Fizikte bazı özel problemlerin eliptik bikuaterniyonlar ile temsili
Representation in terms of elliptic biquaternions of some special problems in physics
Yazar: ZÜLAL DERİN YAQUB
Danışman: PROF. DR. MEHMET ALİ GÜNGÖR
Sakarya Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Ana Bilim Dalı
Derece: Doktora, 2022
Konu: Matematik = Mathematics
Dizin: Bikuaterniyonlar = Biquaternions ; Elektromanyetik alanlar = Electromagnetic fields ; Kuaterniyonlar = Quaternions ; Lorentz dönüşümü = Lorentz transformation ; Maxwell denklemleri = Maxwell equations

Özet:
Bu tez altı bölümden oluşmaktadır. 
Birinci bölümde; ilk alt başlıkta literatür özetine yer verilerek kompleks sayılar, p-kompleks sayılar, kuaterniyonlar gibi matematikte çok önemli sayı sistemleri, fizikteki Lorentz dönüşümleri, açısal momentum ve Dirac denklemi, rölativistik elektromanyetizma ve Maxwell denklemleri, Proca-Maxwell denklemleri gibi bazı önemli çalışmaların genel bir değerlendirmesi yapılmıştır ve ikinci alt başlıkta bu tezin amacı verilmiştir. 
İkinci bölümde, eliptik bikuaterniyonlar cebri tanıtılarak eliptik bikuaterniyonlar ile ilgili temel kavramlar verilmiştir.
Üçüncü bölümde, özel görelilik ile uyum içerisinde olan Lorentz dönüşümleri ilk defa eliptik bikuaterniyonlar ile incelenmiştir. Eliptik bikuaterniyonlar açısından özel matrisler tanımlanarak bu özel matrisler yardımıyla 4x4  tipinde eliptik ve 8x8  tipinde reel matrisler verilmiştir. Ayrıca sol Hamilton operatörüne karşılık gelen matris temsilleri sayesinde eliptik bikuaterniyonların cebirsel yapısında olmayan değişme özelliği sorunu ortadan kalkmıştır. Daha sonra uzay-zamanı ilişkilendiren eliptik bikuaterniyon ifade edilmiştir. Ayrıca bu ifadenin eliptik matris temsili verilmiştir. Bu sayede rölativistik dönüşüm bağıntısının sonucunda elde edilen eliptik bikuaterniyonun uzay ve zaman bileşenleri kolaylıkla görülebilmektedir. Eliptik bikuaternionların cebirsel yapısının bir özelliği olarak I^2=p<0  olduğundan p=-1  alarak eliptik bikuaterniyonların kompleks yapıyı kapsadığı da gösterilmiştir. Daha sonra rölativistik dönüşüm bağıntısının yardımıyla elde edilen Lorentz dönüşümlerinin, eliptik bikuaterniyonlar ile de ifade edilebileceği görülmüştür. Ayrıca elde edilen bu matematiksel denklemlerin matris temsilleri verilmiştir. Daha sonra elektrik ve manyetik alanların rölativistik dönüşüm bağıntısı, eliptik bikuaterniyonlar vasıtasıyla incelenmiştir ve rölativistik elektromanyetizma bağıntıları elde edilmiştir. Bu bağıntıları elde etmek için iki farklı yöntem kullanılmıştır. Burada ilk yöntem eliptik Lorentz dönüşümleri altında elde edilen elektrik alan ve manyetik alandır. Diğer yöntem, eliptik bikuaterniyonik rölativistik dönüşüm bağıntısı yardımıyla elde edilen elektrik ve manyetik alanın rölativistik biçimleridir. Elde edilen bu sonuçlar karşılaştırılarak hangi yöntemin kullanışlı olduğu araştırılmıştır.
Dördüncü bölümde ise eliptik bikuaterniyonlar ile açısal momentum ve Dirac denklemi ve çözümleri araştırılmıştır. Bu düşünceyle eliptik bikuaterniyonların eliptik Pauli matrisleri ve eliptik taban matrisleri verilmiştir. Daha sonra eliptik bikuaterniyonik açısal momentum ifade edilmiştir. Ayrıca serbest parçacık için Dirac denkleminin eliptik bikuaterniyonik yeni formülasyonu ifade edilerek bu denklemin çözümleri ele alınmıştır. Daha sonra rotasyonel parçacık için eliptik bikuaterniyonik Dirac denklemi ifade edilmiştir.
Beşinci bölümde ise eliptik bikuaterniyonik rölativistik esnek çarpışma problemi ele alınmıştır. İlk olarak bir parçacığın eliptik bikuaterniyonik momentumu, hızı gibi nicelikleri tanımlanmıştır. Bu ifadelerin eliptik bikuaterniyonlar açısından matris temsilleri verilmiştir. Daha sonra Lorentz dönüşümleri için rölativistik dönüşüm bağıntısı verilmiştir. Ayrıca rölativistik dönüşüm denklemi ile eliptik bikuaterniyonik rölativistik esnek çarpışma problemi çözülerek gerekli sonuçlar verilmiştir.
Altıncı bölümde ise tezin sonuçlarının bir değerlendirmesi yapılarak bundan sonra yapılacak olan araştırmalara yönelik önerilerde bulunulmuştur.

Abstract:
This thesis consists of six chapters.
In the first subsection of the first chapter, a summary of the literature has been given, and a general evaluation of some important studies such as complex numbers, p − complex numbers, quaternions in mathematics, Lorentz transformations in physics, angular momentum, and Dirac equation, relativistic electromagnetism and Maxwell equations, Proca-Maxwell equations has been made. In the second subsection, the aim of this thesis has been given.
In the second chapter, the algebra of elliptic biquaternions has been introduced., definition of the elliptic biquaternions sentence related to this algebraic structure, quaternionic product, quaternionic inner product, conjugate definitions of quaternion, norm of an elliptic biquaternion, modulus of an elliptic biquaternion, modulus of elliptic biquaternion, inverse of an elliptic biquaternion expressions such as theorem that elliptic biquaternion can also be expressed with the help of hyperbolic functions are given. Given the concepts in this chapter form the basic for other chapters. 
In the third chapter, Lorentz transformations, which are in accordance with special relativity, have been investigated for the first time with elliptic biquaternions. As it is known, Lorentz transformation relations form the basic of the theory of relativity. Lorentz transforms are important in that they explain how to observe the speed of light independent of the reference frame, how the measurements of space and time measured by two observers are related and are in harmony with special relativity. In addition, it is important to give matrix representations in order to make mathematical expressions more descriptive. In this respect, elliptic Pauli spin matrices were first defined. Then, 44 type matrix representations and right and left Hamiltonian matrix representations are expressed. In addition, elliptic biquaternionic special matrices are defined and the elliptic 44 and type real matrix 8 8  have been given with the help of these special matrices. In addition, thanks to the matrix representations corresponding to the left Hamilton operator, the problem of the nonexistence of the commutative property in the algebraic structure of elliptic iquaternions has been eliminated. Then, the elliptic biquaternion R = ct + Ir , which relates space-time, has been expressed. The elliptic number " I " plays an important role in expressing physical quantities more clearly and in easily distinguishing quantities with different physical structures from each other. In addition, the elliptic matrix representation of this expression has been given. Many physical quantities can be expressed in up to eight dimensions with elliptic biquaternions. In this way, the space and time components of the elliptic biquaternion obtained as a result of the relativistic transformation relation can be easily seen. It has also been shown that elliptic biquaternions include complex structure by taking p = −1, since I^2 = p <0 is a feature of the algebraic structure of elliptic biquaternions. Thus, it has been seen that the Lorentz transformations obtained with the help of the relativistic transformation equation can also be expressed with elliptic biquaternions. It was possible to describe the electric field E and the magnetic field H with a single elliptic biquaternion. Here, the real component represents the magnetic field and the elliptic component represents the electric field. In addition, matrix representations of these mathematical equations have been given. Then, the relativistic transformation relation of electric and magnetic
fields has been investigated by means of elliptic biquaternions, and relativistic electromagnetism relations have been obtained. Two different methods have been used to obtain these relations. Here, the first method, is the electric field and magnetic field obtained under elliptic Lorentz transformations. The other method is the relativistic forms of the electric and magnetic fields obtained with the help of the elliptic biquaternionic relativistic transformation relation. As a result of the relativistic transformation relation, the magnetic and electric field components are simply distinguishable. These results have been compared, and it has been investigated which method is more useful. Moreover, the elliptic biquaternionic differential operator D has been defined. Thanks to the effect of the conjugate of this operator on the elliptic biquaternion A, which combines the electric and magnetic fields in the elliptic biquaternionic sense, it has been possible to combine all of Maxwell's equations into a single equation. From the point of view of special relativity in electrodynamics, it has been possible to summarize Maxwell's equations and write them in a relativistic way with elliptic biquaternions. Thanks to elliptic biquaternions, it is possible to obtain an elegant electrodynamic formulation in which Maxwell's equations are reduced to a single elliptic biquaternion equation. Then, generalized gravity, which includes the terms Proca-type and gravitomagnetic monopole, has been investigated through
elliptic biquaternions, a new algebraic structure. In addition, the Proca-Maxwell equations, a more compact and useful formulation of the most general form of the Klein Gordon equation for the gravitational mass-bearing particle, have been discussed in terms of elliptic biquaternions. For the first time, elliptic biquaternions and Proca-Maxwell equations have been investigated and the results have been discussed. Compared to others, the mathematical structure of elliptic biquaternions provides more interesting, useful and elegant formulations for realizing many alternative representations in physics, such as gravity and electromagnetism. Moreover, it has been proposed to write the electromagnetic energy conservation by means of elliptic biquaternions together with the magnetic monopole. All field equations of gravity have
been expressed as an elliptic biquaternionic equation. In addition, thanks to elliptic biquaternions, it has a different perspective and importance in terms of being an additional term to Maxwell-like equations. Proca-type generalized gravitational wave equation has been obtained in an elegant and compact manner. Moreover, the Klein-Gordon equation for the graviton has been developed with the help of elliptic biquaternions. An alternative formulation in terms of elliptic biquaternions has been proposed for gravitoelectromagnetic energy conservation. The imaginary part of the conservation of electromagnetic energy by means of elliptic biquaternions represents Poynting's theorem. The Poynting vector for the energy density together with the graviton mass is directly dependent on the scalar and vector potentials. On the other
hand, physical interpretations have been made for the vector and elliptic vector parts of the Proca-Maxwell equations for the scalar part, and it was seen that the vector part has been related to the energy-momentum.
In the fourth chapter, elliptic biquaternions, angular momentum and Dirac equation and their solutions have been investigated. In these regards, elliptic Pauli matrices and elliptic base matrices of elliptic biquaternions have been given. We have defined new elliptic Dirac matrices with these matrices that we have defined for elliptic
biquaternions. We have associated elliptic Dirac matrices with elliptic biquaternionic bases. We have obtained the solutions of the Dirac equation with these matrices. Then, the elliptic biquaternionic new formula of the known Dirac equation for the free particle, which gives the energy-momentum relations depending on the motion of the electron in space-time, has been described. In addition, an elliptic biquaternionic mass has been defined. Here, the scalar part corresponding to the unit bases of the elliptic biquaternion has been associated with the rest mass and the vector part as the moving mass. Moreover, the Dirac equation for the rotational particle has been written in a simpler and compact form that includes the elliptic biquaternionic rotational energy and angular momentum. Later, the elliptic biquaternionic spinor has been defined. Thanks to the elliptic biquaternionic definition of this wave function, elliptic
biquaternionic energy and angular momentum solutions have been expressed. With these solutions, the elliptic behavior of the quantum wave spinor function associated with the interaction between elliptic biquaternionic spin and orbital angular momentum has been expressed. The obtained these expressions represent the positive and negative energy solution of the rotational particles. These expressions are very useful for fields such as quantum mechanics, general and special relativity. In the fifth chapter, elliptic biquaternionic relativistic elastic collision problem has been discussed. First, quantities such as elliptic biquaternionic momentum and velocity of a particle have been defined. Matrix representations of these expressions in terms of elliptic biquaternions have been given. Then, the relativistic transformation relation has been given for Lorentz transformations. The main problem here is to find the total elliptic biquaternionic momentum in two different reference frames before and after the collision: First, it is to find the rest reference frame of the target S and then the rest reference frame of the bullet S'. Maintaining the relative velocity and then using a Lorentz transform then reduces the problem to solving a linear equation. As a result, the elliptic biquaternionic relativistic elastic collision problem has been solved by the relativistic transformation equation and the necessary results have been given. Afterwards, found expressions in a table have been analyzed and this table has
been given in the results section. Finally, in the sixth chapter, we have present some results of our study, to which the interested reader has been sincerely invited. We have given a table from which inferences have been made about the velocity 1 v obtained as a result of the relativistic elastic collision problem. Here, obtained results about the velocity v1 for different real values of the real number 2 0 I p =  such as -4, -1 ve -0, 25, respectively have
been given. 
An assesment of these results has been made. Moreover, an evaluation of the results of the thesis was made and suggestions has been advised for future research.