The probability in the relativistic m+1 space-time

7 Pages·0.08 MB·English

by  Gunn A. Quznetsov

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THE PROBABILITY IN THE RELATIVISTIC
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n µ + 1 SPACE-TIME
a
J

1
Gunn Quznetsov
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[email protected]

]
h
p February 9, 2008
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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
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the relativistic µ+1-vector of the probability density..
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3 2 SIMPLICES
0
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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)  
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−→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 proba-
bility 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
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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)
−→
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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 )ˆe(S)))
R 0 −→0 · 0 −→0
dx p(t + u(t ,x ) (x x ), x)
−→ 0 −→ 0 −→0 −→ −→0 −→
= · · − . (2)
cos(u(t ,x )ˆe(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 coordi- nates system Rµ+1 by the
Lorentz transformations with the velo- city u. That is:
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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.
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