The evolution of this four vector is given by the three equations 79 00 0 0, m 2 1 c t. The basic starting point is that the axioms of special relativity on the one hand and those of quantum mechanics on the other, should be combined into one theory. Quaternionic quantum mechanics is the proposal to consider the principles of quantum mechanics but with the ground field of complex numbers systematically replaced by the ring of quaternions instead references. Quantum mechanics qm also known as quantum physics, or quantum theory is a branch of physics which deals with physical phenomena at microscopic scales, where the action is on the order of the planck constant. Relativistic quantum field theory is a mathematical scheme to describe the subatomic particles and forces. This space should reduce to the usual hilbert space of complex quantum mechanics with separate spin, isospin plus one extra spinor degree of freedom at the level of the combined symmetry breaking. Translations between quaternion and complex quantum mechanics s. Informal notes on distributions and the fourier transform. Quaternionic quantum mechanics and quantum fields stephen l. Lecture notes relativistic quantum field theory i physics. Theory of quanta and topics of advanced quantum mechanics. Continuity equation for quaternionic quantum fields. It is not immediately clear how to write down such a.
In this post i want to try and paint a picture of what it means to have a field that respects the laws of quantum mechanics. The business of quantum field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accommodates the observations of quantum mechanics. Clearly, this result, due to 24, is related to the problem of the tensor product of the quaternionic hilbert spaces e. This is equivalent to spinor representa tion of ordinary quantum mechanics. In other words, can do quantum mechanics and special relativity lie on the same footing as the classical physics of newton, maxwell, einstein, or schr. Quaternionic quantum mechanics and quantum fields,by stephen l. Recent work on algebraic chromodynamics has indicated the importance of a systematic study of quaternion structures in quantum mechanics. A eld of a classical eld theory, such as electromagnetism. Quantum field theory is the result of the combination of classical field theory, quantum mechanics, and special relativity xi a brief overview of these theoretical precursors is in order.
We have recently developed a quaternionic quantum mechanics dealing with such a four vector 79. Continuity equation for quaternionic quantum fields by ir j. We have found time and space transformations under which diracs. Deformation quantization in quantum mechanics and quantum field theory hirshfeld, allen, 2003.
A wave function of a particle in quantum mechanics. After six decades, quaternion quantum mechanics is coming out of. As earlier conjectured by several authors and much later established by soler, from the latticetheory point of view, quantum mechanics may be formulated in real, complex or quaternionic hilbert spaces only. We embed quaternionic quantum theory into the framework of usual complex quantum information theory. In this formalism, the continuity equation mixes the kleingordon and schrodinger probability density while keeping the kleingordon and schrodinger current unaltered. General framework of quaternionic quantum mechanics3. Historically, the idea of using this tool to express quantum mechanics originates with work by kaneno, finkelstein et al. What are two good books about quantum optics and quantum. Jan 17, 2007 in quantum mechanics of fundamental systems 1 edited by claudio teitelboim, on p. Quaternions, quantum mechanics, dirac equation, kleingordon. Observations of the ramsauertownsend effect in quaternionic. We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal. This investigation grew out of my analysis 1, 3 of the problem of taking the step from quaternionic quantum mechanics to quaternionic quantum. Do you have any interest in quaternionic quantum mechanics.
Modern quantum field theory a concise introduction quantum. Consider a particle described by the four vector, 0. However, it is unclear what advantages quaternionic theory offers over the complex one, and analytic issues do not work out very well. Translations between quaternion and complex quantum mechanics. One should mention right at the start that one still does not understand whether quantum mechanics and special relativity are compatible at a fundamental level in our minkowski fourspace world. Informal notes dirac delta function as a distribution. Rotelli dipartimento di fisica universitii di leece infnsezione di leece while in general there is no onetoone correspondence between complex and quatemion quantum mechanics qqm, there exists at least one version of qqm in which a partial set of translations may be. A new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic and thus noncommutative instead of complex is developed. It is shown how to obtain a unique representation of a quaternion function by a convergent positive power series. Preface this is an update of the paper that was published under the. Free quantum field theory books download ebooks online. The book also provides an introduction to the problem of formulating quantum field theories in quaternionic hilbert space.
From classical mechanics to quantum field theory type. Adler, international series of monographs on physics 88, oxford university press, 1995. The guiding principle in choosing the topics presented apart to serve as introductions to later. Oxford university press, 2001 evans road, cary nc 275. Introduction to quantum optics from the semiclassical approach to quantized light. In conclusion, quaternion quantum mechanics appears. The quaternionic schrodinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic schrodinger equation is reduced to the study of quaternionic eigenequation a. What is the difference between quantum physics, quantum. Authored by a worldrenowned theoretical physicist, this book signals a major conceptual advance and gives a detailed development and exposition of quaternionic quantum. As far as i know, there are some people have interest in research a new formulation of quantum mechanics using quaternionic, rather than complex, numbers. Both quantum logic and the corresponding separable hilbert space offer no place for fields. From classical to quantum mechanics quantum mechanics at. Aug 20, 2015 in this post i want to try and paint a picture of what it means to have a field that respects the laws of quantum mechanics.
Dec 26, 2004 the twistor spaces of a paraquaternionic kahler manifold alekseevsky, dmitri and cortes, vicente, osaka journal of mathematics, 2008. The latter corresponds to a vector boson in spacetime, interpreted as a su2 gauge field. The book concludes with a chapter devoted to discussions on where quaternionic quantum mechanics may fit into the physics of unification, experimental and measurement theory issues, and the many open questions that still. In this part, in the rst three chapters i write about scalar elds, elds with spin, and nonabelian elds. Preface this is an update of the paper that was published under the title quaternionic continuity equation for charges. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. This leads us to conclude that quantum mechanics on quaternionic hilbert spaces is unable to describe compound systems as formalized in terms of the recognition maps described above. Pdf schwinger algebra for quaternionic quantum mechanics. The algebraic consistency of spin and isospin at the level. Here, a the term \ eld refers to one of the following. The second part is dedicated to topological field theories. The ramsauertownsend effect is the expression of what happens in the scattering of low energy electrons 0. The following chapters are dedicated to quantum electrodynamics and quantum chromodynamics, followed by the renormalization theory.
Schwinger algebra for quaternionic quantum mechanics. In quantum mechanics of fundamental systems 1 edited by claudio teitelboim, on p. Adler wrote a book quaternionic quantum mechanics and quantum fields, where the details are worked out. Although a cursory acquaitance with th subject of quantum field theory is helpful, the only requirement to follow the lectures it is a working knowledge of quantum mechanics and special relativity. Quantum eld theory is the quantum theory of elds just like quantum mechanics describes quantum particles. May 30, 20 as far as i know, there are some people have interest in research a new formulation of quantum mechanics using quaternionic, rather than complex, numbers. Stephen adler, quaternionic quantum mechanics and quantum fields, international series of monographs on physics 88 oxford university.
This is why qft is sometimes called \second quantisation. It has been known since the 1930s that quantum mechanics can be formulated in quaternionic as well as complex hilbert space. We have found a wave equation representing the fermionic particle. Doubleslit experiment in quaternion quantum mechanics.
Millard in which we show that standard complex quantum field. The conditions imposed on the form of the corresponding quantum field theory are studied, and the quantum fields are constructed. But systematic work on the quaternionic extension of standard quantum mechanics has scarcely begun. Some are fundamental, but many are just technical aspects of an unfamiliar form of perturbation theory. Further general results in quaternionic quantum mechanicspart ii. Translations between quaternion and complex quantum. Adler,who has been working on quaternionic quantum mechanics for the last fourteen years. Quantum mechanics departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. See also arbabs recent paper quaternionic quantum mechanics.
We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second order wave equation. Quaternionic quantum mechanics has aso shown potential of possible uni. Oct 06, 20 quaternionic quantum mechanics and quantum fields quantum mechanics factorization method in quantum mechanics quo vadis quantum mechanics. A childrens picturebook introduction to quantum field theory. Adler, international series of monographs on physics 88, oxford. Dec 22, 2004 a new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic and thus noncommutative instead of complex is developed. The conditions imposed on the form of the corresponding quantum field theory are. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic hilbert space. In a previous post, i introduced the idea of fields and, in particular, the allimportant electric field by making an analogy with ripples on a pond or water spraying out from a hose. Path integrals, greens functions, and generating functions. In fact, there is belief in some schools of thought that the conventional quantum mechanics in complex spacetime is an asymptotic version of the quaternionic quantum mechanics. Quaternionic quantum mechanics and quantum fields quantum mechanics factorization method in quantum mechanics quo vadis quantum mechanics.
Adler gave a comprehensive treatment of the rules of quaternionic quantum mechanics in. Why cant quantum field theory be quaternion instead of. Scattering of an electron by square well has been one of the important problems in quantum mechanics that has been mentioned in many quantum physics books 1,2,3,4,5,6. In quaternionic quantum mechanics, the dirac transition.
The theory takes the form of either a functional integral with quaternionimaginary lagrangian, or a schrodinger equation and transformation theory for quaternionvalued wave. Pdf the quaternionic quantum mechanics researchgate. A new formulation of quantum mechanics based on differential commutator brackets is developed. The existence of quaternionic spinor fields implies in a quaternionic hilbert space and its necessary mathematical analysis. Quaternionic quantum mechanics and noncommutative dynamics. Quaternionicquantummechanics school of natural sciences. A quaternionic hilbert module, a closed linear vector space with many of the properties of a hilbert space is studied. Stephen adler, quaternionic quantum mechanics and quantum fields, international series of monographs on physics 88 oxford university press. The algebraic consistency of spin and isospin at the level of. Quaternionic quantum mechanics and quantum fields book.
One can say that qft is quantum mechanics qm for systems with many sometimes in nitely many degrees of freedom. An algebraic method for schrodinger equations in quaternionic. Quantum mechanics, principles of relativistic quantum field theory, free field theory, interacting field theories gauge theories, the smatrix and lsz reduction formulas, functional methods in scalar field theory, perturbative renormalization, functional integrals for fermi fields, quantum electrodynamics, 2 qed. Quaternionic quantum mechanics is the proposal to consider the principles of quantum mechanics but with the ground field of complex numbers systematically replaced by the ring of quaternions instead. Quaternionic quantum mechanics and quantum fields book, 1995. In this quaternionic measurement algebra some of the notions of quaternionic quantum mechanics are clarified. Jun 24, 2017 this form of quantum mechanics is based on quaternions.
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