  We propose a new Wigner inequality suitable for test of the hypothesis of
realism. We show that this inequality is not identical neither to the
well-known Wigner inequality nor to the Leggett-Garg inequality in Wigner form.
The obtained inequality is suitable for test of realism not only in quantum
mechanical systems, but also in quantum field systems.
  Also we propose a mathematically consistent derivation of the Leggett--Garg
inequality in Wigner form, which was recently presented in the literature, for
three and $n$ distinct moments of time. Contrary to these works, our rigor
derivation uses Kolmogorov axiomatics of probability theory. We pay special
attention to the construction and studies of the spaces of elementary outcomes.
Basing on the the Leggett--Garg inequality in Wigner form for $n$ distinct
moments of time we prove that any unitary evolution of a quantum system
contradicts the concept of macroscopic realism. We show that application of the
concept of macroscopic realism to any quantum system leads to ``freezing'' of
the system in the initial state.
  It is shown that for a particle with an infinite number of observables the
probability to find a pair of the observables in some defined state is zero,
even if the operators of these observables commute. This fact might serve as an
additional logical argument for the contradiction between quantum theory and
classical realism.

Quantum Physics