Complex quaternion functions

The quaternions were discovered by William Rowan Hamilton in 1843^[1]. He had long searched for an algebra that was to three dimensions what the complex numbers are to two dimensions. He sought to multiply and divide these sought after numbers for many years before, in a flash of insight, the solution came to him. The problem was that there are no such numbers in three dimensions, only in four dimensions.

A quaternion Q can be written as ${\textbf {Q}}=a\,+b\,{\textbf {I}}+c\,{\textbf {J}}\,+d\,{\textbf {K}}$ where ${\textbf {I}}\;{\textbf {I}}={\textbf {J}}\;{\textbf {J}}={\textbf {K}}\;{\textbf {K}}={\textbf {I}}\;{\textbf {J}}\;{\textbf {K}}=-1$ From these, using associativity, it follows that ${\textbf {I}}\;{\textbf {J}}=-{\textbf {J}}\;{\textbf {I}}={\textbf {K}}\quad {\textbf {J}}\;{\textbf {K}}=-{\textbf {K}}\;{\textbf {J}}={\textbf {I}}\quad {\textbf {K}}\;{\textbf {I}}=-{\textbf {I}}\;{\textbf {K}}={\textbf {J}}\quad$

This was the first non-commutative algebra. In hindsight, non-commutativity is to be expected since rotations about the origin in three dimensions do not in general commute but in two dimensions they do. For Hamilton a,b, c, and d were all real. Define the norm as

${\textbf {N}}({\textbf {Q}})=a^{2}+b^{2}+c^{2}+d^{2}$

It is easily verified that

${\textbf {Q}}^{-1}={\frac {a-b\,{\textbf {I}}-c\,{\textbf {J}}-d\,{\textbf {K}}}{{\textbf {N}}({\textbf {Q}})}}$

Since N(Q) is always positive for a non-zero real quaternion, the inverse always exists. This makes the algebra a division algebra. Also, the norm of a product is the product of the norms. This makes the algebra a composition algebra^[2].

The real quaternions can be used to do spatial rotations^[3] , but not to do Lorentz transformations with a boost. But if a, b, c, and d are allowed to be complex, they can^[4]^[5] . This is what motivated the study of functions of a complex quaternion or biquaternion, such as how they are to be computed, when they are defined, and what their multiplicities are.

Since there are non-zero complex quaternions with zero norm, the inverse does not always exist. So they are not a division algebra. But they almost are. And there are zero divisors, as evidenced by (1 + i I) (1 - i I) = 0. In a way, the need for complex quaternions is not surprising, since in special relativity the Minkowski invariant, which is the norm of a 4-vector, can be any real number, including zero (null rays).

A 4-vector $(c\,t,x,y,z)$ is represented by the complex quaternion $c\,t+i\,x\,{\textbf {I}}+i\,y\,{\textbf {J}}+i\,z\,{\textbf {K}}$ , which is called a Minkowski quaternion. Its scalar time-like component is real and its spatial vector component is pure imaginary. This is the convention used by P. A. M. Dirac,^[6] which gives the metric $\eta _{00}>0$ . Choosing the time-like component imaginary and the spatial vector component real has also been done and gives the metric $\eta _{00}<0$ .

The basis quaternions I, J, and K can be represented in terms of the Pauli spin matrices as $-i\,\sigma _{1}$ , $-i\,\sigma _{2}$ , and $-i\,\sigma _{3}$ , respectively^[7], as one possibility. . These have the same multiplication table. The Pauli spin matrices are used in particle physics for Lorentz transformations of 2-spinors and can do Lorentz transformations of a 4-vector by representing it as a 2x2 matrix, which is obtained from its equivalent Minkowski quaternion by replacing I, J, and K by their Pauli spin matrix representations and adding. The scalar term $c\,t$ is replaced by multiplying it by the 2x2 identity matrix. Working with complex quaternions is simpler and more transparent and intuitive than working with matrices. But there is a one-to-one correspondence between the two viewpoints.

The exponential function with a complex quaternion argument is used to generate finite Lorentz transformations and the square root function with a complex quaternion argument is used to express a Lorentz transformation either as a pure boost followed by a spatial rotation or vice versa.

Functions with a complex quaternion argument are treated first, followed by a discussion of their application to Lorentz transformations.

Exp, Sqrt, Log, Trig and Hyperbolic with Inverses

How to compute, when defined, multiplicities

A technique that will be used frequently in the following discussion is that given a power series in a complex quaternion X = a + b V where V V = -1, we can instead work with X = a + b I and, after evaluating the power series, replace I by V. Since V V is always a complex number, any quaternion whose vector part has non-zero norm can be put in this form. Let V = b I + c J + d K and suppose ${\textbf {V}}\,{\textbf {V}}=-(b^{2}+c^{2}+d^{2})$ is non-zero. Then ${\textbf {V}}={\sqrt {b^{2}+c^{2}+d^{2}}}\left[{\frac {\textbf {V}}{\sqrt {b^{2}+c^{2}+d^{2}}}}\right]$ The factor in square brackets has norm -1. The square root of the norm can be pulled out and absorbed into b. A vector quaternion whose norm is -1 will be called basis-like.

A second technique that will be used frequently is to realize that any power series in X = a + b V sums to the form c + d V where c and d are complex numbers, provided that the power series converges.

Exponential function

The exponential function is well-defined by its power series, which converges over the entire domain, even for complex quaternions. Since the basic circular and hyperbolic functions cos, sin, cosh, sinh are linear combinations of exponential functions, they too are well-defined.

A simple expression for exp(X) is sought. Now exp(a + b I + c J + d K) = exp(a) exp(b I + c J + d K) since the complex number a commutes with the complex quaternions. The vector part V = b I + c J+ d K has the property that its square is the complex scalar -(b² + c² + d²), which is the negative of its norm. Let the complex s be the square root of -V V, or, equivalently, the square root of the norm of V.

Any power series in the vector part V can be expressed as a sum of an even power series in s and another even power series in s which multiplies the vector part V. If these two power series are those of common functions, then this gives a simple way to evaluate. For s unequal to zero, $\exp({\textbf {V}})=\cos(s)+{\frac {\sin(s)}{s}}\,{\textbf {V}}$

For the case for which the norm is positive real, s can be chosen to be positive real and we can let V=s I without loss of generality and get De Moivre's formula:

$\exp(s\,{\textbf {I}})=\cos s+{\textbf {I}}\;\sin s$

The quaternion basis element I acts as a square root of -1.

For the case for which the norm of V is negative, we can take s to be a positive pure imaginary. We then get

$\exp(s\,i\,{\textbf {I}})=\operatorname {cosh} \,s+\,i\,{\textbf {I}}\;\operatorname {sinh} \,s\$

For the case for which the vector part V=N, where N has zero norm, $\exp(a+b\,{\textbf {N}})=\exp(a)(1+b{\textbf {N}})$

Square root function

Usually there are four values for the square root of a complex quaternion, but there can be infinitely many or none. We consider the special cases first.

For the special case of the square root of a complex number, any multiple by a quaternion with square +1 is also a solution. This includes -1 and vector quaternions of norm -1 such as i I, i J, i (I+J) / √2 and infinitely many more possibilities.

For the special case of the square root of a complex quaternion with a non-zero vector component N having zero norm, the square root does not exist except for special values

${\sqrt {a\,({\frac {1}{4}}+\,{\textbf {N}})}}=\pm {\sqrt {a}}\,({\frac {1}{2}}+\,{\textbf {N}})$

For the special case of a vector quaternion with non-zero norm, we have ${\sqrt {a\,{\textbf {I}}}}={\sqrt {a}}\;{\frac {1+{\textbf {I}}}{\sqrt {2}}}$ Multiplying by -1, i I, or -i I give other values.

Having considered the special cases first, consider the complex quaternion a + b I with $a\neq 0{\text{ and }}b\neq 0$ . Since a can be factored out we only need to consider X = 1 + A I. We find ${\sqrt {\textbf {X}}}={\tfrac {1}{2}}{\big (}{\sqrt {1+i\,A}}+{\sqrt {1-i\,A}}\,{\big )}-{\tfrac {1}{2}}i\,{\big (}{\sqrt {1+i\,A}}-{\sqrt {1-i\,A}}\,{\big )}\;{\textbf {I}}$ The two square roots in the equation need to be taken on the same branch with a branch cut from 0 to $-\infty$ . This means their phases must be between -90° and 90°. Multiplying by -1, i I, -i I give other possible square roots.

Logarithm function

The logarithm function log X sometimes does not exist, and, when it does, is multi-valued. The logarithm of a complex number is particularly multi-valued. Consider log(1). Some possible values are 2m π i + 2n π I and (2m+1) π i + (2n+1) π I. For I, any vector quaternion of norm +1 may be substituted.

We are to find log(X) such that exp(log(X))=X. We first do the case for which both a and b are non-zero and for which the vector part has a non-zero norm so that it can be scaled to be basis-like with norm +1 and be represented by I. Define the complex number θ by ${\frac {b}{a}}=\tan \theta \;$ Let α be a complex number. Then $\exp(\alpha +\theta \,{\textbf {I}})=\exp(\alpha )(\cos \theta +\sin \theta \;{\textbf {I}})=\left({\frac {\exp \alpha \,\cos \theta }{a}}\right)(a+b\,{\textbf {I}})$

Choose α so that the complex multiplier of a + b I on the right is one. Then a solution is $\log(a+b\,{\textbf {I}})=\log \left({\sqrt {a^{2}+b^{2}}}\right)+\tan ^{-1}({\frac {b}{a}})\,{\textbf {I}}$ Adding 2m π i + 2n &pi I or adding (2m+1) π i + (2n+1) &pi I also gives a solution.

Next find log(X) where X = a + N and N is a null non-zero vector quaternion. As easily verified $\log(a+{\textbf {N}})=\log(a)+{\frac {1}{a}}{\textbf {N}}$ Adding 2m π i also gives a solution.

The log of a quaternion that is a null vector quaternion N does not exist. The above equation diverges as $a\rightarrow 0$

The case of b=0 was discussed in the first paragraph. Lastly, log 0 is undefined.

Circular and hyperbolic functions and their inverses

The sine and cosine trigonometric functions and the hyperbolic sine and hyperbolic cosine hyperbolic functions are defined in terms of the exp function. Their inverses can be expressed in terms of the log and sqrt functions.

Application to Lorentz Transformations

The exp and sqrt functions find application here. Using the exp function we find the complex quaternion representations of a pure boost and of a spatial rotation. The ability to perform Lorentz transformations using complex quaternions is not new.^[8]^[6]^[4]^[5] It is much easier than working with matrices. The discussion here is simple and concise.

Any Lorentz transformation can be expressed as a pure spatial rotation followed by a pure boost or as a pure boost followed by a pure spatial rotation. Given the complex quaternion representing a Lorentz transformation, the square root function is used to find the complex quaternions representing either of these. All has been verified numerically.^[9]

Minkowski quaternions

A Minkowski 4-vector ${\textbf {x}}=(c\,t,x,y,z)$ can be space-like, null, or time-like according to whether its norm is negative, zero, or positive. Real non-zero quaternions always have a real positive norm. Real quaternions can represent spatial rotations but only the complex quaternions have the richness to represent a general Lorentz transformation. The Minkowski quaternion ${\textbf {X}}=c\,t+i\,x\,{\textbf {I}}+i\,y\,{\textbf {J}}+i\,z\,{\textbf {K}}$ representing this 4-vector ${\textbf {x}}$ has the form ${\textbf {X}}=a+ib\,{\textbf {N}}$ and ${\textbf {N}}=n_{1}{\textbf {I}}+n_{2}{\textbf {J}}+n_{3}{\textbf {K}}$ , where $a.b,n_{1},n_{2},n_{3}$ are real numbers and ${\textbf {N}}$ is a real vector quaternion of unit norm ${n_{1}}^{2}+{n_{3}}^{2}+{n_{3}}^{2}=1$ .

The norm of X is $a^{2}-b^{2}$ . This has the correct metric signature since there is only the time-like component associated with $a$ but there are three spatial components associated with $b\,{\textbf {N}}$ .

A Lorentz transformation is a linear transformation which takes a Minkowski quaternion into another Minkowski quaternion having the same norm. Let q = a + b I + c J + d K be any complex quaternion of unit norm and define q = a* - b* I - c* J - d*K. Here * represents complex conjugation. We call q the conjugate transpose of q. The conjugate transpose of a Minkowski quaternion is itself. Any complex quaternion having this property is necessarily a Minkowski quaternion. Consider

${\textbf {X}}'={\textbf {q}}\,{\textbf {X}}\,{\bar {\textbf {q}}}$

The norm is preserved because the norm of a product is the product of the norms. We have the identities ${\overline {{\textbf {Q}}_{1}{\textbf {Q}}_{2}}}={\bar {\textbf {Q}}}_{2}\;{\bar {\textbf {Q}}}_{1}\qquad {\text{and}}\qquad {\bar {\bar {\textbf {Q}}}}={\textbf {Q}}$ The conjugate transpose of a product is the product of the conjugate transposes in reverse order. Therefore X' equals its conjugate transpose and is a Minkowski quaternion. We will show up to a sign how q is determined by the Lorentz transformation.

Relation of complex quaternions to Pauli spin matrices

The complex quaternions are closely related to the Pauli spin matrices, which in the particle physics of spin ${\tfrac {1}{2}}$ massless fermions act on 2-spinors.^[10] The basis quaternions can be represented by

${\textbf {I}}=-i\,\sigma _{1}\qquad {\textbf {J}}=-i\,\sigma _{2}\qquad {\textbf {K}}=-i\,\sigma _{3}$ The multiplication table is the same.

A Minkowski quaternion can be represented in terms of the Pauli matrices as ${\textbf {X}}=c\,t\,+i\,x\,{\textbf {I}}\,+i\,y\,{\textbf {J}}\,+i\,z\,{\textbf {K}}$ $\rightarrow c\,t\,\sigma _{0}+x\,\sigma _{1}+y\,\sigma _{2}+z\,\sigma _{3}={\begin{pmatrix}c\,t+z&x-i\,y\\x+i\,y&ct-z\end{pmatrix}}$ Here $\sigma _{0}=\left({\begin{smallmatrix}+1&{\phantom {+}}0\\{\phantom {+}}0&+1\end{smallmatrix}}\right)$ is the 2x2 identity matrix.

These 2x2 matrices for I, J, and K can be replaced by 4x4 real anti-symmetric matrices by replacing $i$ in each imaginary matrix element by $({\begin{smallmatrix}{\phantom {-}}0&+1\\-1&{\phantom {+}}0\end{smallmatrix}})$ or by its transpose and by replacing 1 in each real matrix element by $\sigma _{0}$ . That is why the term transpose was chosen for the operation ${\textbf {I}}\rightarrow -{\textbf {I}}\quad {\textbf {J}}\rightarrow -{\textbf {J}}\quad {\textbf {K}}\rightarrow -{\textbf {K}}$

This representation by the Pauli spin matrices is not unique. One can change the signs of any two of them and preserve the multiplication table. So another possible representation is

${\textbf {I}}=i\,\sigma _{1}\qquad {\textbf {J}}=-i\,\sigma _{2}\qquad {\textbf {K}}=i\,\sigma _{3}$

Also, they can be cyclically permuted and preserve the multiplication table. More generally, any similarity transform of these 2x2 matrices preserves the multiplication table.

In the equations for the Lorentz transformations that will be given, the quaternion basis elements can be replaced by their 2x2 matrix representations.

Finite Lorentz transformations using the exp function

The complex quaternion function exp can be used to perform Lorentz boosts. A real non-zero quaternion always has positive norm. A 4-vector in special relativity has the form X = t + x i I + y i J + z i K, where t, x, y, and z are real. Its norm is the Minkowski invariant t² - x² - y² - z².

Let a ^T superscript denote a "transpose" operation taking I into -I, J into -J, and K into -K. The reason for the "transpose" name is that I, J, and K can be represented as 4x4 real anti-symmetric matrices and that is what the matrix transpose does. The transpose of a product is the product of the transposes in reverse order, as in matrix algebra.

Let q* denote complex conjugation of the quaternion q. This operation does not do anything to I, J, and K. This definition is again reasonable since I, J, K can be expressed as 4x4 anti-symmetric real matrices.

Let an overbar denote the complex conjugate transpose. If q = a + b I + c J + d K, then q = a* - b* I - c* J - d* K. Note that X = X for the Minkowski quaternion X. Let q have norm 1. Consider X' = q X q. If q has norm 1, then since the complex quaternions are a composition algebra, X' and X have the same norm. Also X' has the same form as X with real scalar part and pure imaginary spatial part and so is also a Minkowski quaternion.

Complex quaternions representing boosts and rotations

In spacetime algebra, a boost in the x direction is done by ${\textbf {X}}'={\textbf {B}}{\textbf {X}}{\bar {\textbf {B}}}\;{\text{ where }}\;{\textbf {B}}=\exp(-i\,{\frac {\alpha }{2}}\;{\textbf {I}})=\operatorname {cosh} {\frac {\alpha }{2}}-\operatorname {sinh} {\frac {\alpha }{2}}\,i\,{\textbf {I}}$ Note that ${\overline {i\,{\textbf {I}}}}=i\,{\textbf {I}}\;$ and $(i\,{\textbf {I}})^{2}=1$ Let v be the velocity and define as usual in special relativity $\beta ={\frac {v}{c}}\quad \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\quad {\text{where}}\;c={\text{light speed}}$

From this we identify $\operatorname {cosh} \alpha =\gamma$ and $\operatorname {sinh} \alpha =\beta \,\gamma$ and easily verify that this gives the usual formulas for the Lorentz boost $x'=\gamma \,(x-v\,t)\;{\text{and}}\;t'=\gamma \,(t-{\frac {v}{c^{2}}}\,x)$

A spatial rotation about the x axis by angle θ is done by ${\textbf {X}}'={\textbf {R}}{\textbf {X}}{\bar {\textbf {R}}}\;{\text{ where }}\;{\textbf {R}}=\cos {\frac {\theta }{2}}+\sin {\frac {\theta }{2}}\;{\textbf {I}}$

These are verified by observing that these give the usual Lorentz transformations. We have the relations ${\textbf {B}}={\bar {\textbf {B}}}\;{\text{ and }}{\textbf {R}}\,{\bar {\textbf {R}}}={\bar {\textbf {R}}}\,{\textbf {R}}=1$

Representing a Lorentz transformation as a rotation followed by a boost or vice versa

One question remains: Suppose q has norm one. What Lorentz transformation is it associated with? It always represents some Lorentz transformation since X' = q X q is a linear transformation of the Minkowski quaternion X into another Minkowski quaternion X' of the same norm.

Any proper Lorentz transformation can be represented either as a rotation followed by a boost or vice versa. Once either of these forms is obtained, we can easily read off what the boost and rotation are.

In the first case we have ${\textbf {q}}={\textbf {B}}\;{\textbf {R}}{\text{ and }}{\textbf {q}}\,{\bar {\textbf {q}}}={\textbf {B}}\,{\bar {\textbf {B}}}={\textbf {B}}\,{\textbf {B}}$ So ${\textbf {B}}={\sqrt {{\textbf {q}}\,{\bar {\textbf {q}}}}}\;{\text{ and }}\;{\textbf {R}}={\textbf {B}}^{-1}\,{\textbf {q}}$

In the second case we have ${\textbf {q}}={\textbf {R}}\;{\textbf {B}}{\text{ and }}{\bar {\textbf {q}}}\,{\textbf {q}}={\bar {\textbf {B}}}\,{\textbf {B}}={\textbf {B}}\,{\textbf {B}}$ So ${\textbf {B}}={\sqrt {{\bar {\textbf {q}}}\,{\textbf {q}}}}\;{\text{ and }}\;{\textbf {R}}=\,{\textbf {q}}\,{\textbf {B}}^{-1}$

References

^ Stillwell, John (2010). Mathematics and Its History Third Edition. Undergraduate Texts in Mathematics. New York Dordrecht Heidelberg London: Springer. p. 422. ISBN 978-1-4419-6053-5.
^ Conway, John H; Smith, Derek A (2003). On Quaternions and Octonions. Boca Raton, Florida: CRC Press. ISBN 978-1-56881-134-5.
^ Kuipers, Jack B (1999). Quaternions and Rotation Sequences. pp. 127–138.
^ ^a ^b Shah, Alam M; Sabar, Bauk (June 2011). "Quaternion Lorentz Transformation". Physics Essays. 24 (2): 158–162.
^ ^a ^b Shah, Alam M; Sabar, Bauk (June 2011). "Quaternion Lorentz Transformation".
^ ^a ^b Dirac, P. A. M. (November 1945). "Application of Quaternions to Lorentz Transformations". Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. 50(1944/1945). Royal Irish Academy: 261–270.
^ Stillwell, John (2010). Mathematics and Its History Third Edition. Undergraduate Texts in Mathematics. New York Dordrecht Heidelberg London: Springer. p. 426. ISBN 978-1-4419-6053-5.
^ Synge, J. L. (1972). "Quaternions, Lorentz Transformations, and the Conway-Dirac-Eddington Matrices" (PDF).
^ Phillion, Donald W (March 10, 2025). "Quaternion Lorentz".
^ Zuber, Kai (2020). Neutrino Physics Third Edition. Series in High Energy Physics, Cosmology, and Gravitation. Boca Raton, Florida: CRC Press. ISBN 978-1-138-71889-0.

[1] Stillwell, John (2010). Mathematics and Its History Third Edition. Undergraduate Texts in Mathematics. New York Dordrecht Heidelberg London: Springer. p. 422. ISBN 978-1-4419-6053-5.

[Conway-2] Conway, John H; Smith, Derek A (2003). On Quaternions and Octonions. Boca Raton, Florida: CRC Press. ISBN 978-1-56881-134-5.

[Kuipers-3] Kuipers, Jack B (1999). Quaternions and Rotation Sequences. pp. 127–138.

[Shah_journal-4] Shah, Alam M; Sabar, Bauk (June 2011). "Quaternion Lorentz Transformation". Physics Essays. 24 (2): 158–162.

[Shah_web-5] Shah, Alam M; Sabar, Bauk (June 2011). "Quaternion Lorentz Transformation".

[Dirac-6] Dirac, P. A. M. (November 1945). "Application of Quaternions to Lorentz Transformations". Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. 50(1944/1945). Royal Irish Academy: 261–270.

[Stillwell-7] Stillwell, John (2010). Mathematics and Its History Third Edition. Undergraduate Texts in Mathematics. New York Dordrecht Heidelberg London: Springer. p. 426. ISBN 978-1-4419-6053-5.

[8] Synge, J. L. (1972). "Quaternions, Lorentz Transformations, and the Conway-Dirac-Eddington Matrices" (PDF).

[9] Phillion, Donald W (March 10, 2025). "Quaternion Lorentz".

[10] Zuber, Kai (2020). Neutrino Physics Third Edition. Series in High Energy Physics, Cosmology, and Gravitation. Boca Raton, Florida: CRC Press. ISBN 978-1-138-71889-0.

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