Path Integration and the Functional Measure* Abstract
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Funaional Integration for Solving the Schroedinger Equation
Comitato Nazionale Energia Nucleare Funaional Integration for Solving the Schroedinger Equation A. BOVE. G. FANO. A.G. TEOLIS RT/FIMÀ(7$ Comitato Nazionale Energia Nucleare Funaional Integration for Solving the Schroedinger Equation A. BOVE, G.FANO, A.G.TEOLIS 3 1. INTRODUCTION Vesto pervenuto il 2 agosto 197) It ia well-known (aee e.g. tei. [lj - |3| ) that the infinite-diaeneional r«nctional integration ia a powerful tool in all the evolution processes toth of the type of the heat -tanefer or Scbroediager equation. Since the 'ieid of applications of functional integration ia rapidly increasing, it *«*ae dcairahle to study the «stood from the nuaerical point of view, at leaat in siaple caaea. Such a orograa baa been already carried out by Cria* and Scorer (ace ref. |«| - |7| ), via Monte-Carlo techniques. We use extensively an iteration aethod which coneiecs in successively squa£ ing < denaicy matrix (aee Eq.(9)), since in principle ic allows a great accuracy. This aetbod was used only tarely before (aee ref. |4| ). Fur- chersjore we give an catiaate of che error (e«e Eqs. (U),(26)) boch in Che caaa of Wiener and unleabeck-Ometein integral. Contrary to the cur rent opinion (aae ref. |13| ), we find that the last integral ia too sen sitive to the choice of the parameter appearing in the haraonic oscilla tor can, in order to be useful except in particular caaea. In this paper we study the one-particle spherically sysssetric case; an application to the three nucleon probUa is in progress (sea ref. |U| ). St»wp«wirìf^m<'oUN)p^woMG>miutoN«i»OMtop«fl'Ht>p>i«Wi*faif»,Piijl(Mi^fcri \ptt,mt><r\A, i uwh P.ei#*mià, Uffcio Eanfeai SaMfiM», " ~ Hot*, Vi.1. -
Chiral Gauge Theories Revisited
CERN-TH/2001-031 Chiral gauge theories revisited Lectures given at the International School of Subnuclear Physics Erice, 27 August { 5 September 2000 Martin L¨uscher ∗ CERN, Theory Division CH-1211 Geneva 23, Switzerland Contents 1. Introduction 2. Chiral gauge theories & the gauge anomaly 3. The regularization problem 4. Weyl fermions from 4+1 dimensions 5. The Ginsparg–Wilson relation 6. Gauge-invariant lattice regularization of anomaly-free theories 1. Introduction A characteristic feature of the electroweak interactions is that the left- and right- handed components of the fermion fields do not couple to the gauge fields in the same way. The term chiral gauge theory is reserved for field theories of this type, while all other gauge theories (such as QCD) are referred to as vector-like, since the gauge fields only couple to vector currents in this case. At first sight the difference appears to be mathematically insignificant, but it turns out that in many respects chiral ∗ On leave from Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany 1 νµ ν e µ W W γ e Fig. 1. Feynman diagram contributing to the muon decay at two-loop order of the electroweak interactions. The triangular subdiagram in this example is potentially anomalous and must be treated with care to ensure that gauge invariance is preserved. gauge theories are much more complicated. Their definition beyond the classical level, for example, is already highly non-trivial and it is in general extremely difficult to obtain any solid information about their non-perturbative properties. 1.1 Anomalies Most of the peculiarities in chiral gauge theories are related to the fact that the gauge symmetry tends to be violated by quantum effects. -
Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics Via the Dyson-Schwinger Equations
University of Adelaide School of Chemistry and Physics Doctor of Philosophy Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics via the Dyson-Schwinger Equations by Tom Sizer Supervisors: Professor A. G. Williams and Dr A. Kızılers¨u March 2014 Contents 1 Introduction 1 1.1 Introduction................................... 1 1.2 Dyson-SchwingerEquations . .. .. 2 1.3 Renormalization................................. 4 1.4 Dynamical Chiral Symmetry Breaking . 5 1.5 ChapterOutline................................. 5 1.6 Notation..................................... 7 2 Canonical QED 9 2.1 Canonically Quantized QED . 9 2.2 FeynmanRules ................................. 12 2.3 Analysis of Divergences & Weinberg’s Theorem . 14 2.4 ElectronPropagatorandSelf-Energy . 17 2.5 PhotonPropagatorandPolarizationTensor . 18 2.6 ProperVertex.................................. 20 2.7 Ward-TakahashiIdentity . 21 2.8 Skeleton Expansion and Dyson-Schwinger Equations . 22 2.9 Renormalization................................. 25 2.10 RenormalizedPerturbationTheory . 27 2.11 Outline Proof of Renormalizability of QED . 28 3 Functional QED 31 3.1 FullGreen’sFunctions ............................. 31 3.2 GeneratingFunctionals............................. 33 3.3 AbstractDyson-SchwingerEquations . 34 3.4 Connected and One-Particle Irreducible Green’s Functions . 35 3.5 Euclidean Field Theory . 39 3.6 QEDviaFunctionalIntegrals . 40 3.7 Regularization.................................. 42 3.7.1 Cutoff Regularization . 42 3.7.2 Pauli-Villars Regularization . 42 i 3.7.3 Lattice Regularization . 43 3.7.4 Dimensional Regularization . 44 3.8 RenormalizationoftheDSEs ......................... 45 3.9 RenormalizationGroup............................. 49 3.10BrokenScaleInvariance ............................ 53 4 The Choice of Vertex 55 4.1 Unrenormalized Quenched Formalism . 55 4.2 RainbowQED.................................. 57 4.2.1 Self-Energy Derivations . 58 4.2.2 Analytic Approximations . 60 4.2.3 Numerical Solutions . 62 4.3 Rainbow QED with a 4-Fermion Interaction . -
НОВОСИБИРСК Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
ИНСТИТУТ ЯДЕРНОЙ ФИЗИКИ им. Г.И. Будкера СО РАН Sudker Institute of Nuclear Physics P.G. Silvestrcv CONSTRAINED INSTANTON AND BARYON NUMBER NONCONSERVATION AT HIGH ENERGIES BIJ'DKERINP «292 НОВОСИБИРСК Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia P.G.SUvestrov CONSTRAINED INSTANTON AND BARYON NUMBER NON-CONSERVATION AT HIGH ENERGIES BUDKERINP 92-92 NOVOSIBIRSK 1992 Constrained Instanton and Baryon Number NonConservation at High Energies P.G.Silvestrov1 Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia Abstract The total cross section for baryon number violating processes at high energies is usually parametrized as <r«,toi oe exp(—.F(e)), where 5 = Л/З/EQ EQ = у/^ттпш/tx. In the present paper the third nontrivial term of the expansion is obtain^. The unknown corrections to F{e) are expected to be of 8 3 2 the order of c ' , but have neither (mhfmw) , nor log(e) enhance ment. The .total cross section is extremely sensitive to the value of single Instanton action. The correction to Instanton action A5 ~ (mp)* log(mp)/</2 is found (p is the Instanton radius). For sufficiently heavy Higgs boson the pdependent part of the Instanton action is changed drastically, in this case even the leading contribution to F(e), responsible for a growth of cross section due to the multiple produc tion of classical Wbosons, is changed: ©Budker Institute of Nuclear Physics 'етаЦ address! PStt.VESTBOV®INP.NSK.SU 1 Introduction A few years ago it was recognized, that the total crosssection for baryon 1БЭ number violating processes, which is small like ехр(4т/а) as 10~ [1] 2 (where a = р /(4тг)) at low energies, may become unsuppressed at TeV energies [2, 3, 4]. -
Supersymmetric Path Integrals
Communications in Commun. Math. Phys. 108, 605-629 (1987) Mathematical Physics © Springer-Verlag 1987 Supersymmetric Path Integrals John Lott* Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Abstract. The supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the Wiener integral. It is then pushed forward to a compact Riemannian manifold by means of a Malliavin-type construction. The relation to index theory is discussed. Introduction An interesting new branch of mathematical physics is supersymmetry. With the advent of the theory of superstrings [1], it has become important to analyze the quantum field theory of supersymmetric maps from R2 to a manifold. This should probably be done in a supersymmetric way, that is, based on the theory of supermanifolds, and in a space-time covariant way as opposed to the Hamiltonian approach. Accordingly, one wishes to make sense of supersymmetric path integrals. As a first step we study a simpler case, that of supersymmetric maps from R1 to a manifold, which gives supersymmetric quantum mechanics. As Witten has shown, supersymmetric quantum mechanics is related to the index theory of differential operators [2]. In this particular case of a supersymmetric field theory, the Witten index, which gives a criterion for dynamical supersymmetry breaking, is the ordinary index of a differential operator. If one adds the adjoint to the operator and takes the square, one obtains the Hamiltonian of the quantum mechanical theory. These indices can be formally computed by supersymmetric path integrals. For example, the Euler characteristic of a manifold M is supposed to be given by integrating e~L, with * Research supported by an NSF postdoctoral fellowship Current address: IHES, Bures-sur-Yvette, France 606 J. -
The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties
The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties Chen Sagiv1, Nir A. Sochen1, and Yehoshua Y. Zeevi2 1 Department of Applied Mathematics University of Tel Aviv Ramat-Aviv, Tel-Aviv 69978, Israel [email protected],[email protected] 2 Department of Electrical engineering Technion - Israel Institute of Technology Technion City, Haifa 32000, Israel [email protected] Abstract. The uncertainty principle is a fundamental concept in the context of signal and image processing, just as much as it has been in the framework of physics and more recently in harmonic analysis. Un- certainty principles can be derived by using a group theoretic approach. This approach yields also a formalism for finding functions which are the minimizers of the uncertainty principles. A general theorem which associates an uncertainty principle with a pair of self-adjoint operators is used in finding the minimizers of the uncertainty related to various groups. This study is concerned with the uncertainty principle in the context of the Weyl-Heisenberg, the SIM(2), the Affine and the Affine-Weyl- Heisenberg groups. We explore the relationship between the two-dimensional affine group and the SIM(2) group in terms of the uncertainty minimizers. The uncertainty principle is also extended to the Affine-Weyl-Heisenberg group in one dimension. Possible minimizers related to these groups are also presented and the scale-space properties of some of the minimizers are explored. 1 Introduction Various applications in signal and image processing call for deployment of a filter bank. The latter can be used for representation, de-noising and edge en- hancement, among other applications. -
Functional Integration and the Mind
Synthese (2007) 159:315–328 DOI 10.1007/s11229-007-9240-3 Functional integration and the mind Jakob Hohwy Published online: 26 September 2007 © Springer Science+Business Media B.V. 2007 Abstract Different cognitive functions recruit a number of different, often over- lapping, areas of the brain. Theories in cognitive and computational neuroscience are beginning to take this kind of functional integration into account. The contributions to this special issue consider what functional integration tells us about various aspects of the mind such as perception, language, volition, agency, and reward. Here, I consider how and why functional integration may matter for the mind; I discuss a general the- oretical framework, based on generative models, that may unify many of the debates surrounding functional integration and the mind; and I briefly introduce each of the contributions. Keywords Functional integration · Functional segregation · Interconnectivity · Phenomenology · fMRI · Generative models · Predictive coding 1 Introduction Across the diverse subdisciplines of the mind sciences, there is a growing awareness that, in order to properly understand the brain as the basis of the mind, our theorising and experimental work must to a higher degree incorporate the fact that activity in the brain, at the level of individual neurons, over neural populations, to cortical areas, is characterised by massive interconnectivity. In brain imaging science, this represents a move away from ‘blob-ology’, with a focus on local areas of activity, and towards functional integration, with a focus on the functional and dynamic causal relations between areas of activity. Similarly, in J. Hohwy (B) Department of Philosophy, Monash University, Clayton, VIC 3800, Australia e-mail: [email protected] 123 316 Synthese (2007) 159:315–328 theoretical neuroscience there is a renewed focus on the computational significance of the interaction between bottom-up and top-down neural signals. -
Arxiv:0812.3578V2 [Hep-Ph] 20 Aug 2017 A
arXiv:0812.357 Regularization, Renormalization, and Dimensional Analysis: Dimensional Regularization Meets Freshman E&M ∗ Fredrick Olness & Randall Scalise Department of Physics, Southern Methodist University, Dallas, TX 75275-0175, U.S.A. (Dated: August 22, 2017) We illustrate the dimensional regularization (DR) technique using a simple problem from elementary electrostatics. This example illustrates the virtues of DR without the complications of a full quan- tum field theory calculation. We contrast the DR approach with the cutoff regularization approach, and demonstrate that DR preserves the translational symmetry. We then introduce a Minimal Sub- traction (MS) and a Modified Minimal Subtraction (MS) scheme to renormalize the result. Finally, we consider dimensional transmutation as encountered in the case of compact extra-dimensions. PACS numbers: 11.30.-j Symmetry and conservation laws 11.10.Kk Field theories in dimensions other than four 11.15.-q Gauge field theories 11.10.Gh Renormalization Keywords: Renormalization, Dimensional Regularization, Regularization, Gauge Symmetries Contents VI. Renormalization 6 A. Connection to QFT 6 I. Dimensional Regularization 2 VII. The Renormalization Group Equation 6 A. Introduction and Motivation 2 A. Physical Observables: 6 B. Relating Perturbative & Non-Perturbative II. Dimension Analysis: The Pythagorean Functions 7 Theorem 2 VIII. Extra Dimensions 7 III. An Infinite Line of Charge 3 A. E and V in arbitrary dimensions 7 A. Statement of the Problem 3 B. Scale invariance: 3 IX. Conclusions 8 IV. Cutoff Regularization: 4 Acknowledgment 8 arXiv:0812.3578v2 [hep-ph] 20 Aug 2017 A. Cutoff Regularization Computation 4 X. Appendix 8 B. Computation of E and δV 4 A. 3-Dimensions 8 C. -
Introduction to Supersymmetric Theory of Stochastics
entropy Review Introduction to Supersymmetric Theory of Stochastics Igor V. Ovchinnikov Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 90095, USA; [email protected]; Tel.: +1-310-825-7626 Academic Editors: Martin Eckstein and Jay Lawrence Received: 31 October 2015; Accepted: 21 March 2016; Published: 28 March 2016 Abstract: Many natural and engineered dynamical systems, including all living objects, exhibit signatures of what can be called spontaneous dynamical long-range order (DLRO). This order’s omnipresence has long been recognized by the scientific community, as evidenced by a myriad of related concepts, theoretical and phenomenological frameworks, and experimental phenomena such as turbulence, 1/ f noise, dynamical complexity, chaos and the butterfly effect, the Richter scale for earthquakes and the scale-free statistics of other sudden processes, self-organization and pattern formation, self-organized criticality, etc. Although several successful approaches to various realizations of DLRO have been established, the universal theoretical understanding of this phenomenon remained elusive. The possibility of constructing a unified theory of DLRO has emerged recently within the approximation-free supersymmetric theory of stochastics (STS). There, DLRO is the spontaneous breakdown of the topological or de Rahm supersymmetry that all stochastic differential equations (SDEs) possess. This theory may be interesting to researchers with very different backgrounds because the ubiquitous DLRO is a truly interdisciplinary entity. The STS is also an interdisciplinary construction. This theory is based on dynamical systems theory, cohomological field theories, the theory of pseudo-Hermitian operators, and the conventional theory of SDEs. Reviewing the literature on all these mathematical disciplines can be time consuming. -
Chapter 3 Feynman Path Integral
Chapter 3 Feynman Path Integral The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications: As well as the canonical examples of a quantum particle in a single and double potential well, we discuss the generalisation of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we will temporarily leave the arena of many–body physics and second quantisation and, at least superficially, return to single–particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of functional field integral methods for many–body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various techniques and concepts of field theory, viz. stationary phase analyses of functional integrals, the Euclidean formulation of field theory, instanton techniques, and the role of topological concepts in field theory will be motivated and introduced in this chapter. 3.1 The Path Integral: General Formalism Broadly speaking, there are two basic approaches to the formulation of quantum mechan- ics: the ‘operator approach’ based on the canonical quantisation of physical observables Concepts in Theoretical Physics 64 CHAPTER 3. -
The Quantum Vacuum and the Cosmological Constant Problem
The Quantum Vacuum and the Cosmological Constant Problem S.E. Rugh∗and H. Zinkernagely To appear in Studies in History and Philosophy of Modern Physics Abstract - The cosmological constant problem arises at the intersection be- tween general relativity and quantum field theory, and is regarded as a fun- damental problem in modern physics. In this paper we describe the historical and conceptual origin of the cosmological constant problem which is intimately connected to the vacuum concept in quantum field theory. We critically dis- cuss how the problem rests on the notion of physically real vacuum energy, and which relations between general relativity and quantum field theory are assumed in order to make the problem well-defined. 1. Introduction Is empty space really empty? In the quantum field theories (QFT’s) which underlie modern particle physics, the notion of empty space has been replaced with that of a vacuum state, defined to be the ground (lowest energy density) state of a collection of quantum fields. A peculiar and truly quantum mechanical feature of the quantum fields is that they exhibit zero-point fluctuations everywhere in space, even in regions which are otherwise ‘empty’ (i.e. devoid of matter and radiation). These zero-point fluctuations of the quantum fields, as well as other ‘vacuum phenomena’ of quantum field theory, give rise to an enormous vacuum energy density ρvac. As we shall see, this vacuum energy density is believed to act as a contribution to the cosmological constant Λ appearing in Einstein’s field equations from 1917, 1 8πG R g R Λg = T (1) µν − 2 µν − µν c4 µν where Rµν and R refer to the curvature of spacetime, gµν is the metric, Tµν the energy-momentum tensor, G the gravitational constant, and c the speed of light. -
Mathematical Aspects of Functional Integration
Pro gradu –tutkielma MATHEMATICAL ASPECTS OF FUNCTIONAL INTEGRATION Aku Valtakoski 22.11.2000 Ohjaaja: Christofer Cronström Tarkastajat: Christofer Cronström Keijo Kajantie HELSINGIN YLIOPISTO FYSIIKAN LAITOS TEOREETTISEN FYSIIKAN OSASTO PL 9 (Siltavuorenpenger 20 D) 00014 Helsingin Yliopisto ÀÄËÁÆÁÆ ÄÁÇÈÁËÌÇ ÀÄËÁÆÇÊË ÍÆÁÎÊËÁÌÌ ÍÆÁÎÊËÁÌ Ç ÀÄËÁÆÃÁ ÌÙÒØ»Ç××ØÓ ÙÐØØ»ËØÓÒ Ý ÄØÓ× ÁÒרØÙØÓÒ ÔÖØÑÒØ Ý Ó ÔÖØÑÒØ Ó È Ì ÓÖØØÖ ÙØÓÖ Ù ÎÐØÓ× ÌÝÓÒ ÒÑ Ö ØØ× ØØÐ ÌØÐ ×Ô Ó ÁÒØÖØÓÒ ÇÔÔÒ ÄÖÓÑÒ ËÙ Ô ÌÝÓÒ Ð Ö ØØ× ÖØ ÄÚÐ ØÙÑ ÅÓÒØ Ò ÝÖ ËÚÙÑÖ ËÓÒØÐ ÆÙÑ Ö Ó Ô× Å×ØÖ³× Ø×× ÆÓÚÑ Ö ¾¼¼¼ ÌÚרÐÑ ÊÖØ ÁÒ Ø Ðר ¬ØÝ ÝÖ× ÒØÖØÓÒ × Ò ÑÔ ÓÖØÒØ ØÓ ÓÐ Ò Ô Ò ÑØ¹ ÒØÖØÓÒ ÒÕÙ× Ö ÚÖÝ ÚÖ×ØÐ Ò ÔÔÐ ØÓ ÚÖÓÙ× ØÝÔ × Ó Ô ÔÖÓÐÑ׺ ÒØÖØÓÒ × Ð×Ó ÑÓÖ ØÒ Ùר ÑØÓ ÓÖ ÜÑÔи Ø ÝÒÑÒ ÔØ ÒØÖÐ × Ò Ò ÒÔ ÒÒØ ÓÖÑÙÐØÓÒ Ó ÕÙÒØÙÑ ÒØÖØÓÒ Ú× ÒØÓ ØÛÓ ×Ù Ø ÏÒÖ ÒØÖÐ ØÖ Ò ÛØ «Ù×ÓÒ Ò Ø ÝÒÑÒ ÔØ ÒØÖÐ × Ù× ØÓ ÕÙÒØÙÑ ÔÒÓÑÒº ÁÒ Ø× ÛÓÖ Û ÖÚÛ Ø ÔÖÓÔ ÖØ× Ó Ø ØÛÓ ÒØÖÐ× Ò ØÑ ØÓ ÓØÖº Ï ¬Ò ØØ ÚÒ ØÓÙ ØÝ Ú ÐÑÓר Ø ×Ñ Ø ØÛÒ ØÑ Ö ÔÖÓÓÙÒº ËÔ ÑÔ×× × ÚÒ ØÓ Ø ÝÒÑÒ ÔØ ÒØÖк Ï Ñ Ø Ó×ÖÚØÓÒ ØØ Ø × ÒÓØ ØÖÙ ÒØÖÐ ÓÚÖ Ó Ð Ø ÏÒÖ ÒØÖк ÁØ × ÓÒÐÝ ×ÓÖØÒ ÒÓØØÓÒ ÓÖ ÐÑØ Ó ÑÙÐØÔÐ ÒØÖÐ׺ ÌÖ Ú Ò ×ÚÖÐ ØØÑÔØ× ØÓ ÓÖÑÙÐØ ¬ÒØÓÒ Ó ÝÒÑÒ ÔØ ÒØÖÐ× ØØ ÛÓÙÐ ×ÓÙÒº ËÓÑ Ó Ø× Ö ÖÚÛ Ò Ø× ÛÓÖº ÚÒ ØÓÙ ØÝ Ú ×ÓÑ ÒÓÒ Ó ØÑ × Ú Ø ÒØÙØÚÒ×× Ó Ø ÓÖÒÐ ¬ÒØÓÒ Ý ÝÒÑÒº Ï Ð×Ó ¬Ò ØÑ ÓØÒ ØÓ Ó ØÓ Ù×Ùк ÅÓÖ ×ÓÙÐ ØÙ× Ñ Ø ¬ÒÒ ÔÖÓÔ Ö ¬ÒØÓÒ ÓÖ ÝÒÑÒ ÔØ ÒØÖÐ׺ Ì× × ÔÖÓÑÔØ Ý ØÖ Û×ÔÖ Ù× ÓØÒ Ø ÔÖÓÐÑ× Ó Ø ¬ÒØÓÒº ÁÒ ØÓÒ ØÓ Ø ÓÒ Ø Ó ÒØÖØÓÒ Û ×ÓÑ ÑÓÖ ×Ô ×Ù × ×ØÓ ÒØÖØÓÒ¸