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2015 | Buch

Space-Time Algebra

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This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future.

At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.

These same techniques, in the form of the ‘Geometric Algebra’, can be applied in many areas of engineering, robotics and computer science, with no changes necessary – it is the same underlying mathematics, and enables physicists to understand topics in engineering, and engineers to understand topics in physics (including aspects in frontier areas), in a way which no other single mathematical system could hope to make possible.

There is another aspect to Geometric Algebra, which is less tangible, and goes beyond questions of mathematical power and range. This is the remarkable insight it gives to physical problems, and the way it constantly suggests new features of the physics itself, not just the mathematics. Examples of this are peppered throughout ‘Space-Time Algebra’, despite its short length, and some of them are effectively still research topics for the future.

From the Foreward by Anthony Lasenby

Inhaltsverzeichnis

Frontmatter
Chapter 1. Geometric Algebra
Abstract
To every n-dimensional vector space \( \fancyscript{V}_{n} \) with a scalar product there corresponds a unique Clifford algebra \( \fancyscript{C}_{n} \). In this section we give an intuitive discussion of how \( \fancyscript{C}_{n} \) arises as an algebra of directions in \( \fancyscript{V}_{n} \). In the next section we proceed with a formal algebraic definition of \( \fancyscript{C}_{n} \).
David Hestenes
Chapter 2. Electrodynamics
Abstract
The electromagnetic field can be written as a single bivector field F in \( \fancyscript{D} \).
David Hestenes
Chapter 3. Dirac Fields
Abstract
Let \( \fancyscript{I} \) be a subspace of an algebra \( \fancyscript{A} \) with the property that the sum of elements in \( \fancyscript{I} \) is also in \( \fancyscript{I} \): \( \fancyscript{I} \) is called a two-sided ideal if it is invariant under multiplication on both the left and the right by an arbitrary element of \( \fancyscript{A} \): \( \fancyscript{I} \) is called a left (right) ideal if it is invariant under multiplication from the left (right) only.
David Hestenes
Chapter 4. Lorentz Transformations
Abstract
Let p be a vector tangent to a point x in space-time. The scalar p 2 is a natural norm for p. We call p timelike if p 2 > 0, spacelike if p 2 < 0, or lightlike if p 2 = 0.
David Hestenes
Chapter 5. Geometric Calculus
Abstract
To complete the space-time calculus developed in chapter I, we must define the gradient operator □ for curved space-time. Evidently □ is an invariant differential operator, but we will not attempt to define it without reference to local coordinate systems. This approach simplifies comparison with tensor analysis. However, once □ is defined, manipulations can be carried out in a coordinate-independent manner. To simplify our discussion, we will ignore all questions about differentiability. Such questions can be answered in the same way as in tensor analysis. We wish to emphasize the special algebraic features of our geometric calculus.
David Hestenes
Backmatter
Metadaten
Titel
Space-Time Algebra
verfasst von
David Hestenes
Copyright-Jahr
2015
Electronic ISBN
978-3-319-18413-5
Print ISBN
978-3-319-18412-8
DOI
https://doi.org/10.1007/978-3-319-18413-5