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Titles and abstracts of lectures
Monday 6th February:
M. Da Luz Foundational issues in natural and social sciences (keynote lecture)
“Complexity and path integrals made easy: The quantum dynamics on graph structures”
Green's function are a powerful tool for solving distinct (classical or not) physical problems. In the
quantum realm they can be constructed through the path integral formalism.
Among paradigmatic features, usual in systems belonging to the so called area of complex phenomena,
very common ones are related to graph-like structures. In this presentation we review many aspects and
distinct phenomena associated to quantum dynamics in networks. For so, we consider an energy
domain Green's function (G) approach. This method is particularly interesting (and surprisingly simple)
because G can be written as a sum over classical-like trajectories, where local quantum effects are taken
into account through the scattering matrix elements (basically transmission and reflection amplitudes)
defined on each one of the graph vertices. The exact G has the functional form of a generalized
semiclassical formula, which through different calculation techniques can be cast into a closed analytic
expression. Concrete examples are given and different applications considering the Green's functions
framework for quantum graphs are outlined.
A.Abanov Complex dynamical systems and Econophysics (keynote lecture)
“Path integrals in condensed matter physics: from partition functions to interference”
The use of path integration in condensed matter physics is ubiquitous and can hardly be reviewed in a
single talk. Therefore, in this talk I will focus on very particular applications of path integrals where the
quantum nature of condensed matter systems is essential. We will start with few examples of path
integrals illustrating how quantum interference may manifest itself in condensed matter physics. Then
we introduce the concept of effective action, discuss topological terms in effective actions and show
how one can use symmetries to derive relations between transverse transport coefficients in quantum
Hall systems.
Z. Kakushadze Complex dynamical systems and Econophysics (keynote lecture)
“Path Integral and Asset Pricing”
We discuss how Euclidean path integral arises in pricing of financial assets such as options, bonds,
etc. In some cases path integral provides an advantageous angle. As an example we discuss an
application to bond pricing in short-rate models. In option pricing, empirical data suggest a deviation
from Gaussian distributions (fat tails). A relativistic extension of Brownian motion provides such a
distribution and suggests that stochastic description of such models (with jumps) is inadequate and a
quantum field theory description is in order, further expanding path integral applications in asset
pricing.
F. Dowker Quantum gravity (keynote lecture)
“The Path Integral Approach to Quantum Gravity”
For ordinary quantum mechanics it is often stated that canonical quantisation and the path integral are
equivalent formulations of the same physics. I will argue that for Quantum Gravity choosing one over
the other makes a difference in the direction pursued in quantum gravity research. I will describe the
“Problem(s) of Time” that arise in the attempt to find a theory or quantum gravity and argue that these
indicate that the path integral is the more promising choice at this “fork in the road”. I will point out that
this fork was identified by Dirac who in 1932 stated that the lagrangian approach to classical mechanics
is probably more fundamental than the hamiltonian approach because it is relativistically invariant and
showed it leads to the path integral. Finally, I will argue that even for non relativistic quantum
mechanics, the choice is physically meaningful in that it leads to different pictures of the physical world.
L. Schulman Topology and the path integral
Using a path integral as well as a conventional approach to dealing with identical particles, I explore
ways to define operators that cannot be defined in the usual formalism, for example the relative
momentum of a pair of electrons. The operator found for this quantity cannot be made self-adjoint,
although the quantity is measurable.
Tuesday 7th February:
D. Garlaschelli Complex dynamical systems and Econophysics
“Network reconstruction for systemic risk estimation”
Recent financial crises have highlighted the need to correctly estimate the so-called "systemic risk", i.e.
the risk of collapse of an entire financial system, as opposed to the risk of failure of individual banks or
firms. Systemic risk is a collective effect arising from a possible "financial contagion" process spreading
over all possible paths in a financial network of, say, banks connected by credit linkages. Estimating
systemic risk therefore requires the knowledge of the entire financial network, which is however
empirically unaccessible due to confidentiality. It is therefore necessary to devise methods that can
reliably reconstruct the structure of real networks from partial, publicly available information. One
successful class of such methods is based on the maximum entropy principle and generates a canonical
ensembles of networks, each with a different probability, consistent with the partial information
available. Systemic risk estimation then becomes a combined problem of integration along all possible
paths of a network, while averaging over all networks in the probabilistic ensemble.
I will describe possible steps towards achieving this goal, with particular emphasis on the construction of
the maximum-entropy ensemble of networks.
J. Klauder Foundational issues in natural and social sciences
“Enhanced Quantization: The Right Way to Quantize Everything”
Canonical quantization solves many problems but fails on others, e.g., covariant scalar fields in 5 and
higher spacetime dimensions. Enhanced quantization gives the same answers for problems that
canonical quantization gets right and offers acceptable results for those for which it fails. A basic
example illustrates the superior properties of enhanced quantization even for traditional functional
integration.
C. Schubert Quantum gravity
“The worldline path integral approach to quantum field Theory”
Although Feynman shortly after his seminal work on the quantum mechanical path integral in 1948 also
showed how to represent the S-matrix in quantum electrodynamics in terms of first-quantized path
integrals, for several decades path integrals were used in quantum field theory primarily at the field
level; particle path integrals were not consider an efficient tool for the calculation of either scattering
amplitudes or effective actions. It is only during the nineties that, triggered by dervelopments in string
theory, such “worldline path integrals” were found to have certain computational and conceptual
advantages over the standard formalism based on Feynman diagrams. After a historical overview, I will
present the three major methods that are presently used for the evaluation of worldline path integrals:
1. The “string-inspired method” using worldine Green's functions. Applications include QED in external
fields, off-shell gluon amplitudes and certain amplitudes in Einstein-Maxwell theory.
2. The semiclassical “worldline instanton” approximation, used for Schwinger pair creation in electric
and gravitational fields.
3. The numerical “worldline Monte Carlo” approach, which has been found particularly suitable for the
calculation of Casimir energies.
M. Grifoni Hard and soft condensed matter theory and biophysics
“A path-integral approach to the strongly driven spin-boson system”
The spin-boson system is an archetypal model to study the impact of a thermal reservoir on the
coherent dynamics of a two-level quantum particle (qubit ). When the coupling between qubit and
environment crosses a threshold, a transition from coherent to incoherent tunneling between the two
qubit eigenstates occurs. At even larger coupling, the dynamics is fully quenched, signaling a strong
entanglement of the qubit with the reservoir’s continuum. When the qubit is additionally driven by an
intense cw-field, it can further hybridize with the radiation field.
In my talk a real time path-integral approach capable to capture both large qubit-reservoir coupling and
strong driving will be presented. Furthermore, I will discuss how such complex physics can be studied in
a strongly driven superconducting qubit coupled to the electromagnetic continuum of a onedimensional waveguide.
We shall show that measurements of the transmitted signal give direct access to the renormalization
effects from the reservoir. When a strong pump tone is added to the probe signal, the appearance of
photon-assisted resonances in the transmission reveals a further entanglement between the damped
qubit and the radiation field.
G.Junker Foundational issues in natural and social sciences
“On the power-law duality in Feynman's path integral”
We review the method of space and time transformations in classical mechanics, quantum mechanics
and path integrals mapping the dynamics of system A onto that of system B. This is utilized to formulate
a duality between power-law potentials in D dimension and its generalization to multi-term power laws.
As an explicit application we discuss the case of a two-term potential of quark confinement.
Wednesday 8th February:
K. Fujikawa Foundational issues in natural and social sciences
“Path integrals in particle and condensed matter physics: anomalies and related topics”
In this talk I would like to discuss 3 topic lines of my recent research:
1. Possible generalization of quantization to theories non-local in time,
which may have some connection with operator ordering problem. I discuss
this issue in connection with my recent attempt to quantize a Lorentz invariant
CPT violating theory which is inevitably non-local.
2. As an illustration of the path integral treatment of anomalies, I would like to discuss
my recent analysis of the bosonization in two-dimensional theory.
3. In view of the recent interest in the chiral anomaly and Berry's phase in
condensed matter physics, I would like to illustrate the basic differences
between these two notions mainly using path integral.
Q. Si Hard and soft condensed matter theory and biophysics
“Effects of Berry phase and topological defects in antiferromagnetic quantum phase transitions”
Quantum condensed matter physics is especially rich in systems with strong correlations. A unifying
principle that may underlie a variety of strongly correlated systems is quantum criticality, which often
develops near metallic antiferromagnetic orders. This has raised the interest in understanding the
nature of the quantum crtiical point and, relatedly, the types of ground states that develop upon the
suppression of the antiferromagnetic order. Some of the important questions are how the phases go
beyond Landau’s description, which is rooted in spontaneous symmetry breaking and order parameters.
In this talk, I will discuss theoretical studies on these issues in a prototype setting of Kondo lattice
models, which describe a lattice of quantum spins coupled to a band of conduction electrons. Within an
approach based on a quantum non-linear sigma model representation of the spin system, I will describe
the effects of Berry phase and topological defects in the antiferromagnetic quantum phase transitions.
The implications for experiments in heavy fermion metals will be briefly discussed.
V. Zatloukal Complex dynamical systems and Econophysics
“Local time path integrals and their application to Lévy random walks”
Local time measures the amount of time that the sample paths x(t) in the Feynman path integral spend
in the vicinity of an arbitrary point x. This gives rise to a stochastic process indexed by x, and an ensuing
local-time path integral whose explicit form is obtained for the case of Brownian motion in external
potential, using field-theoretic path-integral methods. The local-time path integral representation is
argued to be a powerful alternative to the traditional Feynman path integral, especially in the regime of
long evolution times. Local times of more generic processes will also be considered. In particular, I shall
focus on processes driven by the Levy stable distributions, which are frequently used to study the
behavior of financial markets.
A.Inomata Foundational issues in natural and social sciences
“Category-theoretic Aspects of Path Integrals”
In recent years, the category-theoretic approach has attracted attention in field theory and computer
science. Here we study how the path integral in quantum mechanics can be seen from the categorytheoretic aspect. Also, we discuss the relation between the Poisson sum rule and the role of
fundamental groupoids. As an example, we discuss the path integration in the field of topological
defects.
R. Loll Quantum gravity
“Non - perturbative path integrals for gravity: Successes and pitfalls”
Nonperturbative quantum gravity should describe the behaviour of spacetime and gravitational
interactions at Planckian scales, where both are subject to large quantum fluctuations. A key difficulty is
how to set up a consistent formulation in the absence of a preferred background geometry, which - if
everything goes well -- emerges only in a suitable classical, large-scale limit. More specifically, I will
outline the challenges that are present in a path integral formulation for gravity. A program that has
advanced significantly along these lines is that of (Causal) Dynamical Triangulations. I will highlight
generic "pitfalls" it has exhibited, explain how it addresses the challenges of the nonperturbative path
integral, and describe to what extent it has already succeeded.
R. Rivers Hard and soft condensed matter theory and biophysics
“When are two fermions a simple boson? New Gross-Pitaevskii equations for cold Fermi gases”
In this talk I shall discuss condensates of cold Fermi gases controlled by a Feshbach resonance in the BEC
regime where the condensates are dominated by diatoms/dimers. The intention is to see to what extent
such dimers behave as elementary bosons. I shall address the question by means of two different kinds
of Gross-Pitaevskii equations. As an example, I shall show how vortex equations differ in the two cases.
Our results are only valid for systems both with narrow Feshbach resonances and in the deep BEC
regime.
Thursday 9th February:
D. Oriti Quantum gravity
“The group field theory formulation of the quantum gravity path integral”
In the first part of the talk, we introduce the group field theory formulation of quantum gravity. In
particular, we will discuss how the Feynman expansion of suitable group field theory models defines
quantum gravity in terms of a discrete path integral for gravity discretised on a simplicial complex,
complemented by a sum over such simplicial complexes of arbitrary topology. In the second part of the
talk, we report on recent results aimed at identifying the continuum limit and continuum phase diagram
of such models, by taking advantage of the quantum field theory tools made available by such GFT
formulation and by applying to it functional renormalization group methods.
P. Ribeiro Hard and soft condensed matter theory and biophysics
“The non-equilibrium Peierls transition in thermodynamic unbalanced system”
Spin- and charge density waves are a common phenomenon in condensed matter physics. Charge
density waves were predicted by R. Peierls who showed that a one-dimensional lattice can become
unstable and undergo a transition due to the electron-phonon coupling.
This Peierls transition has been well studied and can be conveniently described within a path integral
representation.
Much less, however, is known about its fate away from thermal equilibrium.
In this talk, we address the fate of this instability under non-equilibrium conditions created by imposing
a finite voltage across the system.
We will give a brief introduction into non-equilibrium Green’s functions using the path integral on the
Keldysh contour.
Then, using this technique, we establish that the finite voltage drop across the system changes the
ordering wave function away from its equilibrium position at $2k_F$. The amount of this shift depends
on the applied voltage and on the properties of the contacts. Finally, we discuss the nature of the nonequilibrium steady-state obtained at zero temperature in the incommensurate case.
J. Korbel Complex dynamical systems and Econophysics
“Space - time fractional diffusion and its applications in finance”
Diffusion belongs to very important concepts not only in physics, but also in all stochastic systems.
Financial markets are one important example, where diffusion found its crucial role. On the other hand,
ordinary diffusion based on normal distribution cannot describe the complex phenomena of financial
markets, including bubbles, large jumps and memory effects. We discuss possible generalisations of
diffusion equation which could suitably describe these phenomena. One possible way is to introduce
anomalous diffusion equation based on fractional, i.e. non-natural derivatives. We show that the class of
diffusion equations fractional in space and/or time variable has many advantageous properties and it is
possible to use it in financial applications. We also introduce diffusion-equation with timedependent derivative operators, which can be successfully used for modeling of temporally abnormal
periods, as e.g. financial crisis. We mainly focus on applications to option pricing and connected
problems. Finally, we mention several open problems and possible directions for future work.
L. Smaldone Foundational issues in natural and social sciences
“Signatures of inequivalent representations in path integrals”
A careful non perturbative study of flavor mixing reveals an interesting structure of the flavor vacuum.
This is deeply related to the existence of unitarily inequivalent representations of field algebra in
Quantum Field Theory. We have recently studied the possibility of a dynamical generation of fermion
mixing by using one-loop effective action with the help of path-integral techniques. The analysis of this
problem, which leads to gap-equations for the dynamical generation of masses and mixing, evokes two
immediate questions: i) Does path integral know about inequivalent representation? ii) Is it the standard
generating functional of Green's functions capable of distinguishing among different inequivalent vacua?
In this talk I will put forward some plausible replies to these questions.
V. Fleurov Hard and soft condensed matter theory and biophysics
“Collective States in Optical Microcavities”
We study a model describing competition of interactions between N two-level systems (TLSs)
against decoherence. We apply it to describe dye molecules in an optical microcavity, where molecular
vibrations provide a local source for decoherence. Most interesting is the case when decoherence
strongly affects a single TLS, e.g. via broadening of emission lines as well as vibrational satellites,
however its influence is strongly suppressed for large N due to the interactions between TLSs. In
this interaction dominated regime we find unique signatures in the emission spectrum, including
strong O(sqrt(N)) level shifts, as well as 1/N suppression of both the decoherence width and of the
vibrational satellites
Friday 10th February:
A.Pelster Hard and soft condensed matter theory and biophysics
“Functional Integral Approach for Trapped Dirty Bosons”
The notoriously difficult dirty boson problem amounts to understanding the emergence of coherence
and order for ultracold bosonic atoms in the presence of a quenched disorder potential. It
appears either naturally like in current carrying wire traps, or artificially like in laser speckle fields.
Theoretically it is intriguing because of the competition of localization and interaction as well as
of disorder and superfluidity.
The talk reviews how to deal with the dirty boson problem within the functional integral approach.
At first we report on the perturbative treatment of Huang and Meng for a weak disorder potential.
In particular, we discuss disorder corrections of both condensate and superfluid density as well
as of collective excitation frequencies and the critical temperature. Afterwards, we work out a
Hartree-Fock mean-field theory by invoking Parisi replica symmetry for delta-correlated disorder.
In case of a homogeneous BEC we locate the superfluid phase with a global condensate, the
Bose-glass phase with localized BEC droplets in the minima of the random potential, and the
normal phase within a phase diagram, which is spanned by temperature and disorder strength
[1]. Applying the Thomas-Fermi approximation we extend then the non-perturbative theory to an
additional isotropic harmonic confinement and focus on the question how the spatial distribution
of the fragmented BEC droplets changes with increasing disorder strength [2]. In contrast to the
case of quasi-1d we find in 3d at zero temperature a first-order quantum phase transition from the
superfluid phase to the Bose-glass phase at a critical disorder strength [3,4]. The obtained results
are relevant for a quantitative analysis of ongoing experiments with dirty bosons in harmonic
traps.
[1] R. Graham and A. Pelster, Int. J. Bifurc. and Chaos 19, 2745 (2009)
[2] T. Khellil and A. Pelster, J. Stat. Mech. 063301 (2016)
[3] T. Khellil, A. Balaz, and A. Pelster, New J. Phys. 18, 063003 (2016)
[4] T. Khellil and A. Pelster, arXiv:1512.04870
R. Frésard Hard and soft condensed matter theory and biophysics
“Slave bosons in radial gauge: Foundations and applications to the Hubbard Model and beyond”
The slave-boson representations of the most prominent models of strongly correlated electrons will be
reviewed [1,2]. Their gauge symmetries naturally lead to the concept of radial slave-bosons [3,4], on
which this talk is centered. Their main characteristics will be addressed, including the exact calculation
of functional integrals for a toy model [5]. As applications it will be shown that they help reconciling the
Hubbard split band Physics with Fermi liquid theory, and to predict capacitance enhancement in
capacitors made of a dielectric surrounded by correlated metallic plates [6].
[1] S. E. Barnes, J. Phys. F 6, 1375 (1976); 7, 2637 (1977).
[2] G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986); R. Frésard and P. Wölfle, Int. J. of
Mod. Phys. B 6, 685 (1992).
[3] N. Read, D. M. Newns, J. Phys. C 16, L1055 (1983); 16, 3273 (1983).
[4] R. Frésard and T. Kopp, Nucl. Phys. B 594, 769 (2001).
[5] R. Frésard, H. Ouerdane, and T. Kopp, Nucl. Phys. B 785, 286 (2007).
[6] K. Steffen, R. Frésard, and T. Kopp, Phys. Rev. B 95, 035143 (2017).
N. Kumano-go Foundational issues in natural and social sciences
“Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation”
We give two general classes of functionals for which the phase space Feynman path integrals have a
mathematically rigorous meaning. More precisely, for any functional belonging to each class, the time
slicing approximation of the phase space path integral converges uniformly on compact subsets with
respect to the starting point of momentum paths and the endpoint of position paths. Each class is
closed under addition, multiplication, translation, real linear transformation and functional
differentiation. Therefore, we can produce many functionals which are phase space path integrable.
Furthermore, though we need to pay attention for use, the interchange of the order with the integrals
with respect to time, the interchange of the order with some limits, the semiclassical approximation of
Hamiltonian type, the natural property under translation, the integration by parts with respect to
functional differentiation, and the natural property under orthogonal transformation are valid in the
phase space path integrals.
Reference:
1.N. Kumano-go, Phase space Feynman path integrals with smooth
functional derivatives by time slicing approximation, Bull. Sci. math. 135 (2011).
2.N. Kumano-go and D. Fujiwara, Phase space Feynman path integrals via
piecewise bicharacteristic paths and their semiclassical approximations, Bull. Sci. math. 132 (2008).
3.N. Kumano-go, Phase space Feynman path integrals - Calculation
examples via piecewise bicharacteristic paths, RIMS Kokyuroku, 1797 (2012).
F. Bastianelli Quantum gravity
“Path integrals in curved space and trace anomalies”
Path integrals for particles in curved spaces can be used to study properties of quantum fields coupled
to gravity in first quantization. As an example we describe the calculation of trace anomalies in quantum
field theories, analyzing in particular the case of a Weyl fermion, whose anomaly has recently attracted
some interest. Finally, we discuss a simplified version of the path integral in spaces of maximal
symmetry. It may be used to identify the so-called type-A trace anomalies in a simple way.
J. Zinn-Justin Closing talk
“Functional integration: an essential tool of modern physics”
In the twentieth century has started a systematic study of large scale fluctuating systems: thermal
fluctuations and then an important topic is phase transitions, quantum fluctuations as first recognized
by Feynman with the representation of quantum evolution by a path integral, and where the main topic
is quantum field theory as applied to particle physics (one can also study finite temperature quantum
field theory). This explains why functional integration has become such an essential mathematical tool in
physics. I will illustrate here the role of functional integrals with a few important examples.