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If we take a spacetime with one spatial dimension, we can write a vector as $A^\mu=(t, x)$. This is a contravariant vector, and we can calculate the covariant vector by multiplying it with the Minkowski metric: $$ A_\mu= \begin{bmatrix} 1& 0\\0&-1 \end{bmatrix} A^\mu =\begin{bmatrix} t\\-x \end{bmatrix} $$

(I'm not sure if this is notationally valid, since I might be mixing up tensor notation and matrix notation, but you get the idea. )

We can then calculate the proper distance using the einstein summation convention: $A^\mu A_\mu$.

We can see that coindidentally, the conjugate of a complex number transforms in the same way. If we have a complex number written as a $2\times 1$ matrix $\alpha = \begin{bmatrix} a\\bi \end{bmatrix}$

Then the complex conjugate of $\alpha$ is found by

$$ \alpha^*= \begin{bmatrix} 1& 0\\0&-1 \end{bmatrix} \alpha $$

So we see that the relation between contravariant and covariant vectors in spacetime is similar to the relation between complex conjugates.

Is there a deeper connection between the two, or is this just a mathematical coincidence with no significance?

Qmechanic
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user56834
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1 Answers1

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I think the thing you're looking for here is the Wick rotation. You start with a real vector space with coordinates $(t,x) \in \mathbb{R}^2$ and replace the real $t$-coordinate with a purely complex coordinate $t=-i\tau$ to switch between Euclidean and Minkowski space.

Is there a deeper connection between the two, or is this just a mathematical coincidence with no significance?

My perception is that this is mostly seen as a mathematical trick. It has a lot of significance however, since this method is used to solve path integrals and derive propagators for quantum particles.

There are some problems however. First, you have to make sure that the analytic continuation of your problem is really valid (e.g. your problem is expandable into the complex plane). Second, this works well for vectors, it is however very complicated to do the same thing for spinors (and maybe other quantities of interest), as seen for example in this discussion.

merzt
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