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an analysis of spin dynamics involves calculations of spin correlators . spin operators do not satisfy the wick theorem , and various methods have been used to still enable the use of perturbative ( diagrammatic ) methods . one of the approaches is based on the majorana - fermion representation of spin operators . this approach has a long history and was applied recently to condensed - matter problems . @xcite    in this approach one introduces three majorana fermions @xmath0 ( per spin ) and expresses the spin ( or pauli ) operators via these fermions : @xmath1 obviously , an analysis of spin - spin correlations based on eq . ( [ eq : majorana_representation_3 ] ) requires calculations of a four - fermion correlator . in this note we show that spin correlators coincide with certain two - fermion ( i.e. , one - particle ) correlators . we discuss under which conditions this identity may simplify the analysis and discuss its particular applications . majorana fermions satisfy the anti - commutation relations : @xmath2 for @xmath3 and @xmath4 , and are real : @xmath5 ( @xmath6 ) . these properties ensure that the representation ( [ eq : majorana_representation_3 ] ) reproduces the commutation relations of the spin algebra . an important feature of the representation ( [ eq : majorana_representation_3 ] ) is the fact that the spin operators are bilinear ( ` bosonic ' ) combinations of dirac ( annihilation / creation ) fermionic operators . one may construct the three majorana fermion operators ( for a spin ) out of three different dirac fermions , @xmath7 , each annihilation operator acting in its own two - dimensional hilbert space . this ensures the anticommutation relations and the property @xmath8 . the whole hilbert space is then 8-dimensional . the dimensionality of the hilbert space may be reduced if two majorana fermions , e.g. @xmath9 and @xmath10 , are constructed as linear combinations of a single dirac fermion @xmath11 and its conjugate @xmath12 : @xmath13 and @xmath14 . another dirac fermion @xmath15 is still needed to construct the third majorana fermion @xmath16 . in this mixed majorana - dirac picture @xmath17 this ` drone - fermion ' representation was used for the analysis of magnetic systems in the 60s ( see refs . the whole hilbert space in this representation is 4-dimensional . depending on the rotational symmetry , this representation may be more convenient than ( [ eq : majorana_representation_3 ] ) . we have described this construction in the hilbert space of two dirac fermions @xmath11 and @xmath18 . alternatively one can view it as two replicas of the original spin . indeed , let us label the basis states in the following way : denote the state without @xmath11- and @xmath18-fermions by @xmath19 , and also @xmath20 , @xmath21 , and @xmath22 . for the state @xmath23 the first index @xmath24 denotes the spin component , while the second @xmath25 labels the spin copy . one notices that the spin operators @xmath26 , @xmath27 , and @xmath28 do not mix the @xmath29- and @xmath30-subspaces , i.e. , they operate in the same way on the two ` copies ' of the spin . further , one may view the index @xmath25 as an isospin and introduce the respective pauli isospin operators @xmath31 . in particular , @xmath32 is the ` copy - switching ' operator : @xmath33 and @xmath34 . the fermionic operators can be expressed as @xmath35 further , @xmath36 , for any @xmath37 . accordingly , in terms of the fermionic operators we have @xmath38 . the operator @xmath32 commutes with ` all other ' operators : with @xmath26 , @xmath27 , @xmath28 , @xmath11 , @xmath39 , and @xmath40 . a physical hamiltonian depends on the spin operators @xmath41 , @xmath42 , @xmath28 ( and on other degrees of freedom , e.g. , the electrons in the kondo problem ) . thus the operator @xmath32 commutes with any hamiltonian and we obtain @xmath43 here we have used the fact that @xmath32 is time - independent , commutes with the majorana fermions , and that @xmath44 . for example , we obtain @xmath45\,[f(t')+f^{\dag}(t')]\rangle \ .\end{aligned}\ ] ]    in certain situations the identity ( [ eq : bubble_to_line_reduction ] ) may simplify the calculations . indeed , if the majorana representation ( [ eq : majorana_representation_3 ] ) , ( [ eq : majorana_representation_2 ] ) is used for a calculation of spin - spin correlations , a four - fermion correlator needs to be evaluated . in a typical situation , the lowest - order contribution is given by a loop - like diagram which involves two fermionic propagators . a straightforward evaluation of higher - order contributions requires an analysis of the self - energy corrections to these propagators as well as of the vertex corrections . the use of the relation ( [ eq : bubble_to_line_reduction ] ) reduces the task to the evaluation of a single self - energy .    in general , evaluation of this self - energy to all orders in the perturbative expansion involves complicated diagrams and , in particular , other self - energies and vertex corrections . hence an involved calculation may still be needed . nevertheless , the relation ( [ eq : bubble_to_line_reduction ] ) may be useful if the needed self - energy part can be estimated reliably , for instance , by calculating the low - order contributions . we discuss two examples in section  [ sec : example ] . note also that the relation ( [ eq : bubble_to_line_reduction ] ) may be straightforwardly generalized to multi - spin correlators , the evaluation of which then reduces to multi - fermion correlators . further , in a problem , that involves many spins represented via majorana fermions , ( e.g. , a lattice spin model ) the @xmath32-operators for different spins ( sites ) may be involved in a relation similar to ( [ eq : bubble_to_line_reduction ] ) . in this case their product does not drop out of the calculation ( unlike @xmath44 for one spin ) . one might be concerned by the fact that eq . ( [ eq : bubble_to_line_reduction ] ) reduces a correlator of physical quantities to a correlator of `` unphysical '' operators . another way of expressing this concern is to invoke the gauge symmetry . indeed the majorana representations ( [ eq : majorana_representation_3 ] ) and ( [ eq : majorana_representation_2 ] ) possess the discrete @xmath46-symmetry @xmath47 . as an operator which realizes the symmetry transformation one may choose @xmath48 ( cf .  ref . for example , @xmath49 . thus , the forth majorana fermion allowed in the hilbert space of the representation ( [ eq : majorana_representation_2 ] ) generates the symmetry transformation . one can as well use as a generator @xmath50 , which just flips the sign of the wave function of the @xmath30-spin , keeping that of the @xmath29-spin intact . consider now a time - dependent gauge transformation , which transforms a wave function @xmath51 to @xmath52 , where @xmath53 and @xmath54 . it transforms the operators @xmath55 , @xmath56 . thus the operator @xmath32 is now time - dependent and no longer commutes with the hamiltonian . indeed , the gauge - transformed hamiltonian reads @xmath57 and the last ( gauge ) term does not commute with @xmath32 . thus we find that @xmath58 i.e. , the single - fermion correlators are not gauge - invariant .    the relation ( [ eq : bubble_to_line_reduction ] ) is written in a fixed gauge , in which @xmath59 . in an arbitrary gauge , the factor @xmath60 should be added on the rhs making the identity gauge - invariant . in path - integral calculations the integration is performed over all configurations ( all gauges ) . if a saddle - point solution breaks the @xmath46-symmetry ( and hence has a counterpart ) fluctuations which involve switching between two mean - field solutions play an important part and need special attention . @xcite however , for perturbative calculations around a trivial saddle point , e.g. above the kondo temperature , the ` fixed - gauge ' relation ( [ eq : bubble_to_line_reduction ] ) might be quite useful . the relation ( [ eq : bubble_to_line_reduction ] ) can simplify calculations , since to evaluate a single - fermion green function one needs only to evaluate a self - energy . in a perturbative regime , when one may restrict oneself to evaluating the lowest - order contribution , eq . ( [ eq : bubble_to_line_reduction ] ) simplifies the task . one example of such a problem is dicussed in ref . @xcite where continuous measurement of the state of a spin ( qubit ) by a quantum detector is analyzed and the ouput noise of the detector as well as the spin dynamics is studied . a calculation of a spin - spin correlator in the majorana representation requires an evaluation of a two - fermion loop and involves the analysis of two self - energy parts and a vertex correction . it turns out that the vertex correction is important already in the lowest order and may even cancel one of the self - energy contributions . this fact may be understood and the calculations are considerably simplified if the relation  ( [ eq : bubble_to_line_reduction ] ) is invoked . the obtained result coincides with an alternative calculation , in which instead of using the majorana representation to enable the use of the wick theorem one follows the dynamics of the spin directly . this approach  @xcite may be useful if the problem involves only a small number of spins ( or other ` non - wick ' degrees of freedom ) . here we consider another example : the dissipative dynamics of a spin coupled to an environment , for instance , the spin - boson model . consider the case of a purely transverse coupling : @xmath61 where @xmath62 is a fluctuating bosonic observable of the bath , whose hamiltonian @xmath63 determines the statistics of fluctuations . we consider gaussian , but not necessarily equilibrium , fluctuations . using the keldysh technique we perform a calculation considering the bath - spin coupling ( the second term on the rhs of eq . ( [ eq : hspin - bath ] ) ) as a perturbation . below we use the notations of ref . let us introduce the matrix green function of the bath : @xmath64 its keldysh component @xmath65 and the difference between the retarded and advanced components @xmath66 are related to the symmetrized and antisymmetrized correlators : @xmath67 , @xmath68 , @xmath69 . in equilibrium they are related by the fluctuation - dissipation theorem : @xmath70 . here @xmath71 and @xmath72 . similarly , for the majorana fermion @xmath73 we define @xmath74 . the bare green functions are @xmath75 and @xmath76 . for the @xmath11-fermion we use the bogolubov - nambu spinors @xmath77 and @xmath78 and define a matrix @xmath79 . calculation of the spin propagators ( in a stationary state ) reduces due to the relation ( [ eq : bubble_to_line_reduction ] ) to the evaluation of the fermion green functions @xmath80 and @xmath81 . these functions can be found from the the dyson ( kinetic ) equations : @xmath82 @xmath83 where @xmath84 and @xmath85 are the self - energies . all the quantities in eq . ( [ eq : dyson_psi ] ) are @xmath86 matrices ( with the nambu and keldysh components ) . in the frequency domain the operator @xmath87 is given by @xmath88 while for @xmath89 we obtain @xmath90 we disregard infinitesimal imaginary terms in eqs . ( [ eq : g_psi_0 ] , [ eq : g_eta_0 ] ) since they are superseded by the self - energy parts in eqs . ( [ eq : dyson_psi ] , [ eq : dyson_eta ] ) . the lowest - order contributions to the self - energies @xmath91 are shown in fig . [ figure : sigma_ferm ] ( the matrix structure after the keldysh rotation is given  @xcite ) . we find that @xmath92 and @xmath93 , where @xmath94 ia a matrix in the nambu space . similarly , we obtain @xmath95 and @xmath96 . while the retarded and advanced components of the bare fermion green functions are known , its keldysh component contains information about the distribution function and is sensitive to the environment ( cf . @xcite ) .    to perform the calculation , one introduces the distribution function @xmath97 via @xmath98 . evaluating @xmath84 from the first diagram in fig . [ figure : sigma_ferm ] , we find @xmath99 \ , \ ] ] @xmath100 \ .\ ] ] the symmetry relation @xmath101 implies @xmath102 . hence without evaluating @xmath103 we find @xmath104 and @xmath105 the real parts of the retarded and advanced self - energies give the non - equilibrium generalization of the lamb shift , i.e. , renormalize the level splitting @xmath106 . this renormalization is small if the noise is weak and has a non - singular spectrum .    substituting the self - energy @xmath84 into the dyson equation ( [ eq : dyson_psi ] ) we obtain @xmath107 where @xmath108 . inverting eq . ( [ eq : g_psi_(-1 ) ] ) we obtain @xmath109 @xmath110    from eqs . ( [ eq : g_psi_ra ] ) and ( [ eq : g_psi_k ] ) we see that , at least in this order of perturbation theory , we can use the relation @xmath111 , where @xmath112 and ( for any pair @xmath113 of nambu indices ) @xmath114 a quantity of the zeroth order in the coupling constant .    using these results we can calculate @xmath85 : @xmath115 \nonumber \\ & & + [ b\to -b]\ , \end{aligned}\ ] ] @xmath116 + [ b\to -b ] \ .\end{aligned}\ ] ] we evaluate the self - energy @xmath117 near the pole @xmath118 of the green function @xmath81 : @xmath119 and @xmath120 \ .\ ] ]    now , with the acquired knowledge of single - fermion green functions , we can evaluate various spin - spin correlators . we start with @xmath121 where the bosonic time - ordering is chosen for the spin components . from eq . ( [ eq : bubble_to_line_reduction ] ) we deduce that @xmath122 since @xmath123 $ ] , we obtain @xmath124 this result describes the peaks in the spectrum of spin corelations at @xmath125 , with the width corresponding to the dephasing rate @xmath126 , thus reproducing the form known , e.g. , from the analysis of the bloch - redfield equations . @xcite    similarly , for the green function of the majorana fermion @xmath127 we obtain @xmath128 where @xmath129 \ .\ ] ] the keldysh component vanishes , @xmath130 , and we find @xmath131 this may be compared to the peak shape obtained in the bloch - redfield approach . @xcite it can be expressed in terms of the average value @xmath132 of the @xmath133-component : @xmath134\,\frac{2t_1^{-1}}{\omega^2 + t_1^{-2 } } \ , \ ] ] where the relaxation time is given by @xmath135 . the result ( [ eq : pi_zz ] ) reproduces the high - frequency behavior of eq . ( [ eq : pi_zz_exact ] ) and the ` width of the peak ' , which may be read off from the high-@xmath136 asymptotics , @xmath137 . this indicates that at lower frequencies higher - order corrections ( contributions to @xmath117 ) are important , which requires a further analysis . in summary , we discussed an identity relating spin correlations to single - particle majorana - fermion propagators and problems where the use of this relation simplifies the analysis . we are grateful to a.  auerbach , t.  costi , y.  gefen , m.  kiselev , j.  kroha , i.  martin , and p.  wlfle for stimulating discussions , and in particular to a.  mirlin , a.  rosch , and m.  vojta for numerous inspiring conversations . y.m . was supported by the humboldt foundation , the bmbf , and the zip programme of the german government .
in the majorana representation of a spin 1/2 we find an identity which relates spin - spin correlators to one - particle fermionic correlators . 
 this should be contrasted with the straightforward approach in which two - particle ( four - fermion ) correlators need to be calculated . 
 we discuss applications to the analysis of the dynamics of a spin coupled to a dissipative environment and of a quantum detector performing a continuous measurement of a qubit s state .