Functional determinant - Wikipedia

In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order ... Functionaldeterminant FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Infunctionalanalysis,abranchofmathematics,itissometimespossibletogeneralizethenotionofthedeterminantofasquarematrixoffiniteorder(representingalineartransformationfromafinite-dimensionalvectorspacetoitself)totheinfinite-dimensionalcaseofalinearoperatorSmappingafunctionspaceVtoitself.Thecorrespondingquantitydet(S)iscalledthefunctionaldeterminantofS. Thereareseveralformulasforthefunctionaldeterminant.Theyareallbasedonthefactthatthedeterminantofafinitematrixisequaltotheproductoftheeigenvaluesofthematrix.Amathematicallyrigorousdefinitionisviathezetafunctionoftheoperator, ζ S ( a ) = tr S − a , {\displaystyle\zeta_{S}(a)=\operatorname{tr}\,S^{-a}\,,} wheretrstandsforthefunctionaltrace:thedeterminantisthendefinedby det S = e − ζ S ′ ( 0 ) , {\displaystyle\detS=e^{-\zeta_{S}'(0)}\,,} wherethezetafunctioninthepoints=0isdefinedbyanalyticcontinuation.Anotherpossiblegeneralization,oftenusedbyphysicistswhenusingtheFeynmanpathintegralformalisminquantumfieldtheory(QFT),usesafunctionalintegration: det S ∝ ( ∫ V D ϕ e − ⟨ ϕ , S ϕ ⟩ ) − 2 . {\displaystyle\detS\propto\left(\int_{V}{\mathcal{D}}\phi\;e^{-\langle\phi,S\phi\rangle}\right)^{-2}\,.} Thispathintegralisonlywelldefineduptosomedivergentmultiplicativeconstant.Togiveitarigorousmeaningitmustbedividedbyanotherfunctionaldeterminant,thuseffectivelycancellingtheproblematic'constants'. Thesearenow,ostensibly,twodifferentdefinitionsforthefunctionaldeterminant,onecomingfromquantumfieldtheoryandonecomingfromspectraltheory.Eachinvolvessomekindofregularization:inthedefinitionpopularinphysics,twodeterminantscanonlybecomparedwithoneanother;inmathematics,thezetafunctionwasused.Osgood,Phillips&Sarnak(1988)haveshownthattheresultsobtainedbycomparingtwofunctionaldeterminantsintheQFTformalismagreewiththeresultsobtainedbythezetafunctionaldeterminant. Contents 1Definingformulae 1.1Pathintegralversion 1.2Zetafunctionversion 2Practicalexample 2.1Theinfinitepotentialwell 2.2Anotherwayforcomputingthefunctionaldeterminant 2.3Theinfinitepotentialwellrevisited 3Seealso 4Notes 5References Definingformulae[edit] Pathintegralversion[edit] Forapositiveself-adjointoperatorSonafinite-dimensionalEuclideanspaceV,theformula 1 det S = ∫ V e − π ⟨ x , S x ⟩ d x {\displaystyle{\frac{1}{\sqrt{\detS}}}=\int_{V}e^{-\pi\langlex,Sx\rangle}\,dx} holds. TheproblemistofindawaytomakesenseofthedeterminantofanoperatorSonaninfinitedimensionalfunctionspace.Oneapproach,favoredinquantumfieldtheory,inwhichthefunctionspaceconsistsofcontinuouspathsonaclosedinterval,istoformallyattempttocalculatetheintegral ∫ V e − π ⟨ ϕ , S ϕ ⟩ D ϕ {\displaystyle\int_{V}e^{-\pi\langle\phi,S\phi\rangle}\,{\mathcal{D}}\phi} whereVisthefunctionspaceand ⟨ ⋅ , ⋅ ⟩ {\displaystyle\langle\cdot,\cdot\rangle} theL2innerproduct,and D ϕ {\displaystyle{\mathcal{D}}\phi} theWienermeasure.ThebasicassumptiononSisthatitshouldbeself-adjoint,andhavediscretespectrumλ1,λ2,λ3,…withacorrespondingsetofeigenfunctionsf1,f2,f3,…whicharecompleteinL2(aswould,forexample,bethecaseforthesecondderivativeoperatoronacompactintervalΩ).Thisroughlymeansallfunctionsφcanbewrittenaslinearcombinationsofthefunctionsfi: | ϕ ⟩ = ∑ i c i | f i ⟩ with  c i = ⟨ f i | ϕ ⟩ . {\displaystyle|\phi\rangle=\sum_{i}c_{i}|f_{i}\rangle\quad{\text{with}}c_{i}=\langlef_{i}|\phi\rangle.} Hencetheinnerproductintheexponentialcanbewrittenas ⟨ ϕ | S | ϕ ⟩ = ∑ i , j c i ∗ c j ⟨ f i | S | f j ⟩ = ∑ i , j c i ∗ c j δ i j λ i = ∑ i | c i | 2 λ i . {\displaystyle\langle\phi|S|\phi\rangle=\sum_{i,j}c_{i}^{*}c_{j}\langlef_{i}|S|f_{j}\rangle=\sum_{i,j}c_{i}^{*}c_{j}\delta_{ij}\lambda_{i}=\sum_{i}|c_{i}|^{2}\lambda_{i}.} Inthebasisofthefunctionsfi,thefunctionalintegrationreducestoanintegrationoverallbasisfunctions.Formally,assumingourintuitionfromthefinitedimensionalcasecarriesoverintotheinfinitedimensionalsetting,themeasureshouldthenbeequalto D ϕ = ∏ i d c i 2 π . {\displaystyle{\mathcal{D}}\phi=\prod_{i}{\frac{dc_{i}}{2\pi}}.} ThismakesthefunctionalintegralaproductofGaussianintegrals: ∫ V D ϕ e − ⟨ ϕ | S | ϕ ⟩ = ∏ i ∫ − ∞ + ∞ d c i 2 π e − λ i c i 2 . {\displaystyle\int_{V}{\mathcal{D}}\phi\;e^{-\langle\phi|S|\phi\rangle}=\prod_{i}\int_{-\infty}^{+\infty}{\frac{dc_{i}}{2\pi}}e^{-\lambda_{i}c_{i}^{2}}.} Theintegralscanthenbeevaluated,giving ∫ V D ϕ e − ⟨ ϕ | S | ϕ ⟩ = ∏ i 1 2 π λ i = N ∏ i λ i {\displaystyle\int_{V}{\mathcal{D}}\phi\;e^{-\langle\phi|S|\phi\rangle}=\prod_{i}{\frac{1}{2{\sqrt{\pi\lambda_{i}}}}}={\frac{N}{\sqrt{\prod_{i}\lambda_{i}}}}} whereNisaninfiniteconstantthatneedstobedealtwithbysomeregularizationprocedure.Theproductofalleigenvaluesisequaltothedeterminantforfinite-dimensionalspaces,andweformallydefinethistobethecaseinourinfinite-dimensionalcasealso.Thisresultsintheformula ∫ V D ϕ e − ⟨ ϕ | S | ϕ ⟩ ∝ 1 det S . {\displaystyle\int_{V}{\mathcal{D}}\phi\;e^{-\langle\phi|S|\phi\rangle}\propto{\frac{1}{\sqrt{\detS}}}.} Ifallquantitiesconvergeinanappropriatesense,thenthefunctionaldeterminantcanbedescribedasaclassicallimit(WatsonandWhittaker).Otherwise,itisnecessarytoperformsomekindofregularization.Themostpopularofwhichforcomputingfunctionaldeterminantsisthezetafunctionregularization.[1]Forinstance,thisallowsforthecomputationofthedeterminantoftheLaplaceandDiracoperatorsonaRiemannianmanifold,usingtheMinakshisundaram–Pleijelzetafunction.Otherwise,itisalsopossibletoconsiderthequotientoftwodeterminants,makingthedivergentconstantscancel. Zetafunctionversion[edit] LetSbeanellipticdifferentialoperatorwithsmoothcoefficientswhichispositiveonfunctionsofcompactsupport.Thatis,thereexistsaconstantc>0suchthat ⟨ ϕ , S ϕ ⟩ ≥ c ⟨ ϕ , ϕ ⟩ {\displaystyle\langle\phi,S\phi\rangle\geqc\langle\phi,\phi\rangle} forallcompactlysupportedsmoothfunctionsφ.ThenShasaself-adjointextensiontoanoperatoronL2withlowerboundc.TheeigenvaluesofScanbearrangedinasequence 0 < λ 1 ≤ λ 2 ≤ ⋯ , λ n → ∞ . {\displaystyle0