Genelleştirilmiş osilatörler
Generalized oscillators
- Tez No: 39664
- Danışmanlar: PROF. DR. METİN ARIK
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Osilatörler, Oscillators
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Bu çalışmada ilk olarak SUq(2) kuantum grubu invaryansını, iki boyutlu bir osilatör sistemi alarak, komütatif q-osilatörleri (veya kovaryant q-osila- törleri) için inceledik, iki boyutlu q-osilatörlerinin daha önce bulunmuş olan SUq(2) invaryansından daha büyük bir invaryansını araştırdık. Daha sonra 2x2 operatör elemanlı matrisler için determinant ve iz tanımını bilinen hallerle karşılaştırarak inceledik. Bu incelemeyi yaparken (t0 A matrisinin izi, A0 A matrisinin determinantı) bağıntısı ve er,., (i = 1, 2, 3) Pauli matrislerinden faydalanarak komütatif hal için en uygun dönüşüm olan A{ = aiAi(7i ifadesinden yararlandık. GLq(2) kuantum grubu ve K-örgü grubu için determinant ve iz tanımlarını verdik. Son olarak genelleştirilmiş osilatör tanımım verip ikinci dereceden genelleştirilmiş fark denklemine karşılık gelen genelleştirilmiş osilatör denklemini yazdık. Homografik osilatörü tanımladıktan sonra, bir boyutlu q-osilatörü- nün, Homografik osilatörün,.su(2) Lie cebrinin, suç(2) deforme Lie cebrinin genelleştirilmiş osilatör şeklinde ifadelerini araştırdık. K-örgü grubundan elde edilen psedo örgü cebrinin genelleştirilmiş osilatör temsilini inceledik.
Özet (Çeviri)
The first study on the generalization of the commutation relations of multi-dimensional harmonic oscillators was done in the 1970 s by Arık, Baker, Coon and Yu. The first model, [1], which was taken into conside ration is given by the following equation: aid* - qa-ai = 8{j, 0 < q < 1 (i,j = 1,2,..,!>). This algebra contains multi-dimensional lowering and raising opera tors and depends on one real parameter. By the study of this model it has been shown that all the elements of this algebra, called the q- algebra, which is generated by the lowering and raising operators and is characterized by the parameter 0 < q < 1, have a finite norm [2]. Later, this model was conceived as a one-dimensional model whose commutation relation is of the following form [3]: * * 1 aa - qa a = 1. This equation gives the definition of the simplest one-dimensional q-oscillator. The Hubert space of this model is represented by the symbol Hq. In mathematical literature, it is not new to generalize hypergeomet- ric functions to generalized hypergeometric functions of one parameter [4]. These functions, are called basic hypergeometric functions whose properties are similar to hyi>ergeometric functions. Later, in the 1980s, these studies played a crucial role in the develop ment of quantum groups, which was theorized by V.Drinfeid[6]. These developments led to the q-deformation of Lie groups and of the corres ponding Lie algebras. The SUq(2) quantum group Uq(su(2)) [9], obtained by the q-defor mation of SU{2) Lie algebra, was first shown by Sklyanin and then, viindependent of Sklyanin, by Kulish and Reshatikhin. in their studies of Yang-Baxter equations. On the other hand, it is also possible to consruct quantum groups by simply starting with q-oscillators[8]. In this thesis the first area of research is the possibility of generalizing two-dimensional q-oscillators to a greater invariance than 5X7,(2). For this purpose, the commutation relations of two q-oscillator algebras which commute with each other are written as follows: «lû* = 1 + q2a\a\ a2o-2 = 1 + q ®2a2 [aua2] = [ai,^] = [«Î»fl2] = [«îı «51 = 0. The covariant q-oscillator which has two operators can be expressed in terms of the operators stated above by the following equations where Ni stands for the number operator of the first oscillator: c\ = a\ r* =<,%,' This transformation can also be expressed by the following matrix: The generalized commutation relations satisfied by the covariant q-oscil lator are as follows [8,10,11]: cyc2 - qc2c.\ cicj = qc*a * * * * c2cl=qc1c2 * 1,2* clci = 1 + 7 CjCj c24 = 1 + q24c2 + {q2 - l)cjci The commutation relations satisfied by ci, c2, c* and c% are invariant un der the SUq(2) quantum group. In other words, for M ? SUq(2) with entries a, /?, 7, S, when the elements of the 2x2 matrix M are considered to be commutative with the operators ci and c2, the generalized commu tation relations satisfied by the covariant q-oscillator are invariant under the following transformation: ft) -ft 0(s) VllNow, for a 2x2 matrix U 6 SUq{2) with entries a,/?, 7, 8, if the following transformation is considered where N\ and JVi are number operators, then it is found that the simple commutation relations do not exist among the elements of this transformation matrix: /«,\ _(\ 0 \fQl fay1 /c!\ \~a2J \0 q~Nl) W Si J \e2J' This leads us to an important result; that two dimensional q-oscillators can not be generalized to a greater invariance than the SUq(2) invariance being considered. Then we investigate the; definitions of determinant and trace for a 2x2 matrix whose elements belong to an associative algebra. Our interest in this type of matrix is that it can be used to express some of the representations of quantum groups and the Artin braid group [7]. Since the commutation relations of a 2x2 quantum matrix [14] is be - ch for M 6 GLq{2) with entries a, b, c, d, b and c can be simultane ously be diagonal in a representation. Therefore, in the relation below, which represents the determinant of the matrix M, ad and da must also be diagonal: Dq(M) = ad - qbc = da - q~ be. Thus if a is considered to be the lowering operator, then d becomes the raising operator. Therefore, the effects of the operators a, b, c and d to an eigenvector <pn are as follows [7]: aıpn - a“_ıv?n_ı top» = bn<pn c(p”= cn<pn d<pn - dnıpn+ı This is the representation of GLq(2) quantum matrix in terms of lowering and raising operators. On the other hand for the Artin braid group the effects of the opera tors a,h,c and d to the eigenvector ipn are found as follows[7]: aifin = any>n Cfn = Cn<fn+1 d(pn - dnipn Then one recognizes that b is the lowering operator, c is the raising operator and a and d are diagonal operators. For a 2x2 matrix A with viiientries a, b, c, d, by using the relation below where t0 =trace A and Ao = detA A2 = t0A - A0. If the elements of the matrix belong to an associative algebra, then to and Ao, both of which are 2x2 diagonal matrices can be written as follows: _ / a + Mh~ ' 0 \ 0 ~ V 0 d + cac-1) l-cb) a _ ( bdb~la - be \ 0 cac When the transformation A{ = OiAoi is made by using Pauli matrices ai(i - 1,2,3), it is seen that it is the most appropriate transformation when compared with the known conditions; that is, f,- = trA{ = a + d and A,- = detAi = ad - be for 2x2 real matrix A. According to this definition t{ and A,- are found to be as follows for the quantum group GLq(2) d + q~[a 0 \ _ fa + qd 0 0 a + qd) ' 3“ I. 0 d + q~la A A _ ( q 1ad - bc 0 \A / qda - be 0 Al-A2~V 0 qda-be)'A3={ 0 q'1 ad - be and for the braid group a + d - da 0 \ 0 a + d - ad _/« + </ - ad 0 3 ~ V 0 a + d-da A - A (ad(l-a)-bc 0 \ Ai~^2~V ° da{l-d)-cb)'. _ fda{l -d)-cb 0 3 - V ° ad0- -a) -be Here ti and A^ are the trace and the determinant for the matrix corres ponding to the Pauli matrices Oi, respectively. Finally we present a definition of a generalized harmonic oscillator. Interesting particular cases of a generalized harmonic oscillator algebra are given by: q-oscillators, homographic oscillators, su(2) Lie algebra and.s«,;(2) deformed Lie algebra. ixThe generalized oscillator algebra is defined so as to satisfy the follow ing relations, where N stands for the number operator and a for lowering operator while a* is the raising operator: a*a = ?(N) = e/v aa* = e(N + 1) = eN+i [a,N] = a [a*,N] = -o* Let's define the effects of the operators aa* and a”a to the eigenvector (pn as follows where <pn is the eigenvector of the number operator N: N<pn = rupn, n : integer a*a<pn = ?n<pn aa*<pn = e“+iy>n Thus, a difference equation consisting of the functions e”can be const ructed. Then we consider the generalized oscillator which can be written as Aaa*aa* + Da* ana“ + Ca*aa*a + Daa* + Ea*a + F = 0. This corresponds to the 2nd order generalized difference equation Ae2n+i + Ben+1en + Ce2n + Den+1 + Een + F = 0. Where A, B, C, D, E and F arc constants. Examining some of the special cases of this last equation, for example A = D = C = 0 D - -F = 1 and E = -q, gives the one-dimensional q-oHC.illutor. Then, the »perimin of one dimensional q-oHfillator is l-qn fn = 1-q We named the oscillator which is defined by ha*a + i aa* - ja*a + k as homographic oscillator where h,i,j,k £ IR, a* is the raising operator and a is the lowering operator. Homographic oscillator corresponds to a generalized oscillator where A = C = 0, B = j,D = k,E = -h and F = -i. The homographic oscillator's normalized spectrum for eo = 0 and e\ - 1 is found as follows: e»-[n]-[n-iy [H]-qi-q2-The su(2) Lie algebra can be expressed as follows in terms of generalized oscillator equation where I is the identity element of the algebra: [a, a*]2 = I + u{a,a*}, u : constant {a, a*} = aa* +a*a is the anti commutator. This last equation gives the Lie algebra su(l,l) for u > 0, the harmonic oscillator algebra for « = 0 and the su(2) Lie algebra for u < 0. Thus, su(2) Lie algebra corresponds to the generalized oscillator equation where A = C = 1, B = -2, D = E = -u, and F = -I. Similarly, sug(l, 1) or sug(2) deformed Lie algebra is expressed as follows in terms of generalized oscillator equation where -1\2 G=(q-q + C2 ^ + <7_V l(*7-<7-1)2'.-i\2 H = - 2C + {q-q~l)2\ ' q-q,q + q~\ [a,a*]2 = G + H{a,a*} + J{a,a*}2. Therefore, suq{2) deformed Lie algebra corresponds to the generalized oscillator equation where A = C=l-j, # = -2(l+j), D = E = -tfandF = -G. The spectrum of SU(,(2) is found as follows where Ci,C2 and at are con stants: *n=Clqln+c2q-2n + -. We obtain the following relations by using commutation relations ob tained for the psedo-braid algebra [7] be = (1 - a) - (1 - a)d{l - a) cb = (1 - d) - (1 - d)a(l - d). If c is chosen to be the raising operator, then c - 6* can be written. In this case, the relations above* take the following form: W = (l-a)-(l-a)d(l-a) 6*6 = (l-d)-(l-d)a(l-d). Let us define (1 - a){\ - d) = q when a and d operators are eliminated in terms of bb* and 6*/), it follows: [(6”6)(66*)-92]2-(H-9)2(6*6-f7)[(6*6)(66')-ç2]+(H-5)2(6*6-9)266*=0. This oscillator commutation equation is solvable. In this case the effect of the operator b* to an eigenvector <pn is as follows: r q(l + cqn+l)(l + eg"'1) nbn~ {1 + cq*? where bn is the eigenvalue of 6* [7]. XI
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