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9 9 9 THE PROBABILITY IN THE RELATIVISTIC 1 n µ + 1 SPACE-TIME a J 1 Gunn Quznetsov 2 [email protected] ] h p February 9, 2008 - n e g Abstract . s Theprobabilitybehaviorintheµ+1relativisticspace-timeisconsid- c ered. The probability, which is defined by the relativistic µ+1-vector of i s theprobability density,is investigated. y h p 1 INTRODUCTION [ 2 There we will consider the probability behavior in the µ+1 dimensional rela- v tivistic space-time for to find the conditions, when the probability is defined by 5 the relativistic µ+1-vector of the probability density.. 3 0 3 2 SIMPLICES 0 8 9 Let be the µ-dimensional euclidean space. Let S be the set of the couples ℜ / a= a , a , for which: a is the realnumber and a . Thatis if R is the set s h 0 −→i 0 −→∈ ℜ c of the real numbers then S =R . i If a S, a= a , a , Rµ is t×heℜcoordinates system on and a ,...,a are s 0 −→ 1 µ ∈ h i ℜ y the coordinates of a in Rµ then let us denote: −→ h p a Rµ+1 = a , a Rµ+1 = a ,(a ,...,a ) , 0 −→ 0 1 µ : h i h i v (cid:0) (cid:1) (cid:0) (cid:1) i ”x +x =x ” : ”for all i: if 0 i µ then X 1 2 3 ≤ ≤ x +x =x ”, 1,i 2,i 3,i r and for every real number k: a ”k x =x ” : ”for alli: if 0 i µ then x =k x ”, 1 2 2,i 1,i · ≤ ≤ · ” dx”: ” dx ... dx ”. −→ 1 µ ∫ ∫ ∫ The set M of the points of S is the n-simplex with the vertices a ,a ,...,a 0 1 n (denote: M = a ,a ,...,a ) if for all a: if a M then the real numbers 0 1 n ∈ k ,k ,...,k exist, for which: for all i: if 0 i n then 0 k 1,and 1 2 n i (cid:6) (cid:7) ≤ ≤ ≤ ≤ 1 n r a=a + k a a . 0 i n−r+1 n−r !· − Xr=1 iY=1 (cid:16) (cid:17) Let ∆(a ,a ,...,a ,a ) Rµ+1 be the determinant: 0 1 µ µ+1 (cid:0) (cid:1) a a a a a a µ+1,0 0,0 µ+1,1 0,1 µ+1,µ 0,µ − − ··· − a a a a a a (cid:12) µ,0− 0,0 µ,1− 0,1 ··· µ,µ− 0,µ (cid:12) (cid:12) (cid:12) (cid:12) ··· ··· ··· ··· (cid:12) (cid:12) a1,0 a0,0 a1,1 a0,1 a1,µ a0,µ (cid:12) (cid:12) − − ··· − (cid:12) . (cid:12) (cid:12) (cid:12) (cid:12) Let v be so(cid:12)me real number, for which: v 1. (cid:12) | |≤ Let Rµ+1′ is obtained from Rµ+1 by the Lorenz transformations: For some k, for which 1<k <µ: a v a a v a a′ = k− · 0,a′ = 0− · k k √1 v2 0 √1 v2 − − and for all other r for which 1 r µ and r =k: a′ =a . ≤ ≤ 6 r r In this case: ∆(a ,a ,...,a ,a ) Rµ+1′ =∆(a ,a ,...,a ,a ) Rµ+1 . 0 1 µ µ+1 0 1 µ µ+1 (cid:0) (cid:1) (cid:0) (cid:1) Let the µ+1-measure of the µ+1-simplex a ,a ,...,a ,a be: 0 1 µ µ+1 l m 1 a ,a ,...,a ,a = ∆(a ,a ,...,a ,a ). 0 1 µ µ+1 0 1 µ µ+1 (µ+1)! · (cid:13) (cid:13) (cid:13) (cid:13) The µ+(cid:13)1-measure of µ+1(cid:13)-simplex is invariant for the complete Poincare group transformations. If M is the subdeterminant of 1,k ∆(a ,a ,...,a ,a ), 0 1 µ µ+1 obtained from ∆(a ,a ,...,a ,a ) 0 1 µ µ+1 by the crossing out of the first line and the column of number k then the µ-measure of the µ-simplex a ,a ,...,a 0 1 µ l m in the coordinates system R is: 2 µ+1 0.5 1 a ,a ,...,a (Rµ+1)= (M )2 . 0 1 µ 1,k µ! · ! (cid:13) (cid:13) Xk=1 (cid:13) (cid:13) If Rµ+1′ is o(cid:13)btained by th(cid:13)e Lorentz transformations from Rµ+1 and for all k and s, for which 0 k µ, 0 s µ: ≤ ≤ ≤ ≤ a =a , k,0 s,0 then a ,a ,...,a (Rµ+1′)= a ,a ,...,a (Rµ+1) 1 v2. 0 1 µ 0 1 µ · − (cid:13) (cid:13) (cid:13) (cid:13) p Let S#(cid:13)(cid:13)be the relate(cid:13)(cid:13)d to S vecto(cid:13)(cid:13)r space. Tha(cid:13)(cid:13)t is the couple (S,S#) is the affine space. In the coordinates system Rµ+1 : If n S# then the modulus of n is: ∈ n = n , n =(n2+n2+...+n2). | | |h 0 −→i| 0 1 µ The scalar product of the vectors n and n is denoted as: 1 2 n n = n ,n n ,n =n n +n n +...+n n . 1 2 1,0 −→1 1,0 −→2 1,0 2,0 1,1 2,1 1,µ 2,µ · h i·h i · · · The vector n for which: n S# and n = ( 1)1=k M .is the normal k−1 1,k ∈ − · vector for the µ-simplex a ,a ,...,a . (denote: n a ,a ,...,a ). 0 1 µ 0 1 µ ⊥ Let us denote: l m l m ∂ ∂ ∂ = ,∂ = =∂ . k t 0 θx θt k Let # be the related to the vector space. That is ( #, ) is the affine ℜ ℜ ℜ ℜ space. Let us denote: for n # : n2 =n2+n2+...+n2. −→∈ℜ −→ 1 2 µ Letusdenotethezerovectorof # asthefollowing: for−→n #: if−→n =−→0 ℜ ∈ℜ then for all k: if 1 k µ then n =0. k ≤ ≤ Letusdenotethevectore(S)asthebasicvectorofS ife(S)= 1,→−0 S#. ∈ TRACKS (cid:10) (cid:11) Let a differentiable real vector function −→f (t) (−→f (t) #) be denoted as ∈ ℜ the track in Rµ+1 . Let the distance between the tracks −→f1 and −→f2 be denoted as the following: µ 0.5 −→f1,−→f2 =sup (f1,i(t) f2,i(t))2 . t  − !  (cid:13) (cid:13) Xi=1 (cid:13) (cid:13) (cid:13) (cid:13)   3 −→f1,−→f2 fulfilles to all three metric space axioms: (cid:13) (cid:13) (cid:13)1) −→f1,(cid:13)−→f2 =0 and if −→f1 =−→f2 then −→f1,−→f2 >0; (cid:13) (cid:13) 6 (cid:13) (cid:13) (cid:13) (cid:13) 2) (cid:13)−→f1,−→f2(cid:13)= −→f2,−→f1 ; (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 3) (cid:13)By the(cid:13)Cau(cid:13)chy-Sch(cid:13)warz inequality: (cid:13) (cid:13) (cid:13) (cid:13) −→f1(cid:13),−→f2 +(cid:13) −→f2(cid:13),−→f3 (cid:13) −→f1,−→f3 . ≥ I(cid:13)nthis(cid:13)case(cid:13)theset(cid:13)T o(cid:13)fthetr(cid:13)acksinRµ+1 isthemetricspace. Thetopology (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) on t(cid:13)he set(cid:13)T ca(cid:13)n be c(cid:13)onst(cid:13)ructed(cid:13)by the following way: Let the set Oε(−→f0) (Oε(−→f0) T) be the ε-vicinity of −→f0 if for all −→f : if ⊂ −→f Oε(−→f0) then −→f ,−→f0 <ε. ∈ The track−→f is(cid:13)the in(cid:13)terior pointof setM (M T)if −→f M andfor some (cid:13) (cid:13) ⊆ ∈ ε-vicinity Oε(−→f ) o(cid:13)f −→f : O(cid:13)ε(−→f ) M. ⊆ ThesetM isthe opensetifallelementsofM aretheinteri-orpointsofM. In this case B can be the minimum σ-field (The Borel field) contained all open subsets of T. Let Ptr be thbe probability measure on B. That is (T,B,Ptr) is the probability space. Thevector-function[w(t , a ,a ,...,a b)],whichhasgobttherangeofvalues 0 0 1 µ in #, is l m ℜ the averagevelocityofthe tracksdensity onthe µ-simplex a ,a ,...,a in 0 1 µ the moment t0 if l m [w(t , a ,a ,...,a )]= 0 0 1 µ l m dy y Ptr −→f :∂t−→f (t0)=y −→f :−→f (t0) a0,a1,...,aµ · · ∩ ∈ . (cid:16)n o n l mo(cid:17) R Ptr −→f :−→f (t0) a0,a1,...,aµ ∈ (cid:16)n l mo(cid:17) Thevector-functionw(t, x),whichhasgotthe domaininS andhasgotthe −→ rangeofvalues in #,is the velocityofthe tracksdensity,if forallk, forwhich ℜ 0 k µ: if a t ,x then k 0 −→0 ≤ ≤ →h i w(t ,x )=lim[w(t , a ,a ,...,a )]. 0 −→0 0 0 1 µ l m Let the real function ptr(n,t , x) be the tracks probability density for the 0 −→ directionofnforRµ+1 ifforalla ,a ,...,a ,forwhich: n a ,a ,...,a ,the 0 1 µ 0 1 µ ⊥ following condition is fulfilled: l m if for all i (0 i<µ): a t ,x then i 0 −→0 ≤ →h i 4 Ptr −→f :−→f (t) a0,a1,...,aµ ∈ ptr(n,t ,x )=lim . 0 −→0 (cid:16)n l mo(cid:17) a ,a ,...,a 0 1 µ (cid:13) (cid:13) (cid:13) (cid:13) Let (n1ˆn2) be the angle between n1 (cid:13)and n2 . Tha(cid:13)t is: cos(n1ˆn2) = (n1 · n )/( n n ). In this case, if 2 1 2 · w(t, x) = w (t, x).w(t, x) S# , w (t, x)= 1, w(t, x) is the velocity of the(cid:12)(cid:12)tr−→a(cid:12)(cid:12)ck(cid:12)(cid:12)s d(cid:12)(cid:12)hen0sity−→and−→ π−→(iw∈(t, x)ˆn)0 π−→then −→ −→ −2 ≤ −→ ≤ 2 ptr(n,t, x)=ptr(w(t, x),t, x) cos(w(t, x)ˆn). (1) →− −→ −→ −→ · 3 TRACKELIKE PROBABILITY Let the σ-field B on S be obtained from the set of the µ+1-simplices. Let the probability measure P on B be defined as the following: the realfuncetion p(t, x) (the absolute probability density) exists for which: −→ if D B then e ∈ e0 dt ... dx dx dx p(t,x x , ,x ) 1 1 2··· µ 1 2 µ ≤ · ··· ≤ Z Z Z Z(D) and dt ... dx dx dx p(t,x x , ,x )=1; 1 2··· µ 1 2 µ · ··· Z Z Z Z(Rµ+1) in this case: P(D)= dt ... dx dx dx p(t,x x , ,x ). 1 2··· µ 1 2 µ · ··· Z Z Z Z(D) Because for the Lorentz transformations x v t t v x x′ = k− · , t′ = − · k, k √1 v2 √1 v2 − − for r=k: x′ =x (v <1), 6 r r | | the Jacobian: ∂(t′,x′) J = k =1 ∂(t,x ) k then p′(t′,−→x′)=p(t, x) −→ 5 That is the absolute probability density is the scalar function. Let g(n,t, x) be the conditional probability density for the direction of −→ −→ n, if n #, n= n , n S#, n = 1 and for all points x , for which −→ −→ 0 −→ 0 0 ∈ ℜ h i ∈ x = t ,x S: 0 0 −→0 h i∈ p(t ,x ) cos(e(S)ˆn) 0 −→0 g(n,t, x)= · . −→ −→ dx p(t + n (x x ), x) −→ 0 −→ −→ −→0 −→ · · − TheprobabilitymeasurePRisthetrackelikeprobabilitymeasureinthepoint a(a= t, x S)ifthe vector u(t, x)exists,forwhich u(t, x) #,andthe −→ −→ −→ −→ −→ h i∈ ∈ℜ following condition is fulfilled: for all vectors n (n= n , n S# , n =1): 0 −→ 0 h i∈ if u(t, x)= u (t, x), u(t, x) S#, u (t, x)=1 −→ 0 −→ −→ −→ 0 −→ and π (uh(t, x)ˆn) π i∈ −2 ≤ −→ ≤ 2 then (see (1)): g(n,t, x)=g(u(t, x),t, x) cos(u(t, x)ˆn). −→ →− −→ −→ −→ −→ · Inthiscase u(t, x)isdenotedasthevelocityoftheprobabilityinthepoint −→ −→ t, x . −→ h i If P is the trackelike probability measure in the point a (a S and a = 0 0 0 ∈ t ,x ) and u =1=n then 0 −→0 0 0 h i dx p(t + u(t ,x ) (x x ), x) −→ 0 −→ 0 −→0 −→ −→0 −→ · · − = (cos(u(t ,x )ˆn) cos(u(t ,x )ê(S))) R 0 −→0 · 0 −→0 dx p(t + u(t ,x ) (x x ), x) −→ 0 −→ 0 −→0 −→ −→0 −→ = · · − . (2) cos(u(t ,x )ê(S)) R 0 −→0 If q(t,−→x) = g(−→0,t,−→x) then ρ(t,−→x) is the density function in the moment t. If−→u(t,−→x)isthevelocityoftheprobability,thenthefunction−→j (t,−→x),which has got the domain in S and has got the range of values in #, is denoted as ℜ the probability current if −→j (t,−→x)=ρ(t,−→x) −→u(t,−→x). · These function are fulfilled to the continuity equation: ∂ ρ(t, x)+∂ j (t, x)+ +∂ j (t, x)=0. t −→ 1 1 −→ µ µ −→ ··· Let u be the velocity of the probability in the point t ,x and the coor- 0 −→0 h i dinates system Rµ+1′ be obtained from the coordinates system Rµ+1 by the Lorentz transformations with the velocity u. That is: 6 t u x x t u t′ = −−→·−→ and x′ = −→− ·−→. −→ 1 u2 1 u2 −→ −→ − − In this case ρ′(t′, x′) ips denoted as the localpprobability −→ density (ρ ). This function is the scalar function: (cid:13) ρ′(t′, x′)=ρ(t, x); −→ −→ and ρ (t, x) (cid:13) −→ ρ(t, x)= . −→ 1 u2(t, x) −→ −→ − q Hence for any velocity v, for which v <1: if | | t v x x t v t′ = −−→·−→ and x′ = −→− ·−→ −→ 1 v2 1 v2 −→ −→ − − then p p ρ′(t′, x′)= ρ(t,−→x)−−→v ·−→j (t,−→x), −→ 1 v2 −→ − p −→j ′(t′,−→x′)= −→j (t,−→x)−ρ(t,−→x)·−→v . 1 v2 −→ − p Therefore ρ(t, x) is not the scalar function but: −→ ρ2(t,−→x)−−→j 2(t,−→x)=ρ2(cid:13)(t,−→x). 4 RESUME Inordertotheprobabilityisdefinedbytherelativisticµ+1-vectorofthedensity, the probabilitydistribution function mustfulfil to the odd globalcondition(2), which is expressed by the integrals on all space. 7