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the simplest relativistic generalization of the schrdinger equation is the `` square root klein - gordon equation '' : @xmath2 in which @xmath3 is the wave function in momentum space representation and @xmath4 is the relativistic hamiltonian ; @xmath5 .
this equation can be rewritten in position space representation as follow : @xmath6 the above equation , sometimes called `` salpeter '' equation .
this equation and its integral representations can be uniquely derived from following simple assumptions : ( 1 ) the equation has the first order of time . ( 2 ) the equation is linear . ( 3 ) the equation leads to relativistic dispersion relation for plane waves ( in other words , being consistent with de broglie relations ) .
the history of the square root klein - gordon equation ( srkg equation ) goes back to the early years of relativistic quantum mechanics . in the 1927 weyl
proposed the operator @xmath7 for the formulation of relativistic quantum mechanics @xcite .
but he did not expand his idea as a comprehensive theory and also other pioneers of quantum mechanics used different methods for the formulation relativistic quantum mechanics which led to other wave equations such as dirac and klein - gordon equations . on the other hand , although the srkg equation lacks proper theoretical characteristics and even gets rid of some difficulties that other relativistic wave equations have ( such as klein paradox and `` zitterbewegung '' ) , but higher order derivatives in this equation makes it look more complicated than the other quantum wave equations ; therefore it attracted less attention during the historical development of relativistic quantum mechanics .
but after derivation of this equation from bethe - salpeter formalism @xcite , it was more attention . in this regard , in the recent years , theoretical characteristics and integral representations of this equation have been attracting special attention @xcite .
also , this equation has been successfully used in description of several relativistic problems and phenomena ; such as relativistic harmonic oscillator @xcite , waves in relativistic quantum plasma @xcite , relativistic ( quark ) bound states @xcite , and the relativistic bohmian mechanics @xcite .
we particularly note the consistency of the results of this equation with the experimental spectrum of mesonic atoms @xcite .
in addition , since this equation is of first order with respect to time , there is the possibility of using the born interpretation , @xmath0 , for the interpretation of the wave function and as a result it avoid of negative probability problem of the klein - gordon equation . in this regard , k. kowalski and j. rembielinski derived , the position born s probability density and its corresponding current density in the momentum representation as follow @xcite :    @xmath8    @xmath9    in which the `` velocity '' , @xmath10 defined as @xmath11 . this very possibility of a probabilistic interpretation of the wave function has been often the stimulus to using this equation . on the other hand ,
the lorentz invariance of the srkg equation has always been under discussion because of its high derivatives , in fact checking lorentz invariance of this equation is complicated and can not be specified easily .
therefor sometimes because of the inequality of time and space derivatives , this equation has been mistakenly accepted as a frame dependent equation and incompatible with special relativity @xcite .
but it has been shown that if the wave function is scalar this equation is lorentz invariant ( in the absence of interactions ) @xcite .
but we need to pay attention that born rule , @xmath12 , is inconsistent with assuming the wave function as scalar .
because the position probability density is the first component of the probability current four - vector and can not be scalar . on the other word , because of the relativistic length contraction , the probability density can not be scalar .
hence , born rule ( eq.([3 ] ) and eq.([4 ] ) ) can not be a proper interpretation for the scalar wave function .
in fact in the rest of this article we will show even if the wave function is not scalar , born rule will still not be proper for the probabilistic interpretation of the wave function .
it means that no transformation will be found for the wave function in a way to cause born probability density and born current densities(eq.([3 ] ) and eq.([4 ] ) ) become altogether as one four - vector .
so , the born rule must be modified to conform with lorentz symmetry .
in this regard , we find a proper relativistic modification of the born rule for scalar wave function ( spin 0 particles ) in the section ( i ) and for spin 1/2 particles in the section ( ii ) .
we propose the relativistic generalization of the born rule as follows : @xmath13 in which @xmath14 , is defined as : @xmath15 the expression ( [ 5 ] ) is the simplest covariant generalization of the eq .
( [ 3 ] ) and eq.([4 ] ) .
in fact by means of direct ( and also long ) calculation , we can see that @xmath16 is actually a four - vector and as a result,@xmath17 is a four - vector as well - of course if the wave function be scalar(see section ii ) .
it can also be simply shown that @xmath17 ( as was defined above ) satisfies the continuity equation .
it is obvious that the first component of the @xmath17specify the relativistic generalization of the born rule in the momentum representation : @xmath18 the question that arises at this point is whether the definition ( [ 7 ] ) lead to a positive definite probability density . for answering this question , we note that the coefficient @xmath19 in the expression ( [ 7 ] ) can be separated as follow : @xmath20 where @xmath21.through the aid of this separation , the position probability density can be derived in position space representation as follow : @xmath22\end{aligned}\ ] ] where @xmath23 is compton wavelength ; @xmath24 .
so eq.([7 ] ) leads to a positive - definite probability density .
it is clear that in non- relativistic limit , @xmath25 , the equation ( 9 ) reduce to born rule , @xmath26 .
in addition , if @xmath27 then @xmath28 and the expression ( [ 7 ] ) gives the non - relativistic expression ( [ 3 ] ) .
it means that whenever the width of wave packet in momentum space is small in comparison with @xmath29 , @xmath30 , deviation of @xmath31 from @xmath32 will be negligible . as a result , based on uncertainty relations , deviation of @xmath31 from @xmath32
is considerable only if the width of wave packet is smaller than the compton wavelength @xmath33 . to see this fact explicitly ,
we plot the relativistic ( eq.([9 ] ) ) and nonrelativistic ( born rule ) probability density for a particle in one dimensional infinite square well potential . from fig .
( 1 ) it is obvious that if the width of the box , @xmath34 , is greater than the compton wavelength of the particle , @xmath23 , the diagram of the relativistic probability density , @xmath31 , will be coincident with the diagram of the born probability density , @xmath35 ; and whenever the width of the box is smaller than the compton wavelength then @xmath31 will deviates from @xmath35 .
in fact if@xmath36 then the probability of the particle s presence in the box will be uniform(see figure.1 ) .     ; red diagrams represent the relativistic probability density,@xmath31 , for the first excited state n=2 of particle in infinite potential well .
the diagram is drawn considering different width of the box : ( a)@xmath37 , ( b)@xmath38 and ( c)@xmath39 . ]
particle and spin @xmath72 particle in one dimensional box .
the ( dashed ) green line represents born probability density , the blue line represent @xmath69 and the red line represent the @xmath82 .
the wave function of particle is considered as @xmath83 which @xmath84 is the width of the box and we consider @xmath85 . ]
in this article we have introduced relativistic generalization for the born rule for particles with zero and half spins .
therefore it seems that the srkg equation with the appropriate interpretation of the wave function can be adapted to use for particles having various spins .
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we have shown that the born rule @xmath0 is inconsistent with lorentz symmetry of the salpeter equation ( square root klein - gordon equation ) . 
 so we find relativistic modification of the born rule as @xmath1 , which is consistent with lorentz symmetry of this equation .