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Papers by Jean Zinn-Justin
From Random Walks to Random Matrices, 2019
Chapter 18 describes a few systems where the classical action has an infinite number of degenerat... more Chapter 18 describes a few systems where the classical action has an infinite number of degenerate minima but, in the quantum theory, this degeneracy is lifted by barrier penetration effects. The simplest example is the cosine periodic potential and leads to the band structure. Technically, this corresponds to the existence of instantons, solutions to classical equations in imaginary time. In all examples, we show that the classical solutions are constrained by Bogomolnyi’s inequalities, which involve topological charges associated to a winding number and defining homotopy classes. In the case of quantum chromodynamics, this leads to the famous strong CP violation problem.
From Random Walks to Random Matrices, 2019
Chapter 4 describes a few important steps which have led from the discovery of infinities in quan... more Chapter 4 describes a few important steps which have led from the discovery of infinities in quantum electrodynamics in the calculation of Feynman diagrams (ultraviolet divergences (UV divergences)) to the concept of renormalization and renormalization groups (RG). The constructions of quantum (or statistical) field theories (QFTs) and the deeply related RG have been some of the major theoretical achievements in physics of the last century. RG today plays an essential role in the understanding of the properties of QFT and of continuous macroscopic phase transitions. The existence of RG fixed points makes it possible to understand universality when there is no scale decoupling. In particle physics, it leads to the notion of effective field theory and the fine tuning problem in the Higgs particle sector.
Schwinger has pointed out that the Yang-Mills vector boson implied by associating a generalized g... more Schwinger has pointed out that the Yang-Mills vector boson implied by associating a generalized gauge transformation with a conservation law (of baryonic charge, for instance) does not necessarily have zero mass, if a certain criterion on the vacuum fluctuations of the generalized current is satisfied. We show that the theory of plasma oscillations is a simple nonrelativistic example exhibiting all of the features of Schwinger’s idea. It is also shown that Schwinger’s criterion that the vector field m 6= 0 implies that the matter spectrum before including the Yang-Mills interaction contains m = 0, but that the example of superconductivity illustrates that the physical spectrum need not. However, the absence of massless Goldstone bosons is justified by the observed properties of superconductivity and not by a theoretical argument. Some discussion and confusion about the relevance of the superconducting example, owing to its non-relativistic character, to particle physics follows (e.g...
We are interested in the question which poles of a (large degree) rational function with real pol... more We are interested in the question which poles of a (large degree) rational function with real poles and positive residuals (a Markov function) are well detected by a small degree rational interpolant, and how we can monitor by the choice of the real interpolation points such approximation properties, especially in the more delicate situation where interpolation points lie within the convex hull of the poles. Using classical orthogonality relations related to rational interpolation, we may rephrase this question as how well the zeros of a rational orthogonal function for a discrete scalar product approach the support of orthogonality. Initially, this question was motivated by the mathematically equivalent question of how well rational Ritz values approach the spectrum of a hermitian matrix [1]. In the present talk we suggest some asymptotic answer in terms of logarithmic potential theory, by generalizing work of Kuijlaars [2] (for Padé approximants at∞ or polynomial Ritz values).
Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in... more Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in quantum mechanics (QM) with Grassmann variables, a representation by field integrals of the statistical operator e−βH for the non-relativistic Fermi gas in the formalism of second quantization, and an expression for the evolution operator. Here, it is first recalled how relativistic fermions transform under the spin group. The free action for Dirac fermions is analysed, the relation between fields and particles explained, an expression for the scattering matrix obtained, and the non-relativistic limit of a model of self-coupled massive Dirac fermions derived. A formalism of Euclidean relativistic fermions is then introduced. In the Euclidean formalism: fermions transform under the fundamental representation of the spin group Spin(d) associated with the SO(d) rotation group (spin 1/2 fermions for d = 4). As for the scalar field theory, the Gaussian integral, which corresponds to a free f...
Chapter 16 deals with the important problem of quantization with symmetries, that is, how to impl... more Chapter 16 deals with the important problem of quantization with symmetries, that is, how to implement symmetries of the classical action in the corresponding quantum theory. The proposed solutions are based on methods like regularization by addition of higher order derivatives or regulator fields, or lattice regularization. Difficulties encountered in the case of chiral theories are emphasized. This may lead to obstacles for symmetric quantization called anomalies. Examples can be found in the case of chiral gauge theories. Their origin can be traced to the problem of quantum operator ordering in products. A non–perturbative regularization, also useful for numerical simulations, is based on introducing a space lattice. Difficulties appear for lattice Dirac fermions, leading the fermion doubling problem. Wilson’s fermions provide a non–chiral invariant solution. Chiral invariant solutions have been found, called overlap fermions or domain wall fermions.
This chapter has two purposes; to describe a few elements of differential geometry that are requi... more This chapter has two purposes; to describe a few elements of differential geometry that are required in different places in this work, and to provide, for completeness, a short introduction to general relativity (GR) and the problem of its quantization. A few concepts related to reparametrization (more accurately, diffeomorphism) of Riemannian manifolds, like parallel transport, affine connection, or curvature, are recalled. To define fermions on Riemannian manifolds, additional mathematical objects are required, the vielbein and the spin connection. Einstein–Hilbert's action for classical gravity GR is defined and the field equations derived. Some formal aspects of the quantization of GR, following the lines of the quantization of non-Abelian gauge theories, are described. Because GR is not renormalizable in four dimensions (even in its extended forms like supersymmetric gravity), at present time, a reasonable assumption is that GR is the low-energy, large-distance remnant of a...
Most quantum field theories (QFT) of physical interest exhibit symmetries, exact symmetries or sy... more Most quantum field theories (QFT) of physical interest exhibit symmetries, exact symmetries or symmetries with soft (e.g. linear) breaking. This chapter deals only with linear continuous symmetries corresponding to compact Lie groups. When the bare action has symmetry properties, to preserve the symmetry it is first necessary to find a symmetric regularization. The symmetry properties of the QFT then imply relations between connected correlation functions, and vertex functions, called Ward–Takahashi (WT) identities, which describe the physical consequences of the symmetry. WT identities also constrain UV divergences, and the counter-terms that render the theory finite are not of most general form allowed by power counting. As a consequence the renormalized action is expected to keep some trace of the initial symmetry. Such an analysis is based on a perturbative loop expansion. More generally, some non-trivial relations survive when to the action are added terms that induce a soft br...
Ondes, matière et Univers
Physical Review D, 1970
It is shown that an approximate summation of the" crossed-ladder" Feynman diagrams for ... more It is shown that an approximate summation of the" crossed-ladder" Feynman diagrams for the scattering of two charged particles leads to the formula s= m 1 2+ m 2 2+ 2 m 1 m 2 [1+ Z 2 α 2/(n-ε j) 2]-1/2 for the squared mass of bound states. This formula neglects radiative ...
Physics World, 1993
Lowell Brown has written this nicely self-contained text for physicists with little or no a prior... more Lowell Brown has written this nicely self-contained text for physicists with little or no a priori knowledge of quantum field theory (QFT) – those interested mainly in particle physics applications or, more generally, in fields where a good understanding of quantum electrodynamics (QED) is required. Like several modern textbooks, functional methods, in particular functional integrals, are systematically used throughout, one of the main differences to similar textbooks written in the 1960s and early 1970s.
Phase Transitions and Renormalization Group, 2007
Path Integrals in Quantum Mechanics, 2004
Phase Transitions and Renormalization Group, 2007
Journal de Physique, 1986
Quantum Field Theory and Critical Phenomena, 2002
From Random Walks to Random Matrices, 2019
Chapter 18 describes a few systems where the classical action has an infinite number of degenerat... more Chapter 18 describes a few systems where the classical action has an infinite number of degenerate minima but, in the quantum theory, this degeneracy is lifted by barrier penetration effects. The simplest example is the cosine periodic potential and leads to the band structure. Technically, this corresponds to the existence of instantons, solutions to classical equations in imaginary time. In all examples, we show that the classical solutions are constrained by Bogomolnyi’s inequalities, which involve topological charges associated to a winding number and defining homotopy classes. In the case of quantum chromodynamics, this leads to the famous strong CP violation problem.
From Random Walks to Random Matrices, 2019
Chapter 4 describes a few important steps which have led from the discovery of infinities in quan... more Chapter 4 describes a few important steps which have led from the discovery of infinities in quantum electrodynamics in the calculation of Feynman diagrams (ultraviolet divergences (UV divergences)) to the concept of renormalization and renormalization groups (RG). The constructions of quantum (or statistical) field theories (QFTs) and the deeply related RG have been some of the major theoretical achievements in physics of the last century. RG today plays an essential role in the understanding of the properties of QFT and of continuous macroscopic phase transitions. The existence of RG fixed points makes it possible to understand universality when there is no scale decoupling. In particle physics, it leads to the notion of effective field theory and the fine tuning problem in the Higgs particle sector.
Schwinger has pointed out that the Yang-Mills vector boson implied by associating a generalized g... more Schwinger has pointed out that the Yang-Mills vector boson implied by associating a generalized gauge transformation with a conservation law (of baryonic charge, for instance) does not necessarily have zero mass, if a certain criterion on the vacuum fluctuations of the generalized current is satisfied. We show that the theory of plasma oscillations is a simple nonrelativistic example exhibiting all of the features of Schwinger’s idea. It is also shown that Schwinger’s criterion that the vector field m 6= 0 implies that the matter spectrum before including the Yang-Mills interaction contains m = 0, but that the example of superconductivity illustrates that the physical spectrum need not. However, the absence of massless Goldstone bosons is justified by the observed properties of superconductivity and not by a theoretical argument. Some discussion and confusion about the relevance of the superconducting example, owing to its non-relativistic character, to particle physics follows (e.g...
We are interested in the question which poles of a (large degree) rational function with real pol... more We are interested in the question which poles of a (large degree) rational function with real poles and positive residuals (a Markov function) are well detected by a small degree rational interpolant, and how we can monitor by the choice of the real interpolation points such approximation properties, especially in the more delicate situation where interpolation points lie within the convex hull of the poles. Using classical orthogonality relations related to rational interpolation, we may rephrase this question as how well the zeros of a rational orthogonal function for a discrete scalar product approach the support of orthogonality. Initially, this question was motivated by the mathematically equivalent question of how well rational Ritz values approach the spectrum of a hermitian matrix [1]. In the present talk we suggest some asymptotic answer in terms of logarithmic potential theory, by generalizing work of Kuijlaars [2] (for Padé approximants at∞ or polynomial Ritz values).
Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in... more Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in quantum mechanics (QM) with Grassmann variables, a representation by field integrals of the statistical operator e−βH for the non-relativistic Fermi gas in the formalism of second quantization, and an expression for the evolution operator. Here, it is first recalled how relativistic fermions transform under the spin group. The free action for Dirac fermions is analysed, the relation between fields and particles explained, an expression for the scattering matrix obtained, and the non-relativistic limit of a model of self-coupled massive Dirac fermions derived. A formalism of Euclidean relativistic fermions is then introduced. In the Euclidean formalism: fermions transform under the fundamental representation of the spin group Spin(d) associated with the SO(d) rotation group (spin 1/2 fermions for d = 4). As for the scalar field theory, the Gaussian integral, which corresponds to a free f...
Chapter 16 deals with the important problem of quantization with symmetries, that is, how to impl... more Chapter 16 deals with the important problem of quantization with symmetries, that is, how to implement symmetries of the classical action in the corresponding quantum theory. The proposed solutions are based on methods like regularization by addition of higher order derivatives or regulator fields, or lattice regularization. Difficulties encountered in the case of chiral theories are emphasized. This may lead to obstacles for symmetric quantization called anomalies. Examples can be found in the case of chiral gauge theories. Their origin can be traced to the problem of quantum operator ordering in products. A non–perturbative regularization, also useful for numerical simulations, is based on introducing a space lattice. Difficulties appear for lattice Dirac fermions, leading the fermion doubling problem. Wilson’s fermions provide a non–chiral invariant solution. Chiral invariant solutions have been found, called overlap fermions or domain wall fermions.
This chapter has two purposes; to describe a few elements of differential geometry that are requi... more This chapter has two purposes; to describe a few elements of differential geometry that are required in different places in this work, and to provide, for completeness, a short introduction to general relativity (GR) and the problem of its quantization. A few concepts related to reparametrization (more accurately, diffeomorphism) of Riemannian manifolds, like parallel transport, affine connection, or curvature, are recalled. To define fermions on Riemannian manifolds, additional mathematical objects are required, the vielbein and the spin connection. Einstein–Hilbert's action for classical gravity GR is defined and the field equations derived. Some formal aspects of the quantization of GR, following the lines of the quantization of non-Abelian gauge theories, are described. Because GR is not renormalizable in four dimensions (even in its extended forms like supersymmetric gravity), at present time, a reasonable assumption is that GR is the low-energy, large-distance remnant of a...
Most quantum field theories (QFT) of physical interest exhibit symmetries, exact symmetries or sy... more Most quantum field theories (QFT) of physical interest exhibit symmetries, exact symmetries or symmetries with soft (e.g. linear) breaking. This chapter deals only with linear continuous symmetries corresponding to compact Lie groups. When the bare action has symmetry properties, to preserve the symmetry it is first necessary to find a symmetric regularization. The symmetry properties of the QFT then imply relations between connected correlation functions, and vertex functions, called Ward–Takahashi (WT) identities, which describe the physical consequences of the symmetry. WT identities also constrain UV divergences, and the counter-terms that render the theory finite are not of most general form allowed by power counting. As a consequence the renormalized action is expected to keep some trace of the initial symmetry. Such an analysis is based on a perturbative loop expansion. More generally, some non-trivial relations survive when to the action are added terms that induce a soft br...
Ondes, matière et Univers
Physical Review D, 1970
It is shown that an approximate summation of the" crossed-ladder" Feynman diagrams for ... more It is shown that an approximate summation of the" crossed-ladder" Feynman diagrams for the scattering of two charged particles leads to the formula s= m 1 2+ m 2 2+ 2 m 1 m 2 [1+ Z 2 α 2/(n-ε j) 2]-1/2 for the squared mass of bound states. This formula neglects radiative ...
Physics World, 1993
Lowell Brown has written this nicely self-contained text for physicists with little or no a prior... more Lowell Brown has written this nicely self-contained text for physicists with little or no a priori knowledge of quantum field theory (QFT) – those interested mainly in particle physics applications or, more generally, in fields where a good understanding of quantum electrodynamics (QED) is required. Like several modern textbooks, functional methods, in particular functional integrals, are systematically used throughout, one of the main differences to similar textbooks written in the 1960s and early 1970s.
Phase Transitions and Renormalization Group, 2007
Path Integrals in Quantum Mechanics, 2004
Phase Transitions and Renormalization Group, 2007
Journal de Physique, 1986
Quantum Field Theory and Critical Phenomena, 2002