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1 The Thermodynamic Limit of Quantum Coulomb Systems: A New Approach

Christian Hainzl, Mathieu Lewin B, Jan Philip Solovej

Published Jan 1, 2008• 2 citations
OpenAlex

Abstract

We present recent works [15,16] on the thermodynamic limit of quantum Coulomb systems. We provide a general method which allows to show the existence of the limit for many different systems. Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via Coulomb forces. The potential between two particles of charges z and z ′ located at x and x ′ in R 3 is zz ′ |x − x ′ |. There are two difficulties which occur when trying to describe systems composed of electrons and nuclei. Both have to do with the physical problem Talk given by M.L. at QMath10, 10th Quantum Mathematics International Conference, Moeciu, Romania, September 10–15, 2007. c○2008 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. 2 of stability of quantum systems. The first is due to the singularity of 1/|x | at 0: it is necessary to explain why a particle will not rush to a particle of the opposite charge. One of the first major triumphs of the theory of quantum mechanics is the explanation it gives of the stability of the hydrogen atom (and the complete description of its spectrum) and of other microscopic quantum Coulomb systems, via the uncertainty principle. Stability means that the total energy of the considered system cannot be arbitrarily negative. If there was no such lower bound to the energy it would be possible in principle to extract an infinite amount of energy. One often refers to this kind of stability as stability of the first kind [19,20]. If we denote by E(N) the ground state energy of the system under consideration, for N particles stability of the first kind can be written E(N)> −∞. (1) In proving (1) for Coulomb systems, a major role is played by the uncertainty principle which for nonrelativistic systems is mathematically expressed by the critical Sobolev embedding H 1 (R 3) ֒ → L 6 (R 3). The latter allows to prove Kato’s inequality

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THE THERMODYNAMIC LIMIT OF QUANTUM COULOMB SYSTEMS: A NEW APPROACH

CHRISTIAN HAINZL, MATHIEW LEWIN, JAN PHILIP SOLOVEJ

We present two recent works on the thermodynamic limit of quantum Coulomb systems, in which we provided a general method allowing to show the existence of the limit for many different models.

arXiv:0806.2436•Mathematical Results in Quantum Mechanics (QMath)•Aug 1, 2008•0 citations

Stability and condensation of dilute Bose gases

François Louis Antoine Visconti

This thesis is devoted to the rigorous study of dilute Bose gases with two-body and three-body interactions at zero temperature in two and three dimensions. Bose gases are many-body quantum systems composed of bosons, which notably do not obey the Pauli exclusion principle. As a consequence, at extremely low temperatures, a macroscopic fraction of the particles can occupy the same one-body quantum state. This phenomenon, known as Bose–Einstein condensation, was predicted in the 1920s by Bose and Einstein for noninteracting Bose gases, and first realised experimentally by the groups of Cornell and Wieman, and Ketterle, earning them the Nobel Prize for Physics in 2001. However, proving the occurrence of Bose–Einstein condensation in general interacting bosonic systems remains a major open problem and constitutes a central theme in this thesis. Another important property of bosonic matter is that it is generally more unstable than fermionic matter. In particular, for systems interacting via the Coulomb force, the Pauli exclusion principle is essential for explaining the stability of the second kind, which states that the energy and volume scale linearly with the number of particles. Nonetheless, bosonic systems with attractive but less singular interactions can also be stable, and the rigorous proof of this fact forms a second focus of this thesis. In three dimensions, we consider predominantly repulsive interactions. Firstly, we prove a leading order upper bound on the thermodynamic ground state energy of Bose gases with hard-core three-body interactions, which matches an existing lower bound. In addition, we obtain a simpler proof for integrable potentials. Secondly, we derive the leading order of the thermodynamic ground state energy of Bose gases with combined two-body and three-body interactions. More specifically, we show that, at the leading order, the ground state energy is entirely characterised by the scattering length of the two-body potential and the scattering hypervolume of the three-body potential. Thirdly, we study the time evolution of Bose–Einstein condensates with three-body interactions in the so-called Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation. In particular, we show that the effective coupling constant is universal in the sense that it depends only on the scattering hypervolume of the interaction potential. As a by-product of our proof, we derive a stability of the second kind result for Bose gases with attractive two-body interactions and repulsive three-body interactions. In two dimensions, we consider trapped Bose gases with mean-field-type interactions, including the purely attractive case. We prove the stability of the second kind of the system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger energy functional. As a consequence of the stability of the second kind, we also obtain Bose–Einstein condensation for the many-body ground states.

•Aug 4, 2026•0 citations

Quantum Long-Range Systems

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Abstract This chapter opens with the notion of stable interaction in quantum systems, somewhat different from classical systems. It is shown that an unstable classical system can become stable when treated quantum mechanically. The remaining of the chapter is devoted to a brief survey of the main results concerning the stability of Coulomb systems. Explicit evaluations of the total energy show that the exclusion principle is essential for their stability. The final part of the chapter deals with the thermodynamic limit of globally neutral Coulomb systems. The limit exists in spite of the long-range character of the interaction, and contrary to the stability property, the existence of the limit does not require quantum mechanics, being valid also in classical Coulomb systems.

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The Quantum Many-Body Problem: Methods and Analysis

Michael Lindsey

This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classical field theory. In this setting we also provide a rigorous non-perturbative construction of the LW functional via convex duality and prove several key properties, including continuity up to the boundary of its domain and asymptotics in the limit of large interaction.Quantum embedding methods, meanwhile, view a large system as being composed of smaller fragments that are treated with high accuracy and embedded in the larger system in a mutually consistent way. Inspired by a connection between the boundary analysis of the LW functional and embedding, we perform similar analysis for the 1-RDM theory for fermionic systems, which is also developed via convex duality, illustrating a relation to fermionic embedding methods such as the density matrix embedding theory (DMET).Another embedding method of note is the dynamical mean-field theory (DMFT), which is at the same time a Green's function method that can be understood in terms of the LW functional. DMFT relies on the solution of impurity problems, which specify the embedding of an interacting system into a non-interacting bath. Underlying DMFT is a result about the sparsity pattern of the self-energy matrix for impurity problems, which to our knowledge has not been proved in the literature. We provide a rigorous proof of this result in various classical and quantum settings. We go on to investigate the fermionic DMFT in depth, identifying the key mathematical structures that appear in the algorithmic loop for solving it and using these to prove the well-posedness of this loop, in a certain sense.Finally, we introduce a suite of new approaches to the quantum many-body problem that provide variational lower bounds to the ground-state energy. These methods, which combine the themes of convexity and embedding, are based on novel convex relaxations of the variational principles for the ground-state energies of many-body systems. To begin, we recover a second-quantized version of the formalism of strictly correlated electrons (SCE), which yields an exact expression for the exchange-correlation functional in Kohn-Sham density functional theory in the limit of infinite Coulomb repulsion in terms of the solution of a multi-marginal optimal transport problem. We introduce a semidefinite relaxation method for approximately solving this problem and obtaining a lower bound for the ground-state energy. The ideas underlying this relaxation are generalized considerably, outside the context of SCE, to yield much tighter lower bounds, which we validate numerically for both quantum spin systems and fermionic systems. We also describe how these relaxation methods can be interpreted as embedding methods via convex duality.

•Jan 1, 2019•0 citations

The thermodynamic limit of quantum Coulomb systems Part I. General theory

Christian Hainzl, Mathieu Lewin B, Jan Philip Solovej

arXiv:0806.1708•Advances in Mathematics•Jan 24, 2009•23 citations

On Certain Questions Related to Information and Symmetries - In Physics From Certain View of Philosophy of Science

György Darvas

Introduction My presentation aims at answering a part of the questions posted in the invitation to the session. The questions arose in and originated from the FIS discussion on Physical informatics … in October 2014. No doubt, they were formulated in provocative manner. The goal was to challenge discussion. I plan to illustrate my personal answers with a few examples quoted from the history of 20th c. physics. My answers to the questions are not intended to be enunciations and to provide final solutions, rather they serve as arguments and indicate that nothing is closed, the discussion is open. Methods 1 What do we consider physical information? Can one speak about physical information when there is no live percipient to accept, evaluate and use it? Can one speak about physical information (e.g., signal exchange) between inanimate physical objects (cf., e.g., Feynman diagrams)? And if so, what is it for? Is (physical) information a passive phenomenon, or its existence presumes activity? Interpretation of ‘activity’ plays important role in the possible answers. One of the interpretations says that activity is an antropomorph phenomenon. It is a privilege of the human mind, which is able to perceive and process information, able to teleologically evaluate its possible consequences and (re)act accordingly. Another interpretation says that there is inanimate activity, that means, reception of information between physical agents and their reaction to it. I argue for the latter concept. Let us see the example of the interaction between two (electric) charges. There were two approaches in the classical age of developing these theories (1928-33). When two electric charges interact, there appear two types of interaction. One is a Coulomb-type repulsion/attraction (according to their mutual signs) governed by their scalar potential, and the other is a Lorentz-type one governed by their vector potential depending on their relative velocity to each other. One type of the theories considered first the interaction between the scalar potentials and calculated the effect of the vector potentials as perturbation. It considered two charges approaching to each other from the infinity, when in first approximation they have got information on the amount of the Coulomb charge of the other, but not its velocity and the caused Lorentz force. The latter was considered in the perturbation process. The other type of the theories considered first the effect of the Lorentz force of the approaching charges, and took into consideration the effect of the Coulomb force in course of the perturbation. Representatives of both types of theories agreed that the roles of the interacting electric charges must be symmetrical, but Christian Møller. Møller [7], who belonged to the latter type of theoreticians, applied scattering matrices (1931). He showed that there appeared a component among the matrix elements that was asymmetric in respect to the two interacting charges. H. Bethe (that time a doctoral student of E. Fermi, 1932) could not accept this asymmetry and ‘corrected’ Møller’s theory. He ‘symmetrised’ those matrix elements artificially [1]. That was a rough and unjustified involvement in Møller’s equation, but due to the later attained high authority of both Bethe and Fermi, the physics community accepted the apparently convenient symmetrisation of the theory without discussion and has not treated it until the recent years. Thus any possible distinction between (roles and properties of) interacting charges were unrevealed until the past decade. However, the question can be formulated so: how do the charges, ready to interact, get information from each other? All theories agree that interaction between two charges take place by the exchange of a boson. In case of electromagnetic interaction this boson is a photon. Which of them emits the first photon towards the other? Did this emitter receive any information from the direction of the partner charge prior to the photon emission? What else is this if not an asymmetry between the roles, and possibly between properties of the two interacting charges? The distinction between the interacting charges was introduced by the isotopic field-charge theory, and the notion of ‘isotopic field-charge’ made the distinction between the properties of the two charges [2],[3],[4],[5]. The latter led to the proof of the gauge invariance under rotation of a newly introduced property, the isotopic field-charge spin, in an abstract field, and its conservation. This conservation is a result of a symmetry. Neverthe-less, this mechanism argues for activity between physical objects without an animated (human) agent. 2 What are the limits between (closed and open) systems, from the aspects of information and of symmetries? Further, if so, how wide can we extend the meaning of activity to be still accepted for generating information? What are the roles of different appearances of symmetries in taking a stand in the mentioned questions? What kinds of symmetry (or their absence) may play a role in making decision in the listed problems? The ‘classical’ (20th c.) relativity theories demanded that all physical laws were invariant under the Lorentz transformation. This was established first in the special theory of relativity that was formulated for electromagnetic interactions [6]. Lorentz invariance of physical laws was in fact a symmetry principle (conservation of the form of the laws during reference frame change) [9]. This invariance proved to hold for many other physical laws, so later also the symmetry principle was extended to other physical laws as well. The Lorentz invariant relativity theory included another consequence: there is no distinct (odd) reference frame in nature. In other words, all reference frames are equivalent. Aren’t they? Based on Noether’s theorems [8], one can show that conservation laws hold in all reference frames. However, the quantity of the conserved property (e.g., mass, charge, etc.) may change in the different reference frames. E.g., the amount of mass of matter in a closed system depends on the velocity of the observer relative to that system where the mass is to be measured. One can always find a reference frame from which the amount of the measured mass is minimal. This fact contradicts to the absence of a distinct (odd) reference frame. And there are more. Is there any invariance that compensates this lost equivalence of all reference frames? The observer can be not only a human agent who measures with instruments and reads the records. It can be another inanimate mass that perceives information about the mass from the observed system. Its ‘activity’ is that this information can be obtained by experiencing a force. This force can be gravitational or inertial. According to the general theory of relativity the ‘observer mass’ is unable to make distinction whether the experienced force is of ‘inertial’ or ‘gravitational’ origin. At the same time, the inertial mass changes its value according to its relative velocity to the observer, while the gravitational mass does not. Thus the ‘observer mass’ in different reference frames will experience different inertial forces originating from the ‘observed mass’. Something similar distinction can be made between the Coulomb charges and the Lorentz-type (current) charges. They are sources of the Coulomb force and the Lorentz force, respectively, and similar to the two types of masses, originate from the scalar potential and from the vector potential of the Hamiltonian of the charged object, respectively. The two types of masses and the two types of electric charges are called isotopic field-charge pairs, respectively. They are subject of the same gauge invariance. As such, they can exchange their roles (switch into each other) by the exchange of a gauge boson (in addition to the graviton and the photon, respectively), called delta bosons in the theory. As a consequence of the additional invariance and the corresponding additional mediating gauge boson, the respective systems of the two interacting isotopic field-charges (masses or electric charges) are not subjects of the Lorentz invariance alone. They are subjects of a convolution of the Lorentz- and this additional invariance. One can conclude two things. First, along with the development of physics, there is no more enough to demand invariance under the Lorentz transformation. At extended conditions, one should demand the invariance under a combination of the Lorentz invariance and an additional invariance. In short, we demand invariance under (the applicable) transformations. Secondly, when two charges (let they be either gravitational, electric, or other field-charges) interact, they make a distinction between each other. The system, composed of the interacting two field-charges, follow the Pauli principle. That means, the two interacting field-charges must be in different quantum states. Since this state in which they differ cannot be characterized by any of the earlier known properties, it must be a characteristic of the newly introduced property. It must be one of the two stable positions of the isotopic field-charge that are rotated into each other by the SU(2) symmetry group in the isotopic field-charge field. These two stable positions are called, by an analogically given name, the isotopic field-charge spin (not identical either with the angular momentum spin or the isotopic spin). According to its proven invariance, the isotopic field-charge spin is a conserved property. When two field-charges interact, they must be in the opposite isotopic field-charge spin states. The information that they exchange about each other is about this state: they check whether the partner is in the opposite state. Otherwise they are ‘not allowed’ to interact (Pauli’s exclusion principle). The information exchange takes place by the exchange of a delta boson (called also dion) between them, in addition to the exchange of the traditional mediating bosons (like graviton, photon, weak charged and neutral bosons, or gluons). That delta boson switches the emitting charge from inertial to potential state, and the absorbing charge from potential to inertial state. Conclusions The asymmetry of the interacting charges has been explained. It was subject of information exchange between the interacting particle partners. In order to meet the Pauli principle, physical objects should exchange information about the (opposite) states of each other before getting into active interaction. The explanation led to the loss of an invariance property. However, this loss has been restored by introducing a new physical property (isotopic field-charge spin), by proving its conservation, and completing the Lorentz invariance with the respective invariance attributed to the newly proven conservation. References Bethe, H., Fermi, E., Über die Wechselwirkung von zwei elektronen, Zeitschrift für Physik, 77 (1932), 5-6, 296-306. Darvas, G., The Isotopic Field Charge Spin Assumption, International Journal of Theoretical Physics, 50 (2011), 10, 2961-2991.DOI: 10.1007/s10773-011-0796-9 Darvas, G., The Isotopic Field-Charge Assumption Applied to the Electromagnetic Interaction, Int J Theor Phys, 52 (2013a), 11, 3853-3869, DOI 10.1007/s10773-013-1693-1. Darvas, G., A symmetric adventure beyond the Standard Model – Isotopic field-charge spin conservation in the electromagnetic interaction, Symmetry: Culture and Science, 24 (2013b), 1-4, 17-40. Darvas, G., Electromagnetic Interaction in the Presence of Isotopic Field-Charges and a Kinetic Field, Int J Theor Phys, 53 (2014), 1, 39-51, DOI 10.1007/s10773-013-1781-2. Einstein, A., Zur Elektrodynamik bewegter Körper, Annalen der Physik, 17:891 (1905); English translation: On the electrodynamics of moving bodies, in: The Principle of Relativity, (1923), London: Methuen and Co. Møller, C., Über den Stoß zweier Teilchen unter Berücksichtigung der Retardation der Kräfte, Zeitschrift für Physik, 70 (1931), 11-12, 786-795. Noether, E. A., Invariante Variationsprobleme, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physikalische Klasse, (1918) 235-257. Wigner, E.P. , Events, laws of nature and invariance principles, pp. 38-50, in: Wigner, Symmetries and Reflections, Bloomington: Indiana University Press (1967).

ISIS Summit: The Information Society at the Crossroads•Jun 30, 2015•0 citations
We present recent works [15,16] on the thermodynamic limit of quantum Coulomb systems. We provide a general method which allows to show the existence of the limit for many different systems. Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via Coulomb forces. The potential between two particles of charges z and z ′ located at x and x ′ in R 3 is zz ′ |x − x ′ |. There are two difficulties which occur when trying to describe systems composed of electrons and nuclei. Both have to do with the physical problem Talk given by M.L. at QMath10, 10th Quantum Mathematics International Conference, Moeciu, Romania, September 10–15, 2007. c○2008 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. 2 of stability of quantum systems. The first is due to the singularity of 1/|x | at 0: it is necessary to explain why a particle will not rush to a particle of the opposite charge. One of the first major triumphs of the theory of quantum mechanics is the explanation it gives of the stability of the hydrogen atom (and the complete description of its spectrum) and of other microscopic quantum Coulomb systems, via the uncertainty principle. Stability means that the total energy of the considered system cannot be arbitrarily negative. If there was no such lower bound to the energy it would be possible in principle to extract an infinite amount of energy. One often refers to this kind of stability as stability of the first kind [19,20]. If we denote by E(N) the ground state energy of the system under consideration, for N particles stability of the first kind can be written E(N)> −∞. (1) In proving (1) for Coulomb systems, a major role is played by the uncertainty principle which for nonrelativistic systems is mathematically expressed by the critical Sobolev embedding H 1 (R 3) ֒ → L 6 (R 3). The latter allows to prove Kato’s inequality
This thesis is devoted to the rigorous study of dilute Bose gases with two-body and three-body interactions at zero temperature in two and three dimensions. Bose gases are many-body quantum systems composed of bosons, which notably do not obey the Pauli exclusion principle. As a consequence, at extremely low temperatures, a macroscopic fraction of the particles can occupy the same one-body quantum state. This phenomenon, known as Bose–Einstein condensation, was predicted in the 1920s by Bose and Einstein for noninteracting Bose gases, and first realised experimentally by the groups of Cornell and Wieman, and Ketterle, earning them the Nobel Prize for Physics in 2001. However, proving the occurrence of Bose–Einstein condensation in general interacting bosonic systems remains a major open problem and constitutes a central theme in this thesis. Another important property of bosonic matter is that it is generally more unstable than fermionic matter. In particular, for systems interacting via the Coulomb force, the Pauli exclusion principle is essential for explaining the stability of the second kind, which states that the energy and volume scale linearly with the number of particles. Nonetheless, bosonic systems with attractive but less singular interactions can also be stable, and the rigorous proof of this fact forms a second focus of this thesis. In three dimensions, we consider predominantly repulsive interactions. Firstly, we prove a leading order upper bound on the thermodynamic ground state energy of Bose gases with hard-core three-body interactions, which matches an existing lower bound. In addition, we obtain a simpler proof for integrable potentials. Secondly, we derive the leading order of the thermodynamic ground state energy of Bose gases with combined two-body and three-body interactions. More specifically, we show that, at the leading order, the ground state energy is entirely characterised by the scattering length of the two-body potential and the scattering hypervolume of the three-body potential. Thirdly, we study the time evolution of Bose–Einstein condensates with three-body interactions in the so-called Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation. In particular, we show that the effective coupling constant is universal in the sense that it depends only on the scattering hypervolume of the interaction potential. As a by-product of our proof, we derive a stability of the second kind result for Bose gases with attractive two-body interactions and repulsive three-body interactions. In two dimensions, we consider trapped Bose gases with mean-field-type interactions, including the purely attractive case. We prove the stability of the second kind of the system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger energy functional. As a consequence of the stability of the second kind, we also obtain Bose–Einstein condensation for the many-body ground states.
Abstract This chapter opens with the notion of stable interaction in quantum systems, somewhat different from classical systems. It is shown that an unstable classical system can become stable when treated quantum mechanically. The remaining of the chapter is devoted to a brief survey of the main results concerning the stability of Coulomb systems. Explicit evaluations of the total energy show that the exclusion principle is essential for their stability. The final part of the chapter deals with the thermodynamic limit of globally neutral Coulomb systems. The limit exists in spite of the long-range character of the interaction, and contrary to the stability property, the existence of the limit does not require quantum mechanics, being valid also in classical Coulomb systems.
This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classical field theory. In this setting we also provide a rigorous non-perturbative construction of the LW functional via convex duality and prove several key properties, including continuity up to the boundary of its domain and asymptotics in the limit of large interaction.Quantum embedding methods, meanwhile, view a large system as being composed of smaller fragments that are treated with high accuracy and embedded in the larger system in a mutually consistent way. Inspired by a connection between the boundary analysis of the LW functional and embedding, we perform similar analysis for the 1-RDM theory for fermionic systems, which is also developed via convex duality, illustrating a relation to fermionic embedding methods such as the density matrix embedding theory (DMET).Another embedding method of note is the dynamical mean-field theory (DMFT), which is at the same time a Green's function method that can be understood in terms of the LW functional. DMFT relies on the solution of impurity problems, which specify the embedding of an interacting system into a non-interacting bath. Underlying DMFT is a result about the sparsity pattern of the self-energy matrix for impurity problems, which to our knowledge has not been proved in the literature. We provide a rigorous proof of this result in various classical and quantum settings. We go on to investigate the fermionic DMFT in depth, identifying the key mathematical structures that appear in the algorithmic loop for solving it and using these to prove the well-posedness of this loop, in a certain sense.Finally, we introduce a suite of new approaches to the quantum many-body problem that provide variational lower bounds to the ground-state energy. These methods, which combine the themes of convexity and embedding, are based on novel convex relaxations of the variational principles for the ground-state energies of many-body systems. To begin, we recover a second-quantized version of the formalism of strictly correlated electrons (SCE), which yields an exact expression for the exchange-correlation functional in Kohn-Sham density functional theory in the limit of infinite Coulomb repulsion in terms of the solution of a multi-marginal optimal transport problem. We introduce a semidefinite relaxation method for approximately solving this problem and obtaining a lower bound for the ground-state energy. The ideas underlying this relaxation are generalized considerably, outside the context of SCE, to yield much tighter lower bounds, which we validate numerically for both quantum spin systems and fermionic systems. We also describe how these relaxation methods can be interpreted as embedding methods via convex duality.
Introduction My presentation aims at answering a part of the questions posted in the invitation to the session. The questions arose in and originated from the FIS discussion on Physical informatics … in October 2014. No doubt, they were formulated in provocative manner. The goal was to challenge discussion. I plan to illustrate my personal answers with a few examples quoted from the history of 20th c. physics. My answers to the questions are not intended to be enunciations and to provide final solutions, rather they serve as arguments and indicate that nothing is closed, the discussion is open. Methods 1 What do we consider physical information? Can one speak about physical information when there is no live percipient to accept, evaluate and use it? Can one speak about physical information (e.g., signal exchange) between inanimate physical objects (cf., e.g., Feynman diagrams)? And if so, what is it for? Is (physical) information a passive phenomenon, or its existence presumes activity? Interpretation of ‘activity’ plays important role in the possible answers. One of the interpretations says that activity is an antropomorph phenomenon. It is a privilege of the human mind, which is able to perceive and process information, able to teleologically evaluate its possible consequences and (re)act accordingly. Another interpretation says that there is inanimate activity, that means, reception of information between physical agents and their reaction to it. I argue for the latter concept. Let us see the example of the interaction between two (electric) charges. There were two approaches in the classical age of developing these theories (1928-33). When two electric charges interact, there appear two types of interaction. One is a Coulomb-type repulsion/attraction (according to their mutual signs) governed by their scalar potential, and the other is a Lorentz-type one governed by their vector potential depending on their relative velocity to each other. One type of the theories considered first the interaction between the scalar potentials and calculated the effect of the vector potentials as perturbation. It considered two charges approaching to each other from the infinity, when in first approximation they have got information on the amount of the Coulomb charge of the other, but not its velocity and the caused Lorentz force. The latter was considered in the perturbation process. The other type of the theories considered first the effect of the Lorentz force of the approaching charges, and took into consideration the effect of the Coulomb force in course of the perturbation. Representatives of both types of theories agreed that the roles of the interacting electric charges must be symmetrical, but Christian Møller. Møller [7], who belonged to the latter type of theoreticians, applied scattering matrices (1931). He showed that there appeared a component among the matrix elements that was asymmetric in respect to the two interacting charges. H. Bethe (that time a doctoral student of E. Fermi, 1932) could not accept this asymmetry and ‘corrected’ Møller’s theory. He ‘symmetrised’ those matrix elements artificially [1]. That was a rough and unjustified involvement in Møller’s equation, but due to the later attained high authority of both Bethe and Fermi, the physics community accepted the apparently convenient symmetrisation of the theory without discussion and has not treated it until the recent years. Thus any possible distinction between (roles and properties of) interacting charges were unrevealed until the past decade. However, the question can be formulated so: how do the charges, ready to interact, get information from each other? All theories agree that interaction between two charges take place by the exchange of a boson. In case of electromagnetic interaction this boson is a photon. Which of them emits the first photon towards the other? Did this emitter receive any information from the direction of the partner charge prior to the photon emission? What else is this if not an asymmetry between the roles, and possibly between properties of the two interacting charges? The distinction between the interacting charges was introduced by the isotopic field-charge theory, and the notion of ‘isotopic field-charge’ made the distinction between the properties of the two charges [2],[3],[4],[5]. The latter led to the proof of the gauge invariance under rotation of a newly introduced property, the isotopic field-charge spin, in an abstract field, and its conservation. This conservation is a result of a symmetry. Neverthe-less, this mechanism argues for activity between physical objects without an animated (human) agent. 2 What are the limits between (closed and open) systems, from the aspects of information and of symmetries? Further, if so, how wide can we extend the meaning of activity to be still accepted for generating information? What are the roles of different appearances of symmetries in taking a stand in the mentioned questions? What kinds of symmetry (or their absence) may play a role in making decision in the listed problems? The ‘classical’ (20th c.) relativity theories demanded that all physical laws were invariant under the Lorentz transformation. This was established first in the special theory of relativity that was formulated for electromagnetic interactions [6]. Lorentz invariance of physical laws was in fact a symmetry principle (conservation of the form of the laws during reference frame change) [9]. This invariance proved to hold for many other physical laws, so later also the symmetry principle was extended to other physical laws as well. The Lorentz invariant relativity theory included another consequence: there is no distinct (odd) reference frame in nature. In other words, all reference frames are equivalent. Aren’t they? Based on Noether’s theorems [8], one can show that conservation laws hold in all reference frames. However, the quantity of the conserved property (e.g., mass, charge, etc.) may change in the different reference frames. E.g., the amount of mass of matter in a closed system depends on the velocity of the observer relative to that system where the mass is to be measured. One can always find a reference frame from which the amount of the measured mass is minimal. This fact contradicts to the absence of a distinct (odd) reference frame. And there are more. Is there any invariance that compensates this lost equivalence of all reference frames? The observer can be not only a human agent who measures with instruments and reads the records. It can be another inanimate mass that perceives information about the mass from the observed system. Its ‘activity’ is that this information can be obtained by experiencing a force. This force can be gravitational or inertial. According to the general theory of relativity the ‘observer mass’ is unable to make distinction whether the experienced force is of ‘inertial’ or ‘gravitational’ origin. At the same time, the inertial mass changes its value according to its relative velocity to the observer, while the gravitational mass does not. Thus the ‘observer mass’ in different reference frames will experience different inertial forces originating from the ‘observed mass’. Something similar distinction can be made between the Coulomb charges and the Lorentz-type (current) charges. They are sources of the Coulomb force and the Lorentz force, respectively, and similar to the two types of masses, originate from the scalar potential and from the vector potential of the Hamiltonian of the charged object, respectively. The two types of masses and the two types of electric charges are called isotopic field-charge pairs, respectively. They are subject of the same gauge invariance. As such, they can exchange their roles (switch into each other) by the exchange of a gauge boson (in addition to the graviton and the photon, respectively), called delta bosons in the theory. As a consequence of the additional invariance and the corresponding additional mediating gauge boson, the respective systems of the two interacting isotopic field-charges (masses or electric charges) are not subjects of the Lorentz invariance alone. They are subjects of a convolution of the Lorentz- and this additional invariance. One can conclude two things. First, along with the development of physics, there is no more enough to demand invariance under the Lorentz transformation. At extended conditions, one should demand the invariance under a combination of the Lorentz invariance and an additional invariance. In short, we demand invariance under (the applicable) transformations. Secondly, when two charges (let they be either gravitational, electric, or other field-charges) interact, they make a distinction between each other. The system, composed of the interacting two field-charges, follow the Pauli principle. That means, the two interacting field-charges must be in different quantum states. Since this state in which they differ cannot be characterized by any of the earlier known properties, it must be a characteristic of the newly introduced property. It must be one of the two stable positions of the isotopic field-charge that are rotated into each other by the SU(2) symmetry group in the isotopic field-charge field. These two stable positions are called, by an analogically given name, the isotopic field-charge spin (not identical either with the angular momentum spin or the isotopic spin). According to its proven invariance, the isotopic field-charge spin is a conserved property. When two field-charges interact, they must be in the opposite isotopic field-charge spin states. The information that they exchange about each other is about this state: they check whether the partner is in the opposite state. Otherwise they are ‘not allowed’ to interact (Pauli’s exclusion principle). The information exchange takes place by the exchange of a delta boson (called also dion) between them, in addition to the exchange of the traditional mediating bosons (like graviton, photon, weak charged and neutral bosons, or gluons). That delta boson switches the emitting charge from inertial to potential state, and the absorbing charge from potential to inertial state. Conclusions The asymmetry of the interacting charges has been explained. It was subject of information exchange between the interacting particle partners. In order to meet the Pauli principle, physical objects should exchange information about the (opposite) states of each other before getting into active interaction. The explanation led to the loss of an invariance property. However, this loss has been restored by introducing a new physical property (isotopic field-charge spin), by proving its conservation, and completing the Lorentz invariance with the respective invariance attributed to the newly proven conservation. References Bethe, H., Fermi, E., Über die Wechselwirkung von zwei elektronen, Zeitschrift für Physik, 77 (1932), 5-6, 296-306. Darvas, G., The Isotopic Field Charge Spin Assumption, International Journal of Theoretical Physics, 50 (2011), 10, 2961-2991.DOI: 10.1007/s10773-011-0796-9 Darvas, G., The Isotopic Field-Charge Assumption Applied to the Electromagnetic Interaction, Int J Theor Phys, 52 (2013a), 11, 3853-3869, DOI 10.1007/s10773-013-1693-1. Darvas, G., A symmetric adventure beyond the Standard Model – Isotopic field-charge spin conservation in the electromagnetic interaction, Symmetry: Culture and Science, 24 (2013b), 1-4, 17-40. Darvas, G., Electromagnetic Interaction in the Presence of Isotopic Field-Charges and a Kinetic Field, Int J Theor Phys, 53 (2014), 1, 39-51, DOI 10.1007/s10773-013-1781-2. Einstein, A., Zur Elektrodynamik bewegter Körper, Annalen der Physik, 17:891 (1905); English translation: On the electrodynamics of moving bodies, in: The Principle of Relativity, (1923), London: Methuen and Co. Møller, C., Über den Stoß zweier Teilchen unter Berücksichtigung der Retardation der Kräfte, Zeitschrift für Physik, 70 (1931), 11-12, 786-795. Noether, E. A., Invariante Variationsprobleme, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physikalische Klasse, (1918) 235-257. Wigner, E.P. , Events, laws of nature and invariance principles, pp. 38-50, in: Wigner, Symmetries and Reflections, Bloomington: Indiana University Press (1967).