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In this paper, we study the argument of weak values of general observables,succeeding to give a geometric description to this argument on the Blochsphere. We apply the Majorana symmetric representation to reach this goal. Theweak value of a general observable is proportional to the weak value of aneffective projector: it arises from the normalized application of theobservable over the initial state, with a constant of proportionality that isreal. The argument of the weak value of a projector on a pure state of an$N$-level system corresponds to a symplectic area in the complex projectivespace $(\text{CP}^{N-1})$, which can be represented geometrically with a sum of$N-1$ solid angles on the Bloch sphere using the Majorana stellarrepresentation. Here, we show that the argument of the weak value of a generalobservable can be described, using Majorana representation, as the sum of $N-1$solid angles on the Bloch sphere, merging both studies. These two approachesprovide two geometrical descriptions, a first one in $\text{CP}^{N-1}$ and asecond one on the Bloch sphere, after mapping the problem from the originalspace $(\text{CP}^{N-1})$ by making use of the Majorana representation. Theseresults can also be applied to the argument of the third-order Bargmanninvariant, the most fundamental order as the argument of any higher orderinvariant can be expressed as a sum of the argument of third-order Bargmanninvariants. Finally, we focus on the argument of the weak value of a generalspin-1 operator when its modulus diverges towards infinity. This divergenceamplifies signals with great usefulness in experiments.

Geometrical interpretation of the argument of Bargmann invariants and  weak values in $N$-level quantum systems applying the Majorana symmetric  representation