You would usually want the principal Arakelov divisors, i.e. those of the form $(\sum_{\mathfrak{p}}{\rm ord}_{\mathfrak{p}}(a), \sum_\sigma -\log|\sigma(a)|)$ for $a\in F^\times$, to be cocompact in the group of degree $0$ divisors - the volume of the quotient should be the familiar product ${\rm Reg}(F)\cdot{\rm h}(F)$, where $h$ denotes the class number. For that, you want $\sigma$ to only run over places, not over all embeddings. Incidentally, you also want to normalise your degree function carefully: the degree of a real place should be $1$, while the degree of a complex place should be $2$.

You would usually want the principal Arakelov divisors, i.e. those of the form $(\sum_{\mathfrak{p}}{\rm ord}_{\mathfrak{p}}(a), \sum_\sigma -\log|\sigma(a)|)$ for $a\in F^\times$, to be cocompact in the group of degree $0$ divisors — the volume of the quotient should be the familiar product ${\rm Reg}(F)\cdot{\rm h}(F)$, where ${\rm h}$ denotes the class number. For that, you want $\sigma$ to only run over places, not over all embeddings. Incidentally, you also want to normalise your degree function carefully: the degree of a real place should be $1$, while the degree of a complex place should be $2$.