Quantum Anomalies and Logarithmic Derivatives of Feynman Pseudomeasures J. E. Gough, T. S. Ratiu, O. G. Smolyanov
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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
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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 1064 5624, 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 infinite dimensional 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 well known 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 infinite dimensional 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 pseudo measures 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, CH 1015 Switzerland the same anomalies proposed in [1]. We also discuss e mail: [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 infinite dimensional e mail: [email protected] versions of both theorems of Emmy Noether. ume element in an infinite dimensional space consid along k, and ν'k is defined by ν'k = ν' ; the τ logarith 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 τ differentiability of a measure along a vector field h and its τ logarithmic 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 τ differentiable 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 vector valued 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 σ algebra of Borel subsets of Е and by ᑧ , τ logarithmic 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 τ differentiable 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 τC dif 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 norm defining vector space of Н smooth 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 С differentiable 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 C loga ε ε Ꮾ ν (– , ) to the set of E measurable self mappings of Е rithmic derivative of the measure along h; if h(x) = h ∈ Е for all х ∈ Е, then, as above, the C logarithmic for which S(0) = id; suppose that τ is a topology on ᑧE 0 compatible with the vector space structure. A measure derivative of ν along h is called the C logarithmic ν ν ∈ ᑧ is said to be τ differentiable along S if the derivative of along h0 (С logarithmic derivatives are E τ S(t) ν := ν(S(t)–1) ∈ (ᑧ , τ) denoted by the same symbols as logarithmic deriva ۋ function f: (–ε, ε) t ∗ E tives introduced above). ν is differentiable at t = 0 (the symbol S(t)∗ denotes the Suppose that a vector field h is determined by ν S image of under the mapping S(t)); in this case, we hS(x) := S '(0)x. Then the following proposition is ν valid. denote f '(0) by S' and call it the derivative of the mea ν τ sure ν along S. If, in addition, f '(0) Ӷ f(0) (the mea Proposition 1. A measure is C differentiable along sure f '(0) is absolutely continuous with respect to S if and only if it is C differentiable along hS. In this case, f(0)), then its density with respect to the measure f(0) is called the τ logarithmic derivative of the measure ν 2 A topological space E is called a Radon space if any countably additive Borel measure ν on E is Radon; this means that, for any βν along S and denoted by S . Borel subset A of E and any ε > 0, there exists a compact set K ⊂ A such that ν(A\K) < ε. If E is a completely regular Radon space, If k ∈ Е and S(t)(x) := х – tk, then a measure ν τ then the space of all bounded continuous functions on E is in ᑧ differentiable along S is said to be τ differentiable natural duality with E.
βν βν βν ν by a transformation of the space on which the test h = S , where h is the C logarithmic derivative of S S functions are defined should be used. βν τ ν along hS and S is the C logarithmic derivative of along S. 2. QUANTUM ANOMALIES Proof. This follows from the change of variable for mula. Suppose that ϕ ∈ С and, as above, f(t) := In fact, quantum anomalies arise because the sec (S(t))∗ν. Then ond term in the relation of Theorem 1 proved above is the same for all measures. Indeed, by virtue of Leibniz’ lim t–1∫ϕ()x ()ft()()dx – ∫ϕ()x ()f()0 ()dx rule, the logarithmic derivative (both over a subspace t → 0 and along a vector field) of the product of a function E E and a measure is the sum of the logarithmic derivatives of the factors; therefore, a measure ν whose logarith –1 ϕ()()()() ν () ϕ()ν() = lim t ∫ x St * dx – ∫ x dx mic derivative along a vector field is given by the t → 0 expression in Theorem 1 can formally be taken for the E E product of a function ψν whose logarithmic derivative –1 over the subspace Н coincides with the logarithmic = lim t ()ϕ()ϕSt() – ()x ν()dx = ϕ'()x S'0()()νx ()dx , ν t → 0 ∫ ∫ derivative over the measure over this subspace and a E E measure η whose logarithmic derivative over the same which implies the required assertion. subspace vanishes. If Е is finite dimensional and Н Corollary 1. Let S be another mapping of the inter coincides with E, then such a function and a measure 1 indeed exist; moreover, η turns out to be the Lebesgue val (–ε, ε) to Ꮾ with the same properties as S. If E measure, and ψν is the density of ν with respect to it. h = h , then the measure ν is τ differentiable along S S S1 C τ But in the infinite dimensional case, there exist no if and only if it is C differentiable along S1. exact counterpart of the Lebesgue measure; neverthe Remark 2. It is natural to say that the measure ν is less, an analogue of density, called the generalized ν invariant with respect to S if β = 0. density of a measure, does exist [3, 10, 12], although S its properties are far from those of usual density, and ν ∈ ᑧ Theorem 1. Suppose that a measure E has a the corresponding distribution can be regarded as an τ C logarithmic derivative over a subspace H, and let h be analogue of the Lebesgue measure. It is this distribu βν tion that should be considered as a formalization of the a vector field on E taking values in H. Then h (х) = ν term “volume element” used in [1, p. 362]. We how β (h(x), x) + tr h'(x), where h' is the derivative of the ever emphasize that the contents of this paper depends mapping h over the subspace H. on the properties of neither this distribution nor the Proof. Suppose that ϕ ∈ С and h is a vector field on generalized density. Е, and let μ be the Е valued measure defined by μ := h(·)ϕ(·)ν. Applying Leibniz’ rule to the derivative of μ Let Q be a finite dimensional vector space being μ ϕ ν ϕ ν ⊗ the configuration space of a Lagrangian system with over the subspace Н, we obtain ' = h'(·) (·) + (·) ' × → ޒ ϕ ⊗ ν Lagrange function L: Q Q defined by L(q1, q2) := h(·) + '(·) h(·) . Each summand in this relation is a η measure whose values are operators on Н; calculating (q1, q2) + b(q2), where b is a quadratic functional (the the traces of these operators, we obtain tr μ' = kinetic energy of the system). We assume that the (ϕ(·)trh'(·))ν + ϕ(·)βν(h(·), ·)ν + ϕ'(·)h(·)ν. Since Lagrange function L is nondegenerate (hyperregular), i.e., the corresponding Legendre transform is a diffeo ∫μ' (dx) = 0 and, therefore, ∫( trμ')(dx) = 0, it follows morphism, so that it determines a Hamiltonian system with Hamiltonian function Ᏼ: Q × P → ޒ, where Р = Q*. E E that ϕ' (x)h(x)ν(dx) = –ϕ (x)[βν(h(x), x) + For t > 0, by Et we denote the set of continuous ∫ ∫ functions on [0, t] taking values in Q and vanishing at E E zero and by H , the Hilbert subspace of E consisting of ν t t trh'(x)] (dx). This means that the required relation absolutely continuous functions on [0, t] with square holds. integrable derivative; the Hilbert norm · H on Et is Both the definitions given above and the algebraic t parts of proofs can be extended to distributions (in the defined by Sobolev–Schwartz sense) defined as continuous lin t ear functionals on appropriate spaces of test functions. 2 ()τ 2 τ f H := f' Q d , The difference is that the integrals of functions with t ∫ respect to measures should be replaced by values of 0 these linear functionals at functions, and instead of the where f ∈ E and · is the Euclidean norm on Q. change of variables formula for integrals, the defini t Qt tion of the transformation of a distribution generated Finally, by (t) we denote the classical action defined
as the functional on Ht determined by the Lagrange sponding Schrödinger equation is not necessarily t invariant with respect to the same transformations. · function L via (t)(f) := ∫L (f(τ), f (τ))dτ. Remark 3. For each family S(α) of transformations of the space E , we can obtain an explicit expression for 0 t the transformations of the pseudomeasure W gener The Schrödinger quantization of the Hamiltonian t ated by the transformations S(α) by solving the equa system generated by the Lagrangian system described ν ˆ tion g· (α) = β g(α) (see [10]). It follows [10] that if above yields the Schrödinger equation iψ· (t) = Ᏼ ψ(t), S()α ψ α∈ α α ≠ ˆ tr(hS(α))'( ) > 0 for [0, 0], then det(S( ))' 1; where Ᏼ is a self adjoint extension of a pseudodiffer this fact does not depend on the classical action. ᏸ ential operator on 2(Q) with symbol equal to the Remark 4. Using the notion of the generalized den Hamiltonian function Н generated by the Lagrange sity of a pseudomeasure (cf. [3, 10, 11], where only function L. The solution of the Cauchy problem for generalized densities of usual measures were consid this equation with initial condition f0 is ered), we can say that the pseudomeasure Wt is deter ψ()t ()q mined by its generalized density being the exponential in the Feynman integral (1). Moreover, as mentioned t ⎛⎞ above, the expression for the transformations of the = exp⎜⎟i ηψτ()τ()+ q, ψ· ()τ d f ()φψ()t + q ()dψ , ∫ ⎝⎠∫ 0 t pseudomeasure contain determinants, and the expres E 0 (1) sions for the corresponding logarithmic derivatives t contain traces, which do not depend on the general where φ is the Feynman pseudomeasure on (the t Et ized densities. This can be interpreted by treating the exponential under the integral sign is well defined on Feynman pseudomeasure as the product of its general the space Ht). ized density and distribution whose transformations Let Wt be the pseudomeasure on Et being the prod are described by the corresponding determinants and uct of the exponential in the above integral and the traces, and the logarithmic derivatives of this distribu φ function t. The following theorem is valid. tion along constant vectors vanish. In turn, this allows Theorem 2. The logarithmic derivative of the us to say that the distribution mentioned above corre pseudomeasure Wt along Ht exists and is determined by sponds to the volume element considered in [1]. t Remark 5. Thus, the determinants and traces men W β t()k ψ, = i [L' ()ψτ()+ q, ψ· ()τ k()τ tioned above cannot be eliminated by any choice of ∫ 1 the integrand and the Feynman pseudomeasure if the 0 corresponding Feynman integrals are required to rep ()ψτ() , ψ· ()τ ·()]τ τ resent the solutions of the corresponding Schrödinger + L2' + q k d ; equation. Clearly, this contradicts [1, p. 362]. ∈ ψ ∈ ' ' here, k Ht, Et, and L1 and L2 are the partial Remark 6. If E is a this superspace, then, instead of derivatives of L with respect to the first and the second traces and determinants, we should use supertraces argument. and superdeterminants. Corollary 2. If h is a vector field on Et taking values in H , then the logarithmic derivative of the pseudomea t ACKNOWLEDGMENTS sure Wt along h is determined by t T.S. Ratiu acknowledges the partial support of W ()ψ β t()ψ ' ⎛⎞()τψ () , dh ()τ ()τψ () Swiss National Science Foundation, grant no. Swiss h = i∫ L1⎝⎠h + q h dτ NSF 200021 140238. J. Gough and O.G. Smolyanov 0 acknowledge the partial support of the Swiss Federal ⎛⎞dh()ψ dh()ψ Institute of Technology in Lausanne (EPFL). + L' h()τψ ()+ q, ()τ ()τ dτ + tr h'()ψ . 2⎝⎠dτ dτ O.G. Smolyanov acknowledges the support of the Russian Foundation for Basic Research, project A similar assertion is valid for the logarithmic no. 14 01 00516. derivative along a family S(α) of transformations of α ∈ ε ε the space Et depending on a parameter (– , ). It follows from Corollary 1 that if the classical REFERENCES α action (t) is invariant with respect to a family S( ), 1. P. Cartier and C. DeWitt Morette, Functional Integra α∈ (–ε, ε) of transformations of the space Q, then the tion (Cambridge Univ. Press, Cambridge, 2006). βW ψ logarithmic derivative S ( ) does not necessarily 2. K. Fujikawa and H. Suzuki, Path Integrals and Quan vanish. In turn, this means that if a Lagrange function tum Anomalies (Oxford Univ. Press, Oxford, 2004; 2nd is invariant with respect to transformations S(α) of the ed., 2013). configuration space, then the solution of the corre 3. A. I. Kirillov, Russ. Math. Surv. 49 (3), 43–95 (1994).
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