Kaku M.- Quantum Field Theory
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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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
you do, you will! take it, easyly!
所有者:闻水国
you do, you will! take it, easyly!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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!
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