Preface at first sight (one needs a quantum description of a huge number of degrees of freedom) we show that it can be reduced to a much simpler form,characterizing the environment in terms of its damping coefficient or dynamical susceptibility.After explaining this proce- dure for the damped oscillator in Chapter 11 and discussing dissipation and fluctuations. in Chapter 12 we extend the picture to a qubit(two-level system)in a dissipative envi- ronment t.We elucidate the roe the ent plays in transitions b ween the two qubit states,and,based on what we find,we provide a very general scheme to classify all possible types of environment. In the last part(and chapter)of the book,we give a short introduction to relativistic quantum mechanics.We explain how relativity is a fundamental symmetry of our world. ally leads us to the Dirac equation.Apart from obeying the relativistic symmetry.the Dirac equation predicted revolutionary new concepts,such as the existence of particles and anti-particles.Since the existence of anti- particles has been experimentally confirmed,just a few years after Dirac had put forward include them into our second quantization framework then explain ho particesanti-par icles and the elec magnetic field constitutes the basis of quantum electrodynamics.We briefly touch on this topic and show how a naive application of perturbation theory in the interaction between radiation and matter leads to divergences of almost all corrections one tries to calculate. The way to handle these divergences is given by the theory of renormalization,of which we discuss the basic idea in the last section of the chapte The book thus takes examples and applications from many different fields:we discuss the laser,the Cooper pair box,magnetism,positrons,vortices in superfluids,and many more examples.In this way,the book gives a very broad view on advanced quantum the- ory.It would be very well suited to serve as the principal required text for a master-level course on advanced quantum mechanics which is not exclusively directed toward elemen- tary particle physics.All m terial in the book could be covered in on depending on the amount of time available per week.The five parts of the book are also relatively self-contained,and could be used separately. All chapters contain many"control questions,"which are meant to slow the pace of the student and make sure that he or she is actively following the thread of the text.These ques- tions ofor instance be s during the At th there are four to ten larger exercis s,0e s,others present ing more interesting physical problems.We decided to provide in this book the solutions to one or two exercises per chapter,enabling students to independently try to solve a serious problem and check what they may have done wrong.The rest of the solutions are available online for teachers,and the corresponding exercises could be used as homework for the stud We hope that many students around the world will enjoy this book.We did our absolute best to make sure that no single typo or missing minus sign made it to the printed version, but this is probably an unrealistic endeavor:we apologize beforehand for surviving errors. If you find one,please be so kind to notify us,this would highly improve the quality of a possible next edition of this book. xiii Preface at first sight (one needs a quantum description of a huge number of degrees of freedom), we show that it can be reduced to a much simpler form, characterizing the environment in terms of its damping coefficient or dynamical susceptibility. After explaining this proce￾dure for the damped oscillator in Chapter 11 and discussing dissipation and fluctuations, in Chapter 12 we extend the picture to a qubit (two-level system) in a dissipative envi￾ronment. We elucidate the role the environment plays in transitions between the two qubit states, and, based on what we find, we provide a very general scheme to classify all possible types of environment. In the last part (and chapter) of the book, we give a short introduction to relativistic quantum mechanics. We explain how relativity is a fundamental symmetry of our world, and recognize how this leads to the need for a revised “relativistic Schrödinger equation.” We follow the search for this equation, which finally leads us to the Dirac equation. Apart from obeying the relativistic symmetry, the Dirac equation predicted revolutionary new concepts, such as the existence of particles and anti-particles. Since the existence of anti￾particles has been experimentally confirmed, just a few years after Dirac had put forward his theory, we accept their existence and try to include them into our second quantization framework. We then explain how a description of particles, anti-particles, and the electro￾magnetic field constitutes the basis of quantum electrodynamics. We briefly touch on this topic and show how a naive application of perturbation theory in the interaction between radiation and matter leads to divergences of almost all corrections one tries to calculate. The way to handle these divergences is given by the theory of renormalization, of which we discuss the basic idea in the last section of the chapter. The book thus takes examples and applications from many different fields: we discuss the laser, the Cooper pair box, magnetism, positrons, vortices in superfluids, and many more examples. In this way, the book gives a very broad view on advanced quantum the￾ory. It would be very well suited to serve as the principal required text for a master-level course on advanced quantum mechanics which is not exclusively directed toward elemen￾tary particle physics. All material in the book could be covered in one or two semesters, depending on the amount of time available per week. The five parts of the book are also relatively self-contained, and could be used separately. All chapters contain many “control questions,” which are meant to slow the pace of the student and make sure that he or she is actively following the thread of the text. These ques￾tions could for instance be discussed in class during the lectures. At the end of each chapter there are four to ten larger exercises, some meant to practice technicalities, others present￾ing more interesting physical problems. We decided to provide in this book the solutions to one or two exercises per chapter, enabling students to independently try to solve a serious problem and check what they may have done wrong. The rest of the solutions are available online for teachers, and the corresponding exercises could be used as homework for the students. We hope that many students around the world will enjoy this book. We did our absolute best to make sure that no single typo or missing minus sign made it to the printed version, but this is probably an unrealistic endeavor: we apologize beforehand for surviving errors. If you find one, please be so kind to notify us, this would highly improve the quality of a possible next edition of this book