  The Bell-Clauser-Horne-Shimony-Holt inequality can be used to show that no
local hidden-variable theory can reproduce the correlations predicted by
quantum mechanics (QM). It can be proved that certain QM correlations lead to a
violation of the classical bound established by the inequality, while all
correlations, QM and classical, respect a QM bound (the Tsirelson bound). Here,
we show that these well-known results depend crucially on the assumption that
the values of physical magnitudes are scalars. The result implies, first, that
the origin of the Tsirelson bound is geometrical, not physical; and, second,
that a local hidden-variable theory does not contradict QM if the values of
physical magnitudes are vectors.
