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recently there has been a renewed interest in the finite temperature dynamics of the one dimensional spin 1/2 heisenberg model , especially on the question of diffusive spin transport@xcite . in particular , it was argued that the integrability of the model implies pathological spin dynamics and presumably the absence of spin diffusion@xcite . the role of conservation laws was pointed out in reference@xcite were it was shown that in several quantum integrable models the uniform ( @xmath0 ) current correlations do not decay to zero at long times . this result , established using the mazur inequality@xcite , suggests pathological finite temperature dynamics . as far as the heisenberg model is concerned , the analysis of conservation laws has shown that the energy current operator commutes with the hamiltonian , suggesting anomalous finite-(@xmath1 ) energy density correlations . however , for zero magnetic field , this method turned out to be insufficient for deciding about the decay of the uniform spin current correlations . this case is closely related to the behavior of the finite temperature conductivity in the one dimensional model of spinless fermions at half - filling interacting with a nearest neighbor interaction ( the  t - v " model@xcite ) .    in this work , we address the issues raised above by the numerical diagonalization of the hamiltonian matrix on finite size lattices . more precisely , we study the implications of the energy current conservation on the ( @xmath1 ) energy density correlations , and , as an alternative route to the analysis of spin diffusion , we investigate the decay of the uniform ( @xmath0 ) spin current correlations . the paper is organized as follows : in section 2 , we recall the heisenberg hamiltonian and define the various quantities studied below . in section 3 we briefly summarize the phenomenological picture of diffusion . there , we also argue that the decay of the uniform spin current correlations to a finite value is incompatible with a diffusive behavior , _ assuming _ continuity in the wave - vector @xmath2 of the correlations at @xmath0 . next , we test these ideas in section 4 in the xy limit , where results can be obtained analytically . turning to the numerical results , in section 5.1 we present the energy density correlations at infinite temperature for the case of the isotropic heisenberg model . a simple ansatz for the observed behavior suggests a logarithmic dependence at low frequencies for the energy autocorrelation function . as far as the spin dynamics is concerned , numerous studies of the @xmath3 spin density correlations exist@xcite . therefore , in section 5.2 , we restrict ourselves to the decay of the uniform spin current correlations for various values of the anisotropy parameter @xmath4 and temperatures interestingly , it turns out that these do not decay to zero for @xmath5 . according to the argument given in section 3 , this result implies non - diffusive spin transport . section 6 contains a short discussion on experimental relevance of these findings and open questions . the anisotropic heisenberg hamiltonian for a chain of @xmath6 sites with periodic boundary conditions is given by : @xmath7 where @xmath8 , @xmath9 are the pauli spin operators with components @xmath10 at site @xmath11 .    for a conserved quantity @xmath12 , @xmath13=0 $ ] , the continuity equation in @xmath14space defines the current @xmath15 :    @xmath16    with @xmath17 and @xmath18 .    setting @xmath19 we find the following spin and energy currents respectively : @xmath20 @xmath21 for the discussion of dynamic correlations at finite temperatures , we chose to analyze the anticommutator form : @xmath22 where @xmath23 is the thermal average at temperature @xmath24 over a complete set of states . further , the frequency dependent correlation function defined by : @xmath25 is symmetric in frequency , @xmath26 . a central point in our approach is the relation between the dynamic correlations of a quantity a and its corresponding current correlations , which we obtain by using the continuity equation ( [ ce ] ) : @xmath27    in particular , we will discuss the asymptotic value of the current correlations @xmath28    a finite value of @xmath29 translates to a @xmath30 peak in @xmath31 and , as we will discuss below , implies restrictions in the behavior of @xmath32 . an important observation is that the energy current @xmath33 of the heisenberg model commutes with the hamiltonian@xcite , so that @xmath34 , whereas the spin current does not . however , it will turn out that @xmath35 for @xmath5 , meaning that the spin current and energy current correlations are similar in the sense that in their frequency representation , they both exhibit a finite weight @xmath30 function . when we consider the @xmath3-dependent correlations of a conserved quantity @xmath36 such as the magnetization , it is usually assumed , largely on phenomenological grounds , that they exhibit a diffusive behavior in the long - time @xmath37 , short wavelength @xmath38 regime@xcite : @xmath39 where @xmath40 is the corresponding diffusion constant , or @xmath41 for @xmath42 . this lorentzian form correctly reduces to a @xmath30 function in the limit @xmath38 , as implied by @xmath13=0 $ ] . further , using the continuity equation ( [ cs ] ) for @xmath38 , we obtain : @xmath43 which gives the diffusion constant @xmath40 when first , the limit @xmath38 and then , @xmath44 are taken . on the other hand , if the current correlations for @xmath0 do not decay to zero at long times , @xmath45 and @xmath46 has a finite weight @xmath30 component which is incompatible with the diffusive form ( [ djbqo ] ) . in this reasoning , we must assume a regular behavior of the correlation functions in the @xmath2 variable .    to summarize the argument , if a quantity @xmath36 is conserved ( @xmath13=0 $ ] ) and its current @xmath47 is either conserved ( @xmath48=0 $ ] ) , or @xmath45 , then continuity in @xmath2 at @xmath0 excludes a diffusive form ( [ db ] ) for the corresponding correlation @xmath49 . a simple model for testing these ideas is the @xmath50 limit ( @xmath51 ) , of the heisenberg model . in this case , both the energy current @xmath33 and the spin current @xmath52 commute with the hamiltonian . the model can be mapped to a free spinless fermion model by using a jordan - wigner transformation which allows us also to evaluate explicitly the spin and energy dynamic correlations at @xmath53 . in the spin case , these are well known results@xcite : @xmath54 @xmath55 these forms are indeed consistent with the conservation of both spin ( energy ) and spin current ( energy current ) as they reduce to a @xmath30 function when the limits @xmath38 , @xmath44 are taken . further , the time decay of the autocorrelations at @xmath53 is not of the form @xmath56 , as predicted by the diffusion hypothesis . indeed , @xmath57 @xmath58 which both behave as @xmath59 for @xmath60 . as we mentioned earlier , the energy current @xmath33 associated with the anisotropic heisenberg model ( [ heis ] ) commutes with the hamiltonian for all values of the parameter @xmath4 . therefore , the time correlations do not decay at all ( @xmath34 ) and according to the argument explained in section 3 , no diffusive energy transport occurs . however , the conservation of @xmath33 does not provide us with any details about the shape of @xmath61 at finite @xmath2 . in the absence of an analytical solution , we investigate this quantity by numerical diagonalization of the hamiltonian matrix on a ring of 16 sites .    in figure 1 , we show @xmath61 for @xmath62 , which is experimenally the most interesting point as it describes isotropic quasi one - dimensional antiferromagnets . we study the high temperature limit @xmath53 , which is the most convenient for a numerical study as it involves the full excitation spectrum , but is also relevant experimentally for spin systems as the magnitude of @xmath63 can be of the order of @xmath64 . the plot is represented as histograms of width @xmath65 , all the frequencies which fall into one interval are summed up . the inset shows the normalized , integrated ( prior to summing nearby frequencies ) quantity @xmath66 which has the advantage of smoothing out the finite size discontinuities . to point out the practically linear integrated behavior of the pure heisenberg model , we also show the same quantity for a more generic case obtained by adding a next - next neighbor ( non - integrable ) interaction @xmath67 .    the simplest way to describe this behavior is by means of  plateaus " given by the following ansatz : @xmath68 which satisfy the first @xmath69 and the second @xmath70 exact moments for @xmath53 . further , this ansatz is compatible with the limit @xmath71 as implied by the conservation of energy . using the continuity equation ( [ cs ] ) , we obtain for small @xmath2 and @xmath72 @xmath73 which correctly reduces to a @xmath30-function for @xmath38 , in agreement with the conservation of the energy current @xmath74=0 $ ] .    using this ansatz we find for the energy autocorrelation function ( obtained by integration over @xmath2 ) : @xmath75 a logarithmic behavior at low frequencies , in contrast to the diffusion form @xmath76 . we should stress that these results are only _ indicative _ , as they are obtained from small size lattices which can provide reliable information only for correspondingly high frequencies and large wave - vectors . nevertheless , the consistency of these results with the arguments presented above against a diffusion form are encouraging .      the spin density dynamic correlations @xmath77 have been the subject of many studies which have not been able to answer the question of spin diffusion unambiguously . here , we revisit this problem by investigating the compatibility between spin density and spin current correlations , which requires that we calculate @xmath78 . in contrast to the energy current , the spin current @xmath52 does not commute with the hamiltonian , so that @xmath79 is different from a pure @xmath30 function . nevertheless , if @xmath80 , which means that @xmath79 has a finite weight @xmath81function at @xmath82 , our previous arguments against diffusion still hold .    in determining @xmath78 , we noticed a peculiar difference in the low frequency behavior of @xmath79 depending on the anisotropy parameter @xmath4 . in figure 2 , we show @xmath83 the corresponding integrated , normalized quantity . we see that for @xmath84 , @xmath85 ( @xmath86 in the figure ) all the low frequency weight of @xmath79 is concentrated in the @xmath87-function at @xmath82 . in contrast , for neighboring values such as @xmath88 or @xmath89 , we observe a shift of weight to a low frequency region whose size decreases as the system grows ( inset ) and eventually vanishes as @xmath90 . we believe that the behavior of this special @xmath4 points is related to the existence of finite length strings ( bound states ) as they appear in the formulation of the thermodynamics of the heisenberg model , within the bethe ansatz method@xcite . it seems that in order to determine @xmath78 from finite size systems for @xmath91 , we should include the weight from these low frequency regions . as an example , doing so for @xmath88 gives us a value of @xmath92 for l=16 ( figure 2 ) . having discussed this technical issue , we can then determine @xmath78 for different size systems , as a function of temperature and @xmath4 . by extrapolating our finite size results to the thermodynamic limit using second order polynomials in @xmath93 for @xmath94 , we obtain the results shown in figure 3 . their striking feature is that for @xmath95 , @xmath78 is finite in the @xmath5 region , and practically zero when @xmath96 . in this regime , according to our previous argument , we expect a non- diffusive behavior . deciding about the behavior of @xmath78 for @xmath96 at finite temperatures is rather subtle . the reason is that in the heisenberg model , @xmath62 corresponds to a point of change of symmetry , from easy plane to easy axis , accompanied by the opening of a gap . in the fermionic version of the model , the  t - v " model , it corresponds to a metal - insulator mott - hubbard type transition , with the charge stiffness changing discontinuously@xcite at zero temperature . we should note that this discontinuity is difficult to reproduce by numerical simulations on small finite size lattices , as the transition corresponds to the divergence of the localization length . considering that at high temperature , @xmath78 behaves similarly to the charge stiffness in the  t - v " model@xcite we understand why it is difficult to decide whether @xmath78 is greater than zero in the region @xmath97 and @xmath98 . for the same reason , we can not exclude that @xmath78 behaves discontinuously at @xmath62 . nevertheless , it seems unambiguous that @xmath99 for @xmath100 and @xmath95 . the results presented are of interest in recent experimental studies@xcite of spin dynamics in quasi - one dimensional materials such as @xmath101 and @xmath102 . particular attention should be paid to the unusually high value of the diffusion constant found in nmr experiments on @xmath102@xcite , perhaps related to the integrability of the heisenberg model as discussed above . furthermore , our results on the behavior of energy density correlations are of interest in the interpretation of the quasi - 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in this paper , we study the spin and energy dynamic correlations of the one dimensional spin 1/2 heisenberg model , using mostly exact diagonalization numerical techniques . in particular , observing that the uniform spin and energy currents decay to finite values at long times , we argue for the absence of spin and energy diffusion in the easy plane anisotropic heisenberg model .