**Preface**

The International School of Physics “Enrico Fermi” on the Foundations of Quantum Theory was organized by the Italian Physical Society in Villa Monastero, Varenna, Italy, during July 8–13, 2016 in collaboration with the Wilhelm und Else Heraeus-Stiftung. In the great tradition of the Fermi Schools the main goal was to provide an overview of the recent theoretical and experimental developments of an active field of research, in this case the foundations of quantum mechanics. The timing is especially appropriate considering the fact that the last “Enrico Fermi” Summer School on this topic took place in 1977.

Quantum mechanics is characterized by a dichotomy of unparalleled agreement between theory and experiment on the one hand, and an enormous variety of interpretations of the underlying mathematical formalism on the other. David Mermin(

See for example: N.D. Mermin, What's wrong with this pillow?, Physics Today, 42 (4) (1989) 9. It is interesting that this quote is often attributed to Richard P. Feynman but we recommend N. D. Mermin, Could Feynman have said this?, Physics Today, 57 (5) (2004) 10.

) proposed a very pragmatic approach to this situation: “Shut up and calculate”!However, triggered by the rapid advance of experimental techniques in quantum optics, and the development of the field of quantum technology which takes advantage of the correlations of entangled quantum systems, the question of the interpretation of quantum mechanics has recently again received a lot of attention. Moreover, the old conundrum of the physical reality of the wave function has now been tested by experiments using single photons.

These two examples are only to serve as an illustration of this active field. Indeed, the topics discussed at our school included but were not limited to the history and interpretations of quantum theory, the principle of complementarity and wave-particle duality, quantum theory from first principles, the reality of the wave function, the concept of the photon, measurement in quantum theory, the interface of quantum theory and general relativity, and quantum optical tests of quantum theory.

The present volume summarizes the lectures presented at our School which was attended by more than 80 participants including students, lecturers and seminar speakers from all over the world. All young scientists presented their research in two poster sessions which were each introduced by a “poster-flash” session. We are most grateful to Europhysics Letters for supporting a prize for the best posters. Dennis Rätzel (First Prize), Yael Avni (Second Prize), and Da-Wei Wang (Third Prize) together with the winners of the Fourth Prize Sven Abend, Lorenzo Catani and Piotr Roztocki were invited to summarize their contributions in this volume.

The Proceedings of the International School of Physics “Enrico Fermi” on the Foundations of Quantum Theory start from a historical perspective by two contributions by Nancy Thorndike Greenspan describing the life and the science of Max Born based on her book entitled The End of the Certain World.

In the spring of 1925 Werner Heisenberg had escaped from Göttingen to the island of Helgoland to cure his hay fever. It was there that he discovered quantum mechanics. Considering the radical changes in the principles it is not surprising that his seminal article in Zeitschrift für Physik is hard to read. Manfred Kleber in his contribution summarizes in a pregnant way the underlying ideas of Heisenberg's transition from Born's Atommechanik to Matrizenmechanik.

Although quantum mechanics originally started from matrix mechanics, today we almost always employ the formulation pioneered by Erwin Schrödinger based on a time-dependent wave function. The history and the derivation of the Schrödinger equation constitute the topic of the lectures by Wolfgang P. Schleich. His main theme is the linearization of the nonlinear wave equation of statistical mechanics.

Gerd Leuchs in his contribution complements the previous more mathematical approach towards the Schrödinger wave equation by a more intuitive one. He employs a combination of the analogy between matter and water waves, and dispersion relations.

In 1935 Schrödinger identified entanglement as the trademark of quantum mechanics. Indeed, it is at the very heart of Bell's inequalities and many alien features of quantum theory can be traced back to it. Edward Fry in his lectures summarizes in an impressive way the early history of this field emphasizing the important role of Grete Hermann. Already in 1935 she discovered the flaw in von Neumann's proof that it is impossible to complete quantum mechanics. Da-Wei Wang extends these two-particles considerations to three particles and proposes a new way to generate mesoscopic Greenberger-Horne-Zeilinger states. In the same spirit Piotr Roztocki employs in his contribution a frequency comb to create scalable quantum states.

The lectures by Marlan O. Scully illuminate the problem of time in the process of a quantum measurement. When and how does the observer change or reduce the state vector? Three frequently employed scenarios illustrate how “before” and “after” arguments can be misleading: The Einstein-Podolsky-Rosen situation, Wigner's friend and the quantum eraser.

John Archibald Wheeler (1911–2008) in his seminal article It from bit has vividly argued that quantum theory is information theory. In this way he can be considered the father of quantum information. He often stated that it should be possible to derive quantum mechanics from information theory. Christopher A. Fuchs has followed this path and has proposed the new interpretation of quantum mechanics, Quantum Bayesianism (QBism). In his lectures he first summarizes the most prominent interpretations of quantum mechanics and then provides an introduction into QBism. In the same spirit Giacomo Mauro D'Ariano derives from elementary principles of information theory free quantum field theories.

Across from Varenna at the West end of Lake Como is the place where Niels Bohr in 1927 introduced the principle of complementarity stimulated by Heisenberg's uncertainty relation. Sabine Wölk in her lecture notes employs simple measurements on quantum systems to compare complementarity and entanglement both of which have their roots in non-commuting operators.

The field of experimental quantum optics has opened new avenues towards tests of the foundations of quantum mechanics. Here the process of spontaneous parametric down conversion (SPDC) plays a central role and has created an avalanche of applications. Ralf Menzel in his lectures emphasizes the role of the mode function of the electromagnetic field as the carrier of the photon, and reviews his experiments on stimulated coherence and complementarity.

Quantum imaging is another product of SPDC. In his lecture Robert W. Boyd summarizes this active field by giving three examples: ghost imaging, imaging based on interaction-free measurements and imaging based on Mandel's induced coherence.

The reality of the wave function is an often debated question. So far it has been part of more philosophical discussions. However, SPDC has moved this realm from Gedanken experiments to real ones. The lectures by Andrew White addressed these issues. Unfortunately, due to time constraints he was not able to provide us with a paper. Likewise Aephraim Steinberg could not contribute. Fortunately, the notes by Lorenzo Catani et al. address some aspects of these questions. In particular, they discuss contextuality as a resource in quantum computation.

We recall that quantum mechanics originated from the analysis of blackbody radiation. In contrast to conventional wisdom Max Planck did not quantize the light in the resonator but the mechanical oscillators in the walls. Quantized electrodynamics had to wait till the Drei-Männer-Arbeit of Born, Heisenberg and Pascal Jordan. They rederived the result of Albert Einstein concerning the fluctuations of the radiation field in the thermal state from a field theoretical approach. The Casimir effect, that is the attraction of two uncharged conducting plates, is another consequence of these fluctuations. Yael Avni in her contribution summarizes her recent work with Ulf Leonhardt on Casimir forces in spherically symmetric dielectric media.

The theories of special and general relativity together with quantum mechanics are rightfully considered the major revolutions in physics of the 20th century. Despite the fact that by now they are almost 100 years old, general relativity and quantum mechanics have not been unified yet. The lectures of Daniel M. Greenberger provide insight into the reasons for the resistance and identify the strange roles of proper time and mass. He also discusses consequences originating from considering them as dynamical variables.

The field of atom optics is perfectly suited to probe this interface of quantum mechanics and general relativity. On the one hand we use the wave nature of atoms for interferometry, on the other hand due to their mass the atoms feel gravity. Ernst M. Rasel in his lectures provides an introduction into atom interferometry, discusses a quantum test of the equivalence principle and gives an outlook to experiments in space. The contribution of Sven Abend et al. expands on this theme and discusses a new avenue based on an atom-chip gravimeter.

Since light represents energy it must also gravitate. Already in 1931 Richard Tolman together with Paul Ehrenfest and Boris Podolsky showed that a pencil of light leads to a curvature of spacetime. The contribution by Dennis Rätzel et al. summarizes gravitational properties of light.

Not included in this volume are two other highlights of our school. Nancy Thorndike Greenspan had discovered a movie taken by the Nobel Prize winner Irving Langmuir at the Solvay Meeting of 1927. It was impressive to see the famous quantum physicists of the time “in action” rather than sitting around a table. Moreover, our school ended with the playing of the Mozart piano concerto A-major KV 488 recorded around 1965 by the Bavarian Radio Symphony Orchestra under the conductor Rudolf Albert. The soloist was Werner Heisenberg. We owed this pleasure to Manfred Kleber who had found this treasure. Many thanks, Manfred, for sharing it with us!

All activities were inspired by the breathtaking beauty of Lake Como, the Villa Monastero and its gardens, and by the rich heritage by the Enrico Fermi International School of Physics. The success of the school measured by the exceptionally large number of interactions between the participants and the extremely lively discussions, during and immediately after the talks, in the park and on several excursions, is also due to the excellent organizational and administrative support provided by the staff of the Italian Physical Society. We are also most grateful to the Wilhelm und Else Heraeus-Stiftung for its generous monetary support.

E.M. Rasel, W.P. Schleich and S. Wölk