  Most of the paradoxical, for the classical intuition, features of quantum
theory were formulated for situations which involve a fixed number of
particles. While one can now find a formulation of Bell's theorem for quantum
fields, a Kochen-Specker-type reasoning is usually formulated for just one
particle, or like in the case of Peres-Mermin square for two. A question
emerges. Is it possible to formulate a contextuality proof for situation in
which the numbers of particles are fundamentally undefined? We address this
problem for bosonic fields. We introduce a representation of the
$\mathfrak{su}(2)$ algebra in terms of boson number states in two modes that
allows us to assess nonclassicality of states of bosonic fields. As a figure of
merit of a nonclassical behaviour we analyze first of all contextuality, and we
show that the introduced observables are handy and efficient to reveal
violation of local realism, and to formulate entanglement indicators. We
construct a method which extends the Kochen-Specker contextuality to bosonic
quantum fields. A form of an inequality is derived using a suitable version of
the Peres-Mermin square. The entanglement indicators use a witness built with
specially defined Pauli-like observables. Finally, Bell-nonclassicality is
discussed: an inequality that involves the expectation values of pairs of the
Pauli-like operators is presented. The introduced indicators are shown to be
effective, e.g. they reveal nonclassicality in situaations involving undefined
boson numbers. This is shown via quantum optical examples of the $2\times 2$
bright squeezed vacuum state, and a recently discussed bright-GHZ state
resulting from multiple three photon emissions in a parametric process.
