The mixing of a neutral unstable particle and its antiparticle has been usually discussed in the framework of quantum mechanics in the literature. In this paper, the particle-antiparticle mixing is discussed fully in the quantum field theory, and it is shown that quantum mechanics is not the proper non-relativistic limit of the quantum field theory in the presence of such mixing. Moreover, it is also shown that a discrepancy indeed exists between the results from quantum mechanics and the quantum field theory, which can sometimes make big differences in theoretical predictions of observables such as decay widths. This result has an implication for the mixing of neutral mesons and the CP violation in their decays.
The mixing of a neutral unstable particle and its antiparticle has been usually discussed in the framework of quantum mechanics in the literature. In this paper, the particle-antiparticle mixing is discussed fully in the quantum field theory, and it is shown that quantum mechanics is not the proper non-relativistic limit of the quantum field theory in the presence of such mixing. Moreover, it is also shown that a discrepancy indeed exists between the results from quantum mechanics and the quantum field theory, which can sometimes make big differences in theoretical predictions of observables such as decay widths. This result has an implication for the mixing of neutral mesons and the CP violation in their decays.
Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles , reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot , in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.
After reviewing the description of an unstable state in the framework of Lee Hamiltonians (valid both for Quantum Mechanics (QM) and Quantum Field Theory (QFT)), we consider some theoretical aspects of non-exponential decays: the case of two decay channels, the broadening of the energy spectrum at short times, the effect of an imperfect measurement, the link to QFT, and the decay of an unstable moving particle with definite momentum. All the presented effects were not confirmed in experiments, hence are at the present stage predictions.
We discuss the quantum mechanics of B-factory experiments, and that of neutral K or B decay. Predictions for the processes to be studied at the B factories can be made through an approach based on amplitudes, rather than on wave functions. This approach avoids the puzzles of the "collapse of the wave function." In the treatment of the decay of a neutral K or B, the contributions of the different mass eigenstate components of the decaying particle must be evaluated at exactly the same spacetime point. Otherwise, the frequency predicted for the oscillation of the decay rate may be incorrect.
As in non-relativistic quantum theory, the simplest problem to solve in relativistic quantum theory is that of describing a free particle. Much can be learned from this case which will be of use in interpreting the topics covered in later chapters. Furthermore, some of the most profound features of relativistic quantum theory are well illustrated by the free particle, so it is a very instructive problem to consider in detail. Another advantage of the free-particle problem is that the mathematics involved in solving it is not nearly as involved as that necessary for solving problems involving particles under the influence of potentials. Firstly we shall look briefly at the formulae for the current and probability density, then we shall go on to examine the solutions of the Dirac equation and investigate their behaviour. This leads us to a discussion of spin, the Pauli limit, and the relativistic spin operator. Next we consider the negative energy solutions and show how relativistic quantum theory predicts the existence of antiparticles. Some of the dilemmas this concept introduces and their resolution are discussed. At the end of the chapter we go back to the Klein paradox, and examine it for an incident spin-1/2 particle. We find that the Klein paradox exists for Dirac particles in exactly the same way as it existed for Klein—Gordon particles and has the same resolution and interpretation.
In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrödinger field theory, obtaining the Schrödinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrödinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.