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THE PROBABILITY IN THE RELATIVISTIC μ+1\mu+1 SPACE-TIME
Gunn Quznetsov
quznets@geocities.com
November 23, 2018
Abstract
The probability behavior in the μ+1\mu+1 relativistic space-time is considered. The probability, which is defined by the relativistic μ+1\mu+1-vector of the probability density, is investigated.
1 INTRODUCTION
There we will consider the probability behavior in the μ+1\mu+1 dimensional relativistic space-time for to find the conditions, when the probability is defined by the relativistic μ+1\mu+1-vector of the probability density…
2 SIMPLICES
Let ℜ\Re be the μ\mu-dimensional euclidean space. Let S‾\underline{S} be the set of the couples a‾=⟨a0,a⃗⟩\underline{a}=\left\langle a_{0}, \vec{a}\right\rangle, for which: a0a_{0} is the real number and a⃗∈ℜ\vec{a} \in \Re. That is if RR is the set of the real numbers then S‾=R×ℜ\underline{S}=R \times \Re.
If a‾∈S‾,a‾=⟨a0,a⃗⟩,Rμ\underline{a} \in \underline{S}, \underline{a}=\left\langle a_{0}, \vec{a}\right\rangle, R^{\mu} is the coordinates system on ℜ\Re and a1,…,aμa_{1}, \ldots, a_{\mu} are the coordinates of a⃗\vec{a} in RμR^{\mu} then let us denote:
a‾(Rμ+1)=⟨a0,a⃗⟩(Rμ+1)=⟨a0,(a1,…,aμ)⟩\underline{a}\left(R^{\mu+1}\right)=\left\langle a_{0}, \vec{a}\right\rangle\left(R^{\mu+1}\right)=\left\langle a_{0},\left(a_{1}, \ldots, a_{\mu}\right)\right\rangle
" x1‾+x2‾=x3‾\underline{x_{1}}+\underline{x_{2}}=\underline{x_{3}} ": "for all ii : if 0≤i≤μ0 \leq i \leq \mu then
x1,i+x2,i=x3,i"x_{1, i}+x_{2, i}=x_{3, i} "
and for every real number kk :
" k⋅x1‾=x2‾k \cdot \underline{x_{1}}=\underline{x_{2}} ": "for alli : if 0≤i≤μ0 \leq i \leq \mu then x2,i=k⋅x1,i"x_{2, i}=k \cdot x_{1, i} ",
" ∫d3→:\int d \overrightarrow{\mathscr{3}}: " ∫dx1…∫dxμ"\int d x_{1} \ldots \int d x_{\mu} ".
The set MM of the points of S‾\underline{S} is the nn-simplex with the vertices a0‾,a1‾,…an‾\underline{a_{0}}, \underline{a_{1}}, \ldots \underline{a_{n}} (denote: M=[a0,a1,…,an0]M=\left[\begin{array}{c}a_{0}, a_{1}, \ldots, a_{n} \\ 0\end{array}\right] ) if for all a‾\underline{a} : if a‾∈M\underline{a} \in M then the real numbers k1,k2,…,knk_{1}, k_{2}, \ldots, k_{n} exist, for which: for all ii : if 0≤i≤n0 \leq i \leq n then 0≤ki≤10 \leq k_{i} \leq 1, and
a‾=a0‾+∑r=1n(∏i=1rki)⋅(an−r+1‾−an−r‾)\underline{a}=\underline{a_{0}}+\sum_{r=1}^{n}\left(\prod_{i=1}^{r} k_{i}\right) \cdot\left(\underline{a_{n-r+1}}-\underline{a_{n-r}}\right)
Let Δ(a0‾,a1‾,…,aμ‾,aμ+1‾)(Rμ+1)\Delta\left(\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right)\left(R^{\mu+1}\right) be the determinant:
∣aμ+1,0−a0,0aμ+1,1−a0,1⋯aμ+1,μ−a0,μaμ,0−a0,0aμ,1−a0,1⋯aμ,μ−a0,μ⋯⋯⋯⋯a1,0−a0,0a1,1−a0,1⋯a1,μ−a0,μ∣\left|\begin{array}{cccc} a_{\mu+1,0}-a_{0,0} & a_{\mu+1,1}-a_{0,1} & \cdots & a_{\mu+1, \mu}-a_{0, \mu} \\ a_{\mu, 0}-a_{0,0} & a_{\mu, 1}-a_{0,1} & \cdots & a_{\mu, \mu}-a_{0, \mu} \\ \cdots & \cdots & \cdots & \cdots \\ a_{1,0}-a_{0,0} & a_{1,1}-a_{0,1} & \cdots & a_{1, \mu}-a_{0, \mu} \end{array}\right|
Let vv be some real number, for which: ∣v∣≤1|v| \leq 1.
Let Rμ+1′R^{\mu+1 \prime} is obtained from Rμ+1R^{\mu+1} by the Lorenz transformations:
For some kk, for which 1<k<μ1<k<\mu :
ak′=ak−v⋅a01−v2,a0′=a0−v⋅ak1−v2a_{k}^{\prime}=\frac{a_{k}-v \cdot a_{0}}{\sqrt{1-v^{2}}}, a_{0}^{\prime}=\frac{a_{0}-v \cdot a_{k}}{\sqrt{1-v^{2}}}
and for all other rr for which 1≤r≤μ1 \leq r \leq \mu and r≠kr \neq k : ar′=ara_{r}^{\prime}=a_{r}.
In this case:
Δ(a0‾,a1‾,…,aμ‾,aμ+1‾)(Rμ+1′)=Δ(a0‾,a1‾,…,aμ‾,aμ+1‾)(Rμ+1)\Delta\left(\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right)\left(R^{\mu+1 \prime}\right)=\Delta\left(\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right)\left(R^{\mu+1}\right)
Let the μ+1\mu+1-measure of the μ+1\mu+1-simplex [a0‾,a1‾,…,aμ‾,aμ+1‾]\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right] be:
∥a0‾,a1‾,…,aμ‾,aμ+1‾∥=1(μ+1)!⋅Δ(a0‾,a1‾,…,aμ‾,aμ+1‾)\left\|\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right\|=\frac{1}{(\mu+1)!} \cdot \Delta\left(\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right)
The μ+1\mu+1-measure of μ+1\mu+1-simplex is invariant for the complete Poincare group transformations.
If M1,kM_{1, k} is the subdeterminant of
Δ(a0‾,a1‾,…,aμ‾,aμ+1‾)\Delta\left(\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right)
obtained from
Δ(a0‾,a1‾,…,aμ‾,aμ+1‾)\Delta\left(\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, \underline{a_{\mu+1}}\right)
by the crossing out of the first line and the column of number kk then the μ\mu-measure of the μ\mu-simplex
[a0‾,a1‾,…,aμ‾]\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right]
in the coordinates system RR is:
∥a0‾,a1‾,…,aμ‾∥(Rμ+1)=1μ!⋅(∑k=1μ+1(M1,k)2)0.5\left\|\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right\|\left(R^{\mu+1}\right)=\frac{1}{\mu!} \cdot\left(\sum_{k=1}^{\mu+1}\left(M_{1, k}\right)^{2}\right)^{0.5}
If Rμ+1′R^{\mu+1 \prime} is obtained by the Lorentz transformations from Rμ+1R^{\mu+1} and for all kk and ss, for which 0≤k≤μ,0≤s≤μ0 \leq k \leq \mu, 0 \leq s \leq \mu :
ak,0=as,0a_{k, 0}=a_{s, 0},
then
∥a0‾,a1‾,…,aμ‾∥(Rμ+1′)=∥a0‾,a1‾,…,aμ‾∥(Rμ+1)⋅1−v2\left\|\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right\|\left(R^{\mu+1 \prime}\right)=\left\|\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right\|\left(R^{\mu+1}\right) \cdot \sqrt{1-v^{2}}
Let S#S^{\#} be the related to S‾\underline{S} vector space. That is the couple (S‾,S#)\left(\underline{S}, S^{\#}\right) is the affine space.
In the coordinates system Rμ+1R^{\mu+1} :
If n‾∈S#\underline{n} \in S^{\#} then the modulus of n‾\underline{n} is:
∣n‾∣=∣⟨n0,n⃗⟩∣=(n02+n12+…+nμ2)|\underline{n}|=\left|\left\langle n_{0}, \vec{n}\right\rangle\right|=\left(n_{0}^{2}+n_{1}^{2}+\ldots+n_{\mu}^{2}\right)
The scalar product of the vectors n1‾\underline{n_{1}} and n2‾\underline{n_{2}} is denoted as:
n1‾⋅n2‾=⟨n1,0,n⃗1⟩⋅⟨n1,0,n⃗2⟩=n1,0⋅n2,0+n1,1⋅n2,1+…+n1,μ⋅n2,μ\underline{n_{1}} \cdot \underline{n_{2}}=\left\langle n_{1,0}, \vec{n}_{1}\right\rangle \cdot\left\langle n_{1,0}, \vec{n}_{2}\right\rangle=n_{1,0} \cdot n_{2,0}+n_{1,1} \cdot n_{2,1}+\ldots+n_{1, \mu} \cdot n_{2, \mu}
The vector n‾\underline{n} for which: n‾∈S#\underline{n} \in S^{\#} and nk−1=(−1)1=k⋅M1,kn_{k-1}=(-1)^{1=k} \cdot M_{1, k}.is the normal vector for the μ\mu-simplex [a0‾,a1‾,…,aμ‾]\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right], (denote: n‾⊥[a0‾,a1‾,…,aμ‾]\underline{n} \perp\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right] ).
Let us denote:
∂k=∂θxk,∂t=∂θt=∂0\partial_{k}=\frac{\partial}{\theta x_{k}}, \partial_{t}=\frac{\partial}{\theta t}=\partial_{0}
Let ℜ#\Re^{\#} be the related to ℜ\Re the vector space. That is (ℜ#,ℜ)\left(\Re^{\#}, \Re\right) is the affine space.
Let us denote: for n⃗∈ℜ#:n⃗2=n12+n22+…+nμ2\vec{n} \in \Re^{\#}: \vec{n}^{2}=n_{1}^{2}+n_{2}^{2}+\ldots+n_{\mu}^{2}.
Let us denote the zero vector of ℜ#\Re^{\#} as the following: for n⃗∈ℜ#\vec{n} \in \Re^{\#} : if n⃗=0→\vec{n}=\overrightarrow{0} then for all kk : if 1≤k≤μ1 \leq k \leq \mu then nk=0n_{k}=0.
Let us denote the vector e‾(S‾)\underline{e}(\underline{S}) as the basic vector of S‾\underline{S} if e‾(S‾)=⟨1,0⃗⟩∈S#\underline{e}(\underline{S})=\langle 1, \vec{0}\rangle \in S^{\#}. TRACKS
Let a differentiable real vector function f⃗(t)(f⃗(t)∈ℜ#)\vec{f}(t)(\vec{f}(t) \in \Re^{\#}) be denoted as the track in Rμ+1R^{\mu+1}.
Let the distance between the tracks f1→\overrightarrow{f_{1}} and f2→\overrightarrow{f_{2}} be denoted as the following:
∥f1→,f2→∥=supt((∑i=1μ(f1,i(t)−f2,i(t))2)0.5)\left\|\overrightarrow{f_{1}}, \overrightarrow{f_{2}}\right\|=\sup _{t}\left(\left(\sum_{i=1}^{\mu}\left(f_{1, i}(t)-f_{2, i}(t)\right)^{2}\right)^{0.5}\right)
∥f1→,f2→∥\left\|\overrightarrow{f_{1}}, \overrightarrow{f_{2}}\right\| fulfilles to all three metric space axioms:
- ∥f1→,f2→∥=0\left\|\overrightarrow{f_{1}}, \overrightarrow{f_{2}}\right\|=0 and if f1→≠f2→\overrightarrow{f_{1}} \neq \overrightarrow{f_{2}} then ∥f1→,f2→∥>0\left\|\overrightarrow{f_{1}}, \overrightarrow{f_{2}}\right\|>0;
- ∥f1→,f2→∥=∥f2→,f1→∥\left\|\overrightarrow{f_{1}}, \overrightarrow{f_{2}}\right\|=\left\|\overrightarrow{f_{2}}, \overrightarrow{f_{1}}\right\|;
- By the Cauchy-Schwarz inequality:
∥f1→,f2→∥+∥f2→,f3→∥≥∥f1→,f3→∥\left\|\overrightarrow{f_{1}}, \overrightarrow{f_{2}}\right\|+\left\|\overrightarrow{f_{2}}, \overrightarrow{f_{3}}\right\| \geq\left\|\overrightarrow{f_{1}}, \overrightarrow{f_{3}}\right\|.
In this case the set TT of the tracks in Rμ+1R^{\mu+1} is the metric space. The topology on the set TT can be constructed by the following way:
Let the set Oε(f0→)(Oε(f0→)⊂T)O_{\varepsilon}\left(\overrightarrow{f_{0}}\right)\left(O_{\varepsilon}\left(\overrightarrow{f_{0}}\right) \subset T\right) be the ε\varepsilon-vicinity of f0→\overrightarrow{f_{0}} if for all f⃗\vec{f} : if f⃗∈Oε(f0→)\vec{f} \in O_{\varepsilon}\left(\overrightarrow{f_{0}}\right) then ∥f⃗,f0→∥<ε\left\|\vec{f}, \overrightarrow{f_{0}}\right\|<\varepsilon.
The track f⃗\vec{f} is the interior point of set M(M⊆T)M(M \subseteq T) if f⃗∈M\vec{f} \in M and for some ε\varepsilon-vicinity Oε(f⃗)O_{\varepsilon}(\vec{f}) of f⃗:Oε(f⃗)⊆M\vec{f}: O_{\varepsilon}(\vec{f}) \subseteq M.
The set MM is the open set if all elements of MM are the interi- or points of MM.
In this case B^\widehat{B} can be the minimum σ\sigma-field (The Borel field) contained all open subsets of TT.
Let PtrP t r be the probability measure on B^\widehat{B}. That is (T,B^,Ptr)(T, \widehat{B}, P t r) is the probability space.
The vector-function [w(t0,[a0‾,a1‾,…,aμ‾])]\left[w\left(t_{0},\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right]\right)\right], which has got the range of values in ℜ#\Re^{\#}, is
the average velocity of the tracks density on the μ\mu-simplex [a0‾,a1‾,…,aμ‾]\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right] in the moment t0t_{0} if
[w(t0,[a0‾,a1‾,…,aμ‾])]=∫dy⋅y⋅Ptr({f⃗:∂tf⃗(t0)=y}∩{f⃗:f⃗(t0)∈[a0‾,a1‾,…,aμ‾]})Ptr({f⃗:f⃗(t0)∈[a0‾,a1‾,…,aμ‾]})\begin{gathered} {\left[w\left(t_{0},\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right]\right)\right]=} \\ \frac{\int d y \cdot y \cdot \operatorname{Ptr}\left(\left\{\vec{f}: \partial_{t} \vec{f}\left(t_{0}\right)=y\right\} \cap\left\{\vec{f}: \vec{f}\left(t_{0}\right) \in\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right]\right\}\right)}{\operatorname{Ptr}\left(\left\{\vec{f}: \vec{f}\left(t_{0}\right) \in\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right]\right\}\right)} \end{gathered}
The vector-function w(t,x⃗)w(t, \vec{x}), which has got the domain in S‾\underline{S} and has got the range of values in ℜ#\Re^{\#}, is the velocity of the tracks density, if for all kk, for which 0≤k≤μ0 \leq k \leq \mu : if ak‾→⟨t0,xˉ0s⟩\underline{a_{k}} \rightarrow\left\langle t_{0}, \bar{x}_{0}^{s}\right\rangle then
w(t0,xˉ0s)=lim[w(t0,[a0‾,a1‾,…,aμ‾])]w\left(t_{0}, \bar{x}_{0}^{s}\right)=\lim \left[w\left(t_{0},\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right]\right)\right]
Let the real function ptr(n‾,t0,x⃗)\operatorname{ptr}\left(\underline{n}, t_{0}, \vec{x}\right) be the tracks probability density for the direction of n‾\underline{n} for Rμ+1R^{\mu+1} if for all a0‾,a1‾,…,aμ‾\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}, for which: n‾⊥[a0‾,a1‾,…,aμ‾]\underline{n} \perp\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right], the following condition is fulfilled:
if for all i(0≤i<μ):ai‾→⟨t0,xˉ0s⟩i(0 \leq i<\mu): \underline{a_{i}} \rightarrow\left\langle t_{0}, \bar{x}_{0}^{s}\right\rangle then
ptr(n‾,t0,x⃗0)=limPtr({J⃗:J⃗(t)∈[a0‾,a1‾,…,aμ‾]}∥a0‾,a1‾,…,aμ‾∥\operatorname{ptr}\left(\underline{n}, t_{0}, \vec{x}_{0}\right)=\lim \frac{\operatorname{Ptr}\left(\{\vec{\mathcal{J}}: \vec{\mathcal{J}}(t) \in\left[\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right]\right\}}{\left\|\underline{a_{0}}, \underline{a_{1}}, \ldots, \underline{a_{\mu}}\right\|}
Let (n1‾∧n2‾)\left(\underline{n_{1}}^{\wedge} \underline{n_{2}}\right) be the angle between n1‾\underline{n_{1}} and n2‾\underline{n_{2}}. That is: cos(n1‾∧n2‾)=(n1‾\cos \left(\underline{n_{1}}^{\wedge} \underline{n_{2}}\right)=\left(\underline{n_{1}}\right.. n2‾)/(∣n1‾∣⋅∣n2‾∣)\left.\underline{n_{2}}\right) /\left(\left|\underline{n_{1}}\right| \cdot\left|\underline{n_{2}}\right|\right). In this case, if
w‾(t,x⃗)=⟨w0(t,x⃗),w⃗(t,x⃗)⟩∈S#,w0(t,x⃗)=1,w⃗(t,x⃗)\underline{w}(t, \vec{x})=\left\langle w_{0}(t, \vec{x}), \vec{w}(t, \vec{x})\right\rangle \in S^{\#}, w_{0}(t, \vec{x})=1, \vec{w}(t, \vec{x}) is the velocity of the tracks density and −π2≤(w‾(t,x⃗)∧n‾)≤π2-\frac{\pi}{2} \leq(\underline{w}(t, \vec{x}) \wedge \underline{n}) \leq \frac{\pi}{2} then
ptr(n‾,t,x⃗)=ptr(w‾(t,x⃗),t,x⃗)⋅cos(w‾(t,x⃗)∧n‾)\operatorname{ptr}(\underline{n}, t, \vec{x})=p t r(\underline{w}(t, \vec{x}), t, \vec{x}) \cdot \cos (\underline{w}(t, \vec{x}) \wedge \underline{n})
3 TRACKELIKE PROBABILITY
Let the σ\sigma-field B~\widetilde{B} on S‾\underline{S} be obtained from the set of the μ+1\mu+1-simplices. Let the probability measure PP on B~\widetilde{B} be defined as the following:
the real function p(t,x⃗)p(t, \vec{x}) (the absolute probability density) exists for which:
if D∈B~D \in \widetilde{B} then
0≤∫dt∬…∫(D)dx1dx2…dxμ⋅p(t,x1x2,⋯ ,xμ)≤10 \leq \int d t \iint \ldots \int_{(D)} d x_{1} d x_{2} \ldots d x_{\mu} \cdot p\left(t, x_{1} x_{2}, \cdots, x_{\mu}\right) \leq 1
and
∫dt∬…∫(Rμ+1)dx1dx2…dxμ⋅p(t,x1x2,⋯ ,xμ)=1\int d t \iint \ldots \int_{\left(R^{\mu+1}\right)} d x_{1} d x_{2} \ldots d x_{\mu} \cdot p\left(t, x_{1} x_{2}, \cdots, x_{\mu}\right)=1
in this case:
P(D)=∫dt∬…∫(D)dx1dx2…dxμ⋅p(t,x1x2,⋯ ,xμ)P(D)=\int d t \iint \ldots \int_{(D)} d x_{1} d x_{2} \ldots d x_{\mu} \cdot p\left(t, x_{1} x_{2}, \cdots, x_{\mu}\right)
Because for the Lorentz transformations
xk′=xk−v⋅t1−v2,t′=t−v⋅xk1−v2x_{k}^{\prime}=\frac{x_{k}-v \cdot t}{\sqrt{1-v^{2}}}, t^{\prime}=\frac{t-v \cdot x_{k}}{\sqrt{1-v^{2}}}
for r≠k:xr′=xr(∣v∣<1)r \neq k: x_{r}^{\prime}=x_{r}(|v|<1),
the Jacobian:
J=∂(t′,xk′)∂(t,xk)=1J=\frac{\partial\left(t^{\prime}, x_{k}^{\prime}\right)}{\partial\left(t, x_{k}\right)}=1
then
p′(t′,x′→)=p(t,x⃗)p^{\prime}\left(t^{\prime}, \overrightarrow{x^{\prime}}\right)=p(t, \vec{x})
That is the absolute probability density is the scalar function.
Let g(n⃗,t,x⃗)g(\vec{n}, t, \vec{x}) be the conditional probability density for the direction of n⃗\vec{n}, if n⃗∈ℜ#,n‾=⟨n0,n⃗⟩∈S‾#,n0=1\vec{n} \in \Re^{\#}, \underline{n}=\left\langle n_{0}, \vec{n}\right\rangle \in \underline{S}^{\#}, n_{0}=1 and for all points x0‾\underline{x_{0}}, for which x0‾=⟨t0,x0→⟩∈S‾:\underline{x_{0}}=\left\langle t_{0}, \overrightarrow{x_{0}}\right\rangle \in \underline{S}:
g(n⃗,t,x⃗)=p(t0,x0→)⋅cos(c‾(S‾)⋅n‾)∫dx⃗⋅p(t0+n⃗⋅(x⃗−x0→),x⃗)g(\vec{n}, t, \vec{x})=\frac{p\left(t_{0}, \overrightarrow{x_{0}}\right) \cdot \cos (\underline{c}(\underline{S}) \cdot \underline{n})}{\int d \vec{x} \cdot p\left(t_{0}+\vec{n} \cdot(\vec{x}-\overrightarrow{x_{0}}), \vec{x}\right)}
The probability measure PP is the trackelike probability measure in the point a‾(a‾=⟨t,x⃗⟩∈S‾)\underline{a}(\underline{a}=\langle t, \vec{x}\rangle \in \underline{S}) if the vector u⃗(t,x⃗)\vec{u}(t, \vec{x}) exists, for which u⃗(t,x⃗)∈ℜ#\vec{u}(t, \vec{x}) \in \Re^{\#}, and the following condition is fulfilled:
for all vectors n‾(n‾=⟨n0,n⃗⟩∈S‾#,n0=1)\underline{n}\left(\underline{n}=\left\langle n_{0}, \vec{n}\right\rangle \in \underline{S}^{\#}, n_{0}=1\right) :
if u‾(t,x⃗)=⟨u0(t,x⃗),u⃗(t,x⃗)⟩∈S‾#,u0(t,x⃗)=1\underline{u}(t, \vec{x})=\left\langle u_{0}(t, \vec{x}), \vec{u}(t, \vec{x}) \rangle \in \underline{S}^{\#}, u_{0}(t, \vec{x})=1\right.
and −π2≤(u‾(t,x⃗)∧n‾)≤π2-\frac{\pi}{2} \leq\left(\underline{u}(t, \vec{x})^{\wedge} \underline{n}\right) \leq \frac{\pi}{2}
then (see (1)):
g(n⃗,t,x⃗)=g(u⃗(t,x⃗),t,x⃗)⋅cos(u‾(t,x⃗)∧n‾)g(\vec{n}, t, \vec{x})=g(\vec{u}(t, \vec{x}), t, \vec{x}) \cdot \cos \left(\underline{u}(t, \vec{x})^{\wedge} \underline{n}\right)
In this case u⃗(t,x⃗)\vec{u}(t, \vec{x}) is denoted as the velocity of the probability in the point ⟨t,x⃗⟩\langle t, \vec{x}\rangle.
If PP is the trackelike probability measure in the point a0‾(a0‾∈S‾\underline{a_{0}}\left(\underline{a_{0}} \in \underline{S}\right. and a0‾=\underline{a_{0}}= ⟨t0,x0→⟩)\left.\left\langle t_{0}, \overrightarrow{x_{0}}\right\rangle\right) and u0=1=n0u_{0}=1=n_{0} then
∫dx⃗⋅p(t0+u⃗(t0,x0→)⋅(x⃗−x0→),x⃗)(cos(u‾(t0,x0→)∧n‾)⋅cos(u‾(t0,x0→)∧c‾(S‾)))==∫dx⃗⋅p(t0+u⃗(t0,x0→)⋅(x⃗−x0→),x⃗)cos(u‾(t0,x0→)∧c‾(S‾))\begin{aligned} & \frac{\int d \vec{x} \cdot p\left(t_{0}+\vec{u}\left(t_{0}, \overrightarrow{x_{0}}\right) \cdot\left(\vec{x}-\overrightarrow{x_{0}}\right), \vec{x}\right)}{\left(\cos \left(\underline{u}\left(t_{0}, \overrightarrow{x_{0}}\right) \wedge \underline{n}\right) \cdot \cos \left(\underline{u}\left(t_{0}, \overrightarrow{x_{0}}\right) \wedge \underline{c}(\underline{S})\right)\right)}= \\ & =\frac{\int d \vec{x} \cdot p\left(t_{0}+\vec{u}\left(t_{0}, \overrightarrow{x_{0}}\right) \cdot\left(\vec{x}-\overrightarrow{x_{0}}\right), \vec{x}\right)}{\cos \left(\underline{u}\left(t_{0}, \overrightarrow{x_{0}}\right) \wedge \underline{c}(\underline{S})\right)} \end{aligned}
If q(t,x⃗)=g(0⃗,t,x⃗)q(t, \vec{x})=g(\vec{0}, t, \vec{x}) then ρ(t,x⃗)\rho(t, \vec{x}) is the density function in the moment tt.
If u⃗(t,x⃗)\vec{u}(t, \vec{x}) is the velocity of the probability, then the function j⃗(t,x⃗)\vec{j}(t, \vec{x}), which has got the domain in S‾\underline{S} and has got the range of values in ℜ#\Re^{\#}, is denoted as the probability current if
j⃗(t,x⃗)=ρ(t,x⃗)⋅u⃗(t,x⃗)\vec{j}(t, \vec{x})=\rho(t, \vec{x}) \cdot \vec{u}(t, \vec{x})
These function are fulfilled to the continuity equation:
∂tρ(t,x⃗)+∂1j1(t,x⃗)+⋯+∂μjμ(t,x⃗)=0\partial_{t} \rho(t, \vec{x})+\partial_{1} j_{1}(t, \vec{x})+\cdots+\partial_{\mu} j_{\mu}(t, \vec{x})=0
Let uu be the velocity of the probability in the point ⟨t0,x0→⟩\left\langle t_{0}, \overrightarrow{x_{0}}\right\rangle and the coordinates system Rμ+1′R^{\mu+1 \prime} be obtained from the coordi- nates system Rμ+1R^{\mu+1} by the Lorentz transformations with the velo- city uu. That is:
t′=t−u⃗⋅x⃗1−u⃗2 and x⃗′=x⃗−t⋅u⃗1−u⃗2t^{\prime}=\frac{t-\vec{u} \cdot \vec{x}}{\sqrt{1-\vec{u}^{2}}} \text { and } \vec{x}^{\prime}=\frac{\vec{x}-t \cdot \vec{u}}{\sqrt{1-\vec{u}^{2}}}
In this case ρ′(t′,x⃗′)\rho^{\prime}\left(t^{\prime}, \vec{x}^{\prime}\right) is denoted as the local probability density (ρ◯)\left(\rho_{\bigcirc}\right). This function is the scalar function:
ρ′(t′,x⃗′)=ρ(t,x⃗)\rho^{\prime}\left(t^{\prime}, \vec{x}^{\prime}\right)=\rho(t, \vec{x})
and
ρ(t,x⃗)=ρ◯(t,x⃗)1−u⃗2(t,x⃗)\rho(t, \vec{x})=\frac{\rho_{\bigcirc}(t, \vec{x})}{\sqrt{1-\vec{u}^{2}(t, \vec{x})}}
Hence for any velocity vv, for which ∣v∣<1|v|<1 : if
t′=t−v⃗⋅x⃗1−v⃗2 and x⃗′=x⃗−t⋅v⃗1−v⃗2t^{\prime}=\frac{t-\vec{v} \cdot \vec{x}}{\sqrt{1-\vec{v}^{2}}} \text { and } \vec{x}^{\prime}=\frac{\vec{x}-t \cdot \vec{v}}{\sqrt{1-\vec{v}^{2}}}
then
ρ′(t′,x⃗′)=ρ(t,x⃗)−v⃗⋅j⃗(t,x⃗)1−v⃗2j⃗′(t′,x⃗′)=j⃗(t,x⃗)−ρ(t,x⃗)⋅v⃗1−v⃗2\begin{aligned} & \rho^{\prime}\left(t^{\prime}, \vec{x}^{\prime}\right)=\frac{\rho(t, \vec{x})-\vec{v} \cdot \vec{j}(t, \vec{x})}{\sqrt{1-\vec{v}^{2}}} \\ & \vec{j}^{\prime}\left(t^{\prime}, \vec{x}^{\prime}\right)=\frac{\vec{j}(t, \vec{x})-\rho(t, \vec{x}) \cdot \vec{v}}{\sqrt{1-\vec{v}^{2}}} \end{aligned}
Therefore ρ(t,x⃗)\rho(t, \vec{x}) is not the scalar function but:
ρ2(t,x⃗)−j⃗2(t,x⃗)=ρ◯2(t,x⃗)\rho^{2}(t, \vec{x})-\vec{j}^{2}(t, \vec{x})=\rho_{\bigcirc}^{2}(t, \vec{x})
4 RESUME
In order to the probability is defined by the relativistic μ+1\mu+1-vector of the density, the probability distribution function must fulfil to the odd global condition (2), which is expressed by the integrals on all space.