Clifford-Algebra

The Clifford algebra ( by William Kingdon Clifford ) is an associative algebra that extends the complex and hypercomplex number systems. It takes place in differential geometry and in quantum physics application. It is used to define the spin group and its representations / bundles the construction of Spinorfeldern, which in turn are used to describe electrons and other elementary particles important, and the determination of invariants on manifolds.

• 3.1 As an initial object
• 3.2 Construction in the tensor algebra

• 5.1 graduation
• 5.2 Filtered algebra

• 7.1 The Pin Group
• 7.2 The spin- group

• 9.1 Real Clifford algebra
• 9.2 Complex Clifford algebra
• 9.3 Low-dimensional examples
• 9.4 Quantum Physics important examples

The question of complex units

Preview

There are in Mathematics Number Systems ( division algebras with unit element ) with complex units, specifically the complex numbers, quaternions and octaves. These elements can each be 1, 3 or 7 are fixed, which satisfy (not only) which together span with the number 1 as a real vector space and room. Sometimes this is not enough of. Any number of structures to be searched containing the real numbers, and elements, and in which a product is defined, which conditions the

Satisfied, where is the Kronecker symbol and. The shortcut icon is allowed to be happy away.

The elements are called the generators or generators of the Clifford algebra. The product of all generators is denoted by. The square of can be 1 or -1.

This structure is, apart from the above Examples, no number system in the above sense, but may be realized as algebra, in which the generatrix. Such an algebra is called a Clifford algebra, by William Kingdon Clifford, who has discovered them in 1878. It is designated, or if

And no other algebraic relationship of the generating applies.

So far, we have established formal calculation rules, but know nothing about the existence, uniqueness, and structure of such an algebra. This problem is instantly solved if one can represent the Clifford algebra as part of a real matrix algebra.

More general consideration

In the mathematical part of the calculation rules are supplemented by a universal property and the Clifford algebra constructed from a tensor algebra. It was initially only noted that the generators of a real ( sub) vector space of dimension n = p q span within the algebra. Adding up the defining property of the coordinate representation of a vector of this vector space, we obtain a coordinate- free ( in physical speech: covariant ) representation of the defining algebraic relation.

A quadratic function is defining a ( pseudo-) scalar product:

The generators then form an orthonormal basis.

Such a pair of real is a vector space and it defined quadratic function is the starting point for the mathematical theory of Clifford algebras.

Examples

• The complex numbers can be understood as a simple Clifford algebra with a single generatrix. The vector space is one-dimensional generated by, ie; the quadratic form. The algebra is two-dimensional with, and as basic elements as real vector space; they can identify with the algebra of 2x2 matrices

• The quaternions arise from the Clifford algebra. The generators have a non-trivial product from the defining properties of the product indicates that it corresponds to the product of quaternions. The vector space is real two-dimensional, the algebra of real four-dimensional. A matrix representation is the subalgebra of the complex 2x2 matrices

• , The algebra of binary numbers, has a generating line with square 1 Therefore, elements of the real 2-dimensional algebra can be split into two terms, the first of which reserves under multiplication by its sign and the second its sign changes. In the multiplication of two elements, these summands multiplied separately, as in the multiplication of two diagonal matrices. The algebra is isomorphic to a direct sum of two copies of.

Definition

With hereinafter the field of real numbers or the complex numbers is denoted. It is a finite-dimensional vector space over the field and a symmetric bilinear form on or a square shape.

Clifford algebra of the square space is algebra over consisting of the elements of the combination with

For. This algebra is denoted by.

Alternative definitions

The Clifford algebra is a natural from a mathematical point of view to construct a vector space with it defined square shape, because they can be characterized as the initial object of a category.

As an initial object

Consider the category of all associative algebras, in which is embedded, that is, all pairs with linear, which additionally the property

Or the equivalent statement

Meet for all, from. The morphisms in this category are Algebrenmorphismen which convert the embedded copies of V into each other, that is not only meets, but also.

An initial object of a category is distinguished by the fact that there is exactly to every other object in the category a morphism. If there are multiple initial objects, then they are isomorphic. Each initial object of the category under consideration, if any one exists, Clifford algebra is called. There is for every other pair of the category that is, with a uniquely determined Algebrenmorphismus.

It is identified in the following by its embedding, that is, the image is not explicitly mentioned.

Construction in the tensor algebra

In the tensor algebra, the ideal is defined. Then, the quotient is a realization of the Clifford algebra.

Special Clifford algebras

• If an n-dimensional vector space using standard scalar is, the Clifford algebra is also designated. The generators are then the canonical basis vectors, the square shape of the sum of squares of the coordinates.

• The space a Minkowski space with dimension of the signature, that is, the quadratic shape is given by

• For each real Clifford algebra can be defined and the complexified algebra, here are all non-degenerate bilinear forms another isomorphic.

Properties

Graduation

The figure

Also satisfies the defining identity, thus there is because of the universal property of a Algebrenisomorphismus with and for all. This breaks down the Clifford algebra in a straight part

And an odd part

This decomposition produces a graduation algebra, products, even-even and odd -odd be straight elements, even-odd products found odd elements. Thus, even products with an even number of factors V, products having an odd number of odd factors V.

Is a bottom algebra Clifford algebra and is also referred to as a second Clifford algebra, a module, only one respect.

Filtered algebra

Since the Clifford algebra can be regarded as the quotient of the tensor algebra and the tensor algebra has a natural filtration, a filtration may also be declared for the Clifford algebra. The picture is the natural projection of the tensor algebra in the quotient space and the filtration of the tensor algebra. If, as the Clifford algebra is also added to a filtered algebra.

Relationship with the Grassmann algebra

The Grassmann algebra of a real vector space is the Clifford algebra with the trivial square shape. Within any Clifford algebra the Grassmann algebra can be constructed by the wedge product as - defined - and, by analogy as an alternating sum of more than two factors.

It may, conversely, every Clifford algebra can be constructed within the Grassmann algebra by this new product is defined as

The dimension of the algebra is preserved, it is said.

This relationship is important, among other things for the quantization of supersymmetric field theories.

Relationship to the orthogonal group

Be a vector space with a nondegenerate symmetric bilinear form and. In the Clifford algebra then reflections can be in. For this, a consequence of the elementary structure of the product is used:

Is a unit vector, then the Figure, the reflection in the direction perpendicular to the hyperplane. Each reflection is an orthogonal map, hence the group generated by the reflections is a subgroup of the orthogonal group.

The Pin Group

Conversely, any orthogonal map into a product of reflections disassemble, see Householder transformation or QR decomposition. The decomposition is not unique, but the Clifford products of the unit vectors of the mirror arrays differ at most in sign.

First, the pin - group is defined as the set of all products of unit vectors:

This quantity is a submonoid of the multiplicative monoid of the Clifford algebra and is used to group by the existence of an inverse. There are products whose factors are different, but they refer to the same element of the pin group, for instance, is orthogonal unit vectors and and each pair

However, is that each element of exactly one orthogonal map

Corresponds to their independence from the chosen factorization of the uniqueness of the inverse follows. It is also known that injective of order 2, i.e., a two-fold overlay. The pre-images of the same orthogonal figure differ only by the sign.

The spin group

But physically and geometrically significant is a subset of the pin group, the spin group

Of the products with an even number of factors ( from the playful reinterpretation of the spin group as " special pin - group " resulted in the term " pin " group). From this it is known that it is a two- fold covering of the special orthogonal group, and that they provided the dimension of the underlying vector space is greater than 2, simply connected, ie is universal cover. Since the matrix group is a representation of the weight of 2, they say in physics also that representations of the spin group of weight 1 Spin - representations of the orthogonal group are.

The complex Clifford algebra

Be the real vector space with the standard scalar product. Then, the complex Clifford algebra is defined as the Complexification of, that is, it is

This definition is independent of the complexified inner product, because on there is exactly one uniquely defined, nondegenerate quadratic form.

Representations

A representation of an algebra is an embedding of this. In the algebra of endomorphisms of a vector space, so ( according to base choice) in a matrix algebra The matrices can be real, complex or quaternionic entries have.

It can be shown that any Clifford algebra is isomorphic to a matrix algebra or the direct sum of two matrix algebras over the real numbers, the complex numbers or the quaternions.

Real Clifford algebra

The allocation and dimension of real Clifford algebras tabulated as follows:

The following general isomorphisms apply:

Complex Clifford algebra

The representation of the complex Clifford algebra is easier than the real. It is namely

In this connection, the applicable isomorphism

Which is also essential for the proof of the representation. Is just so called spinor module with the natural graduation in this context.

Low-dimensional examples

The dimension of a real vector space is 2p q. Thus, the Clifford algebra can be represented by real matrices of this dimension, which describe the multiplication in the algebra. This representation is not minimal, that is, there are matrices of lower dimension, which afford the same, and see the examples below.

Quantum Physics important examples

• Cl (3,0 ) ≅ Cl ( 2,0) ⊗ Cl (0,1) ≅ ( Biquaternionen )

• Cl (2,1 ) ≅ Cl (1,1) ⊗ Cl ( 1,0 ) ≅ M2 ( ℂ )

• Cl (1,2) ≅ Cl (1,1) ⊗ Cl (0,1) ≅ M2 ( ℝ ) ⊕ M2 ( ℝ )

• Cl ( 0.3 ) Cl ≅ ( 0.2 ) ⊗ Cl (1.0 ) ≅

• Cl (4,0 ) ≅ Cl ( 2,0) ⊗ Cl ( 0,2) ≅ M2 ( ℍ )

• Cl (3,1 ) ≅ Cl (1,1) ⊗ Cl ( 2,0 ) ≅ M2 ( ℍ )

• Cl (1,3) ≅ Cl (1,1) ⊗ Cl ( 0,2) ≅ M4 ( ℝ )