This advanced lecture series is part of the Thematic Einstein Semester “Mathematics for Quantum Technologies” (Summer 2024) funded by the Cluster of Excellence MATH+ and the Einstein Foundation Berlin.

**Ulf Leonhardt** is a professor of physics at the Weizmann Institute of Science, Israel. He is known for his contributions to the theory of metamaterials (invisibility cloaking), quantum electrodynamics in media, optical analogues of black holes, quantum levitation, quantum state tomography, and quantum effects of optical phenomena involving Hawking radiation.

Professor Ulf Leonhardt received the* Otto Hahn Medal* from the Max Planck Society and the *Wolfson Research Merit Award* from the Royal Society. In 2006 he received the *Scientific American 50 Award* forhis work on invisibility phenomena in optics. Ulf Leonhardt has been a member of the Royal Society of Edinburgh since March 2009.

The forces of the quantum vacuum appear like an arcane subject, but these forces are all around us, and they do play an increasing role in technology. They are responsible for the capillary forces, without which live would be impossible, and for the van der Waals forces that attract small, electrically neutral particles to each other. These forces arise, because the vacuum – the ground state of quantum fields – is not empty, but full of possibilities. In physics, vacuum fluctuations play the role credit plays in economy: they make things possible by borrowing. For example, an unpolarized molecule may borrow an electromagnetic polarization from the vacuum. Being polarized, it gets attracted to other molecules. After “returning its debt” to the vacuum the molecule is unpolarized again, but the attraction remains.

The course gives a brief, self-consistent introduction into the theory behind the forces of the quantum vacuum. It does not require experience in quantum field theory, but mathematical skills in using complex analysis and conformal mapping for elegant calculations. The goal is to acquire intuition and mathematical procedures for learning how to manipulate vacuum forces in technology.

The lecture series comprises four lectures.

- July 15, 2024 (Monday, 10–12 a.m.)
*Topic: Setting the scene**Quantization of the electromagnetic field in media and in space-time**Normal modes and harmomic oscillators**Zero-point energy**Casimir’s cavity: 1D case, zeta-function renormalization**Casimir’s cavity: 3D case, Casimir’s renormalization* - July 16, 2024 (Tuesday, 10–12 a.m.)
*Topic: Lifshitz theory**The need for a theory in dispersive and dissipative media**Point-splitting method**Fluctuation-dissipation theorem**Kubo-Martin-Schwinger relation**Lifshitz formula in planar, piece-wise homogenous media* - July 18, 2024 (Thursday, 10–12 a.m.)
*Topic: Applications**Imaginary frequencies for capturing the broad band of quantum forces**Thermal Casimir effect**Perfectly conducting cavity**Magnetic mirror and Casimir repulsion**Permittivity hierarchy and repulsive Casimir forces**Negative refraction and quantum levitation**Engineering in the imaginary part of the spectrum* - July 19, 2024 (Friday, 10–12 a.m.)

*Topic: Outlook*

*The quantum vacuum seen by accelerated observers (relativistic acceleration, Rindler coordinates, Unruh effect)*

*Entanglement in the quantum vacuum*

*Classical analogue of the Unruh effect*

*The case for a Casimir cosmology*

*Forces of the Quantum Vacuum: An Introduction to Casimir Physics* by William M. R. Simpson and Ulf Leonhardt.

The lectures will be held in the **Erhardt Schmidt** **Lecture Room** of the **Weierstrass Institute for Applied Analysis and Stochastics (WIAS)**.

**Address:**

Mohrenstraße 39

D-10117 Berlin

**Public Transport:
**U2 Hausvogteiplatz

U5 Museumsinsel

U6 Stadtmitte

S5/S7/S75/S9 Friedrichstraße/ Hackescher Markt

Secretariat: Veronica Bove

Mohrenstraße 39

D–10117 Berlin

Phone: +49 30 20372-539

E-Mail: veronica.bove@wias-berlin.de