Quantum Dynamical $check{R}$- Matrix with Spectral Parameter from Fusion

Dynamics Filter语法1. 引言在Dynamics 365中，Dynamics Filter语法是一种用于定义和控制数据筛选和过滤的语言。

通过使用Dynamics Filter语法，用户可以轻松地对数据进行过滤和排序，以满足特定的业务需求。

无论是在报告、视图还是工作流规则中，Dynamics Filter语法都起着至关重要的作用。

本文将详细介绍Dynamics Filter语法，并提供使用示例和最佳实践，帮助读者更好地理解和应用这一强大工具。

2. Dynamics Filter语法基础2.1 运算符Dynamics Filter语法使用一系列运算符来进行数据的比较和筛选。

常用的运算符包括：•等于（=）•不等于（<>）•大于（>）•小于（<）•大于等于（>=）•小于等于（<=）•包含（CONTAINS）•不包含（NOT CONTAINS）•在范围内（IN）•不在范围内（NOT IN）下面是一些示例：- "statuscode eq 1"：筛选状态码等于1的记录- "createdon gt datetime'2021-01-01T00:00:00Z'"：筛选创建日期晚于2021年1月1日的记录- "name contains 'Apple'"：筛选名称包含"Apple"的记录2.2 逻辑操作符除了基本的比较运算符，Dynamics Filter语法还提供了逻辑操作符来组合多个条件。

常用的逻辑操作符包括：•与（AND）•或（OR）•非（NOT）这些逻辑操作符可以帮助用户构建更复杂、更精确的数据筛选条件。

下面是一些示例：- "statuscode eq 1 AND createdon gt datetime'2021-01-01T00:00:00Z'"：筛选状态码等于1且创建日期晚于2021年1月1日的记录- "name contains 'Apple' OR name contains 'Banana'"：筛选名称包含"Apple"或"Ban ana"的记录- "NOT statuscode eq 1"：筛选状态码不等于1的记录3. 使用示例3.1 筛选活动记录假设我们想要筛选出所有状态为”已完成”的活动记录，可以使用以下Dynamics Filter语法："statecode eq 1 AND activitytypecode eq 'task'"这个语法将返回状态码为1且活动类型为”task”的活动记录。

报错解决V ASP⾃旋轨道耦合计算错误汇总静态计算时，报错：VERY BAD NEWS! Internal内部error in subroutine⼦程序IBZKPT:Reciprocal倒数的lattice and k-lattice belong to different class of lattices. Often results are still useful (48)INCAR参数设置：对策：根据所⽤集群，修改INCAR中NPAR。

将NPAR=4变成NPAR=1，已解决！错误：sub space matrix类错误报错：静态和能带计算中出现警告：W ARNING: Sub-Space-Matrix is not hermitian共轭in DA V结构优化出现错误：WARNING: Sub-Space-Matrix is not hermitian in DA V 4 -4.681828688433112E-002对策：通过将默认AMIX=0.4，修改成AMIX=0.2（或0.3），问题得以解决。

以下是类似的错误：WARNING: Sub-Space-Matrix is not hermitian in rmm -3.00000000000000RMM: 22 -0.167633596124E+02 -0.57393E+00 -0.44312E-01 1326 0.221E+00BRMIX:very serious problems the old and the new charge density differ old charge density: 28.00003 new 28.06093 0.111E+00错误：WARNING: Sub-Space-Matrix is not hermitian in rmm -42.5000000000000ERROR FEXCP: supplied Exchange-correletion table is too small, maximal index : 4794错误：结构优化Bi2Te3时，log⽂件：WARNING in EDDIAG: sub space matrix is not hermitian 1 -0.199E+01RMM: 200 0.179366581305E+01 -0.10588E-01 -0.14220E+00 718 0.261E-01BRMIX: very serious problems the old and the new charge density differ old charge density: 56.00230 new 124.70394 66 F= 0.17936658E+01 E0= 0.18295246E+01 d E =0.557217E-02curvature: 0.00 expect dE= 0.000E+00 dE for cont linesearch 0.000E+00ZBRENT: fatal error in bracketingplease rerun with smaller EDIFF, or copy CONTCAR to POSCAR and continue但是，将CONTCAR拷贝成POSCAR，接着算静态没有报错，这样算出来的结果有问题吗？对策1：⽤这个CONTCAR拷贝成POSCAR重新做⼀次结构优化，看是否达到优化精度！对策2：⽤这个CONTCAR拷贝成POSCAR，并且修改EDIFF（⽬前参数EDIFF=1E-6）,默认为10-4错误：WARNING: Sub-Space-Matrix is not hermitian in DA V 1 -7.626640664998020E-003⽹上参考解决⽅案：对策1：减⼩POTIM: IBRION=0，标准分⼦动⼒学模拟。

a r X i v :m a t h -p h /0106019v 1 20 J u n 2001Free particle on SU (2)group manifoldGeorge Chavchanidze,Levan TskipuriDepartment of Theoretical Physics,A.Razmadze Institute of Mathematics,1Aleksidze Street,Ge 380093,Tbilisi,Georgia e-mails:chaffinch@,tskipa@Abstract.We consider classical and quantum dynamics of a free particle on SU(2)group manifold.The eigenfunctions of the Hamiltonian are constructed in terms of coordinate free objectsgrangian descriptionThe dynamics of a free particle on SU (2)group manifold is described by the LagrangianL = g −1˙g g −1˙g(1)where g ∈SU (2)and denotes the normalized trace =−1dtg −1˙g =0(2)otherwise ,one can notice that our Lagrangian has SU (2)”right”and SU (2)”left”symmetry .It means that it is invariant under the following transformationsg −→h 1g ”left”symmetry (3)g −→g h 2”right”symmetry(4)According to the Noether’s theorem the corresponding conserving quantities areR =g −1˙g ddtL =0”left”symmetry(6)Now let’s introduce the basis of su (2)algebraT 1=i 00−i,T 2=01−10,T 3=0i i 0,(7)The elements of su (2)are traceless anti-hermitian matrices,and any A ∈su (2)can be parameterized in the following wayA =A n T nn =1,2,3R =R n T n(8)Scalar product AB= AB =−1g−1˙g=0⇒g−1˙g=const(12)dtg=e Rt g(0)(13) These are well known geodesics on Lie group.2.Hamiltonian descriptionWorking in afirst order Hamiltonian formalism we construct new Lagrangian which is equivalent to the initial one˜L= R(g−1˙g−˜v) +1R2 (16)2Where function1Where L X f g denotes Lie derivative of g with respect to X f.The skew symmetry ofωprovides skew symmetry of Poisson bracket.Hamiltonian vectorfields that correspond to R n,L m and g functions are the followingX n=X R n=([R,T n],g T n)=(X(R)R n ,X(g)Rn)(21)Y n=X L m=([R,g T m g−1],T m g)=(X(R)L m ,X(g)Lm)(22)Therefore Poisson brackets are{L n,L m}=−2ǫknm L k{R n,R m}=2ǫknmR k(23){R n,L m}=0(24){R n,g}=g T n{L m,g}=T m g(25) the results are natural.R and L that correspond respectively to the”right”and”left”symmetry commute with each other and independently form su(2)algebras.It is easy to write down Hamilton’s equations˙g={H,g}=g R(26)˙R={H,R}=0(27) We consider case of SU(2),but the same constructions can be applied to the other Lie groups.3.QuantizationLet’s introduce operatorsˆR n =i2L Y m(29)They act on the square integrable functions(see Appendix A)on SU(2)and satisfy quantum commutation relations[ˆL n,ˆL m]=iǫknm ˆLk(30)[ˆR n,ˆR m]=iǫknm ˆRk(31)[ˆR n,ˆL m]=0(32)The Hamiltonian is defined asˆH=ˆR2=ˆL2(33)Therefore the complete set of observables that commute with each other is as follows ˆH,ˆRa,ˆL b(34)Where a and b unlike n and m arefiing a simple generalization of a well known algebraic construction(see Appendix B)one can check that the eigenvalues of the quantum observablesˆH,ˆR a andˆL b are as followsˆHψjlr=j(j+1)ψj lr(35) where j takes positive integer and half integer valuesj=0,12,2 (36)ˆRaψj lr=rψj lr(37)ˆLbψj lr=lψj lr(38) with r and l taking values in the following range−j,−j+1,...,j−1,j(39) The main aim of the article is construction of the corresponding eigenfunctionsψj lr.The first step of this construction is proposition1proposition1.The function ˜T g where˜T=(I+iT a)(I+iT b)is an eigenfunction ofˆH,ˆRa andˆLb with eigenvalues,respectively32and1is not invariant under(41)local gauge transformations.To make(43)gauge invariantwe should replace ddt +B)g Where B can berepresented as followsB=b T3∈su(2)(44) with transformation ruleB−→h B h−1−˙h h−1(45) in the other wordsb−→b−˙β(46) The new LagrangianL G= g−1∇g g−1∇g (47) is invariant under(41)local gauge transformations.But this Lagrangian as well as every gauge invariant Lagrangian is singular.It contains additional non-physical degrees of freedom.To eliminate them we should eliminate B using Lagrange equations∂L G4X−1˙X X−1˙X (55)This Lagrangian describes free particle on the sphere.Indeed,since X=x a T a it’s easy to show thatL G=14X˙X X˙X =12(˜u+g−1B g)2 (57)˜L= R g−1˙g −12 R2 − B g R g−1 = R g−1˙g −12R2 −b L3(61) Due to the gauge invariance of(47)we obtain constrained Hamiltonian system,where R g−1dg is symplectic potential,12,1,3of gauge invariant variables up to a constant multiple should coincide with well known spherical harmonicsψj r0∼J jr(64) One can chack the followingψj r0∼ˆL j−ˆR j−r− ˜T g 2j∼ˆR j−r−T+g−1T3g j∼ˆR j−r−sin jθe ijθ∼ˆR j−r−J jj∼J jr(65)This is an example of using large initial model in quantization of coset model.6.Appendix AScalar product in Hilbert space is defined as followsψ1|ψ2 = SU(2)3 a=1 g−1dg T a ψ∗1ψ2(A.66)It’s easy to prove that if scalar product is(A.66)operatorsˆR n andˆL m are hermitian.Indeedψ1|ˆR nψ2 = SU(2)3 a=1 g−1dg T a ψ∗1(i2L X nψ1)∗ψ2(A.67) Where integration by part have been used.It’s easy to check that the additional termcoming from measure3 a=1 g−1dg T a (A.68)vanishes sinceL X n g−1dg T a (A.69)For more transparency one can introduce the following parameterization of SU(2).Forany g∈SU(2).g=e q a T a(A.70)Then the symplectic potential takes the formR g−1dg =R a dq a(A.71)and scalar productψ1|ψ2 = 2π0 2π0 2π0d3qψ∗1ψ2(A.72)that coincides with(A.66)because ofdq a= g−1dg T a (A.73)7.Appendix BWithout loss of generality we can takeˆH,ˆL3andˆR3as a complete set of observables. Assuming that there exist at least one eigenfunctions ofˆH,ˆL3andˆR3operators:ˆHψ=Eψ(B.74)ˆRψ=rψ(B.75)3ˆLψ=lψ(B.76)3It is easy to show that eigenvalues ofˆH are non-negativeE≥0(B.77)andE−r2≥0(B.78)E−l2≥0(B.79)Indeed operatorsˆR andˆL are selfadjoint soψ|ˆH|ψ = ψ|ˆR2|ψ = ψ|ˆR aˆR a|ψ = ψ|ˆR†aˆR a|ψ = ˆR aψ|ˆR aψ = ˆR aψ ≥0(B.80) To prove(B.78)-(B.79)we shall considerˆR21+ˆR22andˆL21+ˆL22operatorsψ|ˆR21+ˆR22|ψ = ˆR1ψ + ˆR2ψ ≥0(B.81)andψ|ˆR21+ˆR22|ψ = ψ|ˆH−ˆR23|ψ =(E−r2) ψ|ψ (B.82) Therefore E−r2≥0Now let’s introduce new operatorsˆR=iˆR1+ˆR2ˆR−=iˆR1−ˆR2(B.83)+ˆL=iˆL1+ˆL2ˆL−=iˆL1−ˆL2(B.84)+These operators are not selfadjoint,butˆR†−=ˆR+andˆL†−=ˆL+and they fulfill thefollowing commutation relations[ˆR±,ˆR3]=±ˆR±(B.85)[ˆL±,ˆL3]=±ˆL±(B.86)[ˆR+,ˆR−]=2ˆR3(B.87)[ˆL+,ˆL−]=2ˆL3(B.88)[ˆR∗,ˆL∗]=0(B.89)where*takes values+,–,3using these commutation relations it is easy to show that ifψλrl is eigenfunction ofˆH,ˆL3andˆR3with corresponding eigenvalues:ˆHψλ=λψλrl(B.90)rlˆL3ψλrl=lψλrl(B.91)ˆR3ψλrl=rψλrl(B.92) thenˆR±ψλrl andˆL±ψλrl are the eigenfunctions with corresponding eigenvaluesλ,l±1,r andλ,l,r±1.Consequently usingˆR±,ˆL±operators we construct a family of eigenfunctions with eigenvaluesl,l±1,l±2,l±3,...(B.93)r,r±1,r±2,r±3,...(B.94) but conditions(B.78)and(B.79)give restrictions on a possible range of eigenvalues. We should haveλ−r2≥0(B.95)λ−l2≥0(B.96) In other words,in order to interrupt(B.93)-(B.94)sequences we should haveˆL+ψλrj=0ˆL−ψλr,−j=0(B.97)ˆR+ψλkl=0ˆR−ψλ−kl=0(B.98) and for some j and k therefore l and r take the following values−j,−j+1,...j−1,j(B.99)−k,−k+1,...k−1,k(B.100) The number of values is2j+1and2k+1respectively.Since number of values should be integer,j and k should take integer or half integer valuesj=0,12,2,...(B.101)k=0,12,2,...(B.102)Now using commutation relations we can rewriteˆH in terms ofˆR±,ˆR3operators:ˆH=ˆR+ˆR−+ˆR23+ˆR3(B.103)(B.103)provides thatλ=j(j+1)=k(k+1)so j=k andλ=j(j+1) References[1]V.I.Arnold,Mathematical methods of classical mechanics.(Springer,Berlin,1978).[2]A.Bohm,Quantum mechanics:foundations and applications.(Springer-Verlag,1986).[3]G.Jorjadze,L.O’Raifeartaigh,I.Tsitsui Quantization of a free relativistic particle on the SL(2,R)manifold based on Hamiltonian reduction.(Physics Letters B336(1994)388-394).[4]G.Jorjadze,Hamiltonian reduction and quantization on symplectic manifolds.(Memoirs onDifferential equations and Mathematical Phisics,Volume13,1998,1-98).[5]N.M.J.Woodhouse,Geometric Quantization.(Claredon,Oxford,1992).。