DOCSLIB.ORG

Mathematical Aspects of Functional Integration

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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× ÁÒ×ØØÙØÓÒ 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 ÌÚ×ØÐÑ ÊÖØ ÁÒ Ø Ð×Ø ¬ØÝ ÝÖ× ÒØÖØÓÒ × Ò ÑÔ ÓÖØÒØ ØÓ ÓÐ Ò Ô Ò Ñع ÒØÖØÓÒ ÒÕÙ× Ö ÚÖÝ ÚÖ×ØÐ Ò ÔÔÐ ØÓ ÚÖÓÙ× ØÝÔ × Ó Ô ÔÖÓÐÑ׺ ÒØÖØÓÒ × Ð×Ó ÑÓÖ ØÒ Ù×Ø 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 ÐÑÓ×Ø 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¸ ÔÖÓ Ó Ø ÝÒÑÒ ÔØ ÒØÖÐ Ò Ø ÝÒÑÒ ÔØ ÒØÖÐ ÓÒ ÛØ ØÙÖº Ï Ð×Ó ÔÖ×ÒØ Ò ÜØÒ×Ú Ð×Ø Ó ÓÒ Ø ×Ù ÚÒ×ÒØ ÐÓÖ ÃÝÛÓÖ× ÒØÖØÓÒ¸ ×ØÓ ÒØÖи ÏÒÖ ÒØÖи ÝÒÑÒ ÔØ ÒØÖÐ ËÐÝØÝ×Ô ÓÖÚÖÒ××ØÐÐ ÏÖ Ô Ó×Ø ÄÖÖÝ Ó È ÍÒÚÖ×ØÝ Ó ÀÐ×Ò ÅÙØ ØØÓ ÓÚÖ ÙÔÔØÖ ØÓÒÐ ÒÓÖÑØÓÒ ÀÄËÁÆÁÆ ÄÁÇÈÁËÌÇ ÀÄËÁÆÇÊË ÍÆÁÎÊËÁÌÌ ÍÆÁÎÊËÁÌ Ç ÀÄËÁÆÃÁ ÌÙÒØ»Ç××ØÓ ÙÐØØ»ËØÓÒ Ý ÄØÓ× ÁÒ×ØØÙØÓÒ ÔÖØÑÒØ Ý×Ò ÐØÓ× ÅØÑØØ×¹ÐÙÓÒÒÓÒØØÐÐÒÒ Ì ÓÖØØÖ ÙØÓÖ Ù ÎÐØÓ× ÌÝÓÒ ÒÑ Ö ØØ× ØØÐ ÌØÐ ×Ô Ó ÁÒØÖØÓÒ ÇÔÔÒ ÄÖÓÑÒ ËÙ ÌÓÖØØÒÒ Ý× ÌÝÓÒ Ð Ö ØØ× ÖØ ÄÚÐ ØÙÑ ÅÓÒØ Ò ÝÖ ËÚÙÑÖ ËÓÒØÐ ÆÙÑ Ö Ó Ô× ÈÖÓ ÖÙ ÅÖÖ×ÙÙ ¾¼¼¼ ÌÚ×ØÐÑ ÊÖØ ÎÑ×Ò ÚÒÝÑÑÒÒ ÚÙÓ Ò Ò ÙÒØÓÒйÒØÖÓÒÒ×Ø ÓÒ ØÙÐÐÙØ ØÖ Ý×Ò ÑØÑØÒ ØÝÓÐÙº ÙÒØÓÒйÒØÖÓÒØ ÚÓ ×ÓÚÐØ ÝÚÒ ÖÐ×Ò Ý×Ð×Ò ÓÒÐÑÒº ÌÑÒ Ð×× ×ÐÐ ÓÒ ÑÙÙØÒ ÑÖØÝ×Ø Ý× ÑÓÐÐ×ÙÙ× ÓÖÑÙÐÓ ÚÒØØÑÒ ÓÒ ÝÒÑÒÒ Ô ÓÐÙÒØÖк ÙÒØÓÒйÒØÖÐØ ÒØÙÚØ ØÒ ÐÐÙÓÒ «ÙÙ×ÓÒ ÝØÝ×× ØÚØØÚÒ ÏÒÖÒ ÒØÖÐÒ ÚÒØØÑÒÒ ÝÒÑÒÒ Ô ÓÐÙÒØÖÐÒº Ì×× ØÝÓ×× ÝÒ ÐÔ ÒÒ Ò ÒØÖÐÒ ÓÑÒ×ÙÙØ ÚÖØÐÐÒ ÒØ ×ÒÒº ËÐÚ¸ ØØ Ú ÒØÖÐÐÐ ÓÒ Ð× ×ÑÒÐÒÒ ÑØÑØØÒÒ ÖÒÒ ÓÒ ÒÒ ÚÐÐÐ ÙÓÑØØÚ ÖÓ º ÖØÝ××Ø ØÖ×ØÐÐÒ ÝÒÑÒÒ Ô ÓÐÙÒØÖк ÀÙÓÑØÒ¸ ØØ Ý××× ÓÐ ØÓ ÐÐÒÒ ÒØÖÐ ÝÐ ÙÒØÓÚÖÙÙÒ ÚÒ ÒÓ×ØÒ ÑÖÒØ ÖÐÐ×ÙÐÓØØ×Ò ÒØÖÐÒ Ö ¹ÖÚÓÐк ÃÓ× ÝÒÑÒÒ Ô ÓÐÙÒØÖÐÒ ÑÖØÐÑ ÓÐ ÑØÑØØ××Ø ÝÚÒ Ô ÖÙ×ØÐØÙ¸ ÓÚØ ÑÓÒ¹ Ø ÝÖØØÒØ ÑÙÙØØ ÑÖØÐÑ ×ØÒ¸ ØØ × ÓÐ× ÑØÑØØ××Ø Ø×ÑÐÐÒÒº Ì×× ØÝÓ×× ×ØÐÐÒ ÑÙÙØÑ ØÐÐ× ÚØÓ ØÓ× ÑÖØÐѺ Î Ò ÓÚØÒ ÑØÑØØ××Ø ÔÖѹ ÑÒ Ô ÖÙ×ØÐØÙ¸ ÓÚØ ÒÑ ÑÖØÐÑØ Ù×Ò ÐÒ ×ØÖØ ÓÐÐ×Ò ÝØ ÝØØÓÐÔ Ó× ÙÒ ÝÒÑÒÒ ÐÙÔ ÖÒÒ ÑÖØÐѺ ÃÓ× Ô ÓÐÙÒØÖÐ ÝØØÒ ØÒ ÔÚÒ ÝÐ××Ø Ð× Ó×ÐÐ Ý×Ò ÐÐÐ ß Ù×Ò ÝÐÐÑÒØÙØ ÑØÑØØ×Ø ÓÒÐÑØ ÙÒÓØÒ ß ØÙÐ× ÝÒÑÒÒ Ô ÓÐÙÒØÖÐÒ ÑØÑØØ×Ø ÑÖØÐÑ ÐÐÒ ØÙغ ÅÖØÐÑÒ Ð×× ØÝÓ×× ×ØÐÐÒ ÑÝÓ× ÖÓ×ØÙÒÑÔ Ø¸ ÙØÒ ×ØÓ×Ø×Ø ÒØÖÓÒظ ÝÒÑÒÒ Ô ÓÐÙÒØÖÐÒ ×ÖØ×ÓÒØ × Ô ÓÐÙÒØÖÐ ÖÚÐÐ ÚÖÙÙ×Ðк ÌÝÓÓÒ ××ÐØÝÝ Ð×× Ð Ð×Ø ÚØØØ ØØ ×ØØÐÚÒ ÖØÐÒ ÖÓÒº ÚÒ×ÒØ ÐÓÖ ÃÝÛÓÖ× ÙÒØÓÒйÒØÖÓÒظ ×ØÓ×ØÒÒ ÒØÖи ÏÒÖÒ ÒØÖи ÝÒÑÒÒ Ô ÓÐÙÒØÖÐ ËÐÝØÝ×Ô ÓÖÚÖÒ××ØÐÐ ÏÖ Ô Ó×Ø Ý×Ò ÐØÓ×Ò Ö×ØÓ ÅÙØ ØØÓ ÓÚÖ ÙÔÔØÖ ØÓÒÐ ÒÓÖÑØÓÒ Contents 1 Introduction 1 1.1 Onnotationandconventions . 3 1.2 Acknowledgements........................ 3 2 Brownian Motion and the Wiener Integral 4 2.1 Definitions............................. 5 2.2 PropertiesofBrownianmotion . 7 2.3 Stochasticintegration . 9 2.3.1 Itointegrals........................ 12 2.3.2 Stratonovichintegrals . 15 2.4 TheWienermeasureandintegral . 17 2.4.1 Diffusion ......................... 19 2.4.2 DiscretizationoftheWienerintegral . 21 3 Feynman Path Integrals 23 3.1 Definition ............................. 24 3.2 Problemsofdefinition. 26 3.2.1 Probabilityamplitude. 27 3.2.2 Hamiltonianformalism. 29 3.3 Alternativedefinitions. 32 3.3.1 Firstconstructs . 32 3.3.2 Pseudomeasures . .. .. .. .. .. .. 34 3.3.3 Fresnelintegrals . 38 3.3.4 Polygonalpaths. 41 3.4 DiscretizationofFeynmanpathintegrals . 44 3.4.1 Connectiontotheoperatororderingproblem . 49 3.5 Feynmanpathintegralonspaceswithcurvature . 53 3.5.1 Thehydrogenatom. 56 4 Conclusions 58 Chapter 1 Introduction Functional integration, also known as path integration or Wiener integration, has become a common tool in physics as well as in some branches of mathemat- ics such as functional analysis and partial differential equations during the last fifty years. It connects the theory of measures and integration with stochastics, particularily with stochastic differental equations. Functional integration is the theory of integration on spaces of functions. It can also be understood as the extension of ordinary integration theory to spaces of infinite dimensions. Since Lebesque integration theory does not work in such spaces functional integration is highly nontrivial. Since Feynman [Fey48] introduced path integration methods to physics over fifty years ago functional integration has proved out to be a valuable tool – both in theoretical considerations and in numerical calculations. Quantum field the- ory, statistical physics as well as analysis of stochastic processes have greatly benefited from the development of functional integration. By “path integration” we shall mean this functional integral of quantum mechanics; the Wiener integral means the functional integral of Brownian motion. The widespread use of path integration has been the motivation for mathe- matical research on functional integration in a more general terms. This research has revealed profound connections between path integrals in quantum physics and the Wiener integral applied to Brownian motion. An early review of of the definitions and properties of Feynman’s path integrals was made by Gel’fand and Yaglom [G&Y60]. Today path integration has developed to the level that we have a path inte- gral solution to every quantum mechanical problem solvable by the Schrödinger equation. A good review of the current available calculational techniques of path integrals is given by Grosche and Steiner [G&S95]. The paper also contains a large amount of references which touch almost every relevant aspect of func- tional integration. 2 Unfortunately most physicist take a very pragmatic view on functional inte- gration and use it as a fool-proof tool. They tend to forget – or neglect – the mathematical subtleties actually involved in the definition and methods of func- tional integration. Besides that, the analysis of functional integration offers more information than just calculational tools; as mentioned in the first paragraph, the functional integral is also an important mathematical abstraction. In fact the mathematical foundation of Feynman’s path integrals has never been soundly established. The alternating nature of eiS in the Feynman “mea- sure” effectively prohibits the use of well-founded methods of measure theory. In fact, the Feynman “measure” isn’t really a measure at all – at least not in the sense of probability theory. The aim of this work is to give a thorough, but not entirely concise review of the mathematical background of functional integration and, in particular, to show why Feynman path integrals are not mathematically justified. To achieve this, we will review the mathematical properties of Brownian motion and compare them to the properties of Feynman path integrals. We will also study what has been done to correct the lack of mathematical soundness in functional integration. Although the context of this work is mathematical we will avoid most of complex details. This choice is made at the expense of mathematical complete- ness but the number of actual mathematical concepts and details involved is so large that it would considerably add to the length of this work. We also wish to keep the material accessible to physicists without an extensive background in mathematics. We do not not attempt to give a concise review of functional integration or its history. Such information can be found in many textbooks, such as Feynman’s own book on the subject with Hibbs [F&H65] or more recent material such as Kleinert’s book [Kle90]. Brief definitions of Brownian motion and Feynman path integrals are included as well as some mathematical methods wherever they have some connection to the underlying problems or ambiguities. We shall not go through the calculational methods of functional integration. We shall also not discuss whether or not we can actually explicitely calculate the value of the integral in a closed form. In chapter 2 we shall study Brownian motion and its mathematical properties, stochastic integration and the Wiener measure and integral. We shall point out the ambiguities that lie in the well-studied theory of Brownian motion and briefly study the implications due to this arbitrariness in the discretization of the Wiener integral. The analysis of Brownian motion also serves as a stepping stone as we proceed to take a closer look at the Feynman path integrals.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Knot Theory and the Heuristics of Functional Integration Louis H
    Physica A 281 (2000) 173–200 www.elsevier.com/locate/physa Knot theory and the heuristics of functional integration Louis H. Kau man∗ Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA Abstract This paper is an exposition of the relationship between the heuristics of Witten’s functional integral and the theory of knots and links in three-dimensional space. c 2000 Elsevier Science B.V. All rights reserved. 1. Introduction This paper shows how the Kontsevich integrals, giving Vassiliev invariants in knot theory, arise naturally in the perturbative expansion of Witten’s functional integral. The relationship between Vassiliev invariants and Witten’s integral has been known since Bar-Natan’s thesis [1] where he discovered, through this connection, how to deÿne Lie algebraic weight systems for these invariants. The paper is a sequel to Ref. [2]. See also the work of Labastida and PÃerez [3] on this same subject. Their work comes to an identical conclusion, interpreting the Kontsevich integrals in terms of the light-cone gauge and thereby extending the original work of Frohlich and King [4]. The purpose of this paper is to give an exposition of these relationships and to introduce diagrammatic techniques that illuminate the connections. In particular, we use a diagrammatic operator method that is useful both for Vassiliev invariants and for relations of this subject with the quantum gravity formalism of Ashtekar et al. [5]. This paper also treats the perturbation expansion via three-space integrals leading to Vassiliev invariants as in Refs.
    [Show full text]
  • Quantum Field Theory II: Gauge Theories
    Quantum Field Theory II: Gauge Theories Lecture in WS 2019/20 at the KFU Graz Axel Maas 2 Contents 1 Introduction 1 2 Path integral 3 2.1 Heuristicintroduction ............................. 3 2.2 Functionalanalysis ............................... 7 2.3 Fermions..................................... 11 2.3.1 Grassmannvariables .......................... 11 2.3.2 Fermionicmatter ............................ 14 2.4 Free particles and generating functionals . .. 15 3 Abelian gauge theories 18 3.1 Globalsymmetries ............................... 18 3.2 ClassicalMaxwelltheory ............................ 21 3.3 QuantizationofMaxwelltheory . 23 3.4 Gauge-dependentcorrelationfunctions . ... 29 3.5 Matterfields................................... 31 4 Classical non-Abelian gauge theories 33 4.1 Somebasicsofgrouptheory .......................... 33 4.2 Orbitstructure ................................. 38 4.2.1 Orbits .................................. 38 4.2.2 (Non-)linear representations . 39 4.2.3 The little group . 40 4.3 Yang-Millstheory................................ 41 4.4 Yang-Millstheoryasgeometry. 45 4.5 Matter, representations, and coupling universality . ..... 47 5 Perturbative quantization of Yang-Mills theory 52 5.1 Quantization .................................. 52 i ii Contents 5.2 BRSTandasymptoticstates. .. .. 55 5.2.1 BRSTsymmetry ............................ 56 5.2.2 Constructingthephysicalstatespace . 59 5.2.2.1 Negative-normstates. 59 5.2.2.2 BRSTalgebra......................... 59 5.2.2.3 Statespace .......................... 61 5.2.2.4 Asymptoticspace....................... 62 5.3 Generalanti-fieldapproach. 65 5.3.1 Structureofgaugesymmetries. 65 5.3.1.1 Yang-Mills type . 65 5.3.1.2 Closure only on-shell . 66 5.3.1.3 Reducible gauge theories . 66 5.3.1.4 Reducibility on-shell . 67 5.3.2 BRSTconstructionrevisited . 67 5.3.3 Observables ............................... 70 5.3.4 Application as gauge-fixing to the path integral . 73 5.3.5 BeyondYang-Millstheory . 75 5.3.6 QuantumMasterEquation.
    [Show full text]
  • Lecture 17 Gauge Theories in the Functional Integral
    Lecture 17 Gauge Theories in the Functional Integral As we saw in the previous lecture, the quantization of gauge fields is more subtle than what we anticipated. The main reason is gauge invariance, or the redundancy that it brings. The problem is more clearly identified in the functional integral formulation. In this part of the course we will restrict ourselves to abelian gauge theories, or U(1) gauge theories. But the functional integral approach will come in handy when quantizing non-abelian gauge theories. 17.1 Gauge Redundancy in the Functional Integral The action for the gauge fields in a U(1) gauge theory is 1 Z S[A ] = − d4x F F µν ; (17.1) µ 4 µν where as usual we have Fµν = @µAν − @νAµ ; (17.2) is the gauge invariant gauge tensor. Writing the action out in terms of Aµ and its deriva- tives we have 1 Z S[A ] = − d4x f@ A @µAν − @ A @νAνg µ 2 µ ν µ ν 1 Z = d4x A gµν@2 − @µ@ν A ; (17.3) 2 ν µ 1 2 LECTURE 17. GAUGE THEORIES IN THE FUNCTIONAL INTEGRAL where to obtain the last equality we integrated by parts both terms in the first equality. So we can write the action as a quadratic form 1 Z S[A ] = d4x A Oµν A ; (17.4) µ 2 ν µ with Oµν ≡ gµν@2 − @µ@ν : (17.5) This means that we can use our usual trick to obtain the generating functional of the theory. The generating functional in the presence of a source linearly coupled to Aµ is Z R 4 µ iS[Aµ]+i d x Jµ(x)A (x) Z[Jµ] = N DAµe : (17.6) Then we know that the fact that the action is quadratic in the fields allows us to make a change of variables in Aµ(x) in order to decouple the source Jµ(x).
    [Show full text]
  • Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion
    Article Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion Matthew Bernard 1,*, Vladislav A. Guskov 2 , Mikhail G. Ivanov 2,*, Alexey E. Kalugin 2 and Stanislav L. Ogarkov 2,* 1 Advanced Computational Biology Center (ACBC), University Avenue 2120, Berkeley, CA 94704, USA 2 Moscow Institute of Physics and Technology (MIPT), Institutskiy Pereulok 9, Dolgoprudny 141701, Moscow Region, Russia * Correspondence: [email protected] (M.B.); [email protected] (M.G.I.); [email protected] (S.L.O.) Received: 26 July 2019; Accepted: 26 August 2019; Published: 29 August 2019 Abstract: Nonlocal quantum field theory (QFT) of one-component scalar field j in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions Z as a functional of external source j, coupling constant g and spatial measure dm is studied. An expression for GF Z in terms of the abstract integral over the primary field j is given. An expression for GF Z in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator Lˆ over the separable HS basis. The classification of functional integration measures D [j] is formulated, according to which trivial and two nontrivial versions of GF Z are obtained. Nontrivial versions of GF Z are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure D [j] over the primary field is suggested.
    [Show full text]
  • Building a Path-Integral Calculus: a Covariant Discretization Approach
    Building a path-integral calculus: a covariant discretization approach Leticia F. Cugliandolo1, Vivien Lecomte2 and Fr´ed´eric van Wijland3 1 Laboratoire de Physique Th´eorique et Hautes Energies,´ Sorbonne Universit´e, UMR 7589 CNRS, 4 Place Jussieu Tour 13 5`eme´etage, 75252 Paris Cedex 05, France 2 Laboratoire Interdisciplinaire de Physique, Universit´eGrenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France 3 Laboratoire Mati`ereet Syst`emesComplexes, Universit´eParis Diderot, UMR 7057 CNRS, 10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France E-mail: [email protected] [email protected] [email protected] Abstract. Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom.
    [Show full text]