Gravitasyon alanında parçacık yaratımı

Abstract

Çalışmamız sonsuz uzunlukta ve dönen bir kozmik sicimin kopması sonucunda yayılan klasik gravitasyon dalgalarının kütlesiz bir parçacıkla etkileşiminin incelenmesini içermektedir. Giriş bölümünde konunun tanıtımından sonra ikinci bölümde düz uzaydaki kuantum alan teorisi, üçüncü bölümde ise onun eğri uzaydaki analoğu incelenmiştir. Bu bölümlerde çalışmamızda kullandığımız Feynman propagatörüne ve sağladığı propagasyon denklemine ağırlık verilmiştir.' Dördüncü bölümde kullandığımız fon uzayı anlatılmış ve bu uzayda Feynman propagatörünün sağladığı propagasyon denklemi oluşturulmuştur. Beşinci bölüm tümüyle propagasyon denkleminin çözümüne ayrılmıştır. Denklemin operatörünün özfonksiyonları pertürbasyon metodu ile bulunmuş ve propagatörümüz birinci mertebe pertürbasyon özfonksiyonlarının açılımı ile elde edilmiştir. Altıncı bölümde ise propagatörden faydalanılarak enerji- momentum tansörünün Tvv bileşeninin vakum beklenen değeri hesaplanmıştır. Bu bileşen bize sicimin kopması sonucu bir parçacık yaratımı olup olmadığını verir. Her ne kadar içinde çalışılan fon uzayında bir parçacık yaratımı gözlenememiş ise de, propagator ifademizin de Sitter uzayındaki yapısını elde etmek suretiyle yaptığımız hesap neticesinde de Sitter uzayında parçacık' yaratımı olarak yorumlanabilecek bir matematiksel ifade bulunmuştur.

We know that there is not a general relativistic quantum field theory for gravitational waves. So it is mostly done semi-classical approximation. Semi-classical approximation is the analysis of interaction of a quantized particle of micro-cosmos in a classical gravitational field. In the beginning of our problem, we had a metric that defines a locally flat, but globally curved space-time. This metric was calculated by Nutku- Penrose [1] and [2] for the snapping of an infinite, rotating cosmic string. After the snapping, the ends of the string were escaping with the speed of light. So this metric was the definition of our background space in which we got our propagation equation. The solution of this equation gives us the propagator. We used Feynman propagator that can be derived by the eigenfunction expansion of the Green's function using the first order perturbation eigenfunctions. The first chapter is the introduction part that defines the problem most generally. After the introduction, in the second chapter we give a summary of quantum field theory in Minkowski space-time. Minkowski space-time is a flat space and has a good formalism of quantum fields except gravitation. Quantum field theoretical equations of curved spaces are mostly the analogs of flat space-time ones. In the second chapter the most important mathematical structure is the Feynman's Green function since we used in our problem. Feynman's propagator is the time-ordered product of fields. iGF(x,x)= = 0(t-t')G+(x,x')+0(t'-t)G-(x,x) G+(x,x)= VI G-(x,x')=<0|^(x')^(jc)|0: And the propagation equation that is satisfied by Feynman's Green function is (Di+mt)Gp(?c,x) = -Sm(?e,x) The third chapter is a short summary of quantum field theory of curved spaces. It is not generally quantized theory but a semi-classical approximation. It is based on writing the propagation equations in the background of a classical gravitational field. So the metric of the space helps us in this analogous writing of the propagation equation. The equation we used in our problem is {^\(-gf28^j] +m2(-g)V2+ÇR(x)(-gyl2}GF(x,x) = -6{x-x) In chapter four we begin the solution of our problem. We have the metric which defines us the background space. ds = Idudv - 2 udÇ + 0(v)v{h;4}dÇ The space-time is explained briefly and especially noted that it is a globally curved but locally flat space-time that has a single curvature at one component of the Weyl tensor. All other quantities that have a geometrical meaning, like Ricci tensor, Ricci scalar curvature etc; are zero. Thus, the only curvature is one component of Weyl tensor on the cosmic string's snapping point. We choose the arbitrary function of the metric //(£) as m= AÇ + B lCğ+D\ and in here e defines a very very little curving of flat space-time. By using the metric, one can find the metric tensor and by using it, one can write the propagation equation that is satisfied by the Feynman's Green function. This equation has the operator form of LGF(x,x') = -â(x,x) vu In chapter five, this equation is solved. First of all we divide this L operator Z = Z,0+eL, This L operator satisfies the eigenvalue equation And if it is made the perturbation expansion, this equation is divided into two different equations. IA = Aft o o o o The first one can easily be solved and the zeroth order eigenfunction can be found. It can be easily seen that \ =0 and we can set the new form of the second equation. -A*0=(4-W This new form is not an easily solvable type of equation. Under the experience of the former similar problems; [4] and [5]; we make the ansatz of *, = <">./ After substituting this ansatz into the new formed equation of the first order perturbation, one can get VU1 It is still not an easy differential equation. So we make the new ansatz of After this new ansatz, the equation is divided into two parts based on equating the v dependent terms and the other ones. These are l}f2 = (some u term components) /, l}fx = (some other x,y,u components). The second one must be solved first. If it is transformed from u,x,y to s,z,z by the transformations of variables u z = x + iy z = x-iy and if it is proposed the form /, (s, z, z) = jfe, (r) + g2 (z)) + h, (z) + h2 (z) the second equation can easily be solved. If the solution is substituted into the first one, it can be solved by the same method. Similarly f2 (s, z,z) = s(q, (z) + q2 (z)) + n, (z) + r^ (z) ansatz was given. All the solutions are substituted in the perturbation expansion And the Green function is found by the eigenfunction expansion IX G(x,x ) = -2^ : After the neglecting of the zeroth order term which corresponds to the flat space-time, we get the Feynman propagator of the e valued curved space. Because of working in continuum basis, we transform from series to integration. Integrations have many mathematical difficulties. But they're evaluated by some tricks. First of all we sent the x,y values to x,y in the limit. It could be done because at the end of the problem, this limit should be done for calculating the energy-momentum tensor. R dependent integrations were not evaluated straightforwardly but Dirac delta functions were created by the help of K integrals. At the end of the calculation only one integration dependent to a dummy parameter was there. These integrals transformed to the integral expressions of the second kind Hankel functions of order one and two with convenient variable transformations. In the mass zero limit we used the expansion formula of second kind Hankel functions and all mass terms vanished. After vanishing of the mass terms, we show the propagator as Dp conventionally. After getting the explicit form of propagator especially depends on 0(v) the unit step function, we must calculate the Tw component of energy- momentum tensor. The propagator has the form A- = a(w0('v)-w'0(v)) F (u-u')2(y-v) b(0(y) + 0(v) + + (u-u)(y-v) c(v0(v)-v'fl(v')) (u-u)(y-v)2 in which a,b,c are different terms contain constants and x,y dependent functions. It is calculated the vacuum expectation value of Tvv tensor component by the formula =\\mâv.âv-Dp v - >v We couldn't get a nonzero and finite result of energy-momentum tensor in our background space. But if the propagator is transformed into de Sitter space, we get a physical meaning result. In the de Sitter space we found a result of vacuum expectation value of Tvv component of energy momentum tensor =kuS(v) in which is constant and x,y dependent terms. This result can be interpreted as a particle creation at v=o point because of the snapping of the cosmic string.