We present an alternative formalism of quantum mechanics tailored to statistical ensemble in phase space. The purpose of our work is to show that it is possible to establish an alternative autonomous formalism of quantum mechanics in phase space using statistical methodology. The adopted perspective leads to obtaining within the framework of its theory the master quantum-mechanical equation without recourse to the other formulations of quantum mechanics, and gives the idea for operators pertaining to dynamical quantities. The derivation of this equation starts with the ensemble in phase space and, as a result, reproduces Liouville's theorem and virial theorem for quantum mechanics. We have explained with the help of this equation the structure of quantum mechanics in phase space and the approximation to the Schrödinger equation. Furthermore, we have shown that this formalism provides reasonable results of quantization by dealing with some simple cases such as the quantization of harmonic oscillation, the two-slit interference and the uncertainty relation, which confirm the validity of this formalism. In particular, we have demonstrated that this formalism can easily give the relativistic wave equation without treating the problem of linearizing the Hamiltonian operator by making the most of the point that the master equation is a first-order partial differential equation with respect to time, position and momentum variables, and makes use of the phase velocity. The ultimate outcome this formalism produces is that primary and general matters of quantum mechanics can be studied reasonably within the framework of statistical mechanics. phase space functions of classical mechanics by mathematical representations in configuration space.On the other hand, since the advent of quantum mechanics, some attempts have been made to modify the standard interpretations and mathematical formalism of quantum mechanics or to replace them by any other theories. From the point of view of interpretation, the causal theory of quantum mechanics should be noted. The causal theory of quantum mechanics aims to clarify the dynamical causes of quantum-mechanical movements. It furnishes the methods of analysis and interpretation for solving quantum dynamical problems, reproducing the concepts of classical mechanics even for quantum mechanics [6,7,8,9,10]. De Broglie, Madelung, Bohm and others representative of Bohmian mechanics, and Groenewold, Moyal, Takabayasi and others representative of quantum mechanics in phase space (abbreviated as QMPS) had established the foundations of the causal theory of quantum mechanics

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

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