The meaning of ##(ict, x_1,x_2,x_3)##

[mma]

TL;DR Summary

What is this beyond a calculation trick to calculate the pseudo-norm as if it were an Euclidean norm?

In relativity theory, it's a common habit to use a quadruplet $x=(x_0, x_1, x_2, x_3)$ with $x_0=ict$ (or with $c=1$, $x_0=it$ ) instead of $(t,x_1,x_2,x_3)\in \mathbb R^4$, and to use the formal Euclidean metric $\|x\|^2=\sum_{i=0}^3x_i^2$ instead of the Minkowski pseudo-metric $-t^2 + \sum_{i=1}^3x_i^2$. But what is $(it,x_1,x_2,x_3)$ by itself? For a long time I didn't think there was any real mathematics behind this formal computation trick. Until I came across F. F. Eberlein's "The Spin Model of Euclidean 3-space"¹ and "Models of spacetime"². From them I learned that it is possible to make mathematically well-defined sense of this.

Eberlein's Euclidean 3-space $\mathfrak E_3$ consists of the traceless, Hermitian $2\times 2$ complex matrices with scalar product $A\cdot B:=\frac{1}{2}(AB+BA)/I$ where $AB$ means the product of the matrices $A$ and $B$, $I$ is the $2\times 2$ identity matrix and for a diagonal matrix $D=\lambda I$, D$/I:=\lambda$. Eberlein proves that for any $A, B\in\mathfrak E_3$, $AB$ is always diagonal, $(AB+BA)/I$ is real-valued, and the above-defined $\mathfrak E_3^2\to \mathbb R: (A,B)\mapsto A\cdot B$ function is a symmetric, positive definite nondegenerate bilinear form, i.e. and Euclidean scalar product. In $\mathfrak E_3$, the Pauli-matrices form an orthogonal basis. Furthermore, $A\cdot A=-\mathrm{det}A$, and $i\mathfrak E_3=\mathfrak{su}(2)$

The space of quaternions, $\mathbb H:=\mathbb RI\oplus\mathfrak{su}(2) = \mathbb RI + i\mathfrak E_3$ is also an Euclidean space with Euclidean norm $\|q\|^2=\mathrm{det}(q)$. For $q = tI + iA\in\mathbb H$ ($A\in\mathfrak E_3$), $\mathrm{det}(q) = t^2 - \mathrm{det}(A)$, so, taking an orthonormal basis $(\xi_1, \xi_2, \xi_3)$ in $\mathfrak E_3$ and taking any $q = tI + i\sum_{i=1}^nx_i \xi_i\in \mathbb H$ ($t,x_i\in\mathbb R$), $\|q\|^2=t^2+\sum_{i=1}^3x_i^2$.

In contrast of this, the space $\mathbb RI \oplus \mathfrak E_3$ with the pseudo-norm $\|v\|^2=\mathrm{det}(v)$ is a Minkowski space with signature $(1,3)$, that is, the space $M:=i\mathbb RI \oplus i\mathfrak E_3=i\mathbb RI \oplus \mathfrak{su}(2)$ with pseudo-norm $\|v\|^2 = \mathrm{det}(v)$ is a Minkowski space with the usual signature $(3,1)$. Taking any $v = itI + i\sum_{i=1}^nx_i \xi_i\in M$ ($t,x_i\in\mathbb R$, $(\xi_1, \xi_2, \xi_3)$ is an orthonormal basis in $\mathfrak E_3$), $\|v\|^2=(it)^2+\sum_{i=1}^3x_i^2$

Now, if we take an orthonormal basis $\{I,e_1,e_2,e_3\}$ in the Euclidean space $\mathbb H$, and take a vector $(t,x_1,x_2,x_3):=tI+\sum_{i=1}^3x_ie_i$, then $(it,x_1,x_2,x_3):=itI+\sum_{i=1}^3x_ie_i$ is a vector in the Minkowski space $M$. In short, $(it,x_1,x_2,x_3)$ are the coordinates of a vector of $M$ with respect to an orthonormal basis of $\mathbb H$. This makes sense, since both $\mathbb H$ and $\mathbb M$ consist of $2\times 2$ complex matrices.

The above mathematization of $(it,x_2,x_2,x_3)$ heavily relies on the specific model of the Euclidean space and the Minkowski space. I wonder if is there another way to make $(it,x_2,x_2,x_3)$ mathematically meaningful?

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¹Am. Math. Mon. 69, 587-598 (1962). A crucial correction. Ibid. 69, 960 (1963).
²Bull. Amer. Math. Soc. 71 (1965), 731-736

 

[renormalize]

In relativity theory, it's a common habit to use a quadruplet $x=(x_0, x_1, x_2, x_3)$ with $x_0=ict$ (or with $c=1$, $x_0=it$ ) instead of $(t,x_1,x_2,x_3)\in \mathbb R^4$, and to use the formal Euclidean metric $\|x\|^2=\sum_{i=0}^3x_i^2$ instead of the Minkowski pseudo-metric $-t^2 + \sum_{i=1}^3x_i^2$.

This has not been a "common habit" since sometime before the 1970's. As Misner, Thorne and Wheeler say in Box 2.1 FAREWELL TO "ict" in Gravitation (1973):
"...no one has discovered a way to make an imaginary coordinate work in the general curved spacetime manifold. If "$x^4=ict$" cannot be used there, it will not be used here. In this chapter and hereafter, as throughout the literature of general relativity, a real time coordinate is used..."

 

[Paul Colby]

TL;DR Summary: What is this beyond a calculation trick to calculate the pseudo-norm as if it were an Euclidean norm?

I wonder if is there another way to make (it,x2,x2,x3) mathematically meaningful?

This depends on what one means by meaningful. I take it as leads to a better more comprehensive understanding of the subject. For either mathematics or physics, I don't see this adding much understanding especially given the general metric of general relativity. But, this is just my opinion.

 

[mma]

This has not been a "common habit" since sometime before the 1970's. As Misner, Thorne and Wheeler say in Box 2.1 FAREWELL TO "ict" in Gravitation (1973):
"...no one has discovered a way to make an imaginary coordinate work in the general curved spacetime manifold. If "$x^4=ict$" cannot be used there, it will not be used here. In this chapter and hereafter, as throughout the literature of general relativity, a real time coordinate is used..."

OK. Nevertheless, Eberlein's spin model provides a remarkably simple way to derive the Pauli and Dirac equations. And in the MTW, later (chapters §41.3., §41.4.), the same spinor algebra is used which is the heart of Eberlein's model. And after all, the whole thing is connected to $SL(2,\mathbb C)$ which is the double cover of $SO^+(3,1)$, and is a subset of $\mathbb H \oplus i\mathbb H$. So I had a feeling that $i$ is inherently present in relativity.

 

[renormalize]

And in the MTW, later (chapters §41.3., §41.4.), the same spinor algebra is used which is the heart of Eberlein's model. And after all, the whole thing is connected to $SL(2,\mathbb C)$ which is the double cover of $SO^+(3,1)$, and is a subset of $\mathbb H \oplus i\mathbb H$. So I had a feeling that $i$ is inherently present in relativity.

Indeed, using the biquaternions $\mathbb H \oplus i\mathbb H$ to implement Lorentz transformations was discussed here on PF just a month ago: https://www.physicsforums.com/threads/quaternions-and-special-relativity.1064558/#post-7107578.
That said, I think the following quote summarizes the attitude of most modern theoretical physicists:
"... quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist."
— Simon L. Altmann (1986)