Generalized hopf fibration and geometric so3 reduction. Hopf fibrations and hurwitzradon numbers springerlink. Such a representation is not only concise and elegant, it also yields a very e. It is a hopf fibration generated by charge movements, according to maxwells equations. If we apply stereographic projection from s3 to r3 f1g, have r3 is completely lled by disjoint circles and a line circle through 1.
Discovered by heinz hopf in 1931, it is an influential early example of a fiber bundle. Geometric spinors, relativity and the hopf fibration dr. Hopf fibrationsconstruction and quaternion interpretations. The bloch sphere and hopf fibrations david wakeham. Tested quaternion visualiza tion techniques uses alternatively. Generating uniform incremental grids on so 3 using the hopf. Survey of hopf fibrations and rotation conventions in mathematics. The algebra h of quaternions 249 circle s1, we need to consider the sphere s3 in r4,andu1 is replaced by su2. The solution to a variational equation in depends on a real parameter. An elementary introduction to the hopf fibration niles johnson. Note that so3 is homeomorphic to the real projective. Generating uniform incremental grids on so3using the hopf fibration 5 a b c d fig.
New tools for visualization of human motion trajectory in quaternion. The hopf fibration also known as the hopf bundle or hopf map, named after german geometer and topologist heinz hopf 7 is a foundation stone in the theory of lie groups. Whats really amazing is that this fiber bundle is very nontrivial. In particular, no vector calculus, abstract algebra, or topology is needed. It was proven by heinz hopf that the an be constructed as a nontrivial fiber bundle. In this paper, geometric relation of complex and quaternionic hopf fibration is studied using base space map. In particular, no vector calculus, abstract algebra or topology is needed. Similarly, s2 refers to the sphere that we are most familiar with in 3d space. We summarize some notable properties about quaternions. On the hopf fibration s 7 s 4 over z nagoya mathematical. Hopf proved that the hopf map has hopf invariant 1, and therefore is not nullhomotopic. Linking base spaces of hopf fibration for twoqubit state.
For this purpose we will introduce quaternions in section 3. The classical hopf fibrations the hopf fibration of s3 the. The hopffibration is an aspect of quaternion geometry. Power series and derivatives for quaternions 25 15. The concepts introduced culminate in an application to the hopf fibration, a projectionlike. However, the surface measure is not straightforwardly expressed using quaternions. We prove the result that points on the 3sphere correspond to circles on the sphere, and from this we are able to construct the hopf fibration, s1. Differential and twistor geometry of the quantum hopf fibration.
Our approach uses the algebra of quaternions and illustrates some of the algebraic and. Lyons using quaternions to give an undergraduateaccessible exposition of the hopf bundle. In the mathematical field of differential topology, the hopf fibration describes a 3 sphere in. This is one of the earliest examples of locally trivial fibrations, introduced by h.
Restricting this action to the unit sphere s3 c2 given by the equation jz. As opposed to spherical coordinates for hy perspheres, the hopf coordinates are unique for the 3sphere, and thus for uniform incremental grids on so3 using the hopf fibration 5 rp3. The antecedents of our geometric algebras can be found in the works of w. Geometric spinors, relativity and the hopf fibration.
The hopf bration and the mathematics that are developed along the way makes for some very interesting visual images. Pdf linking base spaces of hopf fibration for twoqubit. The hopf fibration, named after heinz hopf who studied it in a 1931 paper 9, is an important object in mathematics and physics. Lyons gives an elementary introduction to quaternions to elucidate the hopf fibration as a mapping on unit quaternions. The lie group su2 hopf fibration and the fourier equation. This shows a 3d print of a mathematical sculpture i produced using. The quaternions and the spaces s3, su2, so3, and rp3 8. These groups are known to be very complex and in particular they are nonzero for infinitely many values of. It was a landmark discovery in topology and is a fundamental object in the theory of lie groups. Generating uniform incremental grids on so3 using the. The serre fibration so2 so3 s 2 comes from the action of the rotation group so3 on the 2sphere s 2. The paintings that have been included here are all the work of lunyi tsai, an artist, a mathematician, and a good friend.
Any rotation invariant surface measure on s3 naturally preserves the haar measure for so3 and can be used for quaternions. Note that using quaternions is not strictly necessary here, but they provide an elegant way to put the calculations into a compact form, and have by nature an easy geometrical interpretation 2. This project will introduce the hopf bration from the points of view of the quaternions and of the complex numbers. I also like penrose, hopf fibration and clifford parallels clifford parallels and today, at the end of this post you will see the same structures reappearing when quaternions are used for rotating the winged nut in space. Hopf a graphical program for exploring the hopf fibration. Confusingly at least for me, s3 actually refers to the sphere or hypersphere in 4d space. The dimension of a unit sphere is the number of parameters we need to describe it. We will also examine geodesics on sn, namely showing that geodesics lie on great circles. Uniform incremental grids on so3 using the hopf fibration 5 rp3.
Hopf fibrations in the language of 3d geometric algebra. An elementary introduction to the hopf fibration david w. In quantum mechanics, the riemann sphere is known as the bloch sphere, and the hopf fibration describes the topological structure of a quantum mechanical twolevel system or qubit. Group structure on spheres and the hopf bration hopf fibration classical hopf fibration lets now look at s1. As it happens, many of my publications have used quaternions in an essential way, so i. Lyons wrote the fibers of the hopf map are circles in s 3 page 95. Quantum computing through quaternions research india. In the mathematical field of topology, the hopf fibration also known as the hopf bundle or hopf map describes a 3sphere a hypersphere in fourdimensional space in terms of circles and an ordinary sphere. Lyons department of mathematical sciences lebanon valley college 101 n. Throughout this project, the link of various concepts to quaternions is discussed, in particular that unit quaternions are isomorphic to the 3sphere. Herman gluck, frank warner, wolfgang ziller, the geometry of the hopf fibrations, lenseignement mathematique, t.
The hopf fibration has a wide variety of physical applications including magnetic monopoles 14, rigid body. We explain the intimate connection of the hopf fibra. Quaternion representation is also convenient for calculating the composition of rotations, which is expressed as multiplication of quaternions. For example, a single qubit can be described using the picture of bloch sphere, which could be obtained by s3 hopf. The fiber here is the group of unit quaternions sp 1. Recall that the 3sphere s3 is the set of points x,y,z,t. Pdf an elementary introduction to the hopf fibration. We proceed in a similar fashion, but replace quaternions with an even stranger object called octonions, with seven imaginary units. Hopf fibrations and hurwitzradon numbers valentin ovsienko. So ive been working on understanding the hopf fibration in terms of quaternions for the past few months, following along with the investigations in david lyons paper, an elementary introduction to.
There are several fun ways to explore the hopf fibration and other beautiful mathematics on the university of tokyo mathematics department page of foliations animations. The second consists in the presentation of an example of a map from s to s that has hopf invariant 1 and thus represents an element of infinite order o 75f. The hopf bration describes rp3 in terms of a circle, s1, and an ordinary 2sphere, s2. Ken shoemake hopf topology archive, revised version. Versors have been used to represent rotations of the bloch sphere with quaternion multiplication. It may be projected down to 3dimensional space, and then the 4th dimension is compactified as you said. They correspond to three noncompact hopf maps that represent topological maps between hyperboloids in different dimensions with hyperboloid bundle. Newest hopffibration questions mathematics stack exchange. Thus the electromagnetic waves are then modeled as a hopf fibration. C 2, by the hopf fibration, and by lie group structure s 3 of unit quaternions coincide. Through the hopf fibrations, we identify geometric. Our approach uses the algebra of quaternions and illustrates some of the algebraic and geometric properties of the hopf fibration.
These mappings induce trivial mappings in homology and cohomology. Spheres, hyperspheres and quaternions university of warwick. May 18, 2009 we develop a noncompact version of the hopf maps based on the split algebras. In section 5 we will show that, after choosing an appropriate real representation, this hopf map can be considered as an orbit map for an appropriately chosen symplectic so3 action, and.
Hopf fibration amsi vacation research scholarships australian. They naturally describe the intrinsic structure of both the 3sphere and rp3 and provide a natural tool for obtaining uniform distributions on these spaces. As i understand it, the hopffibration is a broad area of research with a lot of unknowns. The hopf fibration the hopf bration is an important object in elds of mathematics such as topology and lie groups and has many physical applications such as rigid body mechanics and magnetic monopoles. Generalized hopf fibration and geometric so3 reduction of. The resulting fibration has this circle as a base b and line segment as fiber f.
Single qubit, bloch sphere and the s3 hopf fibration 2. Infinitedimensional quaternionic projective space the space is the classifying space bs3. Op2 and the g2 to b3 to d4 to b4 to f4 magic triangle vixra. The hopf fibration is a mapping from s3 to s2 discovered by heinz hopf in 1931. Quantum information, hopf fibration, complex projective space, transition function, principal bundle 1 introduction geometry enters quantum mechanics in a very profound level. This paper describes the key sphere h7 in vedic physics and then attempts to draw isomorphic relationships between the structures. Our approach uses the algebra of quaternions and illustrates some of the. Here, the base space is and the fiber space is the circle. Hopf fibration and e8 in vedic physics by john frederick sweeney abstract a parallel construction exists in vedic nuclear physics which appears to be the exceptional lie algebra e8 and the hopf fibration. Visualization of the spherical and hopf coordinates on so3 using angle and axis representation. Not only that, but all these circles are pairwise \linked. Pdf generating uniform incremental grids on so3 using. Our approach allows us to validate the hopf fibration model that we carried out in paragraphs 2. Pdf generating uniform incremental grids on so3 using the.
Using quaternions, we study the geometry of the single and two qubit states of quantum computing. This representation corresponds to a projection of the s3 onto the equatorial solid sphere which we draw in r3. Pdf two and threequbits geometry, quaternionic and. In this part we briefly recall the wellknown hopf fibration and hopf map in the complex plane, which can be shown to be an orbit map and a reduction map for a 2dof harmonic oscillator. Below are some links about william rowan hamiltons quaternion algebra and about the hopf fibration of the 3sphere by linked circles some of the history of the quaternions, including a picture of broom bridge in dublin hamiltons original paper on the quaternions a wonderful paper by david w. Hopf fibration picture i just found the hopf topology archive on the web, which it appears you handle. Generating uniform incremental grids on so3 using the hopf. Jun 01, 2011 it was shown in that subriemannian geometry on s 3 induced by crstructure of s 3. At the top of the page i read that the logo will eventually be a visualization of the hopf fibration of the three sphere over the two sphere. Hopf map p is related to the projection of the third hopf. You should check that this is indeed a map from s3 to s2. Normalization reduces this to a 15sphere of states.
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