## Copernicus Center for Interdisciplinary Studies

Copernicus Center is an autonomous faculty of Jagiellonian University. It carries out advanced interdisciplinary studies in various domains of sciences and humanities, including mathematics, physics, philosophy and theology.

The secondary role of the Copernicus Center consists in the popularisation of science and education. It regularly organises events for general audience, most notably the Copernicus Festival, and publishes books with the help of the Copernicus Center Press.

## Programme

## Abstracts

#### Henryk Arodź (Jagiellonian University)

### Ehrenfest's theorem revisited

The beginning of the talk is devoted to the question what is the essence of the distinction between classical and quantum theories. Next, I point out certain important feature of quantum mechanical wave packets, which alleviates the need for the often postulated sharp divide into the classical and the quantum. In the main part of the talk several versions of Ehrenfest's theorem are outlined, including a relativistic one.

#### Klaus Fredenhagen (University of Hamburg)

### The operator product of quantum observables

The operator product in quantum theory can be decomposed into the Jordan product and a Lie product in terms of the commutator. Both products have a physical interpretation. Given such products, they can be combined to an associative product if and only if their associators are proportional. It is shown that this relation can be derived from the assumption that independent physical systems can be considered as parts of a larger system.

#### Ryszard Horodecki (University of Gdańsk)

### Quantum Darwinism versus spectrum broadcast structure

Together with birth of quantum mechanics a touchy question arose: How does Nature makes a food-bridge from quantum substratum to the objective everyday world. The fact that this quantum-to-classical transition as yet not fully understood may mean that we have not grasped all the possible implications of the quantum formalism. I will consider the above problem in this context of two different approaches - quantum Darwinism and the so called spectrum broadcast structure which describe the emergence of classical objectivity from the quantum subjective world. Unfortunately the two approaches sometimes can lead to different predictions. However, an interesting attempt to relate the two frameworks in the common perspective has recently been made.

#### Stanisław Krajewski (University of Warsaw)

### What makes the universal and existential quantifiers classical?

What makes the universal and existential quantifiers classical? These standard quantifiers are assumed to be the absolutely basic ones. In this sense, and perhaps other ones too, they can be called classical. However, many other quantifiers have been considered in contemporary logic. Some, like ‘many’ and numerus other counterparts of natural language quantifier expressions, are context-sensitive. The context-independent ones can usually be defined mathematically, possibly with reference to set theory or second order logic.

Thesis: The universal quantifier is the simplest one that suffices to define all context- independent quantifiers in the framework of logic.

Logic is understood here as classical logic – both first order and higher order. The higher
order logics assume an absolute concept of set, that is, the concept which forms the basis
of so-called classical mathematics.

So classical quantifiers are the ones that distinguish context-independent ones among all
quantifiers by making it possible to define them in the framework of classical logic or
classical mathematics.

#### Jerzy Król (University of Silesia)

### Himalayan topology for parameters of physics

There are various free parameters in the (extension of) Standard Model of particle physics
(e.g. masses of electron and neutrinos) and in the Standard Model of cosmology with
inflation (e.g. the cosmological constant, rate and timing of the inflation, tensor - scalar
ratio, the α parameter in the Starobinsky R

We observe (with surprise) that spacetime manifold when embedded in some compact smooth 4-manifold K serves as a method for generating many of these undetermined so far parameters. The point is to take K3 surface as K and its suitable smooth structure. Then we obtain two topology changes of the 3-dimensional space during the cosmic evolution leading to the existence of two different energy scales. The scales are calculated based on the topology of the underlying manifolds like Chern–Simons invariants of certain homology 3-spheres. As the result we determine (topologically) cosmological parameters: the cosmological constant, the number of e-folds and duration of inflation or the neutrino masses. Whole approach is a part of the project "smooth quantum gravity" where quantum realm is rather by-product of choosing certain nonstandard smoothness structure of spacetime.

#### Marek Kuś (Center for Theoretical Physics PAS and International Center for Formal Ontology, Warsaw University of Technology)

### Quantum entanglement and emergence of the classical Leibnitz’s principle of the identity of indiscernibles

Treating classical mechanics as an emergent theory (with quantum mechanics as the basic one), we may enumerate several elements that could be treated as “emergent”. In quantum mechanics, thus, some “elements of reality” do not exist. In general these are properties that can be revealed only in measurements of physical quantities such as position, momentum, angular momentum, etc.. An assumption that particular values of such quantities “are carried” by physical objects during their evolution, or that they “exist in a system” prior to a measurement leads to a contradiction. On the other hand, they are well defined in classical worlds and one can safely assume that they exist in a system independently of any measurements. In a similar vein, on the quantum level completely indiscernible particles (seem to) violate the Leibniz’s Principle of the Identity of Indiscernibles, that emerges on the classical level. By introducing an operational measure of the effective indistinguishability exhibited in a measurement process, we will show how the “emergent phenomenon” of Leibniz’s Principle and its dependence on purely quantum phenomenon of entanglement can be quantified.

#### Józef Spałek (Jagiellonian University)

### Macroscopic wave function: Decoherence versus spontaneous symmetry breakdown

In my talk I address the question to what extent the so-called macroscopic wave function can be regarded as a quantum entity and in what sense it reflects classical properties of a condensed system. This type of wave function appears when describing either superfluid or superconducting condensates and is defined as statistical average of a quantum field. Additionally, such a wave function plays role of the order parameter. After describing the fundamental concepts I ask why such a wave function normally does not decohere for the macro condensed system, but may show such a feature for small (nanoscopic) system size, an important factor when storing densely quantum-state information. If time allows, I shall provide two remarks about Einstein’s statistical interpretation of the wave function and related problems.

The work was supported by the grant MAESTRO No. DEC-2012/04/A/ST3/00342 from the National Science Centre of Poland.

#### Karol Życzkowski (Jagiellonian University)

### On the classical limit of quantum theory: Why we do not see Schrödinger cats everyday?

The world is inherently quantum: non-intuitive effects of quantum interferrence do exist. However, they are not easy to observe in the macro scale: Due to inevitable interaction with an environment a quantum state suffers decoherence and becomes classical, so quantum effects are suppressed. On the other hand, in specially favorable conditions (e.g. small system size, low temperature, systems well isolated from the environment) quantum effects can become decisive and they allow one to design genuine quantum technologies.