Dirac fields in a Bohm-Aharonov background and spectral boundary conditions

a r X i v :h e p -t h /9811242v 1 27 N o v 1998Dirac fields in a Bohm-Aharonov background and spectralboundary conditionsC.G.Beneventano,M.De Francia and E.M.Santangelo Departamento de F´ısica -Facultad de Ciencias Exactas -Universidad Nacional de La Plata,ArgentinaWe study the problem of a Dirac field in the background of an Aharonov-Bohm fluxstring.We exclude the origin by imposing spectral boundary conditions at a finite radius then shrinked to zero.Thus,we obtain a behaviour of the eigenfunctions which is compatible with the self-adjointness of the radial Hamiltonian and the invariance under integer translations of the reduced flux.After confining the theory to a finite region,we check the consistency with the index theorem,and discuss the vacuum fermionic number and Casimir energy.1Setting of the problemWe study the Dirac equation for a massless particle in four dimensional Minkowski space,in the presence of a flux tube located at the origin,i.e.,(i ∂−A )Ψ=0H= ∇∧ A =κ2πis the reduced flux.As the gauge potential is z −independent,equation (1)can be decoupled into two uncoupled two-component equations,1by choosing:γ0=σ300σ3 γ1= iσ200iσ2 γ2= −iσ100iσ1 γ3= 0iσ1iσ10 (2)In order to avoid singularities,we will consider that only r >r 0is accessi-ble,and take the limit r 0→0,which is equivalent to having a punctured planewith the removed point corresponding to the string position.By taking A z =A r =0,A θ=κr∂θ−κs =−1can be studied in a similar way,and explicit reference will be made to it whenever necessary).In this case,we can write:H += 0L †L 0 ,with L =−ie iθ(−∂r +B)L †=ie −iθ(∂r +B)(4)and its eigenfunctions:ΨE =ϕE (r,θ)χE (r,θ),satisfy :LϕE =EχE L †χE =EϕE(5)Now,the two components in ΨE have different θdependence.In order to makethis fact explicit,and to discuss boundary conditions at r =r 0,we introduce 2ΨE =1r e−i θ2χ1E (r,θ) L 1=−∂r +B L †1=∂r +B (6)so thatL 1ϕ1E =iEχ1EL †1χ1E =−iEϕ1E(7)We expand ϕ1E and χ1E in terms of eigenfunctions of B,which are of theform:e n =e i (n +12−κ2)θχ1E (r,θ)=∞n =−∞g n (r )e i (n +1E(A n J n +1−κ(|E |r )−B n J κ−n −1(|E |r ))e iθ(10)(Of course,for integer κ,a linear combination of Bessel and Neumann functions must be taken).Finally,for s =−1,the upper and lower components of ΨE interchange,and E →−E .22Boundary conditions at the originAs is well known,3theradialDiracHamiltonianin the background of anAharonov-Bohm gaugefield requires a self-adjoint extension for the critical subspace n=k.In fact,imposing regularity of both components of the Dirac field at the origin is too strong a requirement,except for integerflux.Rather, one has to apply the theory of Von Neumann deficiency indices,which leads to a one parameter family of allowed boundary conditions,characterized by i limr→0(Mr)ν+1g n(r)sin π2 =lim r→0(Mr)−νf n(r)cos π2 (11)withνvarying between−1and0(ν=−αfor s=1;ν=α−1for s=−1). Here,Θparametrizes the admissible self-adjoint extensions,and M,a mass parameter,is introduced for dimensional reasons.Which of these boundary conditions to impose depends on the physical situation under study.One possibility is to take afiniteflux tube,ask for continuity of both components of the Diracfield atfinite radius and then let this radius go to zero.1 Thus,one of the possible self-adjoint extensions is obtained,which corresponds toΘ=πEJ k+α−n−1(|E|r)e i(n+1)θ +3+∞n=k+1A n J n−k−α(|E|r)e inθi|E|2(13)ΨE(r,θ)=k−1n=−∞B n J k+α−n(|E|r)e inθ−i|E|EJ n+1−k−α(|E|r)e i(n+1)θ α<1 2).It corresponds to the values of the parameterΘ:Θ= −π2π2(15)As shown in,7Θ=±π2.This is due to the already commentedpresence of a constant zero mode on the cylinder.However,for the repre-sentation s=−1of2×2Dirac matrices,APS boundary conditions must bereversed,forλ=0,as compared to(12),since the operator B changes into −B.Forλ=0the lower component will be taken to be zero at r0which,as we will show later,allows for charge conjugation to be a symmetry of the wholemodel.In this case,the resulting extension corresponds toΘ= π2−π2(16) It is worth pointing that,for integerκ=k,our procedure leads(both for s=±1)to the requirement of regularity of both components at the origin.3The theory in a bounded regionFrom now on,we will confine the Diracfields inside a bounded region,byintroducing a boundary at r=R,and imposing there boundary conditions ofthe APS type,complementary to the ones considered at r=r0.For s=1f n(R)=0,forλn(R)>0g n(R)=0,forλn(R)≤0(17)4We start by studying the zero modes of our theory,which are of the form Ψ0(r,θ)= e−iθ2)θe iθ2)θ (18)It is easy to see that no zero mode remains after imposing the boundary conditions at both the internal and external radius,even without taking r0→0.This is in agreement with the APS index theorem.5In fact,according to such theoremn+−n−=A+b(r0)+b(R)(19) where n+(n−)is the number of chirality positive(negative)zero energy solu-tions,A is the anomaly,or bulk contribution,and b are the surface contribu-tions coming from both boundaries2,8b(R)=12(η(r0)−h r)(20)withη(r)the spectral asymmetry of the boundary operator B and h r the dimension of its kernel.In our case,the boundary contributions cancel.As regards the volume part,it also vanishes for the gaugefield configuration under study,and we have n+−n−=0,which is consistent with the absence of zero modes.For s=−1both boundary contributions interchange,and identical conclusionshold regarding the index.The nonzero energy spectrum can be determined by imposing(for s=1) the boundary conditions(17)at r=R on the eigenfunctions in eqs.(13)and (14).Thus,one gets:E n,l= ±j n−α,l R,n≥−1,α≥1R,n≥0±j n+α,l2(21) where jν,l is the l-th positive root of Jν.The same spectrum results for s=−1.For both s values,the energy spectrum is symmetric with respect to zero. 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