This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. It just means i will doubt whether you know anything about quaternion algebras. Arithmetic of quaternion algebra 2012 wesleyan university. A function as above is also called an inner product. With this convention, a vector is the same as an element of the vector space r3. A quaternion with a 0 is called a pure quaternion, and the square of a. The model selects the most versatile division ring. An introductory theory of frames on finite dimensional left quaternion hilbert spaces is demonstrated along the lines of their complex counterpart. This is a compilation of quaternionic number systems, quaternionic function theory, quaternionic hilbert spaces and gelfand triples.
Quaternions, quaternionic hilbert spaces, sspectrum, fredholm operator. Similarly, for the caseofquaternionicsignals,wecanworkonaquaternion hilbert space. In this regard, we shall examine some functional analytic properties of finite dimensional quaternion hilbert spaces as required for the development of the frame theory. Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield a unit quaternion. Discrete frames on finite dimensional left quaternion hilbert spaces. Q, the lattice of subspaces has a symmetry group which is. Among other things we prove that every wachs space is reflexive. This is clear since every element of fis a square in f. I in particular, multiplication by a unit complex number. In assisting the study of frames the functional analytic properties of the underlying hilbert space are essential. Hilbert spaces, for the left quaternion hilbert spaces called wachs spaces. Introduction into quaternions for spacecraft attitude.
Quaternionvalued hilbert spaces to introduce the concept of quaternion hilbert space, we. Moreover, in a one sided quaternionic hilbert space, given a linear. Pdf on the inductive limit of quaternion hilbert spaces. Berberian extension and its sspectra in a quaternionic hilbert space. By the way, each infinite dimensional separable hilbert space owns a gelfand triple. We also show how to rotate objects forward and back using quaternions, and how to concatenate several rotation operations into a single quaternion. As the quaternions are noncommutative in nature therefore there are two dif ferent types of quaternionic hilbert spaces, the left quaternionic hilbert space and the. But while a unit quaternion q is e ectively a point on a 3sphere, a qubit is often represented as a point on a 2sphere the bloch sphere. Euler angles quaternions and transformation matrices. Also, the space of unit quaternions is flat in any infinitesimal neighborhood of a given unit quaternion.
And how the one of time, of space the three, might in the chain of symbols girdled be. Introduction into quaternions for spacecraft attitude representation dipl. It is proved that the so obtained set forms a measurable field of hilbert spaces and their direct integral appears again as a reproducing kernel hilbert space for a bigger hilbert space over the whole quaternions. Sspectrum and associated continuous frames on quaternionic. Quaternions, quaternionic hilbert spaces, sspectrum, berberian. This work challenges that assumption, proposing instead an integrated set of equations which are all based on the same 4. Eulerangles,quaternions,andtransformationmatricesworkingrelationships byd. The hilbert book model applies number systems that can serve its base model, which is based on a combination of an infinite dimensional separable hilbert space and its unique nonseparable companion that embeds its separable partner. A left vector space h over the noncommutative field of quaternions q, complete in respect to the quaternionvalued inner product, is said to be left quaternionic. Great resemblance exists between maxwelllike equations and quaternionic. We say that a quaternion algebra over fsplits if it splits over f.
A subset cof a vector space xis said to be convex if for all x,y. A vector space over the field of complex or real numbers equipped with a scalar product, satisfying the following conditions. Hilbert book model projectquaternionic hilbert spaces. In conclusion, quaternion quantum mechanics appears. Hilbert symbols there is a wellknown way to construct quaternion algebras. The mathematical quaternion partakes of both these elements. M 2k then we can take x 0 1 0 0 0 and the other choices are obvious.
In this regard, we shall examine some functional analytic properties of. The functional analytic properties of the underlying hilbert space are crucial to the study of frames. It is proved that the so obtained set forms a measurable field of hilbert spaces and their direct integral appears again as a reproducing kernel hilbert space for a bigger hilbert space over the. Additive isometries on a quaternionic hilbert space. N in h is a tight frame if it is there exist r1, r2 satisfying inequality 1 with r1 r2 and a parseval frame if it is a tight frame with r1 r2 1. The grain chains also have an equivalent in this rigged hilbert space and this fact can be used to attach a position in the background coordinate system to each of the grains. A quaternionic inner product on a quaternionic vector space x is a mapping. We now introduce some basic denitions needed for quaternionic numerical signal processing over a quaternion hilbert space. Discrete frames on finite dimensional left quaternion. Such dimensional reduction results from ignoring the global phase of the spinor j. If the hilbert space represents both dual spaces, then the scalar product is also called an inner product. In this paper the physical implications of quaternion quantum mechanics are further explored. One must then include a lot of parentheses to dictate order. Singular value decomposition ofquaternion matrices.
Pdf additive isometries on a quaternionic hilbert space. Suppose that f and g is an arbitrary pair of orthogonal vectors in the left quaternion hilbert space v. Abstract the main topics of this tec hnical rep ort are quaternions, their mathematical properties, and ho w they can b e used to rotate ob jects. X,0 in order to change from wrist watch time the interval in spacetime to the norm of a hilbert space does not require any. Quaternionvector classically, in numerical signal processing, signals are represented in a hilbert space.
On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic hilbert space. As we will see later, this is the case of the classical mod. The scalar part of a quaternion is always real, and the vector part is always pure imaginary. The inequality 1 is called the frame inequality for the frame xnn. The difference between quaternionic differential calculus and maxwell based differential calculus is explained. Pdf in this paper, we introduce and study the frames in separable quaternionic hilbert spaces. W ein tro duce quaternion mathematics and discuss wh y quaternions are a b etter c hoice for implemen ting rotation than the w ellkno wn. Dual space of a quaternion hilbert space facta universitatis. Pdf frames in quaternionic hilbert spaces researchgate. Hilbert spaces and banach spaces, one may refer to 5,6,8,9,10,17. Quaternion algebras let bbe a quaternion algebra over a eld k. The completion of with respect to this norm is a hilbert space comments. Pdf frame of operators in quaternionic hilbert spaces. Fredholm operators and essential sspectrum in the quaternionic.
An overview of doing physics with quaternions it has been said that one reason physics succeeds is because all the terms in an equation are tensors of the same rank. Introducing the quaternions the complex numbers i the complex numbers c form a plane. Zizung yoon technical university of berlin department of astronautics and aeronautics berlin, germany may 31, 2012 abstract the purpose of this paper is to provide a straightforward and practical introduction to quaternion. I their operations are very related to twodimensional geometry. 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. Discrete frames on finite dimensional left quaternion hilbert. Fortunately, the rigged hilbert space has an uncountable number of dimensions and can easily deliver a gps system that can act as a continuum background coordinate system. Even though every quaternion can be viewed as a vector in a fourdimensional vector space, it is common to define a vector to mean a pure imaginary quaternion. Lower values of the real part of the eigenvalue define the historical part of the hilbert space. Despite an apparent similarity in the formulations, noncommutativity in the quaternion case brings in.
A set of reproducing kernel hilbert spaces are obtained on hilbert spaces over quaternion slices with the aid of coherent states. Further, due to the noncommutativity there are two types of hilbert spaces on quaternions, called right quaternion hilbert space and left quaternion hilbert space. An introduction to quaternions with application to rotations 3 this property is important because without it the product of two or more terms, such as vwzq, is ambiguous. The adjective quaternionic indicates that the inner products of vectors and the eigenvalues of operators are taken from the number system of the quaternions. The completion of with respect to this norm is a hilbert space. Pdf for a bounded right linear operators a, in a right quaternionic hilbert. Matrix multiplication is associative, but cross product is.
We will specify the characteristics of a generalized quaternionic infinite dimensional separable hilbert space. Quaternion wavelets for image analysis and processing wai lam chan, hyeokho choi, richard baraniuk department of electrical and computer engineering, rice university houston, texas, usa abstract using the concepts of twodimensional hilbert transform and analytic signal, we construct a new quaternion wavelet transform qwt. X,0 in order to change from wrist watch time the interval in spacetime to the norm of a. The positive real constants r1 and r2, respectively, are called lower and upper frame bounds for the frame xnn. Higher values define the future part of the hilbert space. Quaternions, quaternionic hilbert spaces, sspectrum, berberian extension, commutator. Despite an apparent similarity in the formulations, noncommutativity in the quaternion case brings in a specific peculiarity, often hidden and unexpected.
Some authors write the products as ibinstead of bi, and therefore they get the property k ji ij, which results in ijk 1 and a lefthanded quaternion. Random vectors with values in quaternion hilbert spaces. If the hilbert space re prese nts both dual spaces, then the scalar product is als o called an inner. We give a simple definition of quaternions, and show how to convert back and forth between quaternions, axisangle representations, euler angles, and rotation matrices.
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