In this blog post, Terry Tao discusses the $n$-fold tensor product of a one-dimensional vector space $V^L$ ($L$ is just a non-numeric label, not an exponent). He claims that

With a bit of additional effort (and taking full advantage of the one-dimensionality of the vector spaces), one can also define spaces with fractional exponents; for instance, one can define $V^{L^{1/2}}$ as the space of formal signed square roots $\pm l^{1/2}$ of non-negative elements $l$ in $V^L$, with a rather complicated but explicitly definable rule for addition and scalar multiplication. ... However, when working with vector-valued quantities in two and higher dimensions, there are representation-theoretic obstructions to taking arbitrary fractional powers of units.

What is the "rather complicated but explicitly definable rule for addition and scalar multiplication"? Is it easy to see why it doesn't work in higher than one dimension? Could one extend the construction to include irrational exponents?

And what properties does the field need to satisfy in order for this construction to work? (Tao claims that the 1D vector space needs to be totally ordered. I'm not sure if this is exactly the same requirement as the field's being totally ordered. Presumably this construction doesn't work for arbitrary ordered fields, because you certainly can't define a square root function $\mathbb{Q} \to \mathbb{Q}$. Does it only work for real vector spaces?)

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    $\begingroup$ I suggest asking Terry. If he has something like generators and relations for an actual space, that could be really interesting to the Deligne representations community. $\endgroup$ – darij grinberg Oct 20 '18 at 2:45

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