Geometric Algebra, much like Bayesian probability, is an emerging field (although it’s been around for a while) with a change in perspective that comes with great power and elegance. This is a short overview of some concepts that it can reframe. At the end, there will be a short list of references for further reading.
Geometric Algebra and Geometric Calculus: A Unified Framework for Mathematics
In mathematics, we often deal with various systems—vectors, matrices, complex numbers, calculus—each with its own rules and limitations. Geometric Algebra (GA), along with its extension Geometric Calculus (GC), provides a unified framework that encompasses all these structures. Through GA, we can simplify and generalize concepts across multiple fields, offering a powerful language for both mathematics and physics.
In this post, we’ll explore the key concepts of GA and GC, highlighting their notational elegance and how they help unify ideas from linear algebra, vector calculus, complex numbers, and more.
1. What is Geometric Algebra?
Geometric Algebra (GA) generalizes familiar vector operations and introduces a new kind of product, the geometric product, which combines both the dot product and wedge product. This forms the backbone of GA, enabling us to treat scalars, vectors, and higher-dimensional entities within a single system.
The Geometric Product
If $ \mathbf{a} $ and $ \mathbf{b} $ are vectors, the geometric product is defined as:
$$ \mathbf{a} \mathbf{b} = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \wedge \mathbf{b} $$- The dot product $ \mathbf{a} \cdot \mathbf{b} $ returns a scalar, capturing the magnitude of projection.
- The wedge product $ \mathbf{a} \wedge \mathbf{b} $ returns a bivector, representing the oriented area spanned by $ \mathbf{a} $ and $ \mathbf{b} $.
This product simplifies how we handle geometric objects, combining familiar operations into a single, flexible tool.
2. Fundamental Objects in Geometric Algebra
Scalars, Vectors, and Multivectors
In GA, objects are built hierarchically:
- Scalars $ s $: 0-dimensional quantities.
- Vectors $ \mathbf{v} $: 1-dimensional quantities.
- Bivectors $ \mathbf{v}_1 \wedge \mathbf{v}_2 $: 2-dimensional quantities representing oriented planes.
- Trivectors $ \mathbf{v}_1 \wedge \mathbf{v}_2 \wedge \mathbf{v}_3 $: 3-dimensional quantities representing volumes.
These combine into multivectors, which can represent scalars, vectors, planes, and higher-dimensional objects simultaneously.
Clifford Algebras
Geometric Algebra is a realization of Clifford algebras, where the geometric product extends the traditional vector space into more complex algebraic structures. The geometric product allows vectors to be combined in ways that reveal both their magnitude and orientation in space.
3. Expanding Geometric Algebra
Rotations and Reflections Using Rotors
In traditional mathematics, rotations are handled using matrices or quaternions. In GA, rotations and reflections are more naturally managed with rotors. A rotor $ R $ is derived from a bivector and is used to rotate vectors via:
$$ \mathbf{v}' = R \mathbf{v} R^\dagger $$where $ R^\dagger $ is the reverse of $ R $. Rotors, constructed from bivectors, generalize the quaternion approach to any dimension and offer a more intuitive representation of rotations.
Complex Numbers and Quaternions as Special Cases of GA
Complex numbers are a subset of GA. In two dimensions, the unit bivector $ \mathbf{i} = \mathbf{e}_1 \wedge \mathbf{e}_2 $ plays the role of the imaginary unit $ i $ in complex numbers. Any complex number $ z = a + bi $ can be written as:
$$ z = a + b\mathbf{i} $$In GA, this formulation extends to higher dimensions, where the imaginary unit $ i $ generalizes to a bivector. Quaternions, used to describe 3D rotations, are also naturally generalized in GA as bivectors.
4. Revisiting the Cross Product: A More General Approach
The cross product $ \mathbf{a} \times \mathbf{b} $ is a familiar operation in three dimensions, where it returns a vector orthogonal to both $ \mathbf{a} $ and $ \mathbf{b} $. However, the cross product is limited to 3D and does not generalize to higher dimensions.
The Cross Product in 3D
The cross product of two vectors $ \mathbf{a} $ and $ \mathbf{b} $ in 3D is:
$$ \mathbf{a} \times \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \sin\theta \, \mathbf{n} $$where $ \theta $ is the angle between $ \mathbf{a} $ and $ \mathbf{b} $, and $ \mathbf{n} $ is a unit vector perpendicular to the plane of $ \mathbf{a} $ and $ \mathbf{b} $.
The Wedge Product as a Generalization
Geometric Algebra replaces the cross product with the wedge product $ \mathbf{a} \wedge \mathbf{b} $, which represents the oriented plane spanned by $ \mathbf{a} $ and $ \mathbf{b} $ as a bivector. The wedge product generalizes to any number of dimensions and works more naturally in GA.
To recover the traditional cross product from the wedge product, we can take the dual using the negative unit pseudoscalar $ I $ in 3D, where:
$$ \mathbf{a} \times \mathbf{b} = -I (\mathbf{a} \wedge \mathbf{b}) $$Here, $ I $ represents the unit volume element in 3D. This formulation shows that the cross product is just a special case of the wedge product, with the dual operation transforming the bivector into a vector.
Benefits of the Wedge Product
- Works in any dimension: Unlike the cross product, which is specific to 3D, the wedge product is dimension-independent.
- Geometric clarity: It directly represents areas and higher-dimensional objects, offering a more intuitive understanding of vector interactions.
5. Geometric Calculus: Extending GA
Derivatives in Geometric Calculus
In Geometric Calculus (GC), the vector derivative $ \nabla $ generalizes traditional calculus operations (gradient, divergence, curl) into a single operator. For a scalar field $ f(\mathbf{x}) $, the gradient is:
$$ \nabla f = \mathbf{e}_1 \frac{\partial f}{\partial x_1} + \mathbf{e}_2 \frac{\partial f}{\partial x_2} + \mathbf{e}_3 \frac{\partial f}{\partial x_3} $$This same operator can also compute divergence and curl when applied to vector fields, unifying these operations in a cohesive framework.
Integration in Geometric Calculus
GC also unifies line, surface, and volume integrals using multivectors. For instance, the surface integral of a bivector field $ \mathbf{B} $ over a surface $ S $ can be written as:
$$ \int_S \mathbf{B} \cdot d\mathbf{S} $$This formulation simplifies the treatment of integrals across different dimensions and generalizes classical theorems like Stokes’ and Gauss’ Theorems.
6. Unification of Multivariable Calculus with Geometric Calculus
Unifying Gradient, Divergence, Curl, and Laplacian
In traditional vector calculus, gradient, divergence, and curl are distinct operations. In GC, these are all handled by the vector derivative $ \nabla $:
- Gradient: $ \nabla f $
- Divergence: $ \nabla \cdot \mathbf{v} $
- Curl: $ \nabla \wedge \mathbf{v} $
- Laplacian: $ \nabla^2 f $
The Laplacian, for example, is simply the square of the vector derivative, further simplifying complex operations.
Applications in Physics
For instance, Maxwell’s equations in electromagnetism, traditionally split into four separate equations, are unified into a single equation in GA:
$$ \nabla F = J $$where $ F $ is the electromagnetic field bivector, and $ J $ is the current.
7. Complex Numbers and a Note on Complex Analysis
Revisiting Complex Numbers
In GA, complex numbers are represented as combinations of scalars and bivectors in 2D. For example, a complex number $ z = a + bi $ can be written as:
$$ z = a + b\mathbf{i} $$where $ \mathbf{i} = \mathbf{e}_1 \wedge \mathbf{e}_2 $, the unit bivector, generalizes the imaginary unit.
A Note on Complex Analysis
Complex analysis, which studies functions of complex variables, can be viewed through the lens of GA. Operations like multiplication by $ i $ (or $ \mathbf{i} $) represent rotations, and the concepts of holomorphic functions and conformal mappings generalize naturally to higher dimensions using GA.
8. Why Geometric Algebra and Calculus Matter
Advantages Over Traditional Vector Calculus
GA and GC simplify the notation and unify many seemingly disparate concepts, providing an elegant and general framework. This not only reduces the cognitive load but also opens up new possibilities for applications in fields like physics, computer graphics, and robotics.
Geometric Algebra and Geometric Calculus offer a unified, powerful language that integrates linear algebra, vector calculus, complex numbers, and more. By unifying these fields, GA and GC simplify both mathematical reasoning and practical computations, making them indispensable tools for anyone working in mathematics, physics, or engineering.
Further Reading
- https://bivector.net/
- Wikipedia - Geometric Algebra
- Wikipedia - Geometric Calculus
- Eric Chisolm - Geometric Algebra
- Wikipedia - Comparison of vector algebra and geometric algebra
- Wikipedia - Clifford algebra
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