Providing a self contained step by step explanation, this book will guide the reader with a basic knowledge of quantum mechanics, to a sufficiently comprehensive level as well as to the frontier of contemporary physics. For the last two decades there has been a ceaseless growth of the area where the path integral (PI) method plays an important role: the main reasons are its intuitive aspect and ease of handling. However, this has raised questions elsewhere and in this book fundamental issues are resolved by starting from the canonical operator formalism to lead the reader to a more comprehensive level. Containing the most recent topics such as the lattice fermion problem in quantum field theory as well as the quantum Monte Carlo method in statistical mechanics this book will suit graduate students of quantum physics.

In examining the problem of quantization it is customary to proceed by following the order of historical developments, i. e. , to start with the so-called first quantization and then go on to second, or field, quantization. As is well known, the former consists in setting up commutation relations among the canonical variables (of finite degrees of freedom) which are defined originally in classical mechanics, and the latter consists in doing the same among field variables. In our opinion, however, the above order does not necessarily reflect that of physical importance. In fact, a close look at the theoretical that the latter plays a more fundamental role structure immediately reveals than the former in various respects. First, the former can be derived from the latter, and second, the assumptions that have to be made in the former, e. g. the spin-statistics connection, can be justified in the latter on the basis of its own theory. Furthermore, as the history of the positron theory shows, if we remain within the framework of the first-quantized formalism, it will no longer be possible to have a closed theory in the relativistic region where the problem becomes essentially that of an infinite number of particles.