A topological approach to renormalization and its geometrical, dimensional consequences
aa r X i v : . [ phy s i c s . g e n - ph ] A ug A topological approach to renormalization
F. GHABOUSSI
Department of Physics, University of KonstanzP.O. Box 5560, D 78434 Konstanz, GermanyE-mail: [email protected]
Abstract
We show that because the necessity of renormalization arises from the infiniteintegrals caused by the discrepancy between the orders of differential- andintegral operators in four dimensional QFTs. Therefore in view of the factthat finiteness and invariant properties of operators are topological aspects anyessential renormalization tool to extract finite invariant results comparable withexperimental results from those infinities, e. g. regularization, perturbationand radiative corrections follow some topological standards.1he necessity of renormalization arises from the discrepancy between the orders of differential operators/propagators which are up to two/ three and the order of integral operators in the four dimensional(4D) theories which is four. Insofar the renormalization can be considered as a method of adjustmentbetween these different orders. In view of the fact that invariant properties of operators on suitablecompact manifolds are described by their analytical/ topological indices through the dimensions of relatedco/homology groups therefore any invariant treatment of operators properties should be considered as atopological matter [1]. Thus also the values of propagator integrals/ Green’s functions as the inverse ofdifferential operators in the renormalized theory should be topologically invariant in order to be comparedwith the globally invariant experimental values.To justify the topological approach to renormalization let us note that beyond the enormous relevance oftopological methods in physics appeared in the last decades Hermann Weyl already showed one centuryago that even the most empirical aspects of electrodynmaics such as the Kirchhoff laws follow purely topo-logical laws [2]. Thus Kirchhoff followed also early topological considerations to derive them. Moreoverin view of the fact that the Hodge-de Rham theory of differential topology is a geometric generaliza-tion of equations of classical electrodynamics [3] and QED as their quantization needs renormalizationcorrections [4] therefore the renormalized QED is closely connected with the topology.Nevertheless our topological approach is an attempt to make renormalization more intelligible. In otherwords the aim of this work is to understand why renormalization is an admissible method to extract finitevalues from infinite results of QED interactions and how renormalization can be understood by geometricphysical consideration including topology. The relevance of topology to understand the renormalizationarises also from the fact that both renormalization of QED and our topological approach consider physicalquantities of the same dimension (in geometric units), e. g. the 1 L dimensional momentum component ofelectron p µ , its mass M and the gauge potential eA µ as equivalent quantities contributing to the renor-malized mass of electron and to the self energies of electron and photon. Thus for example the extractionof corrective terms for the mass of electron from the radiative corrections shows that photon participatesdirectly to the corrected value of electron mass. Then the mass operator of interacting electron is givenas the sum of two terms M ( x, x ′ ) = m δ ( x − x ′ ) + ie γ µ G ( x, x ′ ) γ µ D + ( x − x ′ ) where G ( x, x ′ ) is theGreen’s function of the Dirac equation in the external field, and D + ( x − x ′ ) is a photon Green’s function24]. Insofar the momentum of electron in the renormalized Dirac equation and the coupled photon fieldact qualitatively equivalent to the mass correction. Thus in view of the equivalent treatment of differ-ential forms of the same order in topology physical quantities such as momentums, masses and gaugepotentials as the components of related differential one forms will be treated equivalently (see below).Also the introduced abstract vectors in the standard renormalization [5] recalls the relevance of abstracttopological methods in renormalization.The main result of these considerations is that from topological stand point due to the invariance oflagrangian all various participants of the same geometric dimension in renormalization relations shouldbe considered as components of differential forms of the same order, i. e. as qualitatively equivalentquantities.It is not so hard to consider the renormalization as a topological problem if one compares and interpretsthe renomalization tools such as Green’s functions as differential topological quantities. Then in thetopology where one treats topological invariants such as the dimensions of co/homology groups of certaindifferential forms, the relations among participants are perfectly adopted to achieve invariants. Thusthe differential forms (forms) are dimensionally invariants in view of the 1 L r dimensionality of their ω m ,...m r , r ∈ Z components and the L r dimensionality of their dx m ∧ ... ∧ dx m r basis according to1 L r .L r = L . Therefore integrals of certain differential r-forms over certain r-chains are invariant integersknown as their periods or dimensions of the related co/homology groups [1]. Nevertheless in 4D QFTswhere the integrals are 4 dimensional R ∞ dp x ∧ dp y ∧ dp z ∧ dp t and the 1 L dimensional propagatorslike 1 p + ... can be considered as components of some two forms 1 p + ... dp x ∧ dp y ∈ ω the result iscoordinate dependent and divergent (see also below). This is as mentioned above due to the discrepancybetween the order of relevant differential operators/ propagators and integral operators in 4D QFT.Note that the regularization of these divergencies results in R Λ0 ( 1 p + ... ) − ( 1 p + ... ) dp x ∧ dp y ∧ dp z ∧ dp t ∼ Z Λ0 ω which are however logarithmic divergent in the limit of Λ → ∞ .Moreover with respect to the invariant properties of differential operators note that their main invariantproperty is their topological/analytical index on a suitable compact manifold [1]. Thus it seems that thecompactification of integration manifold in QFTs by regularization /cut off is related to the requirementof a compact manifold in order to define topologically well behaved differential operators/ propagators3n QFTs (see also below).We will show in the following that essential renormalization tools to extract finite results from the infiniteintegrals, i. e. perturbation, regularization and radiative corrections follow some topological methods inview of the fact that finiteness, invariant properties of operators are topological aspects.To begin note that from topological point of view one may consider all 1 L dimensional physical quantitiessuch as the Hamiltonian H , components of momentum of electron, photon p eµ , k γµ or gauge potentialcomponents A µ and masses M ∼ p µ as vector components of their one forms ω , respectively in thegeometric units where all constants ¯ h, C, e and the velocity v µ are dimensionless. Thus the momentum M.v µ can be considered as the component of a potential one form in the symplectic geometry [8].Therefore from topological stand point it is admissible and reasonable that one renormalizes 1 L mass bycorrections of 1 L Hamiltonian or momentum or gauge potential terms.In general the topological invariant aspects of any 1 L r dimensional physical quantity can be considered asthe experimentally measured invariant aspects of tensor components of some r-form whereby we have todo with quantities up to r = 2, i. e. the field strengths or curvature components. Thus treating physicalfield strengths as differential topological two form F = F µν dx µ ∧ dx ν on a compact oriented manifoldone can describe the experimentally measured integral form of Maxwell equations R D, or ∂ (2 D ) F ∝ , Q, orJ, etc. . Also the only quantum invariant which is experimentally well confirmed, i. e. thequantum of magnetic flux R D F ∝ ¯ h follows the same topological invariant property that integrals ofsuitable two forms over suitable two manifolds are as inner products <, > invariant [1].In the following we explain how any essential renormalization tool follows some topological method.As an example of application of topology in renormalization note that the radiative corrections of elec-tromagnetic potential A µ ∈ ω given by A µ = A µ ⊕ ✷ A µ ⊕ ✷ A µ ⊕ ... [6] follows the topological propertyof Hodge decomposition theorem for connection one form ω = Harm ⊕ dω ⊕ d † ω including thezero form decomposition ω = Harm ⊕ d † ω on a compact oriented manifold without boundary by itsiteration ω = Harm ⊕ ✷ ω ⊕ ✷ ω ⊕ ..., ✷ := dd † + d † d assuming Lorenz gauge as in electrodynamics d † ω ≡ ✷ on a differential form does not change its order ✷ n ω r ∈ ω r [1]. Therefore the structure of radiative corrections according to [6] follows from the Hodgedecomposition theorem for differential forms on a compact oriented manifold without boundary whereby4 µ dx µ represents the harmonic one form Harm . Thus also the counter term technics to complete aconcrete term follows the same decomposition theorem schema for ω r where the decomposition schemeincludes complete relevant terms related to a concrete form on a compact oriented manifold withoutboundary.Furthermore note that as mentioned above the necessary compactness of manifold in topological consid-eration of renormalization to apply the Hodge decomposition for the radiative corrections may be relatedto the regularization scheme of renormalization. Thus the standard regularization by cut off compactifythe domain of integration. Insofar regularization follows the topologically necessary compactification ofthe underlying manifold in order to apply topological methods in renormalization. Then most of essen-tial topological methods, e. g. Hodge decomposition apply only to compact oriented manifolds withoutboundary. Thus the regularization of QFTs is also the preparation step to renormalization in the samemanner that its equivalent compactification of manifold is the preparation of manifold for the applicationof topological methods.Also the general theory of perturbation H = H + H + ... + H n ; { H, H i ∼ L } ∈ ω follows the topologicalmethod of Hodge decomposition ω = d † dω ⊕ ( d † d ) ω ⊕ ... related with the above mentioned iterationof Hodge decomposition theorem ω = Harm ⊕ ✷ ω ⊕ ✷ ω ⊕ ... under the suitable condition d † ω = 0on a compact oriented manifold without boundary [7]. Here H ∈ Harm , H i ∈ ✷ i ω . Thus alsothe subtraction method in regularization technics can be considered topologically as subtraction of twoequivalent components ( 1 p + ... ) − ( 1 p + ... ) of some two forms by ω = d † dω ⊕ ( d † d ) ω ⊕ ... followingthe Hodge decomposition of two forms ω = dω ⊕ Harm that includes the Hodge decomposition ofone forms ω = d † ω ⊕ ... In other words in view of the topological equivalence of all terms in a Hodge decompositions, e. g.among H i ∈ ω ; i = 1 , ...n or ✷ i A µ the degeneracy which arise from each of these terms in any ordercan be compensated by degeneracy arising from the other term(s) in further order and finite resultsarising from one term can be completed by finite results from other(s) as required in renormalization.Accordingly the topological equivalence of terms in iterated Hodge decomposition explains the correctnessof compensation of divergent terms in different orders by each other. Then one may consider the power i in the Hodge decomposition, e. g. in H i ∈ ✷ i ω or in ✷ i A µ as the order of perturbation.5t is worth mentioning that also the usual gauge transformation A µ = A µ + dλ follows the Hodgedecomposition theorem of topology for ω in the absence of d † ω which recalls the absence of the mattercurrent one form J = d † ω , J ∈ ω in the homogenous Maxwell equations. Whereas the continuity ofvector current J µ becomes a differential topological identity according to d † ≡ d † ω = 0 whichapplies on any structurally stable or topologically stable dynamical system [8]. Insofar also these aspectsof physics including classical and QED follow topological standards on the mentioned compact manifolds.Furthermore the structure of radiative correction according to Feynamn diagrams follows also the topo-logical measure of the Euler characteristic of diagrams. Thus the second order radiative correction toreplace the vertex by a triangle diagram [6] is topologically admissible in view of the fact that bothvertex and triangle possess the Euler characteristics +1. Then for a vetex alone the Euler character-istic χ ( v ) = 1 v = 1 and for a triangle χ ( triangle ) = +3 v − e + 1 f = 1 [1] with v , e and f for thevertex, edges and faces. Because from topological stand point you do not need to distinguish betweenthe electron p µ or photon k µ edges according to their above discussed differential topological equivalenceas one form components. Thus also the Euler characteristic of the self energy diagram of electron [6] is2 v − e + 1 f = 1.It is worth mentioning that all diagrams with Euler characteristic 1, e. g. the self energy, vacuumpolarisation and proper vertex part are logarithmically divergent, i. e. divergent of order one forΛ → ∞ [10]. In other words the order of divergence of these diagrams equals their Euler characteristics.Nevertheless if one assume a regularization by compactifying the integration manifold of these interactionsby Λ < ∞ the interaction integrals become convergent. Topologically this would mean that in order torenormalize them their diagrams should be considered on a compact surface where two faces or closedareas exist, one inside and one outside of these diagrams which increase the Euler characteristic of thesediagrams from 1 to 2, i. e. +3 v − e + 2 f = 2 and +2 v − e + 2 f = 2, respectively. Thus also the Eulercharacteristic of the usual compact model of 4D Euclidean space-time manifold S required by Wickrotation of the renormalization theory is 2 [11].Furthermore not that the restriction of topological approach to the abelian case of QED, i.e. the absencea topological approach for non-abelian models is due to the impossibility of a generalization of Hodge-6e Rham theory of differential topology to the non-abelian cases in view of the absence of differentialtopological invariants such as harmonic forms in these cases [12].In the sense of general confirmation of topological approach and a further motivation regarding the useof topology in renormalization note that the forrest method in renormalization manifests the relativeco/homology on a manifold with respect to its sub-manifolds [13]. Thus also the Hopf algebra approachto renormalization by trees [14] follows the topological methods in view of the facts that trees are specialgraphs and that global invariant aspects of any algebra is related to its topological aspects. For examplethe concepts of coproduct and antipode in the Hopf algebra approach according to the admissible cutsrecalls the K¨unneth formulas of standard topology. A model of topological renormalization
The most important achievement of renormaliaztion is known to be the explanation of anomalous mag-netic moment of electron and the Lamb shift related to the interaction of electron with the electromag-netic field strength and vacuum polarization according to the new term added to the Dirac equation [4].We prove in the following that such an additional term can be explained as the Hodge decompositiontheorem of the electromagnetic one form on an oriented compact manifold without boundary.As it is mentioned above the regularization results in the compactness of the integration manifold ofinteractions where one is able to use the Hodge decomposition theorem after requiring its orientability andboundarylessness. Accordingly one may interpret the renormalized Dirac equation with electromagneticpotential and field strength coupling [4] as a component of the Hodge decomposition of a connection oneform
Harm ⊕ dω ⊕ d † ω − ω = 0 multiplied by ω in view of the fact that ω .ω r ∈ ω r .( dω ⊕ ω ⊕ Harm ⊕ d † ω ) ω = 0 ∼ ( − γ [ i∂ + eA ] + m − µ ′ . / σF ) ψ = 0 , (1)where the abstract ⊕ incorporates ± and µ ′ = R ... is a constant value proportional to the fine structureconstant [4].Here ψ ∼ ω , eγA ∼ ω , iγ∂ ∼ d, m ∼ Harm , / σF ∼ ω and hence d † ω ∈ ω = R F... = F . R ... according to d † F := R ω , ω := R ω , ω = dω , F = dA, A ∈ ω and in view of the constancy of F ∈ ω [15]. The main effect is here the contribution of d † F ∼ R ω ∼ σF . R ... ∼ σF . µ ′ of the constantmagnetic field strength F which is not involved in the standard Dirac equation.7he most important point in the Hodge decomposition of one form is that on the suitable compactmanifold there are three terms of decomposition, i. e. the exterior differential of a zero form, the adjointexterior differential of a two form and the harmonic one form. Altogether four terms of the same order.Exactly these four terms appear in the renormalized Dirac equation whereas the usual Dirac equationpossesses only three terms. In other words the Hodge decomposition terms on the mentioned compactmanifold include all necessary terms which are not all considered in differentials like the Dirac equation.Whereas they must be included by renormalization in order that the equation has finite ”compact” valuescomparable with the experimental values. Thus the ”rigorous”, ”explicit” and ”exact solutions” for therenormalized Dirac equation are given just for the constant field strength and plane wave [4].Note also that the standard renormalization calculations to reproduce the experimental result of anoma-lous magnetic moment uses various calculative adjustments to reproduce those result [4]. Neverthelessthe main achievement is as mentioned above the appearance of the additional term proportional to F inthe renormalized Dirac equation which can be founded by the above discussed topological necessity.In the same manner other renormalization corrections such as the self energy corrections can be foundedand understood also topologically using a. o. the Hodge decompositions of zero form for wave functionsand the electromagnetic two form of Photon under suitable conditions.In other words the renormalization/ regularization can be founded and understood topologically as therequirement of a compact oriented manifold without boundary for the space-time/ momentum spaceintegrations where differential equations are described by Hodge decompositions why also Green’s func-tion or propagators are adjusted to dies decompositions. Because also the renormalized Green’s function ie γ µ G ( x, x ′ ) γ µ D + ( x − x ′ ) includes the additional terms proportional to the constant electromagneticfield strength F [4].It is interesting to mention with respect to Feynman’s space-time approach to renormalization that alsothe Tylor expansion with which he founded the achievement of Schroedinger equation and the relevanceof his approach to quantum mechanics [4] follows the mentioned Hodge decomposition theorem for zeroforms if one assumes a one dimensional compact manifold where ω = ∗ ω according to the Hodgeduality. Thus after first iteration ω = d † ω ⊕ Harm → ω = Harm ⊕ ∗ dω ⊕ d † dω ∼ f ( x ) = f (0) + ∂f ( x ) + ∂ f ( x ), respectively. Note that although all such expansions are performed usually up8he second order nevertheless it is possible to achieve all further terms of derivatives from the furtheriteration of Hodge decomposition as above [7]. References ✷ for the d’ Alambertian. Whereas we use as in the modern standard the ✷ for d’Alambertianwhich is also the Laplacian on a flat manifold of Lorentzian signature.[7] With respect to the multiplicity of ω r in the iteration of Hodge decomposition note that n.ω r ∈ ω r , n ∈ Z [1]. Therefore one may have several ω r s under operations of several powers of d † d , etc.98] R. Abraham, J. E. Marsden, ”Foundations of Mechanics”, second edition, AMS 2008. Woodhouse,N.M. J., Geometric Quantization (second edition). Oxford University Press (1991).[9] De Rham theorem implies the duality between the vector spaces H r ( M ) and H r ( M ) [1]. Onemay interpret this as the duality between their L r and 1 L r dimensional cocycle- and cycle basisrespectively or the duality between their inverse dimensional components respectively in view ofthe invariance of inner product 1 L r .L r = 1.[10] K. Huang, https://arxiv.org/pdf/1310.5533.[11] Note that if one considers the Euler characteristic on S as the number of degrees of freedomon S in view of the fact that Euler characteristic is equal to the index of differential operatorsuch as the Laplacian on S counting its independent solutions on S [1]. Further if one describesthe propagators integrals on S4