The worldline formalism has been widely used to compute physical quantities
in quantum field theory. However, applications of this formalism to quantum
fields in the presence of boundaries have been studied only recently. In this
article we show how to compute in the worldline approach the heat kernel
expansion for a scalar field with boundary conditions of Robin type. In order
to describe how this mechanism works, we compute the contributions due to the
boundary conditions to the coefficients A_1, A_{3/2} and A_2 of the heat kernel
expansion of a scalar field on the positive real line.Comment: Presented at 8th Workshop on Quantum Field Theory Under the Influence
  of External Conditions (QFEXT 07), Leipzig, Germany, 16-21 Sep 200

Bastianelli, Fiorenzo

Corradini, Olindo

Pisani, Pablo A. G.

English

arXiv

The worldline formalism has been widely used to compute physical quantities in quantum field theory. However, applications of this formalism to quantum fields in the presence of boundaries have been studied only recently. In this article we show how to compute in the worldline approach the heat kernel expansion for a scalar field with boundary conditions of Robin type. In order to describe how this mechanism works, we compute the contributions due to the boundary conditions to the coefficients A_1, A_{3/2} and A_2 of the heat kernel expansion of a scalar field on the positive real line

BASTIANELLI, FIORENZO

CORRADINI, OLINDO

P. A. G. Pisani

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Scalar Field with Robin Boundary Conditions in the Worldline Formalism

The worldline formalism has been widely used to compute physical quantities
in quantum field theory. However, applications of this formalism to quantum
fields in the presence of boundaries have been studied only recently. In this
article we show how to compute in the worldline approach the heat kernel
expansion for a scalar field with boundary conditions of Robin type. In order
to describe how this mechanism works, we compute the contributions due to the
boundary conditions to the coefficients A_1, A_3/2 and A_2 of the heat kernel
expansion of a scalar field on the positive real line

CORRADINI, Olindo

Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia

Scalar field with Robin boundary conditions in the worldline formalism

Fiorenzo Bastianelli

Olindo Corradini

Pablo A G Pisani

Crossref

The worldline formalism has been widely used to compute physical quantities in quantum field theory. However, applications of this formalism to quantum fields in the presence of boundaries have been studied only recently. In this paper we show how to compute in the worldline approach the heat kernel expansion for a scalar field with boundary conditions of Robin type. In order to describe how this mechanism works, we compute the contributions due to the boundary conditions to the coefficients A1, A3/2 and A2 of the heat kernel expansion of a scalar field on the positive real line.Facultad de Ciencias Exacta

González Pisani, Pablo Andrés

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICALJ. Phys. A: Math. Theor. 41 (2008) 164010 (7pp) doi:10.1088/1751-8113/41/16/164010Scalar field with Robin boundary conditions in theworldline formalismFiorenzo Bastianelli1, Olindo Corradini1 and Pablo A G Pisani2,31 Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, via Irnerio,46 Bologna I-40126, Italy2 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1,07743 Jena, GermanyReceived 20 October 2007Published 9 April 2008Online at stacks.iop.org/JPhysA/41/164010AbstractThe worldline formalism has been widely used to compute physical quantitiesin quantum field theory. However, applications of this formalism to quantumfields in the presence of boundaries have been studied only recently. In thispaper we show how to compute in the worldline approach the heat kernelexpansion for a scalar field with boundary conditions of Robin type. In orderto describe how this mechanism works, we compute the contributions due tothe boundary conditions to the coefficients A1, A3/2 and A2 of the heat kernelexpansion of a scalar field on the positive real line.PACS numbers: 03.70.+k, 11.10.−z1. IntroductionMany problems in quantum field theory can be studied in the worldline formalism. Thisapproach provides an intuitive and, in many cases, more efficient method to carry outcomputations in the 1-loop approximation (see [2–4] and references therein).This formalism makes use of the connection between some quantities in quantum fieldtheory, like the effective action, and the operator e−T A corresponding to some relevantdifferential operator A. The first important step in this approach is to identify a classicalparticle whose Hamiltonian, after first quantization, is this differential operator A. If thisidentification is found, then the heat kernel corresponds to the evolution operator of the particlein Euclidean time T . The kernel hy| e−T A|xi of the operator e−T A is a quantum-mechanicaltransition amplitude and therefore can also be computed in the path integral formulation offirst quantization.It is important to mention that many physical quantities of the quantum field theoryare determined by the small-T asymptotic expansion of the heat kernel hy| e−T A|xi. In this3 Permanent address: IFLP (CONICET) and Departamento de Fı́sica, Facultad de Ciencias Exactas, UniversidadNacional de La Plata, CC 67 La Plata 1900, Argentina.1751-8113/08/164010+07$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1J. Phys. A: Math. Theor. 41 (2008) 164010 F Bastianelli et alsense, the path integral computation of this transition amplitude results particularly appropriatebecause the small-T expansion comes out in a natural way. By this procedure the worldlineformalism has been proved useful in the study of quantum fields of different spins and theircoupling to external gravity.The purpose of the present paper is to describe how the worldline formalism can alsobe applied to quantum field theory on manifolds with boundaries. In order to do this, onehas to deal with the path integral formulation of the quantum mechanics of a point particleon a manifold with boundaries. The main difficulty is to implement the boundary conditionsimposed on the quantum fields, i.e. on the domain of the differential operator A, in thepath integral calculation of the quantum-mechanical transition amplitude hy| e−T A|xi. Thisproblem has been solved in [1] for Dirichlet and Neumann boundary conditions by the methodof images. In this paper we show how to implement Robin boundary conditions in thepath integral calculation and compute the contributions of the boundary conditions to thecoefficients A1, A3/2 and A2 of the small-T asymptotic expansion of the heat kernel for ascalar field on the positive half-line. For this setting, Robin boundary conditions are the mostgeneral boundary conditions determined by the selfadjointness of the differential operator A.The method of images is not straightforwardly applicable to Robin boundary conditions.However, as is shown in the appendix (see also [5]), one can implement Robin boundaryconditions on the heat kernel by using the image construction for Neumann boundaryconditions supplemented by a suitable delta-function potential at the boundary. This delta-function then constrains the paths which contribute to the path integral computation of thetransition amplitude.In the next section, we describe the worldline method by considering the heat kernel tracefor a scalar field on a manifold without boundaries. In section 3, we apply the worldlineapproach to a scalar field on the positive half-line under Robin boundary conditions at theorigin and compute the contributions of the boundary to the first coefficients of the heat kernelexpansion. In section 4, we draw our conclusions and comment on possible generalizationsof our technique.2. Heat kernel expansion without boundariesIn order to describe the worldline approach to quantum field theories let us consider the 1-loopeffective action of a scalar field φ(x) with a self-interaction U(φ(x)) on a flat manifold M.The classical action I [φ(x)] of the scalar field is given byI [φ(x)] =ZMdx½12(∂φ(x))2 + U(φ(x))¾(2.1)and the 1-loop effective action can be represented as0[φ(x)] = I [φ(x)] − h̄ log Det−1/2{−1 + U 00(φ(x))}= I [φ(x)] − h̄2Z ∞0dTTTr(e−T {−1+U00(φ(x))})= I [φ(x)] − h̄2Z ∞0dTTZMdxhx| e−T {−1+U 00(φ(x))}|xi. (2.2)The integrand in the last expression is the heat kernelhy| e−T {−1+U 00(φ(x))}|xi (2.3)of the Schrödinger operator−1 + U 00(φ(x)) (2.4)2J. Phys. A: Math. Theor. 41 (2008) 164010 F Bastianelli et alevaluated at the diagonal x = y. Inspection of expression (2.2) shows that the small-Tasymptotic expansion of this heat kernel contains information about the ultraviolet divergencesof the theory.In the worldline approach, the kernel (2.3) is regarded as the quantum-mechanicaltransition amplitude in Euclidean time T between x and y of a point particle whose Hamiltonianis given by (2.4). In fact, the Schrödinger operator (2.4) can be regarded as the Hamiltonian,after first quantization, of a point particle whose Euclidean classical action isS[x(t)] =Z T0dt½14ẋ2(t) + V (x(t))¾(2.5)whereV (x) := U 00(φ(x)). (2.6)Therefore, we can represent the transition amplitude (2.3) in the path integral approachhy| e−T {−1+V (x)}|xi =Z x(T )=yx(0)=xDx(t) e−R T0 dt{ 14 ẋ2+V (x(t))}. (2.7)With this procedure we can compute some quantities in quantum field theory in terms ofquantum-mechanical path integrals. An essential step in this description is the identificationof the differential operator appearing in the first line of expression (2.2) with the quantizedHamiltonian of a point particle.Next, we show how the path integral in expression (2.7) turns out to be convenientfor the computation of the small-T asymptotic expansion of the transition amplitude. Inorder to do that, we rescale the Euclidean time variable t by introducing τ := t/T and wereplace the integration over trajectories x(t) by an integration over the ‘quantum fluctuations’y(τ) := x(T τ) − (y − x)τ − xhy| e−T {−1+V (x)}|xi = e− (y−x)24TZ y(1)=0y(0)=0Dy(τ) e− 14TR 10 dτ ẏ2(τ ) e−TR 10 dτV ((y−x)τ+x+y(τ))=: e− (y−x)24T­e−TR 10 dτV ((y−x)τ+x+y(τ))® = e− (y−x)24T ∞Xn=0(−T )nn!Z 10dτ1 . . .Z 10dτn×*nYk=1V ((y − x)τk + x + y(τk))+. (2.8)Note that we end up with a perturbative calculation in (0 + 1)-field theory where the numberof loops is related to the power of T and the propagator is proportional to T .For our purposes it suffices to consider the trace of the heat kernelTr e−T {−1+V (x)} =ZMdx­e−TR 10 dτV (x+y(τ))®=∞Xn=0(−T )nn!ZMdxZ 10dτ1 . . .Z 10dτn*nYk=1V (x + y(τk))+. (2.9)If the potential is an analytic function, we can make a Taylor expansion and write the integrandin the previous expression as a sum whose terms consist, in general, of a product betweenthe potential and its derivatives evaluated at x and many-point functions, which, due toWick’s theorem, can be written in terms of two-point functions­y¡τk1¢y¡τk2¢®. Since thesepropagators are proportional to T we obtain an asymptotic expansion in integer powers of Twhose coefficients are integrals on M of products of the potential and its derivatives.3J. Phys. A: Math. Theor. 41 (2008) 164010 F Bastianelli et alFinally, we normalize the expectation value h1i such that for the case V (x) ≡ 0 we obtain:hy| e−T {−1}|xi = e− (y−x)24T · h1i = e− (y−x)24T√4πT(2.10)which is the known value for the transition amplitude for a free particle. Note that thenumerator in the last expression corresponds to the contribution of the classical trajectorywhereas the denominator corresponds to the contributions of the quantum fluctuations.We end this section by recalling that for a Schrödinger differential operator on amanifold with boundaries the trace of the heat kernel admits the following small-T asymptoticexpansion4,Tr e−T {−1+V (x)} = 1(4πT )m/2∞Xn=0An/2 · T n/2 (2.11)where m is the dimension of the base manifold M. The coefficients An/2 for odd n vanishfor manifolds without boundary (see the discussion after equation (2.9)). In the followingsection, it will become clear how these half-integer powers of T appear due to the presence ofboundaries.3. Heat kernel expansion under Robin boundary conditionsIn this section, we consider the small-T asymptotic expansion of the trace of the heat kernelcorresponding to a Schrödinger operator on the positive half-line M = R+. We impose Robinboundary conditions on the functions ψ(x) in the domain of the Schrödinger operator∂xψ(x) + γ · ψ(x)|x=0 = 0 (3.1)where γ ∈ R.This type of boundary conditions can be implemented in the path integral by computingthe contributions of all paths that correspond to Neumann boundary conditions and, at the sametime, introducing a Dirac delta-function in the potential (see the appendix.) The computationof the contribution of all paths that correspond to Neumann boundary conditions can beperformed by the method of images, according to whichhx| e−T {−1+V (x)}|xiN = hx| e−T {−1+VR(x)}|xiR + h−x| e−T {−1+VR(x)}|xiR, (3.2)where the subscript N stands for ‘Neumann’ and the subscript R indicates that thecorresponding transition amplitudes are computed in the whole real line. Accordingly, VR(x)is the extension to the whole real line of the potential V (x), which is defined only on R+, bya reflection about the origin. As just mentioned, in order to obtain the transition amplitudefor Robin boundary conditions we still have to include in the potential a Dirac delta-function.Finally, using equation (2.8) to compute the transition amplitudes in the whole real line, thetrace of the heat kernel for Robin boundary conditions can be written as,Trγ e−T {−1+V (x)} =Z ∞0dxhx| e−T {−1+V γ R(x)}|xiR +Z ∞0dxh−x| e−T {−1+V γ R(x)}|xiR=∞Xn=0(−T )nn!Z ∞0dxZ 10dτ1 . . .Z 10dτn×(*nYk=1V γ R(x + y(τk))++ e−x2T*nYk=1V γ R((1 − 2τk)x + y(τk))+), (3.3)4 This expansion holds under certain regularity assumptions on the manifold, the potential and the (local) boundaryconditions.4J. Phys. A: Math. Theor. 41 (2008) 164010 F Bastianelli et alwhereV γ R(x) = V (x) + θ(−x)(V (−x) − V (x)) − 2γ · δ(x). (3.4)The term with the Heaviside function θ(x) represents the extension of the potential to thenegative half-line whereas the term with the Dirac delta-function is needed in order to obtainthe transition function for the boundary conditions given by (3.1). Note that the factor e−x2/Tin the second term of equation (3.3) is responsible for the appearance of half-integer powersof T .Next, we compute the contributions to the first coefficients An/2 of the heat kernelexpansion (2.11) due to Robin boundary conditions. Since γ = 0 corresponds to Neumannboundary conditions these contributions, which are proportional to positive powers of γ ,must be added to the coefficients An/2 corresponding to Neumann boundary conditions.The coefficients An/2 for Neumann boundary conditions have been already computed in theworldline formalism in [1].It is interesting to note that the inclusion of the Dirac delta-function leads to Dirichletboundary conditions for γ → −∞. Indeed, by the method of images, the transition functionfor these boundary conditions is given by the difference of the terms on the RHS of (3.2).This subtraction amounts to omitting from the computation the contribution of all thosepaths that hit the origin at least once. In accordance, as can be seen from the first line ofequation (2.9), the Dirac delta-function in (3.4) cancels the contributions of the paths that hitthe origin.To compute the contributions of the boundary condition (3.1) to the coefficients An/2(with respect to Neumann boundary conditions) we do not use the procedure described inthe previous section. Due to the singularities introduced in the potential by the Dirac delta-function and also by making the reflection with respect to the origin, one cannot make aTaylor expansion of V γ R in expression (3.3) (see the discussion after (2.9)). Nevertheless,one can deal with the Heaviside and the delta-function introduced in (3.4) by appropriatelyconstraining the paths in the functional integration. We describe this mechanism by computingthe contributions to the coefficients A1, A3/2 and A2.The first non-vanishing contribution due to Robin boundary conditions comes from theterm corresponding to n = 1 in equation (3.3),− TZ ∞0dxZ 10dτ1©h−2γ · δ(x + y(τ1))i + e− x2T h−2γ · δ((1 − 2τ1)x + y(τ1))iª. (3.5)Let us consider the delta-function in the expectation value. For the first term, this means thatone should consider the contributions of only those paths y(τ) beginning and ending at theorigin in Euclidean time T such that at τ = τ1 reach the point −x. This restriction can beimplemented by computing the product of the (free) transition amplitude (see equation (2.10))from the origin to −x in time T τ1 times the (free) transition amplitude from −x back again tothe origin in time T (1 − τ1). The same calculation can be implemented for the second termin (3.5) but considering the intermediate point −(1 − 2τ1)x instead of −x. The result can bewritten as2T γZ ∞0dxZ 10dτ⎧⎨⎩ e− x24T τ1√4πT τ1· e− x24T (1−τ1)√4πT (1 − τ1)+ e−x2Te−(1−2τ1)2x24T τ1√4πT τ1· e− (1−2τ1)2x24T (1−τ1)√4πT (1 − τ1)⎫⎬⎭= T√4πT· 2γ, (3.6)where the result can be easily obtained by interchanging the order of integration. We concludethat the contribution of the boundary conditions to A1 is given by 2γ .5J. Phys. A: Math. Theor. 41 (2008) 164010 F Bastianelli et alNext, we proceed with the term corresponding to n = 2 in equation (3.3). There are twocontributions to be considered: one is proportional to γ and the other one is proportional toγ 2. The contribution proportional to γ 2 is given byT 224γ 2Z ∞0dxZ 10dτ1Z 10dτ2©hδ(x + y(τ1)) · δ(x + y(τ2))i+ e−x2T hδ((1 − 2τ1)x + y(τ1)) · δ((1 − 2τ2)x + y(τ2))iª= 4T 2γ 2Z ∞0dxZ 10dτ1Z 1τ1dτ2⎧⎨⎩ e− x24T τ1√4πT τ11√4πT (τ2 − τ1)e−x24T (1−τ2)√4πT (1 − τ2)+ e−x2Te−(1−2τ1)2x24T τ1√4πT τ1e−(τ2−τ1)x2T√4πT (τ2 − τ1)e−(1−2τ2)2x24T (1−τ2)√4πT (1 − τ2)⎫⎬⎭= T3/2√4πT· √πγ 2. (3.7)Therefore, the contribution of the boundary conditions to A3/2 is given by√πγ 2.The leading order in T of the contribution proportional to γ in the n = 2 term ofequation (3.3) contributes to the coefficient A2 and is given byT 22(−2γ )Z ∞0dxZ 10dτ1Z 10dτ2©h2V (x + y(τ1))δ(x + y(τ2))i+ e−x2T h2V ((1 − 2τ1)x + y(τ1))δ((1 − 2τ2)x + y(τ2))iª. (3.8)The leading order in T of this expression is obtained by making a Taylor expansion of thepotential about the origin−2T 2γV (0)Z ∞0dxZ 10dτ1Z 10dτ2⎧⎨⎩ e− x24T τ2√4πT τ2e−x24T (1−τ2)√4πT (1 − τ2)+ e−x2Te−(1−2τ2)2x24T τ2√4πT τ2e−(1−2τ2)2x24T (1−τ2)√4πT (1 − τ2)⎫⎬⎭ = − T2√4πT· 2γV (0). (3.9)This is a contribution to the coefficient A2 proportional to γ . There is another contributionto this coefficient due to the boundary conditions which is proportional to γ 3. We end thissection by computing this contribution, which comes from the n = 3 term in equation (3.3)and is given by−T33!(−8γ 3)Z ∞0dxZ 10dτ1Z 10dτ2Z 10dτ3©hδ(x + y(τ1))δ(x + y(τ2))δ(x + y(τ3))i+ e−x2T hδ((1 − 2τ1)x + y(τ1))δ((1 − 2τ2)x + y(τ2))δ((1 − 2τ3)x + y(τ3))iª= T2√4πT· 43γ 3. (3.10)Taking into account this last result together with that given in equation (3.9), we conclude thatthe total contribution to A2 due to the boundary conditions is given by 2γV0 + 4/3γ 3. Allthese contributions are the expected ones (see, e.g., [6].)4. ConclusionsWorldline approaches can be successfully used to study QFTs on manifolds with boundaries.After recalling the methods employed in [1], we have considered the inclusion of Robin6J. Phys. A: Math. Theor. 41 (2008) 164010 F Bastianelli et alboundary conditions. These conditions can be enforced by adding a delta-function potentialto the setup for Neumann boundary conditions described in [1]. In this way we have beenable to recover the expected corrections to the heat kernel coefficients up to, and including,A2. We plan to further simplify and extend the methods reviewed in the present paper [7]. Inaddition, it would be interesting to extend them to include curved boundaries, curved spacesand fields with nontrivial spins.AcknowledgmentsPAGP would like to thank the participants and the organizers of QFEXT’07. PAGP alsoacknowledges financial support from CONICET (PIP 6160), UNLP (11/X381), INFN andDAAD.AppendixConsider the following kernel, defined on the half-lineψ(y, x;β) := hy| e−βHγ |xiR + h−y| e−βHγ |xiR (A.1)withHγ = −∂2y − 2γ δ(y). (A.2)By construction, (A.1) is even under y → −y. In other words ψ(y, x;β) can be extendedto the whole line such that it is continuous as a function of y. On the other hand, its firstderivative is odd,∂yψ(y, x;β) = −∂yψ(−y, x;β), (A.3)and is not defined at y = 0 because of the presence of the delta function. In fact, (A.1) satisfiesthe heat equation¡∂β − ∂2y − 2γ δ(y)¢ψ(y, x;β) = 0, (A.4)which is equivalent to the heat equation for a free particle away from the boundary whosewavefunction satisfies the jump condition−∂yψ(0+, x;β) + ∂yψ(0−, x;β) − 2γψ(0, x;β) = 0. (A.5)Using equations (A.3) and (A.5) we obtain∂yψ(0+, x;β) + γ · ψ(0, x;β) = 0. (A.6)Therefore, (A.1) is the heat kernel for a free particle on the half-line subject to Robin boundaryconditions. Note that taking γ → 0 one correctly achieves Neumann boundary conditions.References[1] Bastianelli F, Corradini O and Pisani P A G 2007 J. High Energy Phys. JHEP02(2007)059[2] Fliegner D, Haberl P, Schmidt M G and Schubert C 1995 Discourses Math. Appl. 4 87[3] Schubert C 2001 Phys. Rep. 355 73[4] Bastianelli F and van Nieuwenhuizen P 2006 Path Integrals and Anomalies in Curved Space (Cambridge:Cambridge University Press) p 379[5] Clark T E, Menikoff R and Sharp D H 1980 Phys. Rev. D 22 3012[6] Branson T P, Gilkey P B, Kirsten K and Vassilevich D V 1999 Nucl. Phys. B 563 603[7] Bastianelli F, Corradini O, Pisani P A G and Schubert C (work in progress)7

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