İki boyutlu kuantum alan teorisinde sonsuz konform simetrisi
Infinite conformal symmetry in two dimensional quantum field theory
- Tez No: 22025
- Danışmanlar: PROF. DR. MAHMUT HORTAÇSU
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Konform simetri, Kuantum teorisi, Conformal symmetry, Quantum theory
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Bu tezde, kütlesiz, iki boyutlu etkileşen alan teorileri incelenmiştir. Bunların en önemli özelliği konform dönüşümleri n sonsuz parametreli gurubu altında değişmeziliği (invariance)dır. Operatör cebrini oluşturan yerel alanların, Virasoro cebrinin indirgenemez temsillerine göre sınıflandırılabildiği ; ve korrelasyon fonksiyonlarının konform değişmezlikle belirlenen konform bloklardan elde edilebildiği gösterilmiştir. Dejenere temsillerle ilgili, tamamen çözülebilir konform teoriler incelenmiştir. Bu teorilerde, anomali boyutları ve lineer diferansiyel denklem sistemlerini sağlayan korrelasyon fonksiyonları tam olarak bulunabilmektedir. Bulunan anomali boyutlarına karşılık gelen, ve kapalı bir cebir oluşturan yerel alanlardan minimal konform alan teorisi kurulabilir. Bu teoriler, faz geçişi noktalarında Cphase transition points} iki boyutlu termodinamik sistemlerin kritik davranışlarını belirler. Minimal teorilerin en basiti, iki boyutlu isi ng modelidir. Dotsenko, S. tarafından incelenen diğer bir minimal teori, yani Z Potts modeli de teze eklenmiştir. 9 ÎV
Özet (Çeviri)
SUMMARY INFINITE CONFÜRMAL SYMMETRY IN TWO DIMENSIONAL QUANTUM FIELD THEORY We present an investigation of the massless, two-dimensional, interacting field theories. Their basic property is their invariance under an in-finite-dimensional group of conformal trans-formations. It is shown that the local -fields forming the operator algebra can be classified according to the irreducable representations of Virasoro algebra, and that the correlation functions are built up of the“conformal blocks”which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the dejenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear diferantial equat i ons. It is also shown that there is infinite number of special quantum conformal invariant theories, which contain only a finite number of basic operators. Such minimal theories of whose anomalous dimensions are found from simple algebraic equations, govern the critical behaviour of 2--di mensi onal thermodynamical systems near the phase transition points. The simpliest example of minimal theories is 2-D Ising model, and another example is Z Potts model too. a In more detail, we shall show the following.(i) The components the stress-energy tensor T (Ç) a© represent the generators of the con-formal group *§. Stress energy tensor is trace! ess and symmetric in two dimensional conform quantum field theory. The algebra of these generators is the central extension of the algebra £ and coincides with the Virasoroalgebra £ The value of the central charge c is the parameter of the theory. (ii) Among the fields A.(Ç) forming the operator- algebra, there are some primery fields <p (Ç) which transform in the simpliest way, dÇ ^A d£ ^A v ds J K dz J under the substitution ?? C<z> » z ? £<z>. of coordinate transformation. Here A and A are real n n non-negative parameters. In fact, the combinations d ~A -+A and s ~A -A are the anomalous scale dimension n n n n n n and the spin of the field d> respectively. The spin s of a local field can take an integer or half-integer value only. We sahall often refer to the quantities. A and A as» to the dimensions of the field. The simpliest example of the primary field is the i dendi ty operator I. A nontrivial theory involves more than one primary field and the index n is introduced to distinguish between them. >JI(iii) A copmlete set of the fields A (Ç) consist of J conf ormalf ami 1 iesCjû“1 each corresponding to a certain primary field <p. The primary field <p belongs to the n n conformal family Zd> 1 and, in some sense, serves as the n ancestor of the family. each conformal family also contains infinitely many other secondary fiel ds (descendants). Dimensions of these secondary fields form integer spaced series. < k>., - < k> -r r A = A + k, A = A + k n n n n where k, k- 0, 1, 2,.... Variations of any secondary field A e Ld> 1 under the infinitesimal conformal transformations -> 2 + £ ( Z ) are expressed linearly in terms of representations of the same conformal family Zd> 1. So, each conformal family n corresponds to some representation of the conformal group $. In accordance with.§ ~ r ® r definition, this representation is a direct product Zip 1 = V « V n n n where V and V are representations of the Virasoro n n algebra £. The representation V is known as the Verma c ri modulus over the Virasoro algebra. In general, these representations are irreducable. (iv) Correlation functions of any secondary fields vncan be expressed in terms of the correlators o-f the corresponding primary -fields by means o-f special linear dif f erential operators. Therefore all information about the conformal quantum field theory is accumulated in the correlators of the primary field <p. n (v) The structure constants C. (?) of the operator ».j algebra (Ç)A.(O) = ^ Ck.(Ç)A, (0) J Z vj k A. can, in prinriple, be computed in terms of the coefficients C of the primary field <p, in the operator product expansion of <P 4>. Therefore, the boostrap n xn equations (i. e. the associativity condition for the operator algebra) can be reduced to equations imposing constraints upon these coefficients and the dimensions A n of the primary field. (vi ) At a given value of the charge c there are infinitely many special values of the dimension A such that the representation C0.3 proves to be degenerate. The most important property of the corresponding ”degenerate“ primary field <p. is that the correlation functions involving this field, satisfy special linear differential equations, the simpliest example of which is the hyper-geometry equation. (vii) If the parameter c satisfies the equation: i&-c - / 1 -c Ytllwhere p and q «re positive integers. The ”minimal" conformal quantum field theory con be constructed so that it be exactly solvable in the following sense. <i) A finite number of conformal families C<* D is involved in n the operator algebra, each of them being degenerate, (ii) all anomalous dimensions A are known exactly, (iii) all n correlation functions of the theory can be computed as solutions of special systems of linear partial differential equations. There are infinitely many conformal field theories of this type, each associated with a certain solution of equation given above. The simpliest nontrivial example lc~l/2) describingthe critical theory of the two-dimensional Ising model. We supposed that other minimal conformal theories describe second-order phase transitions in some two-dimensional spin systems with discrete symmetry groups such as 2 Potts model. Apart from second-order phase transitions in two dimensions, there is another application of the conformal quantum field theory. This is the dual theory. From the mathematical point of view dual models are no other than special kinds of the two-dimensional conformal quantum field theory. This is natural in view of their association with the string theory. Quantum fields describe the degrees of freedom associated with the string, the conformal symmetry being a manifestation of the reparametr ization invariance of the world surface swept out by the string. In fact, the dual amplitudes are expressed in terms of correlation functions of some local f i elds (vertex operators). In standart models (like the Veneziano model) vertex operators are related in a simple way to free massless fields. It is supposed that if considerably interacting fields are incorporated into theory, it can produce new types of dual models with more suitable physical properties. IX
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