given Diophantine polynomial can be inferred through some further quantum measurements on this ground state.It is worth noting that we have here a peculiar situation in which the computational com-plexity,that is,the evolution time,might not be known exactly before carrying out the quantum computation–although it can be estimated approximately.4When is a proof a proof?Proof,be it mathematical or general,is the means to an end:a proof is there to explain,convince or persuade the others(and even oneself)the“truthful”value of certain statement(s).A classical proof,based on classical logic,starts with afinite number of axioms from which afinite number of subsequent/intermediate statements can be derived with the help of afinite number of inference rules.And it ends with afinal statement,the“truth”.All thesefiniteness requirements are there to ensure the reproducibility of the proof in afinite time and manner.A classical algorithm is a particular case of such a type of proof,with input being(part of)the axioms and output thefinal“truth”.Deutsch and some others[1,16]see such a classical proof/algorithm as an object,static with intermediate records.On the other hand and in contrast,quantum algorithms are seen as dy-namical processes wherein the intermediate“steps”cannot be recorded without destroying the interference and thus the algorithms themselves.The intermediate quantum“steps”,as a matter of facts,cannot be even made out as clearly defined,discrete steps,expressible in terms of classical propositions.The requirement offiniteness of the number of intermediate steps in a classical proof is no longer relevant(and unobtainable in a quantum proof anyway)if consistency is maintainable and reproducibility is achievable in quantum“proofs”.For such quantum proofs to be credible,they must be consistent:a statement and its negation cannot be proved at the same time under the same conditions.In basing the notion of proof on the physical reality,the condition of consistency should be automatically and implicitly guaranteed,for after all there is only one reality–or at most one which we can perceive.Signs of inconsistency would not be pointing to something intrinsic of the reality,but would be only because of our perception or understanding of reality.In other words,inconsistency,as we see it,cannot refute the reality;it only hints that it is time we need a new theory of Nature,which in turn may or may not affect the“proof.”For such quantum proofs to be of some usefulness,they must be repoducible:reproducible in afinite duration of time,reproducible in at different locations and reproducible at different times.The latter two requirements are ensured by the principles of invariance under spatial and temporal translations,which are some of the most cherished physical principles.These principles can be tested,and have been tested extensively to the hisghest accuracy without any failures,via their consequences in the conservation of energy and linear momentum.But can they,the quantum proofs,be acceptable as proofs?We would prefer an affirmative answer even though the answer to this question might only be a matter of taste.It should be easier to accept a quantum process a proof if the end result of the quantum process can be verified by classical means–for instance,in a factoring problem,the obtained primes can be easily multiplied together to give back the original number as a check.Similarly,a quantum process should also be acceptable as a valid proof even if such direct verification cannot be carried out as a matter of principle but the result somehow can be verified indirectly through some other (physical or mathematical)handles.This is the case of our algorithm for Hilbert’s tenth problem above where we only need to verify that some state is indeed the ground state,a physical attribute only indirectly linked to the mathematical solutions sought.The authors whose quotation is quoted at the beginning of this article have also argued that even when,as a matter of principles,there is no direct nor indirect verifications possible,quantum process should still be considered as valid means of proof,simply because of“our acceptance of the physical laws underlying the computing operations.”5Concluding remarksIn this paper we review the important characters of Quantum Principles and put forward the arguments that these physical principles may help compute some of the classically noncomputable. We also outline a recently proposed quantum algorithm for the Hilbert’s tenth problem,and emphasise the key role of probability distributions in the crucial verification,of a physical nature, for the solution for this classically noncomputable problem.It remains to be seen if and when the quantum algorithm can be physically realised.If not prohibited by any physical principles,and we know none so far,then we trust that it can be implementable and will be realisable.Whatever the case it may turn out to be,our investigation has already opened up new and interesting directions for Mathematics itself.Our quantum algo-rithm has inspired a reformulation of the Hilbert’s tenth problem,a problem in the domain of the discrete integers,in terms of a set of infinitely coupled differential equations over continuous vari-ables[17].This may lead to new insights and/or solution of the problem(recalled that,despite the mathematical noncomputability of Hilbert’s tenth problem,there does exist a general procedure to decide whether any given polynomial with many unknowns and real(continuous)coefficients has real solutions or not[18].)Our decidability study here in the framework of Quantum Mechanics only deals with the property of being Diophantine,which does not cover the property of being arithmetic in general (which could involve unbounded universal quantifiers).This entails that our consideration has no direct consequences on G¨o del’s Incompleteness theorem.AcknowledgementI am indebted to Alan Head and Peter Hannaford for encouragement and discussions;and would like to acknowledge the discussions with Cristian Calude,Jack Copeland,Toby Ord,Boris Pavlov and Andrew Rawlinson,who also helped me with the Figure1.References[1]D.Deutsch,A.Ekert and R.Lupacchini,Machines,logic and quantum physics,Bull.Sym-bolic Logic6,265-283(2000).See also a review of this paper by E.Knill on /cgi/display.cgi?reviewID=knill.bsl.6.265[2]H.R.Lewis and C.H.Papadimitriou,Elements of the Theory of Computation(Prentice Hall,1981).[3]H.T.Siegelmann,Computation beyond the Turing limit,Science268,545-548(1995).[4]C.S.Calude and G.Paun,Computing with Cells and Atoms(Taylor and Francis,2001).[5]M.Stannett,Hypercomputation is experimentally irrefutable,Tech.Report CS-01-04Deptof Computer Science,Sheffield University(2001).[6]T.D.Kieu,Quantum algorithm for the Hilbert’s tenth problem,Archive quant-ph/0110136(2001).See also T.D.Kieu,Computing the noncomputable,Archive quant-ph/0203034 (2002).[7]T.D.Kieu,G¨o del’s Indompleteness,Chaitin’sΩand Quantum Physics,Archivequant-ph/0111062(2001).[8]C.S.Calude and B.Pavlov,Coins,quantum measurements and Turing’s barrier,Archivequant-ph/0112087(2001).[9]R.Geroch and J.B.Hartle,J.Found.Phys.16,533-50(1986).[10]M.Pour-El and I.Richards,Computability in Analysis and Physics(Springer-Verlag,1989).[11]G.J.Chaitin,Algorithmic Information Theory(Cambridge University Press,1992).[12]J.von Neumann,Mathematical Foundations of Quantum Mechanics(Princeton UniversityPress,1983).R.Omnes,The Interpretation of Quantum Mechanics(Princeton University Press,1994).[13]Y.V.Matiyasevich,Hilbert’s Tenth Problem(MIT Press,1993).[14]M.Davis,Computability and Unsolvability(Dover,1982).[15]E.Bernstein and U.Vazirani,Quantum complexity theory,SIAM put.26,1411(1997).[16]D.Deutsch,The Frabric of Reality(The Penguin Press,1977).[17]T.D.Kieu,A reformulation of the Hilbert’s tenth problem through Quantum Mechanics,Archive quant-ph/0111063(2001).[18]A.Tarski,A Decision Method for Elementary Algebra and Geometry(University of CaliforniaPress,1951).。