Relativistic Quantum Fields

所有者:闻水国

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Useful Formulae ® 闻水国 ©

Useful Formulae

= = 6.58 ×10−25 GeV sec = 1 =c = 0.197 GeV F = 1 (1 GeV) −2 = 0.389 mb

α=

e− (− E ) < 0

e+ ≡ E >0

time

e2 1 = 4π 137 ⎛∂ ⎞ p μ = ( E , p) = i ⎜ , −∇ ⎟ = i∂ μ ⎝ ∂t ⎠ 2 2 2 μ p ≡ pμ p = E − p = m 2

x μ = (t , x), xμ = (t , − x) p ⋅ x = Et − p ⋅ x, (,2 + m 2 )φ = 0,

J PC P = −(−1) L C = (−1) L + S

(i γ μ ∂ μ − m)ψ = 0. μ

μ

μ

In an electromagnetic filed, i∂ → i∂ + eA (charge − e)

j μ = −ie(φ * ∂ μφ − φ∂ μφ *),

j μ = −eψ γ μ ψ

γ-Matrices Tra/ 1 " a/ n = 0 (for n odd)

γ μ γν + γν γ μ = 2 g μν , γ μ † = γ 0 γ μ γ 0 . †

γ0 = γ0 ,

†

γ 0 γ 0 = I ; γ k = −γ k ,

γ 5 = i γ 0 γ1 γ 2 γ 3 ,

γ k γ k = − I , k = 1, 2,3. †

γ μ γ 5 + γ 5 γ μ = 0, γ 5 = γ 5 .

(Trace theorems on pages 104 in relativistic quantum mechanics of Bjorken) Standard representation:

⎛I 0 ⎞ ⎛ 0 γ0 ≡ β = ⎜ ⎟ , γ ≡ βα = ⎜ ⎝ 0 −I ⎠ ⎝ −σ

σ⎞

⎛0 I ⎞ 5 ⎟, γ = ⎜ ⎟ 0⎠ ⎝ I 0⎠

⎡0 1 ⎤ ⎡ 0 −i ⎤ ⎡1 0 ⎤ ⎡1 0 ⎤ , σ2 = ⎢ , σ3 = ⎢ , σ0 = ⎢ ⎥ ⎥ ⎥ ⎥. ⎣1 0 ⎦ ⎣i 0 ⎦ ⎣0 −1⎦ ⎣0 1 ⎦

σ1 = ⎢

Spinors ⎧(p/ − m)u = 0 ; ⎨ ⎩u (p/ − m) = 0 †

u ( r ) u ( s ) = 2 Eδ rs ,

u ( r )u ( s ) = 2mδ rs ,

1 (1 − γ 5 )u ≡ uL , 2

1 (1 + γ 5 )u ≡ uR . 2

∑u

s =1,2

(s)

⎧⎪u ≡ u † γ 0 , Γ ≡ γ 0 Γ + γ 0 ⎨ μ / ≡ γ μ Aμ ⎪⎩ p/ ≡ pμ γ , A

u ( s ) = p/ + m = 2mΛ +

1 1 If m = 0 or E  m, then uL has helicity λ = − , u L has λ = + . 2 2

标量 (φφ * )

J P = 0+

赝标 (φγ 5φ * )

J P = 0−

矢量 (φγ μφ * )

J P = 1−

轴矢 (φγ 5γ μφ * )

J P = 1+

Mandelstam variables

p3

p1

p2

p4

s = ( p1 + p2 ) 2 = ( p3 + p4 ) 2 t = ( p1 − p3 ) 2 = ( p2 − p4 ) 2 u = ( p1 − p4 ) 2 = ( p2 − p3 ) 2

p1

p3

p2

p4 s-channel

p1

p3

p1

p3

p2

p4

p2

p4

t-channel

Time

Feynman parameter integral 1 1 dx =∫ ab 0 [ax + b(1 − x)]2

Trace Theorems μ ν a/ ≡ aμ γ μ , ab / / = a ⋅ b − iσ μν a b

Tra/ 1 " a/ n = 0 for n odd;

p μ pν = g μν p 2 / d ;

Trγ 5 = 0;

Tr1 = 4;

Trab / / = 4a ⋅ b;

Trγ 5 ab / / = 0;

α β γ δ Trγ 5 abcd / / / / = 4iε αβγδ a b c d ;

γ μ a/ γ μ = −2a/ ;

μ μ γ μ ab / / / γ = −2cba / / /; / / γ = 4a ⋅ b; γ μ abc

Tr[γ μ γ ν γ λ γ δ ] = 4( g μν g λδ − g μλ gνδ + g μδ gνλ ); Tr[a/ 1a/ 2 a/ 3a/ 4 ] = 4[a1 ⋅ a2 a3 ⋅ a4 − a1 ⋅ a3a2 ⋅ a4 + a1 ⋅ a4 a2 ⋅ a3 ]

u-channel

所有者:闻水国

you do, you will! take it, easyly!

所有者:闻水国

you do, you will! take it, easyly!