# Peter JizbaCzech Technical University in Prague | ČVUT · Faculty of Nuclear Sciences and Physical Engineering (FJFI), Department of Physics

Peter Jizba

PhD

statistical foundations of entropy, complex systems, conformal quantum gravity, generalized uncertainty relations

## About

135

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Introduction

Additional affiliations

September 2008 - December 2011

May 2004 - present

September 2000 - April 2004

## Publications

Publications (135)

In this paper we employ a recent proposal of C. Tsallis and formulate the first law of thermodynamics for gravitating systems in terms of the extensive but non-additive entropy. We pay a particular attention to an integrating factor for the heat one-form and show that in contrast to conventional thermodynamics it factorizes into thermal and entropi...

The generalized uncertainty principle (GUP) is a phenomenological model whose purpose is to account for a minimal length scale (e.g., Planck scale or characteristic inverse-mass scale in effective quantum description) in quantum systems. In this Letter, we study possible observational effects of GUP systems in their decoherence domain. We first der...

Uncovering causal interdependencies from observational data is one of the great challenges of nonlinear time series analysis. In this paper, we discuss this topic with the help of information-theoretic concept known as R\'enyi information measure. In particular, we tackle the directional information flow between bivariate time series in terms of R\...

The generalized uncertainty principle (GUP) is a phenomenological model whose purpose is to account for a minimal length scale (e.g., Planck scale or characteristic inverse-mass scale in effective quantum description) in quantum systems. In this Letter, we study possible observational effects of GUP systems in their decoherence domain. We first der...

During the last few decades, the notion of entropy has become omnipresent in many scientific disciplines, ranging from traditional applications in statistical physics and chemistry, information theory, and statistical estimation to more recent applications in biology, astrophysics, geology, financial markets, or social networks[...]

In this paper, we discuss the statistical coherence between financial time series in terms of Rényi's information measure or entropy. In particular, we tackle the issue of the directional information flow between bivariate time series in terms of Rényi’s transfer entropy. The latter represents a measure of information that is transferred only betwe...

Superstatistics is a well‐known term in the field of non‐equilibrium statistical physics. It describes a system in a local thermodynamic equilibrium. This chapter aims to provide an accurate method for the detection of a transition between superstatistics. Superstatistics is a concept devised by Beck for systems with fluctuating intensive parameter...

In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivati...

In this paper we review dynamical mixing generation in a generic quantum field theoretical model with global SU(2)L × SU(2)R ×U(1)V symmetry. By purely algebraic means we analiyze the vacuum structure for different patterns of symmetry breaking and show explicitly how the non-trivial flavor vacuum condensate characterizes dynamical mixing generatio...

In this paper we review dynamical generation of field mixing after chiral symmetry breaking. We also study the explicit form of discrete transformations of flavor states in a two-flavor scalar model with field mixing. We find that CP T symmetry is spontaneously broken on flavor vacuum because of its dynamically generated condensate structure.

We address the issue of a dynamical breakdown of scale invariance in quantum Weyl gravity together with related cosmological implications. In the first part, we build on our previous work [Phys. Rev. D2020, 101, 044050], where we found a non-trivial renormalization group fixed point in the infrared sector of quantum Weyl gravity. Here, we prove tha...

In this paper we review flavor-energy uncertainty relations for neutrino oscillations in quantum field theory, putting in evidence the analogy with the case of unstable particles. Our study reveals that flavor neutrinos are intrinsically characterized by an energy distribution with a non-vanishing width. In the ultrarelativistic limit, the energy w...

We study the explicit form of Poincaré and discrete transformations of flavor states in a two-flavor scalar model, which represents the simplest example of the field mixing. Because of the particular form of the flavor vacuum condensate, we find that the aforementioned symmetries are spontaneously broken. The ensuing vacuum stability group is ident...

With the help of a functional renormalization group, we study the dynamical breakdown of scale invariance in quantum Weyl gravity by starting from the UV fixed point that we assume to be Gaussian. To this end, we resort to two classes of Bach-flat backgrounds, namely maximally symmetric spacetimes and Ricci-flat backgrounds in the improved one-loop...

Even though irreversibility is one of the major hallmarks of any real-life process, an actual understanding of irreversible processes remains still mostly semi-empirical. In this paper, we formulate a thermodynamic uncertainty principle for irreversible heat engines operating with an ideal gas as a working medium. In particular, we show that the ti...

We propose a unified framework for both Shannon-Khinchin and Shore-Johnson axiomatic systems. We do it by rephrasing Shannon-Khinchine axioms in terms of generalized arithmetics of Kolmogorov and Nagumo. We prove that the two axiomatic schemes yield identical classes of entropic functionals—the Uffink class of entropies. This allows to re-establish...

We study the explicit form of Poincaré and discrete transformations of flavor states in a two-flavor scalar model, which represents the simplest example of the field mixing. Because of the particular form of the flavor vacuum condensate, we find that the aforementioned symmetries are spontaneously broken. The ensuing vacuum stability group is ident...

Even though irreversibility is one of the major hallmarks of any real life process, an actual understanding of irreversible processes remains still mostly semiempirical. In this paper we formulate a thermodynamic uncertainty principle for irreversible heat engines operating with an ideal gas as a working medium. In particular, we show that the time...

We study the non-relativistic limit of Dirac equation for mixed neutrinos. We demonstrate that such a procedure inevitably leads to a redefinition of the inertial mass. This happens because, in contrast to the case when mixing is absent, the antiparticle sector contribution cannot be neglected for neutrinos with definite flavor. We then show that,...

Starting from ultraviolet fixed point we study infrared behavior of quantum Weyl gravity in terms of functional RG flow equation. To do so, we employ two classes of Bach-flat backgrounds, namely maximally symmetric spacetimes and Ricci-flat backgrounds in the improved one-loop scheme. We show, that in the absence of matter fields and with a topolog...

In this paper we review dynamical generation of field mixing after chiral symmetry breaking. We also study the explicit form of Lorentz boosts transformations of flavor states in a two-flavor scalar model with field mixing. We find that Lorentz symmetry is spontaneously broken on flavor vacuum because of its dynamically generated condensate structu...

We propose a unified framework for both Shannon--Khinchin and Shore--Johnson axiomatic systems. We do it by rephrasing Shannon--Khinchine axioms in terms of generalized arithmetics of Kolmogorov and Nagumo. We prove that the two axiomatic schemes yield identical classes of entropic functionals --- Uffink class of entropies. This allows to re-establ...

We use Rényi-entropy-power-based uncertainty relations to show how the information probability distribution associated with a quantum state can be reconstructed in a process that is analogous to quantum-state tomography. We illustrate our point with the so-called “cat states”, which are of both fundamental interest and practical use in schemes such...

In their recent paper [Phys. Rev. E 99, 032134 (2019)], Oikonomou and Bagci have argued that Rényi entropy is ill suited for inference purposes because it is not consistent with the Shore-Johnson axioms of statistical estimation theory. In this Comment we seek to clarify the latter statement by showing that there are several issues in Oikonomou's a...

In their recent paper [Phys. Rev. E 99 (2019) 032134], T. Oikinomou and B. Bagci have argued that R\'enyi entropy is ill-suited for inference purposes because it is not consistent with the Shore{ Johnson axioms of statistical estimation theory. In this Comment we seek to clarify the latter statement by showing that there are several issues in Oikin...

In this Letter, we show that the Shore-Johnson axioms for the maximum entropy principle in statistical estimation theory account for a considerably wider class of entropic functional than previously thought. Apart from a formal side of the proof where a one-parameter class of admissible entropies is identified, we substantiate our point by analyzin...

In this Letter, we show that the Shore-Johnson axioms for the maximum entropy principle in statistical estimation theory account for a considerably wider class of entropic functional than previously thought. Apart from a formal side of the proof where a one-parameter class of admissible entropies is identified, we substantiate our point by analyzin...

In this paper we study dynamical chiral symmetry breaking of a generic quantum field theoretical model with global chiral flavor symmetry. By purely algebraic means we analyze the vacuum structure for different symmetry breaking schemes and show explicitly how the ensuing non-trivial flavor vacuum condensate characterizes the phenomenon of field mi...

In this paper we study dynamical chiral symmetry breaking of a generic quantum field theoretical model with global chiral flavor symmetry. By purely algebraic means we analyze the vacuum structure for different symmetry breaking schemes and show explicitly how the ensuing non-trivial flavor vacuum condensate characterizes the phenomenon of field mi...

In this paper, we have modified one of the simplest multi-level cellular automata – a hodgepodge machine, so as to represent the best match for the chemical trajectory observed in the Belousov–Zhabotinsky reaction (BZR) in a thin layered planar setting. By introducing a noise term into the model, we were able to transform the central regular struct...

In the context of quantum field theory, we derive flavor-energy uncertainty relations for neutrino oscillations. By identifying the nonconserved flavor charges with the “clock observables,” we arrive at the Mandelstam-Tamm version of time-energy uncertainty relations. In the ultrarelativistic limit, these relations yield the well-known condition fo...

In the context of quantum field theory, we derive flavor energy uncertainty relations for neutrino oscillations. By identifying the non conserved flavor charges with the clock observables, we arrive at the Mandelstam Tamm version of time energy uncertainty relations. In the ultrarelativistic limit these relations yield the well known condition for...

In this paper we study a unified formalism for Thermal Quantum Field Theories, i.e., for the Matsubara approach, Thermo Field Dynamics and the Path Ordered Method. To do so, we employ a mechanism akin to the Hawking effect which explores a relationship between the concept of temperature and spacetimes endowed with event-horizons. In particular, we...

We briefly review some results on Green's functions for mixed fermion fields and reinterpret them in terms of Schwinger–Dyson equations. Working with retarded propagators, we identify the self-energy operator for the mixing interaction, and find an exactly solvable expansion for the complete-fermion propagator. Finally, we show that an equation à l...

Superstatistics is a widely employed tool of non-equilibrium statistical physics which plays an important role in analysis of hierarchical complex dynamical systems. Yet, its "canonical" formulation in terms of a single nuisance parameter is often too restrictive when applied to complex empirical data. Here we show that a multi-scale generalization...

The aim of this paper is to study mixed-representation Green's functions from the point of view of coherent-state path integrals. This is achieved by using the machinery of generalized generating functionals of Green's functions, previously introduced by the present authors in the context of standard phase-pace path integrals. The obtained results...

In the framework of the canonical quantization of the electromagnetic field, we impose as primary condition on the dynamics the positive definiteness of the energy spectrum. This implies that (Glauber) coherent states have to be considered for the longitudinal and the scalar photon fields. As a result we obtain that the relation holds which in the...

We discuss the idea that the Tsallis-type (q-additive) entropic chain rule allows for a wider class of entropic functionals than previously thought. In particular, we point out that the ensuing entropy solutions (e.g., Tsallis entropy) can be determined uniquely only when one fixes the prescription for handling conditional entropies. Our point is s...

When one tries to take into account the non-trivial vacuum structure of Quantum Field Theory, the standard functional-integral tools such as generating functionals or transitional amplitudes, are often quite inadequate for such purposes. Here we propose a generalized generating functional for Green's functions which allows to easily distinguish amo...

We examine a simple two-flavor scalar model with a non-diagonal mass matrix. We argue that the conventional definition of the QFT partition function does not allow to evaluate the Green's functions with respect to the flavor vacuum as it favors only the mass vacuum. By using functional-integral techniques, we derive new generating functional for Gr...

We use the concept of entropy power to introduce a new one-parameter class of information-theoretic uncertainty relations. This class constitutes an infinite hierarchy of uncertainty relations, which allows to determine the shape of the underlying information-distribution function by measuring the relevant entropy powers. The efficiency of such unc...

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 w...

We discuss canonical transformations in Quantum Field Theory in the framework of the functional-integral approach. In contrast with ordinary Quantum Mechanics, canonical transformations in Quantum Field Theory are mathematically more subtle due to the existence of unitarily inequivalent representations of canonical commutation relations. When one w...

Recently in [Physica A 411 (2014) 138] Ili\'{c} and Stankovi\'{c} have suggested that there may be problem for the class of hybrid entropies introduced in [P.~Jizba and T.~Arimitsu, Physica A 340 (2004) 110]. In this Comment we point out that the problem can be traced down to the $q$-additive entropic chain rule and to a peculiar behavior of the De...

In this paper we point out that the generalized statistics of Tsallis-Havrda-Charvát can be conveniently used as a conceptual framework for statistical treatment of random chains. In particular, we use the path-integral approach to show that the ensuing partition function can be identified with the partition function of a fluctuating oriented rando...

An important feature of Quantum Field Theory is the existence of unitarily inequivalent representations of canonical commutation relations. When one works with the functional integral formalism, it is not clear, however, how this feature emerges. By following the seminal work of M. Swanson on canonical transformations in phase-space path integral,...

We use the concept of entropy power to derive a new one-parameter class of information-theoretic uncertainty relations for pairs of conjugate observables in an infinite-dimensional Hilbert space. This class constitutes an infinite tower of higher-order statistics uncertainty relations, which allows one in principle to determine the shape of the und...

In this paper we discuss a mechanism for the dynamical generation of flavor mixing, in the framework of the Nambu--Jona Lasinio model. Our approach is illustrated both with the conventional operatorial formalism and with functional integral and ensuing one-loop effective action. The results obtained are briefly discussed.

We combine an axiomatics of R\'enyi with the $q$--deformed version of
Khinchin axioms to obtain a measure of information (i.e., entropy) which
account both for systems with embedded self-similarity and non-extensivity. We
show that the entropy thus obtained is uniquely solved in terms of a
one-parameter family of information measures. The ensuing m...

DOI:http://dx.doi.org/10.1103/PhysRevE.93.029901

Mesoscopic dynamics of self-organized structures is the most important aspect in the description of complex living systems. The Belousov--Zhabotinsky (B--Z) reaction is in this respect a convenient testbed because it represents a prototype of chemical self-organization with a rich variety of emergent wave-spiral patterns. Using a multi-state stocha...

We show that a multifractal analysis offers a new and potentially promising avenue for quantifying the com-plexity of various time series. In particular, we compare the most common techniques used for multifractal scaling exponents estimation. This is done from both a theoretical and phenomenological point of view. In our discussion we specifically...

In this paper we show how the Feynman checkerboard picture for the 1+1-dimensional Dirac equation can be extended to accommodate the neutrino mixing and ensuing neutrino oscillations.

We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the v...

We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the v...

Scaling properties and fractal structure are one of the most important aspects of real systems that point to their complexity. These properties are closely related to the theory of multifractal systems and theory of entropy. Estimation of scaling (or multifractal) exponents belongs to the essential techniques that can reveal complexity and inner st...

In this paper we show how the neutrino oscillations can emerge within the Feynman chessboard picture. Salient issues such as the use of Euclidean metric in treatment of Dirac fermions and a role of relativistic inertial mass in neutrino oscillations are also discussed.

We analyze the functional integral for quantum Conformal Gravity and show
that with the help of a Hubbard-Stratonovich transformation, the action can be
broken into a local quadratic-curvature theory coupled to a scalar field. A
one-loop effective action calculation reveals that strong fluctuations of the
metric field are capable of spontaneously g...

Uncertainty relations based on information theory for both discrete and
continuous distribution functions are briefly reviewed. We extend these results
to account for (differential) R\'{e}nyi entropy and its related entropy power.
This allows us to find a new class of information-theoretic uncertainty
relations (ITURs). The potency of such uncertai...

Measuring information transfer between time series is a challenging task. Classical statistical approaches based on correlations do not provide complete image about sources of the information flow. On the other hand, there have been introduced many sophisticated approaches that enable us to reveal the complex nature of many processes. One of these...

There is a theoretical evidence that relativistically invariant quantum dynamics at (enough) large space-time scales can result from a cooperative process of two inter-correlated non-relativistic stochastic dynamics, operating at different energy scales. We show that the Euclidean transition amplitude for a relativistic particle is identical to the...

We show that a number of realistic financial time series can be well mimicked by multiplicative multifractal cascade processes. The key observation is that the multi-scale behavior in financial progressions fits well the multifractal cascade scaling paradigm. Connections with Kolmogorov’s idea of multiplicative cascade of eddies in the well develop...

We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As shown by its applications to different potentials, the presented expansion turns out to be quite efficient in gene...

In the framework of Multifractal Diffusion Entropy Analysis we propose a
method for choosing an optimal bin-width in histograms generated from
underlying probability distributions of interest. This presented method uses
techniques of Renyi's entropy and the mean square error analysis to discuss the
conditions under which the error in Renyi's entrop...

In this paper, we compare two key approaches used in time series analysis, namely the Multifractal Detrended Fluctuation Analysis and Multifractal Diffusion Entropy Analysis. The comparison is done from both the theoretical and numerical point of view. To put some flesh on bare bones, we illustrate our analysis by applying both methods to three mod...

We present a dynamical mechanism \`a la Nambu--Jona-Lasinio for the
generation of masses and mixing for two interacting fermion fields. The
analysis is carried out in the framework introduced long ago by Umezawa et al.,
in which mass generation is achieved via inequivalent representations, and that
we generalize to the case of two generations. The...

We present a new theoretical evidence that a relativistically invariant
quantum dynamics at large enough space-time scales can be derived from
two inter-correlated genuinely non-relativistic stochastic processes
that operate at different energy scales. This leads to Feynman
amplitudes that are, in the Euclidean regime, identical to transition
proba...

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transit...

We show that the special relativistic dynamics when combined with quantum
mechanics and the concept of superstatistics can be interpreted as arising from
two interlocked non-relativistic stochastic processes that operate at different
energy scales. This interpretation leads to Feynman amplitudes that are in the
Euclidean regime identical to transit...

We propose a method for characterizing the image—multidimensional projection—of complex, self-organising, system. The method is general and may be used for characterisation of any structured, experimentally observable, complex self-organized systems. The method is based on calculation of information gain by which a point contributes to the total in...

We consider the possibility that two-flavor neutrino mixing can be viewed as resulting from the interaction of two non-mixed flavor neutrinos with an external vector field. Two distinct scenarios of the origin of such a vector field are presented. First we argue that the vector field might be understood as an su(2) gauge field. In the second scenar...

We study uncertainty relations as formulated in a crystal-like universe,
whose lattice spacing is of order of Planck length. For Planck energies,
the uncertainty relation for position and momenta has a lower bound
equal to zero. Connections of this result with double special
relativity, and with't Hooft's deterministic quantization proposal, are
br...

We show how a Brownian motion on a short scale can originate a relativistic motion on scales larger than the particle's Compton wavelength. Thus, Lorentz symmetry appears to be not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength, is taken. We also...

Using the concept known as a superstatistics path integral we show that
a Wiener process on a short spatial scale can originate a relativistic
motion on scales that are larger than particle's Compton wavelength.
Viewed in this way, special relativity is not a primitive concept, but
rather it statistically emerges when a coarse graining average over...

We propose a method for characterizing structured, experimentally observable, complex self-organized systems. The method in question is based on the observation that number of self-organized systems can be mathematically perceived as consisting of several interconnected multifractal components. We illustrate our key results with ensuing application...

In the framework of 't Hooft's "deterministic quantization" proposal, we show how to obtain from a composite system of two classical Bateman's oscillators a quantum isotonic oscillator. In a specific range of parameters, such a system can be also interpreted as a particle in an effective magnetic field, interacting through a spin-orbit interaction...

Using the concept known as a superstatistics path integral we show that a Wiener process on a short spatial scale can originate a relativistic motion on scales that are larger than particle's Compton wavelength. Viewed in this way, special relativity is not a primitive concept, but rather it statistically emerges when a coarse graining average over...

The present paper gives a new method of attack on the Nambu-Goldstone
dynamics in spontaneously broken theories. Since the target space of the
Nambu-Goldstone fields is a group coset space, their effective quantum dynamics
can be naturally phrased in terms of generalized coherent-state functional
integrals. As an explicit example of this line of re...

We formulate generalized uncertainty relations in a crystal-like
universe whose lattice spacing is of order of Planck length -- a "world
crystal". For energies near the border of the Brillouin zone, i.e., for
Planckian energies, the uncertainty relation for position and momentum
does not pose any lower bound. We apply these results to micro black
h...

We formulate generalized uncertainty relations in a crystal-like universe whose lattice spacing is of order of Planck length --- a "world crystal". For energies near the border of the Brillouin zone, i.e. for Planckian energies, the uncertainty relation for position and momentum does not pose any lower bound. We apply these results to micro black h...

We study uncertainty relations as formulated in a crystal-like universe, whose lattice spacing is of order of Planck length. For Planck energies, the uncertainty relation for position and momenta has a lower bound equal to zero. Connections of this result with double special relativity, and with 't Hooft's deterministic quantization proposal, are b...

In this paper, we quantify the statistical coherence between financial time series by means of the Rényi entropy. With the help of Campbell’s coding theorem, we show that the Rényi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others. This accentuation is controlled with Rény...

Building on our previous work [Phys.Rev.D82,085016(2010)], we show in this
paper how a Brownian motion on a short scale can originate a relativistic
motion on scales that are larger than particle's Compton wavelength. This can
be described in terms of polycrystalline vacuum. Viewed in this way, special
relativity is not a primitive concept, but rat...

"Physicists believe quantum fields to be the true protagonists of nature in the full variety of its wonderful, manifold manifestations. Quantum field theory is the tool they created to fulfill their visionary dream of describing with a universal, unique language all of nature, be it single particles or condensed matter, fields or many-body objects....

Quantum dynamics underlies macroscopic systems exhibiting some kind of ordering, such as superconductors, ferromagnets and crystals. Even large scale structures in the Universe and ordering in biological systems appear to be the manifestation of microscopic dynamics ruling their elementary components. The scope of this book is to answer questions s...