Chern-Simons terimi olmaksızın anyonların kuantum alanı

Abstract

Bu çalışmamızda, 2+1 boyutlu bir relativistik ayar teorisinde chern-simons terimi kullanmadan anyon alanı kurmaya çalıştık. Zira bu çalışma ref (l)'de ele alınmıştır. Ayrıca referans (6)'da da bu çalışma chern-simons terimi varlığında kurulmuştur. Burda C-S teriminde var olan yenilik topolojik alanın benzer olarak farklı iki korunan dağılımın hesaba katılması ve Q+ 0« N şeklinde gauss kanunu sağlamasına rağmen chern- simons terimi olamadığında bunun Q»N olduğunu görmeye çalıştık bu işi yaparken örgü istatistiğinde, 2+1 boyutta relativistik ayar teorisi içinde yüklü alanları tanımlamak suretiyle yaptık. Topolojik olarak ifade ettiğimiz Q,$ ve x yüklü alanlarının birbirleriyle komintasyon ilişkilerini gösterdik. Bu işi yaparken korunumlu akımları da hesaba katmayı ihmal etmedik. Biz Chera-Simons terimi olmaksızın örgü istatistiklerin, 2+1 boyutlu relativistik kuantum ayar teorisinde geçerli olacağım ve de anyonların ortaya çıkmas 2+1 boyutta hem elektirik hemde magnetik akıyı taşıyan yüklü bölgelerin evrensel özelliği olduğunu göstermeye çalıştık.

Braid-group statistics CS in (2+l)-dimentional quantum field theory represent a striking theoretical which has interesting, although not fully confirmed phenomenological applications to quasi-planar condensed matter systems. A riqorous analysis of braid statistics has been performed [2] in the framework of axiomatic field theory. A comparison of the abstract results with the known quantum field realizations of anyons reveals however a sort of discrepancy. On the hand, all the known examples [9-6] make use at some stage of the Chern-Simon lagrangian -W^p^AVAP Which violates separately parity (P) and time reversal (T) invariance. So the natural question arises if it is possible to contruct anyon quantum fields without using the Cherm- Simons lagrangian. We give an affimative answer to this question. Our strategy for avoiding jCq.s will be to generalize the construction of anyons recantly developed in ref [6]. The fundamental properties of anyon fields proposed are: i) localization in space-like cones ii) appearance of both "electric' and 'magnetic' cones. We first show that these features can be implemented in 2+1 dimentional even in the absence of the Cherm-Simons term. Second we demonstrate that in many cases (i) and (ii) are actually sufficient for achieving anyon statistics. The class of model we are going to consider in the Minkowski space-time is described by the classical action. S=/d3xHl/4)Ff*vF|XV-Ba^H, a(x,b)h>o(x,b) and 3iai(x)=6(x) (i) /d2x£İJ5ibj(x)=l (j) Summarising, p and 0 are BRS-invariant composite field. Whose charge content is determined by eqs (16), (19), (20) As expected the Gauss law Q=N is satisfied. A fundamental role is played in what follows by the composite field; Z(x£)= p(x;a)a(x;b) Where £ is a shorthand notation for set of variables {a, b, e, p} bi= -ey. a1 VI The latter is easily solved; a convenient parametrization of the solutions in momentum space is 2n a'(p)= i J + is) 211 J dOr{0 )=1 o Performing the inverse Fourier transform one finds. «3 2n a'(x}= \dses£ jdÛT(0)rifi(x-sn) 0 0 Summarizing, without using any Cherm-Simons lagrangian. We have shown that £- data exist, such that the classical composite field x(x; £) has the properties (i) and (ii) mentioned in the introducition. The problem now is to determine the statistic of x(x ; £). This is very instructive to compare already on the classical level the canonical PB, M*M>0}1 v =o ".*o-vo with the equal-time PB of x with itself. An easy computation gives where and 8 is the dimensionless parameter 6=eg Now we turn to the canocical quantization of the action (1) in the framework of renormalized perturbation theory. We introduce the quantum fields p(r,a) = ç(x)xcKp('iejdyai(x-y)AM(x0,y)) cr(x;b) = xexp(igfd2ybl(x-y)7ti(xQ,y)), X(x,0 = p(x:a)a(x,b) VII Let's determine the charge content of p and o and the statistic of %. Using the Cherm-Simons, for the fields A» one finds [Q,p(x;a)]=ep(x;a) [,a(x;b)]=go(x;b) We have shown in [1] that the fields % admits a non-trivial monodromy. In agrement with the generalizal spin-statistic theorem [9], the spin of % turns out to be S=(8/2jt)+k k£z since in 2+1 space-time dimensions the spin is not quantized, there are no restrictions on the parameter 9 which can take orbitrary real values. In conclusion, we have shown above braid statistics generally occur in (2+1) dimensional relativistic quantum quage theories even in the absence of a Chern-Simons term. Claerly. our consideration do not rule out the possibility to include this term; we rather demonstrates that the apperance of anyons is a universal feature of charged sectors carrying both electric charge and magnetic flux in 2+1 dimensions. We first show that these features can be implemented İn 2+1 dimentional even in the absence of the Cherm-Simons term. Second we demonstrate that in many cases (i) and (ii) are actually sufficient for achieving anyon statistics. Summarizing, without using any Cherm-Simons lagrangian. We have shown that Xr data exist, such that the classical composite field x(x; 9 has the properties (i) and (ii) mentioned in the introducition. The problem now is to determine the statistic of x(x ; £). Let us analyse finally the behaviour of K - field under space and time reflection and under charge - conjugation. Denote by i|> the field algebra generated by the basic fields. Since the dynamics defined by the action preserves separately P. T. and C invariance. fJp Şc and an anti - linear automorphism Pt which implement these transformations on ty. This is one crucial difference between our present construction and the popular Cherm - Simons approach in the later Pp and Pt do not exist at all, because the dynamics explicitly breaks down space and time - reflection invariance. Further more, the absance of anomalies implies that the perturbative renormalized correlation-functions of the model are separetly P, D and C-invariant. Therefore, the automorphismis Pp, Pt and pc are unitarily implacementable in the perturbative phase i.e. there exist unitary operators c and p and anti-unitary operators T. which leave invariant the perturbative vacuum. In models possessing states whith non-trivial magnetic flux. The operators C, P and T interplate between the sectors with flux <£and -O. In particulary, it is not possible to restrict C, P and T within a sector with fixed magnetic flux. After these general observations, we concentrate on the transformation properties of the field x(x, a, e, g). At this stage the quantum fields p, a, and k can be introduced. vm This thesis is My dependent on the work of AXiguori Mintchev, and MRossi in Ref.(l).