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Vacuum polarization in uranium
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Abstract
We formulate the Hartree–Fock method using a functional integral approach. Then we consider a nonperturbative component of the vacuum polarization. For the Dirac–Coulomb operator the renormalization flow of the vacuum polarization is calculated numerically. For the Hartree–Fock operator the polarization is obtained by integrating an appropriately rescaled flow. The text includes an approximate calculation of the vacuum polarization in Uranium.
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For neutral and positively charged atoms and molecules, we prove the existence of infinitely many Hartree-Fock critical points below the first energy threshold (that is, the lowest energy of the same system with one electron removed). This is the equivalent, in Hartree-Fock theory, of the famous Zhislin-Sigalov theorem which states the existence of infinitely many eigenvalues below the bottom of the essential spectrum of the N-particle linear Schr\"odinger operator. Our result improves a theorem of Lions in 1987 who already constructed infinitely many Hartree-Fock critical points, but with much higher energy. Our main contribution is the proof that the Hartree-Fock functional satisfies the Palais-Smale property below the first energy threshold. We then use minimax methods in the N-particle space, instead of working in the one-particle space.
We develop a Renormalization Group (RG) approach to the study of existence
and uniqueness of solutions to stochastic partial differential equations driven
by space-time white noise. As an example we prove well-posedness and
independence of regularization for the model in three dimensions
recently studied by Hairer. Our method is "Wilsonian": the RG allows to
construct effective equations on successive space time scales. Renormalization
is needed to control the parameters in these equations. In particular no theory
of multiplication of distributions enters our approach.
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Using the Pauli-Villars regularization and arguments from convex analysis, we
construct solutions to the classical time-independent Maxwell equations in
Dirac's vacuum, in the presence of small external electromagnetic sources. The
vacuum is not an empty space, but rather a quantum fluctuating medium which
behaves as a nonlinear polarizable material. Its behavior is described by a
Dirac equation involving infinitely many particles. The quantum corrections to
the usual Maxwell equations are nonlinear and nonlocal. Even if photons are
described by a purely classical electromagnetic field, the resulting vacuum
polarization coincides to first order with that of full Quantum
Electrodynamics.
There is a strong intuitive understanding of renormalization, due to Wilson, in terms of the scaling of effective lagrangians. We show that this can be made the basis for a proof of perturbative renormalization. We first study renormalizability in the language of renormalization group flows for a toy renormalization group equation. We then derive an exact renormalization group equation for a four-dimensional λø4 theory with a momentum cutoff. We organize the cutoff dependence of the effective lagrangian into relevant and irrelevant parts, and derive a linear equation for the irrelevant part. A lengthy but straightforward argument establishes that the piece identified as irrelevant actually is so in perturbation theory. This implies renormalizability. The method extends immediately to any system in which a momentum-space cutoff can be used, but the principle is more general and should apply for any physical cutoff. Neither Weinberg's theorem nor arguments based on the topology of graphs are needed.
We derive a new exact evolution equation for the scale dependence of an effective action. The corresponding equation for the effective potential permits a useful truncation. This allows one to deal with the infrared problems of theories with massless modes in less than four dimensions which are relevant for the high temperature phase transition in particle physics or the computation of critical exponents in statistical mechanics.
1. Relativistic Wave Equation for Spin-0 Particles: The Klein-Gordon Equation and Its Applications.- 2. A Wave Equation for Spin-1/2 Particles: The Dirac Equation.- 3. Lorentz Covariance of the Dirac Equation.- 4. Spinors Under Spatial Reflection.- 5. Bilinear Covariants of the Dirac Spinors.- 6. Another Way of Constructing Solutions of the Free Dirac Equation: Construction by Lorentz Transformations.- 7. Projection Operators for Energy and Spin.- 8. Wave Packets of Plane Dirac Waves.- 9. Dirac Particles in External Fields: Examples and Problems.- 10.The Two-Centre Dirac Equation.- 11. The Foldy-Wouthuysen Representation for Free Particles.- 12. The Hole Theory.- 13. Klein's Paradox.- 14. The Weyl Equation - The Neutrino.- 15. Wave Equations for Particles with Arbitrary Spins.- 16. Lorentz Invariance and Relativistic Symmetry Principles.
Statistical mechanics is the bridge between molecular science and continuum mechanics. The input to statistical mechanics is a force law between particles. The particles can be atoms in a crystal, molecules in a gas or liquid, electrons in a plasma, amino acid units in a protein, elementary constituents in a complex polymer, etc. The forces between particles originate from Coulomb forces between electric charges and from magnetic dipole forces between magnetic moments. The classical force laws may be modified by the quantum mechanics (especially the Pauli exclusion principle) describing the particles. Normally the force laws are quite complicated and are not given by a simple analytical expression. Rather, they are given in one or more of the following four ways: (1) In terms of an unknown function, for which qualitative properties are postulated as laws of physics. (2) As a specific function, such as the Lennard-Jones potential or hard sphere potential. Such functions are chosen because they have representative features in common with the true force laws; for example, they may be asymptotically exact in some limiting region. (3) As a result of numerical calculation, based e.g. on the complicated exact force law and a Hartree-Fock approximation. (4) As a result of experimental measurement.
Tractable nonlinear theories generally have either a small parameter and are close to a linear theory or other exactly soluble theory, or have a definite sign and are characterized by global positivity, monotonicity, or convexity properties. These general facts apply to the study of quantum physics, and just as Section 2.4 was an introduction to expansion methods, the present chapter is an introduction to convexity methods. Generally, expansion methods apply to all interactions but only in a limited parameter range (small parameters), while convexity methods apply to a limited class of interactions (the convex ones) but over all or a large part of their parameter ranges. For statistical mechanical systems, the convexity properties, known as correlation inequalities, are expressed as general inequalities between the expectation values (i.e., the moments or correlation functions) of the system. The Lee-Yang theorem is included here because its proof and usage are closely related.
We consider the four-dimensional space with coordinates xµ (µ = 0, l, 2, 3), which — assuming the most general case — may be complex numbers. The absolute value of the position vector is given by (summation convention) (16.1) and the length s can also take complex values. Examining an orthogonal transformation aµv, which relates each point with coordinates xµ to new ones x′µ: (16.2) the absolute value should remain unchanged by this transformation (this is the fundamental, defining condition for orthogonal transformations), i.e.
A study is carried out of the vacuum polarization in a strong Coulomb field. Radiative corrections are neglected. A perturbation calculation is avoided by making use of the explicit solutions of the Dirac equation in a Coulomb field. The Laplace transform of the polarization charge density times is found and used as a basis for further study. It is proved to be an analytic function of the strength of the inducing charge. It is verified that the first-order term in a power series expansion in the strength of the inducing charge just corresponds to the Uehling potential. The third-order term is studied in some detail. The leading term in the polarization potential close to the inducing charge and the space integral of the induced potential divided by r are found to all orders in the strength of the inducing charge. Ambiguities are handled by a method corresponding to regularization.
The Quantum Theory of Fields, first published in 1996, is a self-contained, comprehensive introduction to quantum field theory from Nobel Laureate Steven Weinberg. Volume II gives an account of the methods of quantum field theory, and how they have led to an understanding of the weak, strong, and electromagnetic interactions of the elementary particles. The presentation of modern mathematical methods is throughout interwoven with accounts of the problems of elementary particle physics and condensed matter physics to which they have been applied. Many topics are included that are not usually found in books on quantum field theory. The book is peppered with examples and insights from the author's experience as a leader of elementary particle physics. Exercises are included at the end of each chapter.
Nobel Laureate Steven Weinberg continues his masterly exposition of
quantum field theory. This third volume of The Quantum Theory of Fields
presents a self-contained, up-to-date and comprehensive introduction to
supersymmetry, a highly active area of theoretical physics that is
likely to be at the center of future progress in the physics of
elementary particles and gravitation. The text introduces and explains a
broad range of topics, including supersymmetric algebras, supersymmetric
field theories, extended supersymmetry, supergraphs, nonperturbative
results, theories of supersymmetry in higher dimensions, and
supergravity. A thorough review is given of the phenomenological
implications of supersymmetry, including theories of both gauge and
gravitationally-mediated supersymmetry breaking. Also provided is an
introduction to mathematical techniques, based on holomorphy and
duality, that have proved so fruitful in recent developments. This book
contains much material not found in other books on supersymmetry, some
of it published here for the first time. Problems are included.
The notion of renormalization is at the core of several spectacular achievements of contemporary physics, and in the last years powerful techniques have been developed allowing to put renormalization on a firm mathematical basis. This book provides a self-consistent and accessible introduction to the sophisticated tools used in the modern theory of non-perturbative renormalization, allowing an unified and rigorous treatment of Quantum Field Theory, Statistical Physics and Condensed Matter models. In particular the first part of this book is devoted to Constructive Quantum Field Theory, providing a mathematical construction of models at low dimensions and discussing the removal of the ultraviolet and infrared cut-off, the verification of the axioms and the validity of Ward Identities with the relative anomalies. The second part is devoted to lattice 2D Statistical Physics, analyzing in particular the theory of universality in perturbed Ising models and the computation of the non-universal critical indices in Vertex or Ashkin-Teller models. Finally the third part is devoted to the analysis of complex quantum fluids showing Luttinger of Fermi liquid behavior, like the 1D or 2D Hubbard model. © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
We investigate the possibility that radiative corrections may produce spontaneous symmetry breakdown in theories for which the semiclassical (tree) approximation does not indicate such breakdown. The simplest model in which this phenomenon occurs is the electrodynamics of massless scalar mesons. We find (for small coupling constants) that this theory more closely resembles the theory with an imaginary mass (the Abelian Higgs model) than one with a positive mass; spontaneous symmetry breaking occurs, and the theory becomes a theory of a massive vector meson and a massive scalar meson. The scalar-to-vector mass ratio is computable as a power series in e, the electromagnetic coupling constant. We find, to lowest order, m2(S)/m2(V)=(3/2π)(e2/4π). We extend our analysis to non-Abelian gauge theories, and find qualitatively similar results. Our methods are also applicable to theories in which the tree approximation indicates the occurrence of spontaneous symmetry breakdown, but does not give complete information about its character. (This typically occurs when the scalar-meson part of the Lagrangian admits a greater symmetry group than the total Lagrangian.) We indicate how to use our methods in these cases.
This chapter describes asymptotic formulae of quantum electrodynamics. The asymptotic constancy of the total cross-section is a characteristic property of scattering processes whose diagrams can be cut across internal photon lines. It occurs even when more than two particles are present in the final state of the reaction. The double-logarithmic corrections occur in cases of two kinds. One kind includes scattering through a fixed finite angle. The cross-sections decrease in the asymptotic high-energy range. In such cases, the double-logarithmic corrections are associated with the infrared divergence. These cases include elastic scattering of electrons in an external Coulomb field. The other class of cases includes reaction cross-sections, which decrease with increasing energy for a given square of the momentum transfer, that is, for scattering angles, which asymptotically approach zero or π.
The renormalized propagation functions DFC and SFC for photons and electrons, respectively, are investigated for momenta much greater than the mass of the electron. It is found that in this region the individual terms of the perturbation series to all orders in the coupling constant take on very simple asymptotic forms. An attempt to sum the entire series is only partially successful. It is found that the series satisfy certain functional equations by virtue of the renormalizability of the theory. If photon self-energy parts are omitted from the series, so that DFC=DF, then SFC has the asymptotic form A[p2/m2]n[iγ·p]-1, where A=A(e12) and n=n(e12). When all diagrams are included, less specific results are found. One conclusion is that the shape of the charge distribution surrounding a test charge in the vacuum does not, at small distances, depend on the coupling constant except through a scale factor. The behavior of the propagation functions for large momenta is related to the magnitude of the renormalization constants in the theory. Thus it is shown that the unrenormalized coupling constant e02/4πℏc, which appears in perturbation theory as a power series in the renormalized coupling constant e12/4πℏc with divergent coefficients, may behave in either of two ways: (a) It may really be infinite as perturbation theory indicates; (b) It may be a finite number independent of e12/4πℏc.
The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic.
A study is carried out of the vacuum polarization in a strong Coulomb field. Radiative corrections are neglected. A perturbation calculation is avoided by making use of the explicit solutions of the Dirac equation in a Coulomb field. The Laplace transform of the polarization charge density times r2 is found and used as a basis for further study. It is proved to be an analytic function of the strength of the inducing charge. It is verified that the first-order term in a power series expansion in the strength of the inducing charge just corresponds to the Uehling potential. The third-order term is studied in some detail. The leading term in the polarization potential close to the inducing charge and the space integral of the induced potential divided by r are found to all orders in the strength of the inducing charge. Ambiguities are handled by a method corresponding to regularization.Some experimental applications are considered. The corrections to the Uehling term in these cases are found to be small.
In this article I present a physically motivated renormalization scheme for Coulomb-gauge QED. This scheme is useful in calculations involving QED bound states. I implement this scheme to one loop by calculating the electron self-energy, the electron self-mass, and the renormalization constants Z/sub 1/ and Z/sub 2/. Formulas for the dimensional regularization of some noncovariant integrals useful in one-loop Coulomb-gauge calculations are given.
By use of the path-integral formulation of quantum mechanics, a series expansion for the effective potential is derived. Each order of the series corresponds to an infinite set of conventional Feynman diagrams, with a fixed number of loops. As an application of the formalism, three calculations are performed. For a set of n self-interacting scalar fields, the effective potential is computed to the two-loop approximation. Also, all loops are summed in the leading-logarithmic approximation when n gets large. Finally, the effective potential for scalar, massless electrodynamics is derived in an arbitrary gauge. It is found that the potential is gauge-dependent, and a specific gauge is exhibited in which all one-loop effects disappear.
Two-dimensional massless fermion field theories with quartic interactions are analyzed. These models are asymptotically free. The models are expanded in powers of 1/N, where N is the number of components of the fermion field. In such an expansion one can explicitly sum to all orders in the coupling constants. It is found that dynamical symmetry breaking occurs in these models for any value of the coupling constant. The resulting theories produce a fermion mass dynamically, in addition to a scalar bound state and, if the broken symmetry is continuous, a Goldstone boson. The resulting theories contain no adjustable parameters. The search for symmetry breaking is performed using functional techniques, the new feature here being that a composite field, say ψ̅ ψ, develops a nonvanishing vacuum expectation value. The "potential" of composite fields is discussed and constructed. General results are derived for arbitrary theories in which all masses are generated dynamically. It is proved that in asymptotically free theories the dynamical masses must depend on the coupling constants in a nonanalytic fashion, vanishing exponentially when these vanish. It is argued that infrared-stable theories, such as massless-fermion quantum electrodynamics, cannot produce masses dynamically. Four-dimensional scalar field theories with quartic interactions are analyzed in the large-N limit and are shown to yield unphysical results. The models are extended to include gauge fields. It is then found that the gauge vector mesons acquire a mass through a dynamical Higgs mechanism. The higher-order corrections, of order 1/N, to the models are analyzed. Essential singularities, of the Borel-summable type, are discovered at zero coupling constant. The origin of the singularities is the ultraviolet behavior of the theory.
The scalar λ0φ4 interaction and the Fermi interaction G0(ψ̅ ψ)2 are studied for space-time dimension d between 2 and 4. An unconventional coupling-constant renormalization is used: λ0=u0Λε (ε=4-d) and G0=g0Λ2-d, with u0 and g0 held fixed as the cutoff Λ→∞. The theories can be solved in two limits: (1) the limit N→∞ where φ and ψ are fields with N components, and (2) the limit of small ε, as a power series in ε. Both theories exhibit scale invariance with anomalous dimensions in the zero-mass limit. For small ε, the fields φ, φ2, and φ∇α1·∇αnφ all have anomalous dimensions, except for the stress-energy tensor. These anomalous dimensions are calculated through order ε2; they are remarkably close to canonical except for φ2. The (ψ̅ ψ)2 interaction is studied only for large N; for small ε it generates a weakly interacting composite boson. Both the φ4 and (ψ̅ ψ)2 theories as solved here reduce to trivial free-field theories for ε→0. This paper is motivated by previous work in classical statistical mechanics by Stanley (the N→∞ limit) and by Fisher and Wilson (the ε expansion).
Several topics involving renormalization group ideas are reviewed. The
solution of the s-wave Kondo Hamiltonian, describing a single magnetic impurity
in a nonmagnetic metal, is explained in detail. ''Block spin'' methods, applied
to the two dimensional Ising model, are explained. A relatively short review of
basic renormalization group ideas is given, mainly in the context of critical
phenomena. The relationship of the modern renormalization group to the older
problems of divergences in statistical mechanics and field theory and field
theoretic renormalization is discussed. The special case of ''marginal
variables'' is discussed in detail, along with the relationship of the modern
renormalization group to its original formulation by Gell-Mann and Low and
others.
An exact renormalization equation is derived by making an infinitesimal
change in the cutoff in momentum space. From this equation the expansion
for critical exponents around dimensionality 4 and the limit n=∞
of the n-vector model are calculated. We obtain agreement with the
results of Wilson and Fisher, and with the spherical model.
URL: http://www-spht.cea.fr/articles/t99/151
The quantum theory of the electron allows states of negative kinetic energy as well as the usual states of positive kinetic energy and also allows transitions from one kind of state to the other. Now particles in states of negative kinetic energy are never observed in practice. We can get over this discrepancy between theory and observation by assuming that, in the world as we know it, nearly all the states of negative kinetic energy are occupied, with one electron in each state in accordance with Pauli's exclusion principle, and that the distribution of negative-energy electrons is unobservable to us on account of its uniformity. Any unoccupied negative-energy states would be observable to us, as holes in the distribution of negative-energy electrons, but these holes would appear as particles with positive kinetic energy and thus not as things foreign to all our experience. It seems reasonable and in agreement with all the facts known at present to identify these holes with the recently discovered positrons and thus to obtain a theory of the positron.(Received February 02 1934)(Accepted March 05 1934)
The last in a series of three papers in which it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. The work presented here is concerned with the reduced Green functions which arise in second-order stationary state perturbation theory. Using a simple linear transformation of the four radial parts of the relativistic reduced Green function it is shown how the non-relativistic and relativistic functions are special instances of the solution of a general second-order differential equation. The general solution of this equation is exhibited in the form of a Sturmian expansion, and complete solutions in both cases are presented. Recursion relations are deduced for the radial parts of both reduced Green functions and their matrix elements are examined in detail. As a test of the given functions the second-order effect of a perturbation of the nuclear charge is calculated and is shown to agree exactly with the value expected from a simple Taylor expansion of the hydrogenic energy formula.
In this and the following two papers in this series it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. The angular part of such calculations, being well understood, is performed in the standard way. In this, the first paper, it is shown how a suitable linear transformation of the two relativistic radial wavefunctions allows the pair of relativistic coupled differential equations to be written as two uncoupled second-order equations which are simple generalizations of the corresponding non-relativistic equation. This transformation is presented in a manner which allows for a simple extension to the Green function problem. The transformed relativistic wavefunctions are explicitly derived and the normalization is presented in a novel and simple way. A new derivation is given for the recursion relations for both non-relativistic and relativistic radial wavefunctions, some of which are new. These relations are required in the subsequent papers.
We propose a new nonperturbative evolution equation for Yang-Mills theories. It describes the scale dependence of an effective action. The running of the nonabelian gauge coupling in arbitrary dimension is computed.
This collection of review lectures on topics in theoretical high energy physics has few rivals for clarity of exposition and depth of insight. Delivered over the past two decades at the International School of Subnuclear Physics in Erice, Sicily, the lectures help to organize and explain material that a the time existed in a confused state, scattered in the literature. At the time they were given they spread new ideas throughout the physics community and proved very popular as introductions to topics at the frontiers of research.
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The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equation. It is possible to write down a complete solution of the problem in terms of boundary conditions on the wave function, and this solution contains automatically all the possibilities of virtual (and real) pair formation and annihilation together with the ordinary scattering processes, including the correct relative signs of the various terms.
These notes are the second part of a common course on Renormalization Theory given with Professor P. da Veiga at X Jorge Andre Swieca Summer School, Aguas de Lindoia, Brazil, February 7-12, 1999. I emphasize the rigorous non-perturbative or constructive aspects of the theory. The usual formalism for the renormalization group in field theory or statistical mechanics is reviewed, together with its limits. The constructive formalism is introduced step by step. Taylor forest formulas allow to perform easily the cluster and Mayer expansions which are needed for a single step of the renormalization group in the case of Bosonic theories. The iteration of this single step leads to further difficulties whose solution is briefly sketched. The second part of the course is devoted to Fermionic models. These models are easier to treat on the constructive level, so they are very well suited to beginners in constructive theory. It is shown how the Taylor forest formulas allow to reorganize perturbation theory nicely in order to construct the Gross-Neveu2 model without any need for cluster or Mayer expansions. Finally applications of this technique to condensed matter and renormalization group around Fermi surface are briefly reviewed.