Quaternion Lorentz transformations

Short description: Linear transformation of spacetime coordinates

In special relativity, a Lorentz transformation is a real linear transformation of the spacetime coordinates $t$, $x$, $y$, $z$ that preserves the space time interval squared of Minkowski space-time

$$c^2{t}^2-x^2-y^2-z^2$$

Here, $c$ is the speed of light. Using the 4 × 4 matrix $\eta$ of the metric tensor, this is

$$X^{\text{T}}\eta X=\left[\begin{array}{cccc}ct& x& y& z\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\\ \end{array}\right]\left[\begin{array}{c}ct\\ x\\ y\\ z\\ \end{array}\right]$$

Here $X^{\text{T}}$ is the matrix transpose of $X$.

One way to implement Lorentz transformations is to let $X^{\prime }=AX$, where $A$ is a 4 × 4 real matrix that makes

$$c^{2}t{\text{'}}^2-x{\text{'}}^2-y{\text{'}}^2-z{\text{'}}^2=c^2{t}^2-x^2-y^2-z^2$$

Here t is time and x,y,z are the Cartesian spatial coordinates. This is so if $A^{\text{T}}\eta A=\eta$.

Another way to do Lorentz transformations is to let the spacetime coordinates be represented by a 2 × 2 hermitian matrix

$$X=\left[\begin{array}{cc}ct+z& x-iy\\ x+iy& ct-z\end{array}\right]$$

Here $i$ is the square root of $-1$. The determinant of $X$ is the spacetime interval squared. Let A now be a 2 × 2 complex matrix with determinant 1 and let $A^{†}$ be the hermitian conjugate of A (the complex conjugate of the transpose of A). Then $X^{\prime }=A^{†}XA$ has the same determinant as $X$ since the determinant of a product is the product of the determinants and since the determinant of $A^{†}$ is the complex conjugate of the determinant of $A$, so is also 1. Thus $X$ and $X^{\prime }$ have the same spacetime interval squared. Also, $X^{\prime }$ is hermitian since the hermitian conjugate of a product is the product of the hermitian conjugates in reverse order and since $(A^{†}{)}^{†}=A$ and since $X^{†}=X$. So this is a Lorentz transformation.

This article treats biquaternion Lorentz transformations without reflections, such as time reversal T or spatial inversion P. One author, in Other Lorentz transformations (§3.2.3), does treat biquaternion Lorentz transformations with reflections. Biquaternions are sometimes called the complex quaternions or the complexified quaternions or even just the quaternions in the literature. The biquaternions differ from the quaternions only in that the coeffficients of the bases $\{1,\mathbf{𝐈},\mathbf{𝐉},\mathbf{𝐊}\}$ can be complex numbers rather than real numbers.

As will be discussed, the biquaternions representing these Lorentz transformations are those of norm +1 and can be represented non-uniquely by the subgroup of 2 × 2 complex matrices having determinant +1. Representations exist for the biquaternion basis $\{1,\mathbf{𝐈},\mathbf{𝐉},\mathbf{𝐊}\}$ in terms of 2 × 2 complex matrices which have the same multiplication table. The 2 × 2 identity matrix always represents 1. As will also be discussed, for any given representation, there is a one-to-one correspondence between all 2 × 2 complex matrices and all biquaternions and a one-to-one correspondence between the biquaternions of norm +1, which are those representing Lorentz transformations, and the 2 × 2 complex matrices of determinant +1. A one-to-one correspondence preserving all the algebraic properties is an isomorphism. One particular isomorphism that will be discussed in detail is $1\to I_{2\times 2},\mathbf{𝐈}\to -i{\sigma }_x,\mathbf{𝐉}\to -i{\sigma }_y,\mathbf{𝐊}\to -i{\sigma }_z$, where $I_{2\times 2}$ is the 2 × 2 identity matrix, $i$ is the square root of $-1$, and ${\sigma }_x,{\sigma }_y,{\sigma }_z$ are the Pauli matrices.

Definition

The biquaternions have the form $\text{<mi fromhbox="1">Q</mi>}=a+b\text{<mi fromhbox="1">I</mi>}+c\text{<mi fromhbox="1">J</mi>}+d\text{<mi fromhbox="1">K</mi>}$ for complex $a$, $b$, $c$, and $d$. The biquaternion basis elements $\text{I}$, $\text{J}$, and $\text{K}$ satisfy $$\text{<mi fromhbox="1">I</mi>}\text{<mi fromhbox="1">I</mi>}=\text{<mi fromhbox="1">J</mi>}\text{<mi fromhbox="1">J</mi>}=\text{<mi fromhbox="1">K</mi>}\text{<mi fromhbox="1">K</mi>}=\text{<mi fromhbox="1">I</mi>}\text{<mi fromhbox="1">J</mi>}\text{<mi fromhbox="1">K</mi>}=-1$$

From these, using associativity, it follows that

$$\text{<mi fromhbox="1">I</mi>}\text{<mi fromhbox="1">J</mi>}=-\text{<mi fromhbox="1">J</mi>}\text{<mi fromhbox="1">I</mi>}=\text{<mi fromhbox="1">K</mi>}\text{<mi fromhbox="1">J</mi>}\text{<mi fromhbox="1">K</mi>}=-\text{<mi fromhbox="1">K</mi>}\text{<mi fromhbox="1">J</mi>}=\text{<mi fromhbox="1">I</mi>}\text{<mi fromhbox="1">K</mi>}\text{<mi fromhbox="1">I</mi>}=-\text{<mi fromhbox="1">I</mi>}\text{<mi fromhbox="1">K</mi>}=\text{<mi fromhbox="1">J</mi>}$$

The real quaternions can be used to do spatial rotations, but not to do Lorentz transformations with boosts, which are transformations from one inertial reference frame to another in uniform relative motion. But if $a$, $b$, $c$, and $d$ are allowed to be complex, they can.

Minkowski biquaternions

We use a biquaternion representing $t$, $x$, $y$, $z$ that was used by P. A. M. Dirac, which has the form:

$$\text{<mi fromhbox="1">X</mi>}=t+ix\text{<mi fromhbox="1">I</mi>}+iy\text{<mi fromhbox="1">J</mi>}+iz\text{<mi fromhbox="1">K</mi>}$$

Here, $i$ is the square root of −1 and $c=1$ henceforth. We will call this the Minkowski biquaternion.

The reason for this is that its norm is the spacetime interval squared $t^2-x^2-y^2-z^2$. The norm is defined as

$$\mathbf{𝐍}(a+b\mathbf{𝐈}+c\mathbf{𝐉}+d\mathbf{𝐊})=a^2+b^2+c^2+d^2$$

and has the important property that the norm of a product is the product of the norms, making the biquaternions a composition algebra. A real non-zero quaternion always has real positive norm, but a non-zero complex quaternion can have a norm with any complex value, including zero.

A biquaternion $\text{<mi fromhbox="1">Q</mi>}=a+b\text{<mi fromhbox="1">I</mi>}+c\text{<mi fromhbox="1">J</mi>}+d\text{<mi fromhbox="1">K</mi>}$ with complex $a$, $b$, $c$, $d$ has two kinds of conjugates:

• The biconjugate is

$$Q^{*}=a-b\mathbf{𝐈}-b\mathbf{𝐉}-d\mathbf{𝐊}$$

• The complex conjugate is

$$\overline{Q}=\overline{a}+\overline{b}\mathbf{𝐈}+\overline{c}\mathbf{𝐉}+\overline{d}\mathbf{𝐊}$$

The overbar $\stackrel{}{¯}$ denotes complex conjugation. The biconjugate of a product is the product of the biconjugates in reverse order. The operations denoted by the asterisk superscript and by the overbar are defined as in the article Biquaternion.

For a Minkowski biquaternion

$$\stackrel{*}{\stackrel{‾}{\mathbf{𝐗}}}=\mathbf{𝐗}$$

As can be seen from the definition, this is a necessary and sufficient condition for a biquaternion $\mathbf{𝐗}$ to be a Minkowski biquaternion.

Also needed is the identity

$$\mathbf{𝐗}{\mathbf{𝐗}}^{*}=\mathbf{𝐗}\overline{\mathbf{𝐗}}=t^2-x^2-y^2-z^2$$

Lorentz transformations

General form

Let $\mathbf{𝐐}$ be a biquaternion of norm one and let $\mathbf{𝐗}$ be a Minkowski biquaternion. Then

$${\mathbf{𝐗}}^{\prime }=\stackrel{*}{\stackrel{‾}{\mathbf{𝐐}}}\text{<mi fromhbox="1">X</mi>}\text{<mi fromhbox="1">Q</mi>}=\stackrel{‾}{(\stackrel{*}{\stackrel{‾}{\mathbf{𝐐}}}\text{<mi fromhbox="1">X</mi>}\text{<mi fromhbox="1">Q</mi>}}{)}^{*}$$

Because of the second equality, ${\mathbf{𝐗}}^{\prime }$ is a Minkowski biquaternion. And if $\mathbf{𝐐}$ has norm 1, then the norm of ${\mathbf{𝐗}}^{\prime }$ equals the norm of $\mathbf{𝐗}$. This is then a linear transformation of one Minkowski biquaternion into another Minkowski biquaternion having the same spacetime interval squared. Therefore it is a Lorentz transformation.

Spatial rotations and Lorentz boosts

Let $\mathbf{𝐧}$ be the real direction biquaternion $$\mathbf{𝐧}=n_1\mathbf{𝐈}+n_2\mathbf{𝐉}+n_3\mathbf{𝐊}\text{ such that }{n}_1^2+n_3^2+n_3^2=1$$

Spatial rotations are represented by

$$\mathbf{𝐑}=\exp (-\frac{\theta }{2}\mathbf{𝐧})=\cos (\frac{\theta }{2})-\mathbf{𝐧}\sin (\frac{\theta }{2})$$

$\mathbf{𝐑}$ has norm 1 and so represents a Lorentz transformation. It does not change the scalar part and so must be a rotation.

Boosts are represented by

$$\mathbf{𝐁}=\exp (-i\frac{\alpha }{2}\mathbf{𝐧})=\cosh (\frac{\alpha }{2})-i\mathbf{𝐧}\sinh (\frac{\alpha }{2})$$

$\mathbf{𝐁}$ also has norm 1 and so also represents a Lorentz transformation. It does not change the vector part normal to $\mathbf{𝐧}$ and so must be a Lorentz boost.

Expressing the exponentials as circular or hyperbolic trigonometric functions is basically De Moivre's formula.

It is immediately seen that $\mathbf{𝐁}$ and $\mathbf{𝐑}$ have the conjugate and norm properties

$$R{R}^{*}=1R=\overline{R}N(\mathbf{𝐑})=1$$

$$B={\overline{B}}^{*}{B}^{*}=\overline{B}B{B}^{*}=1B\overline{B}=1N(\mathbf{𝐁})=1$$

Here $N(\mathbf{𝐑})$ and $N(\mathbf{𝐁})$ are the respective norms of $R$ and $B$. If a biquaternion has one of these sets of conjugate and norm properties, it must have the corresponding form given. Also note that $\mathbf{𝐑}\mathbf{𝐑}$ has the same form as $\mathbf{𝐑}$ except that $\theta /2$ is replaced by $\theta$ and that $\mathbf{𝐁}\mathbf{𝐁}$ has the same form as $\mathbf{𝐁}$ except that $\alpha /2$ is replaced by $\alpha$. Useful identities for representing a Lorentz transformation as a boost followed by a rotation or vice versa are

$$(\overline{RB}{)}^{*}(RB)=B^2(BR)(\overline{BR}{)}^{*}=B^2$$

The general spatial rotations and Lorentz boosts can be worked out by letting $\mathbf{𝐗}=t+i\mathbf{𝐫}$, where $\mathbf{𝐫}=x\mathbf{𝐈}+y\mathbf{𝐉}+z\mathbf{𝐊}$ and then repeatedly using the identity for the product of vectors

$$\mathbf{𝐧}\mathbf{𝐫}=-(\mathbf{𝐧},\mathbf{𝐫})+\mathbf{𝐧}\times \mathbf{𝐫}$$

$$\mathbf{𝐫}\mathbf{𝐧}=-(\mathbf{𝐧},\mathbf{𝐫})-\mathbf{𝐧}\times \mathbf{𝐫}$$

$$\mathbf{𝐧}\mathbf{𝐫}\mathbf{𝐧}=-\mathbf{𝐧}(\mathbf{𝐧},\mathbf{𝐫})-\mathbf{𝐧}\times (\mathbf{𝐧}\times \mathbf{𝐫})$$

Here $(\mathbf{𝐧},\mathbf{𝐫})$ is the scalar product of $\mathbf{𝐧}$ and $\mathbf{𝐫}$ and $\mathbf{𝐧}\times \mathbf{𝐫}$ is their cross product.

Examples

Let $\mathbf{𝐧}=\mathbf{𝐈}$. Then the boost $\mathbf{𝐁}$ in the x-direction gives the familiar coordinate transformations:

$$t^{\prime }=\cosh (\alpha )t-\sinh (\alpha )x$$ $$x^{\prime }=\cosh (\alpha )x-\sinh (\alpha )t{y}^{\prime }=y{z}^{\prime }=z$$

Now let $\mathbf{𝐧}=\mathbf{𝐊}$. The spatial rotation $\mathbf{𝐑}$ is then a rotation about the z-axis and gives the again familiar coordinate transformations:

$$x^{\prime }=x\cos (\theta )-y\sin (\theta )$$

$$y^{\prime }=x\sin (\theta )+y\cos (\theta )$$

$$t^{\prime }=t{z}^{\prime }=z$$

2 × 2 matrices

By a simple identification, we show that Lorentz transformations using biquaternions are equivalent to Lorentz transformations using 2 × 2 matrices. The biquaternions have the advantages of being more transparent and simpler to work with.

The biquaternion basis elements $\mathbf{𝐈},\mathbf{𝐉},\mathbf{𝐊}$ can be represented as the 2 × 2 matrices $-i{\sigma }_x,-i{\sigma }_y,-i{\sigma }_z$, respectively. Here the ${\sigma }_i$ are the 2 × 2 Pauli spin matrices. These have the same multiplication table. This representation is not unique. For instance, without changing the multiplication table, the sign of any two can be reversed, or the ${\sigma }_i$ can be cyclically permuted, or a similarity transformation can be done so that the ${\sigma }_i$ are replaced by $S^{-1}{\sigma }_{i}S$.

Everything that follows is by simple replacement of $\mathbf{𝐈},\mathbf{𝐉},\mathbf{𝐊}$ by $-i{\sigma }_x,-i{\sigma }_y,-i{\sigma }_z$. Except for $X$, lower case letters $q$, $r$, $b$, and ${\sigma }_i$ are used for 2 × 2 matrices.

What we call a Minkowski 2 × 2 complex matrix is that 2 × 2 complex matrix associated with a Minkowski biquaternion. It has the form

$$X=t+x{\sigma }_x+y{\sigma }_y+z{\sigma }_z=\left(\begin{array}{cc}t+z& x-iy\\ x+iy& t-z\end{array}\right)$$

Let an arbitrary 2 × 2 matrix have the form $q=a+b{\sigma }_x+c{\sigma }_y+d{\sigma }_z$, where $a$, $b$, $c$, and $d$ are complex.

• The analog of the biconjugate is $q^{*}=a-b{\mathit{\sigma }}_x-c{\mathit{\sigma }}_y-d{\mathit{\sigma }}_z$
• The analog of the complex conjugate is $\overline{q}=\overline{a}-\overline{b}{\mathit{\sigma }}_x-\overline{c}{\mathit{\sigma }}_y-\overline{d}{\mathit{\sigma }}_z$
• The analog of the biconjugate of the complex conjugate is the hermitean conjugate (conjugate transpose) since the ${\sigma }_i$ are hermitean 2 × 2 matrices:

$${\overline{q}}^{*}=q^{†}=\overline{a}+\overline{b}{\mathit{\sigma }}_x+\overline{c}{\mathit{\sigma }}_y+\overline{d}{\mathit{\sigma }}_z$$

• The analog of the norm is $N(q)=a^2-b^2-c^2-d^2$. This is also its determinant $\left|\begin{array}{cc}a+d& b-ic\\ b+ic& a-d\end{array}\right|$
• The Lorentz transformation is $X^{\prime }={\overline{q}}^{*}Xq=q^{†}Xq$ for a 2 × 2 matrix q that has norm 1 (determinant 1).

A direction can be represented as $\mathbf{𝐧}\cdot \mathit{\sigma }=n_1{\sigma }_x+n_2{\sigma }_y+n_3{\sigma }_z$ where $n_1^2+n_2^2+n_3^2=1$

The spatial rotation is $r=\exp (i\frac{\theta }{2}\mathbf{𝐧}\cdot \mathit{\sigma })$ so ${\overline{r}}^{*}\equiv r^{†}=\exp (-i\frac{\theta }{2}\mathbf{𝐧}\cdot \mathit{\sigma })$

The Lorentz boost is $b=\exp (-\frac{\alpha }{2}\mathbf{𝐧}\cdot \mathit{\sigma })$ so ${\overline{b}}^{*}\equiv b^{†}=\exp (-\frac{\alpha }{2}\mathbf{𝐧}\cdot \mathit{\sigma })$

See also

• Biquaternion
• Quaternion
• Lorentz transformation
• Spacetime algebra