Rotors and Spinors

In this section, we will explore how transformations in Clifford algebra can be represented using mathematical objects called rotors. This leads us to the generalization of spin groups, which are the double cover of the special orthogonal group.

Table of Contents

• Table of Contents
• Rotations in 2D
• Reflections and Rotations in 3D
• Rotations and Boosts in 4D
• Rotors in Higher Dimensions
  • Grade-Involution
• Idempotents
  • Idempotents in Clifford Algebras
• Ideals
• Relationship Between Clifford Algebras and Tensors
• Summary and Next Steps

Rotations in 2D

We shall first consider a simple case of a rotation in the Clifford algebra , which is the algebra of 2D vectors. Recall that this algebra consist of the scalar , the basis vectors and , and the bivector . We know that in 2D, specifically the complex numbers, a rotation by an angle can be represented as a complex number .

The highest-grade element of the algebra is the bivector . We call these elements pseudoscalars because they only have 1 basis and thus act like scalars. We shall call this bivector because it squares to :

Just like complex numbers, we expect to rotate a vector by 90 degrees. This means that a vector rotated by 90 degrees should be given by . We can explicitly check this by multiplying the vector by , which yields

which is indeed the expected result. Multiplying in the other direction, we have

which is a rotation by 90 degrees in the opposite direction.

To generalize this to arbitrary rotations, we can use the exponential map. The exponential map allows us to represent a rotation by an angle as . Note that here is the bivector . To exponentiate a bivector, we can use the Taylor series expansion

which leads to

Reflections and Rotations in 3D

In 3D, we can represent a reflection across a plane using the Clifford algebra . Recall that this algebra consists of the scalar , the basis vectors , , and , and the bivectors , , and .

Recall from the previous sections that a reflection across the -axis can be represented as a negative conjugation with , leading to the transformation

We also know that this works for any arbitrary unit vector , so we can express the reflection across as

So, for example, if we want to rotate by an angle in the -plane, we can choose any two vectors in the -plane that are apart, and reflect along these two vectors. We can just choose the first as , and the second as

Then, the rotated vector is given by

The product is a multivector whose explicit form is

A multivector which can be expressed as a product of invertible vectors, such as this one, is known as a versor.

As squares to , it satisfies Euler's formula

If we use instead of , we get

which matches the form of . Thus, the rotation is

More generally, to rotate any vector by an angle around any unit bivector , we can use the formula

The term is known as a rotor.

Rotations and Boosts in 4D

In 4D, we can represent a rotation or a boost using the Clifford algebra . This algebra consists of the scalar , the basis vectors , , , and , the bivectors , , , , , and , the trivectors , , and , and the quadvector (pseudoscalar) .

The form of transformation using rotors is independent of the dimension of the Clifford algebra. As such, we can use the same formula as in 3D. A rotation in two spatial dimensions and is given by

Similarly, a boost in the time direction and a spatial direction is given by

where is the rapidity of the boost. Note that boosts are slightly different in that squares to instead of . This means that they are elements of the split complex numbers. They satisfy a different version of Euler's formula,

We can leverage this fact to find out if any unit bivector is a rotation or a boost. If squares to , then it is a rotation, and if squares to , then it is a boost.

Rotors in Higher Dimensions

In higher dimensions, we can generalize the concept of rotors to arbitrary dimensions. Specifically, we can define a rotor in the Clifford algebra as

where is a unit bivector in the algebra and is the angle of rotation or rapidity of the boost. These rotors are elements of the group.

If we only consider spatial dimensions, then the rotors are elements of the group, which is the double cover of the special orthogonal group . Another consideration is how to distinguish between rotations and reflections for a given rotor.

A versor's "length" is defined as the number of vectors used to construct it.

For a length-1 versor, we simply get a reflection:

For a length-2 versor, we get a rotation, which is made of two reflections:

For a length-3 versor, we get a rotation followed by a reflection, which is made of three reflections:

For a length-4 versor, we get two rotations made of four reflections:

In general, if the length of the versor is odd, then it represents a reflection, and if it is even, then it represents a rotation.

Note that for any versor, we can always scale it by an arbitrary scalar and the transformation will still hold. This is because the inverse of is , so the scalar cancels out in the transformation. As such, by convention, we enforce that the versor is made up of unit vectors. This leaves us with only two versors for any transformation ( and ).

The group is then defined as the following:

The group is the group of all even-length versors in the Clifford algebra that are made up of unit vectors. They are the double cover of and are used to represent rotations in -dimensional space.

If we allow spatial reflections, then we need the double cover of . As the group is just without the "S", we denote the double cover as (i.e. the spin group without the "S").

If we allow boosts, then we need to consider the Clifford algebra . The groups and are defined as follows:

The group is the group of all even-length versors in the Clifford algebra that are made up of unit vectors (), where timelike vectors square to and spacelike vectors square to .

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The group is the group of all even-length versors in the Clifford algebra that are made up of unit vectors (), where timelike vectors square to and spacelike vectors square to . Additionally, the inverse of such versors must be equal to the reverse of the versor;

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The group is the group of all versors in the Clifford algebra that are made up of unit vectors (), where timelike vectors square to and spacelike vectors square to .

Grade-Involution

We currently have two different transformations—rotations and reflections. These are expressed differently;

When combined with the fact that rotations are even-length versors and reflections are odd-length versors, we can define a new operation that acts on a versor to provide the correct sign;

This operation is known as the grade-involution operator, denoted or . It is defined with the following two properties:

1. For a single vector , .
2. For a product of versors and , .

This means that the grade-involution operator is multiplicative, and it is equal to for even-length versors and for odd-length versors.

Idempotents

Now that we have explored the transformation properties of rotors (spin groups), we also need a way to represent spinors themselves in the Clifford algebra. There are two definitions of spinors that we will explore:

1. A spinor is an element of an even-grade subalgebra of a Clifford algebra.
2. A spinor is a minimal left ideal of a Clifford algebra.

The first definition was pushed forward by David Hestenes, and the second by Fritz Sauter and Marcel Riesz. To understand these definitions, we need to explore the concept of idempotents or projectors.

Intuitively, a projector is an operator that projects a vector onto a subspace. For example, we can project a vector in 2D and project it onto the -axis;

Similarly, we can project a vector onto the -axis;

In 3D, we can project a vector onto the -plane;

These projectors have matrix representations. For example, the projector onto the -plane is given by the matrix

There are two important properties of projectors that arise from this intuition:

1. If we apply a projector twice, it is the same as applying it once. This means that .
2. If we project along two orthogonal axes, then the result is zero. For example, if we project onto the -axis and then onto the -axis, we get zero.

A projector, or idempotent, is a linear operator that satisfies the property . There are two important theorems that describe the properties of idempotents:

1. If is an idempotent, then is also an idempotent, and is orthogonal to .

   This is because , satisfying the idempotent property. They are orthogonal because .

2. If and are orthogonal idempotents (), then is also an idempotent.

   This is because , satisfying the idempotent property.

A minimal idempotent is an idempotent that cannot be expressed as a sum of other idempotents.

Idempotents in Clifford Algebras

Idempotents can also be applied not only to vectors, but also to multivectors in Clifford algebras, such as . In this case, for any unit vector (such that ), then

is an idempotent. This is because

The idempotent

is also an idempotent, and it is orthogonal to .

One such example is where . This leads to the idempotents

We can see in the matrix representations that they add to unity as expected;

When we apply the projectors onto a multivector , we can split into two parts, one transformed by and the other by ;

These two parts are orthogonal to each other and add up to the original multivector .

Ideals

We proceed to the second definition of spinors, which is that a spinor is a minimal left ideal of a Clifford algebra. First, we establish some preliminary definitions.

• A ring is a set of elements where addition and multiplication are defined. We do not need multiplicative inverses for all elements, so division is not required.
• A field is a ring where every non-zero element has a multiplicative inverse. In other words, division is required.

An ideal is a subset of a ring that is closed under addition and multiplication by any element of the ring. An analogy is that an ideal is like a subspace in linear algebra, but it is not necessarily closed under scalar multiplication. Writing it formally, we have:

An ideal of a ring is a subset such that

1. is closed under addition: .
2. is closed under multiplication by any element of the ring: .

To take an example, the set of even integers is an ideal of the ring of integers . This is because it is closed under addition (the sum of two even integers is even) and multiplication by any integer (the product of an even integer and any integer is even).

As multiplication is not necessarily commutative for matrices, we need to specify whether multiplication is cloesd from the left or from the right. For example, consider the set of matrices of the form

This set is closed under addition;

It is closed when multiplying from the left by any matrix;

However, this closure does not hold when multiplying from the right;

This means that this set is a left ideal of the ring of matrices, but not a right ideal. A left ideal is a subset of a ring that is closed under addition and multiplication from the left by any element of the ring. A right ideal is a subset of a ring that is closed under addition and multiplication from the right by any element of the ring. A two-sided ideal is a subset of a ring that is closed under addition and multiplication from both the left and the right by any element of the ring.

Previously I stated that an ideal is similar to a subspace of a vector space. We can extend this analogy further. We can break a ring into ideals, just like we can break a vector space into subspaces. For example, in the ring of matrices, we can break it into the set of matrices and the set of matrices . In a sense, if a ring is visualized as a space, then an ideal is like a line or a plane within that space.

This analogy is further extended when we consider idempotents. Specifically, idempotents of rings create ideals of such rings.

Consider a multivector and an idempotent . We first demonstrate that is a left ideal of the ring of multivectors.

Relationship Between Clifford Algebras and Tensors

So far it might appear that Clifford algebras and tensors fulfill a similar role in mathematics. They are not fundamentally related, although they can be used to represent similar concepts. However, there is a relationship between the two.

A tensor algebra is a vector space where the product is the tensor product . The elements of a tensor algebra form the set , where is the field of scalars, is the vector space, and is the tensor product. In other words, you can have tensors of any rank in the tensor algebra.

A quotient is a set of equivalence classes of elements in a vector space. For example, consider the set of integers and the set of numbers on a clock, i.e. set of integers modulo 12. The equivalence relation is that two integers are equivalent if they differ by a multiple of 12. In other words, if .

An equivalence class is a set of elements that are equivalent to each other. For example, the equivalence class of 0 modulo 12 is . The equivalence class of 1 modulo 12 is . A quotient of a vector space is a vector space where the elements are equivalence classes of the original vector space. So if we take the set of integers and the equivalence relation of integers modulo 12, then the quotient is the set of equivalence classes , which is the set of integers on a clock. This means that the set is equal to

In our case, we will define the equivalence class to be the ideal generated by the expression , where is the metric tensor. We can prove that this is an ideal by showing that it is closed under addition and multiplication by any element of the tensor algebra. Next, we shall define a set to be the quotient of the tensor algebra by this ideal;

Summary and Next Steps

• Spin groups are the double cover of the special orthogonal group and are used to represent rotations and boosts in Clifford algebras. More specifically, the spin group is the group of all even-length versors in the Clifford algebra that are made up of unit vectors, where timelike vectors square to and spacelike vectors square to .

• Rotors are used to represent transformations in Clifford algebras. Their general form is

  where is a unit bivector in the Clifford algebra and is the angle of rotation or rapidity of the boost.

• The grade involution operator is used to distinguish between rotations and reflections, where for rotations and for reflections.

• Idempotents are linear operators that satisfy the property and can be used to project multivectors onto subspaces.

• An ideal is a subset of a ring that is closed under addition and multiplication by any element of the ring.

• The subspaces created by idempotents are ideals of the ring of multivectors.

• Spinors can be defined as minimal left ideals of Clifford algebras (Sauter, Riesz, et al.) or as elements of an even-grade subalgebra (Hestenes).

• The relationship between Clifford algebras and tensors is established through the quotient of the tensor algebra by an ideal generated by the expression , where is a symmetric bilinear form. However, this represents only one formulation of Clifford algebras, and there are other formulations that do not rely on the tensor algebra.