BRST ve antibrst simetrilerinde genel hayalet dekuple teoremi
A General ghost decoupling theorem using BRST and antibrst symmetries
- Tez No: 14136
- Danışmanlar: PROF.DR. MAHMUT HORTAÇSU
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1990
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 54
Özet
ÖZET BRST VE ANTIBRST SİMETRİLERİNDE GENEL HAYALET DEKUPLE TEOREMİ İlk bölümde Hamilton formalizmi ele alxnxp Dirac'ın baglarx sxnxf landirxlmasx incelendi ve klasik problemlerin kuantize edilme yöntemleri kısaca anlatıldı. BRST kuantizasyonunun, teoride ayar değişmezliğine eşdeğer bir simetriyi bırakıp bir ayar saptaması yaptığı ve BRST simetrisinin ilk ortaya çıkışı bölüm 2' de incelen di. Bölüm 3 'de, korunan, hermitiyen ve BRST operatörü içeren genel bir ayar teorisi alxnarak BRST kuantizasyonu nun ilkeleri verildi. Ayrxca BRST simetrisine hermitiyen antiBRST yükü eklenerek, antiBRST kuantizasyonunun ilkele ri de acxklandx. Bölüm 4 "de BRST ve antiBRST kosullarx çözüldü. Sa dece BRST simetrisi için genel hayalet dekuple teoremi is pat edildi. BRST simetrisinin bütün durumlarxnxn hayalet dekuplesi sağladxğx, hatta antiBRST değişmezliği yüklenme sine izin verildiği gösterildi. Bölüm 5 'de, önceki bölümlerde anlatxlanlarx acxkla- yxcx örnekler verildi. Bölüm 5. 3 'deki örnekte kanonik olmayan durum uzaylarında uygun bir kuantizasyon için antiBRST simetrisi eklenmesi gerekti. Böylece antiBRST simetrisi ile fiziksel olmayan hayalet dejenereliği orta dan kalktx. Bölüm 6 'da elde edilen sonuçlar yazxldx. XV
Özet (Çeviri)
SUMMARY A GENERAL GHOST DECOUPLING THEOREM USING BRST AND ANTIBRST SYMMETRIES The BRST symmetry was first considered in the quantization of Yang-Mills theories. It may simply be viewed as a quantization of a constrained hamiltonian system with first class constraints in Dirac's classifica tion. Such a theory may be called a general gauge theory. We pointed out that the virtue of BRST quantization is in its allowing gauge-fixing while keeping a symmetry of the theory manifest, which is equivalent to gauge invariance. The necessity for ghost fields in covariant gauges was first recognized by Feynmann in Yang-Mills theory. Afterwards, Faddeev-Popov showed a concise derivation of the rules based on a path integral method and this gave us a clear insight into the origin of the ghosts. Since then their method has become a standard procedure for gauge fixing in gauges theories. An important step was made by Becchi, Rouet, Stora and Tyutin (BRST) in gauge theories : they noticed that the total lagrangian including the gauge fixing and compen sating Faddeev-Popov (FP) ghost terms is invariant under a super-type transformation, which is called the BRST transformation today. This BRST invariance has been proven very useful in showing renormalizability and unitarity in gauge theories. The trick of the BRST technique is to enlarge the Hilbert space of the gauge theory and the replace the notion of a gauge transformation. Unlike the usual gauge transformation the BRST transformation possesses the unusual property that it is nilpotent, i.e., 62-0. The general trick which allows us to perform gauge fixing is to add to the lagrangian a trivial BRST invariant function, i.e., something which is 6 of something else but which would not be invariant with respect to an ordinary gauge transformation.We have enlarged our Hilbert space and replaced the notion of gauge invariance by that of BRST invariance. This allowed us to add a term to the Lagrangian which is not invariant with respect to ordinary gauge transforma tions but which is trivially BRST invariant, thus accomplishing the same goal achieved by ordinary gauge- fixing without losing the symmetry needed to make the renormalization of the theory. Since we have destroyed the original gauge invariance by our gauge fixing prescription, we no longer know that such a projection commutes with the Hamiltonian. To establish this fact we must turn to the sutdy of an operator which does commute with the Hamiltonian. The obvious candidate for such an operator is the generator of the BRST transformation, Q. The BRST transformation Q is nilpotent (i.e., Q2=0) and mixes operators which satisfy Bose and Fermi statistics.. We have analyzed the BRST quantization of a general class of gauge models with finite number of degrees of freedom. We have studied a general gauge theory for which there exist a conserved hermitian BRST operator. We have furthermore two independent set of constraint generators \b and Pa, a=l,...,m. These are both of the same Grassmann type. The first set forms the Lie algebra G [*a'+b3 = iUabCV The second set consist of the trivial generators P, which are momenta conjugate to the Lagrange multipliers v, i.e. [P,vb] = -xfia We have, for simplicity, assumed that all the generators ib are Hermitian and m finite. The BRST charge has the fundamental property that it is nilpotent Q2 = | [Q»Q]+ = o. The crucial fundamental ingredient in the BRST quantization is that the subtheory, built from the set of BRST invariant states, describes the true physical dynamics of the system. Thus the physical states satisfy VIQ|phys> = O. The nilpotency of Q implies that any state of the form Q|...> is a physical state. This states will be called exact states and they will decouple from all physical states i.e. have inner products equal to zero. The beauty of the BRST formalism is that theory described by the BRST invariant states and operators is a quantum gauge theory, where the gauge transformations have the form |phys >-*|phys>+Q | >. The original state space for the class of models we are considering is built form states which are direct product of matter states and Faddeev-Popov ghost states, i.e., from elements of the from I > = |M>|G>, where M and G denote matter and ghost respectively. The ghost states may be specified exactly. The state space is assumed to consist of states for which all inner products are finite. Finite inner products require all states to be built from vacuum states. A convenient way to represent the ghost states is to define a ghost“vacuum”. The matter state space must be constructed such that the inner products are finite. This implies that we may represent the bra and ket states of the auxiliary variables v and Pa in, e.g., the following way = A(P)|0>A where Val0>A=0 > AA= 1. The use of the antiBRST symmetry in addition of the BRST symmetry has been more a matter of choice rather than of real fundamental importance in standard theories. Still the antiBRST symmetry shares essentialy all the basic features of the BRST symmetry. viiOne may extend the _BRST algebra by introducing the hermitian antiBRST charge Q. The extended algebra is defined by [Q,Q]+ = 0, Q2 = \ [Q,Q]+ = 0. In the extended BRST quantization one imposes in addition to the condition of BRST invariance, also the antiBRST invariance Q |phys> = 0. We have solved the BRST and antiBRST condition. The first theorem we establish is the following: If in a theory the BRST invariance leads to FP ghost decoupling, i.e., all non-decoupling BRST invariant states contains a unique ghost state, then we may always choose a representation of the nontrivial BRST invariant states such that they are also antiBRST invariant. To prove this theorem we first need to solve BRST condition. The state space is assumed to be built from states which are direct products of matter states and Faddeev- Popov ghost states, such that all inner products are finite. A general state may be written in the form m m a,...a. lA> = I lM>a a I“! +k> k=o ai---ak z where the indices on |M>n fl distinguish between diffe rent matte states are ?\... °k rent matter states and are summed over, and the ghost 3. _. _. 3-| 3. a. âı I - -K- + k> =i n...n \~ ~2 ' Consider now for simplicity a state with only one matter state e.g., |M> v~IM> different from zero i.e. 1 2... K |A> = |M>|- | + k>12”#K e |M>|0>( V11XThe dual state is given by and = 0 ; .^:) = 0 where fC^'^) and g(^'y) are polynomials in i|>'a and f ", only have the trivial solutions g=f=0 for any non-zero |M> and
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