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Quantum Anomalies and Logarithmic Derivatives of Feynman Pseudomeasures J Quantum Anomalies and Logarithmic Derivatives of Feynman Pseudomeasures J. E. Gough, T. S. Ratiu, O. G. Smolyanov To cite this version: J. E. Gough, T. S. Ratiu, O. G. Smolyanov. Quantum Anomalies and Logarithmic Derivatives of Feynman Pseudomeasures. / Doklady Mathematics, MAIK Nauka/Interperiodica, 2015, 92 (3), pp.764-768. 10.1134/S1064562415060356. hal-01397731 HAL Id: hal-01397731 https://hal.archives-ouvertes.fr/hal-01397731 Submitted on 22 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ISSN 10645624, Doklady Mathematics, 2015, Vol. 92, No. 3, pp. 764–768. © Pleiades Publishing, Ltd., 2015. Original Russian Text © J. Gough, T.S. Ratiu, O.G. Smolyanov, 2015, published in Doklady Akademii Nauk, 2015, Vol. 465, No. 6, pp. 651–655. MATHEMATICS Quantum Anomalies and Logarithmic Derivatives of Feynman Pseudomeasures J. Gougha, T. S. Ratiub, and O. G. Smolyanovc Abstract—Connections between quantum anomalies and transformations of pseudomeasures of the type of Feynman pseudomeasures are studied. Mathematical objects related to the notion of the volume element in an infinitedimensional space considered in the physics literature [1] are discussed. A quantum anomaly is the violation of symmetry sure. If the action and the initial condition are invari (see [4]) with respect to some transformations under ant with respect to some transformation, then this quantization. At that, the quantization of a classical object is invariant with respect to this transformation Hamiltonian system invariant with respect to some as well. transformations yields a quantum system noninvariant The second object is a determinant, which plays the with respect to the same transformations (see [2]). role of a Jacobian; this determinant may differ from 1 The situation in related literature is fairly unusual. even in the case where the action and the initial condi The points of view presented in monographs [1, 2], tion are invariant with respect to phase transforma whose authors are wellknown experts, contradict tions; of course, in this case, the Feynman integral is each other. On page 352 of the book [1], it is written noninvariant as well. What is said above virtually coin that the description of reasons why quantum anoma cides with what is said in [2]. At the same time, in [1], lies occur which is given in [2] is incorrect. This refers it was proposed to compensate this determinant by to the first, 2004, edition of [2]; however, in 2013, the multiplying the measure with respect to which the second edition of [2] appeared, which did not differ integration is performed (that is, the counterpart of from the first; in the new edition, the authors did not the classical Lebesgue measure, which does not exist cited book [1] containing criticism of their book. In in the infinitedimensional case according to a well this paper, we show that the correct description of the known theorem of Weil) by an additional factor, reasons for the emergence of quantum anomaly is that which, of course, is equivalent to multiplying the inte given in [2]. grand by the same factor. We use the fact that the transformations of a func In this paper, we consider families of transforma tional Feynman integral (i.e., an integral with respect tions of the domain of a (pseudo)measure depending to a Feynman pseudomeasure; its definition is given on a real parameter and show that such a compensa below) are determined by transformations of two tion is impossible; for this purpose, we use differentia objects. One of them is the product of a Feynman tion with respect to this parameter. pseudomeasure and a function integrable with respect The paper is organized as follows. First, we recall to this pseudomeasure, which is, in turn, the product two basic definitions of the differentiability of a mea of the exponential of a part of the classical action and sure and, more generally, a pseudomeasure (distribu the initial condition. The exponential of the other part tion); then, we give explicit expressions for the loga of the action determines the Feynman pseudomea rithmic derivatives of measures and pseudomeasures with respect to transformations of the space on which they are defined. The application of these expressions1 a Institute of Mathematics, Physics, and Computer Sciences, makes it possible to obtain a mathematically correct Aberystwyth University, Great Britain version of results of [2] concerning quantum anoma b Section de Mathématiques and Bernoulli Center, École lies. After this, we discuss the approach to explaining Polytechnique Fédérale de Lausanne, CH1015 Switzerland the same anomalies proposed in [1]. We also discuss email: [email protected] mathematical objects related to the notion of the vol c Mechanics and Mathematics Faculty, Moscow State University, Moscow, 119991 Russia 1 Among other things, these formulas lead to infinitedimensional email: [email protected] versions of both theorems of Emmy Noether. ume element in an infinitedimensional space consid along k, and ν'k is defined by ν'k = ν' ; the τlogarith ered in physics literature (including [1]). We concen S mic derivative of the measure ν along S is called the τ trate on the algebraic structure of problems, leaving logarithmic derivative of ν along k and denoted by aside most assumptions of analytical character. βν(k, ·). The τdifferentiability of a measure along a vector field h and its τlogarithmic derivative along h 1. DIFFERENTIATION OF MEASURES βν (denoted by h ) are defined in a similar way: we set AND DISTRIBUTIONS S(t)(x) := x – tanh(x). This section contains results (in fact, known) about If a measure ν is τdifferentiable along each k ∈ Н, ν'k is ۋ differentiable measures and distributions on infinite then it can be shown that the mapping Н ෉ k dimensional spaces in a form convenient for our pur linear; the corresponding vectorvalued measure ν' : ν τ ۋ ۋ poses. Ꮾ ෉ E A [k ( 'k)(A)] is called the derivative of Ꮾ ν ∈ Given a locally convex space (LCS) Е, by E we over the subspace Н. If, for any k Н, there exists a denote the σalgebra of Borel subsets of Е and by ᑧ , τlogarithmic derivative measure ν along k, then the βν(k, ·) is linear; it is called the τ ۋ E mapping Н ෉ k the vector space of countably additive (complex) mea ν βν sures on Е. We say that the vector space С of bounded logarithmic derivative of over Н and denoted by . Borel functions on Е determines a norm if, whatever a Remark 1. If the measure ν has a logarithmic deriv ative over a subspace and ( ) ∈ for all ∈ , measure ᑧE, its total variation μ ∈ ᑧE satisfies the Н h x Н х Е ν ||μ|| μ ∈ || || ≤ then, against expectation, β (х) ≠ βν(h(x), x) in the condition 1 = sup{∫u d : u C, u ∞ 1}, where S general case (see below). ||u||∞ = sup{|u(x)|: x ∈ E}. Remark 2. If τ is the topology of convergence on all A Hilbert subspace of an LCS Е is defined as a vec sets, then any measure τdifferentiable along S has a tor subspace Н of Е endowed with the structure of a logarithmic derivative along τ (see [12]); for weaker Hilbert space such that the topology induced on Н by topologies, this may be not the case. An example is as the topology of Е is weaker than the topology gener follows. Given an LCS Е being a Radon space,2 let S ated by the Hilbert norm. be the space of bounded continuous functions on E, A mapping F of an LCS Е to an LCS G is said to be and let τC be the weak topology on ᑧE determined by smooth along a Hilbert subspace Н of Е (or Н the duality between С and ᑧE. Then a measure τCdif smooth) if it is infinitely differentiable over Н and both ferentiable along S may have no logarithmic derivative the mapping F and all of its derivatives are continuous along S (even in the case Е = ޒ1). on Е, provided that the spaces in which the derivatives Let С be a normdefining vector space of Нsmooth take values are endowed with the topologies of uni functions on Е bounded together with all derivatives. form convergence on compact subsets of Н. A vector A measure ν is said to be Сdifferentiable along a vec field on an LCS Е is a mapping h: E → E; we denote the ∈ ν' set of vector fields on Е by Vect(E). The derivative tor field h Vect(E) if there exists a measure h along a vector field h ∈ Vect(E) of a function f defined such that, for any ϕ ∈ С, we have ϕ' (x)h(x)ν(dx) = on Е is the function on E denoted by f 'h and defined ∫ by (f 'h)(x) := f '(x)h(x) for х ∈ Е, where f '(x) is the ϕ ν' –(∫ x)(h )(dx). The Radon–Nikodym density of Gâteaux derivative of f at the point х. ν' ν Let ε > 0, and let S be a mapping of the interval h with respect to (if it exists) is called the Cloga ε ε Ꮾ ν (– , ) to the set of Emeasurable selfmappings of Е rithmic derivative of the measure along h; if h(x) = h ∈ Е for all х ∈ Е, then, as above, the Clogarithmic for which S(0) = id; suppose that τ is a topology on ᑧE 0 compatible with the vector space structure.
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