قوانين نظرية المجال الكمي مقدمة :- Overture t'=γ(t - vx/ | 𝑴𝒂𝒙 𝑷𝒍𝒂𝒏𝒄𝒌 𝒏𝒆𝒘𝒔𝒑𝒂𝒑𝒆𝒓

قوانين نظرية المجال الكمي

مقدمة :- Overture

t'=γ(t - vx/c²) , x'=γ(x - vt) , y'=y , z'=z (1)

γ=(1- β²)⁻½ and β=v/c

∂ᵩ =∂/∂χᵠ=(∂/∂t , ∇)=(∂/∂t , ∂/∂χ , ∂/∂y , ∂/∂z)

a'ᵠ =Σᵦ₌₀³(∂x'ᵠ/∂xᵝ) aᵝ

a'ᵠ =(∂x'ᵠ/∂xᵝ) aᵝ (3)

∂Φ/∂x'ᵠ =(∂xᵝ/∂x'ᵠ) (∂Φ/∂xᵝ) (4)

It'I=Iγ -βγ 0 0I ItI
Ix'I I-βγ γ 0 0I IxI
Iy'I I0 0 1 0I IyI
Iz'I I0 0 0 1I IzI (5)

x'ᵠ=Λᵠᵦ xᵝ (6)

p'ᵠ=Λᵠᵦ pᵝ (7)

a'ᵩ=Λᵝᵩ aᵦ (8)
x|²=x . x=(x⁰) - (x¹)² - (x²)² - (x³)² (10)

a · b = a⁰b⁰ - a• · b• (11)

a · b = gᵩᵦ aᵠbᵝ (12)

gᵩᵦ =I1 0 0 0I
I0 -1 0 0I
I0 0 -1 0I
I0 0 0 -1I (13)

aᵩ= gᵩᵦ aᵝ (14)

a⁰= a₀ , aⁱ= -aᵢ (15)

a · b=gᵩᵦ aᵠbᵝ=aᵩbᵠ=aᵠbᵩ (16)
p . p= (E ، p• ) · (E ، p• ) = E² - p• ² = m² (17)

a . b=gᵠᵝ aᵩbᵦ

∂²=∂ᵠ∂ᵩ=∂²/∂t² - ∂²/∂x² - ∂²/∂y² - ∂²/∂z² (18)

∂²=∂ᵠ∂ᵩ=∂²/∂t² - ∇² (19)

Tᵠ'···ᵝ' ᵥ،...ₖ،=(∂xᵠ'/∂xᵠ)...(∂xᵝ'/∂xᵝ)(∂xᵛ/∂xᵛ')...(∂xᵏ/∂xᵏ')Tᵠ···ᵝ ᵥ...ₖ (20)

δ'ᵠᵦ=(∂x'ᵠ/∂xᵅ)(∂xᵞ/∂x'ᵝ)δᵅᵧ (21)

f'(k)dk =∫d⁴x eⁱᵏ · ˣ f(x) (22)

∫d⁴x = ∫dx⁰dx¹dx²dx³ (23)

f(x)dx =∫(d⁴k/(2π)⁴) e⁻ⁱᵏ · ˣ f'(k) (24)

f(w ، k• ) = ∫d³xdt exp[i(ωt - k• . x• )] f(t , x) (25)

∫ dᵈx δ⁽ᵈ⁾(χ)=1 (26)

∫ dᵈx f(x) δ⁽ᵈ⁾(χ)= f(0) (27)

δ'⁽ᵈ⁾(k)= ∫ dᵈx e⁻ⁱᵏ · ˣn δ⁽ᵈ⁾(χ)=1 (28)

δ⁽⁴⁾(x)= ∫ (d⁴k/(2π)⁴) e⁻ⁱᵏ · ˣ (29)

∇ . E=ρ/ε₀ , ∇ x E = -∂B/∂t

∇. B = 0 , ∇ x B = µ₀ J + (1/c²) ∂E/∂t (30)

V(x)=q/(4π|x|) (31)

∇ . E=ρ , ∇ x E = -(1/c)∂B/∂t

∇. B = 0 , ∇ x B = (1/c)[J+ ∂E/∂t] (32)

α = e²/(4π) (33)

α = e/(4πħc)=1/137

q=Q|e|

الفصل الاول : اللاغرانجيان :- Lagrangians .
قوانين الفصل الاول:
F = mx·· (1-1)

Tₐᵥ=(1/τ) ∫₀ᵞ (1/2)m[x·(t)]² dt (1-2)

Vₐᵥ= (1/τ) ∫₀ᵞ V[x(t)] dt (1-3)

F[f]= ∫₀¹ f(x) dx (1-4)


F[f]= ∫₀¹ x² dx=(1/3)|x³|₀¹=1/3 (1-5)

G[f]= ∫₋ₐᵃ 5[f(x)]² dx (1-6)

G[f]= ∫₋ₐᵃ 5[x²]² dx = ∫₋ₐᵃ 5 x⁴ dx =5(1/5)|x⁵|₋ₐᵃ=2a⁵ (1-7)

Fₓ[f]=∫₋ₐᵃ f(y) δ(y - χ) dy=f(x) (1-8)


df/dx= limₑ→₀ [f(x + e) ​​- f(x)]/e (1-9)


δF/δf(χ)= limₑ→₀ [F{f(x') + eδ(χ - χ')} ​​- F{f(x')}]/e (1-10)

δJ[f]/δf(x)=limₑ→₀ (1/e) ∫ [f(y) + eδ(y - x)]ᵖ Φ(y)dy + ∫ [f(y)]ᵖ Φ(y)dy]
δI[f]/δf(x₀)=limₑ→₀ ∫₋₁¹ δ(χ - x₀) dx ​​

δI[f]/δf(x₀)=1 if -1≤ x₀ ≤ 1
or=0 otherwise (1-11)

δJ[f]/δf(x)= p[f(x)]ᵖ⁻¹ Φ(x) (1-12)

δH[f]/δf(x)= g'[f(x₀)] (1-13)

Vₐᵥ(x)=(1/T)∫₀ᵀ V[x(t)] dt

F[Φ]=∫(∂Φ/∂y)² dy

δVₐᵥ(x)/δχ(t)=(1/T)V'[x(t)] dt (1-14)

δg[f(χ)]/δf(x)=-(d/dx)(dg(f')/df') (1-18)

δF[Φ(χ)]/δΦ(x)= -2(∂²Φ/∂y²) (1-19)

Τₐᵥ(x)=(1/T)∫₀ᵀ(1/2) m[x·(t)]² dt

δTₐᵥ(x)/δχ(t)= (1/T)(1/2)(δt/δχ(t))(δ/δt)[m x·] , δVₐᵥ(x)/δχ(t)= - (1/T)(δ/δχ(t))V(x)

δTₐᵥ(x)/δχ(t)= (1/T)(1/2)(1/x·)[2m x·x··] , δVₐᵥ(x)/δχ(t)= - (1/T)(δ/δχ(t))V(x)

δTₐᵥ(x)/δχ(t)= (1/τ)(δ/δχ(t))[(1/2)m x·²] , δVₐᵥ(x)/δχ(t)= - (1/τ)(δ/δχ(t))V(x)

δTₐᵥ(x)/δχ(t)=m x··/T , δVₐᵥ(x)/δχ(t)=- V'(x)/T (1-21)

δTₐᵥ(x)/δχ(t)= - m x··/T
[δTₐᵥ(x)/δχ(t)]T= - [- δVₐᵥ(x)/δχ(t)]T

δTₐᵥ(x)/δχ(t)= δVₐᵥ(x)/δχ(t) (1-22)

δ/δχ(t))[Tₐᵥ(x) - Vₐᵥ(x)]=0 (1-23)

δTₐᵥ(x)/δχ(t) - δVₐᵥ(x)/δχ(t)=0 (1-24)

L=T - V (1-25)

S = ∫ L dt (1-25)

S = ∫ (T - V) dt =τ(Tₐᵥ[a] - Vₐᵥ[x])

δS/δχ(t)= 0 (1-26)

δS/δx(u)=δL/δχ(t) - (d/dt)(δL/δχ·(t)) (1-27)

δL/δχ(t) - (d/dt)(δL/δχ·(t)) =0 (1-28)

L = ∫ dx L° (1-29)

S= ∫ dt L= ∫dt dx L° (1-30)