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# IntroductiontoFieldTheory AppendixA

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## Introduction to Field Theory

(2)

·

π '

2

π

π

2

·

'

2

2

2

2

e

2

µ

2

(3)

p

2

0

e

2

×

−8









e

e

e

e

µ

µ









Matrix

ij

ji

### . (A.2.3)

Differential Operator

−i

µ

T

hψ|

Z

−∞

−i Z

−∞

µ

h

−i

6= ˆ

(4)

Field

Scalar Product

r

1

2

3

p

x

y

z

1

2

3

r·p

X3

k=1

k

k

k

k

Vector Product

r×p≡

2

3

3

2

3

1

1

3

1

2

2

1

×p)i

ijk

j

k

ijk

123

231

312

132

213

321

−1,

µ

0

r),

µ

x

y

z

0

p)

·

−r·p

0

0

k

k

(5)

µ

µ

µ

0

0

1

1

2

2

3

3

−r·p

·

µ

µ

µ

0

−r),

µ

0

−p).

µν

µν

µν



−1

−1

−1



·

µ

ν

µν

−r·p.

µ

µ

µ

µ

µ

0

1

2

3

µ

µ

µ

µ

µ

µ

µ

−∇

µ

µ

µ

p) =

µ

−i∇

µ

µ

µ

j),

µ

µ

∇·j

µ

µ

(6)

2

≡∇·∇

2

2

2

2

2

2

2≡

µ

µ

2

2

x

1

µ

y

2

µ

−i

z

3

µ

−1

Hermiticity

1

1

2

2

3

3

### .

Complex Conjugate

1

1

2

−σ2

3

3

Transposed

1T

1

2T

−σ2

T3

3

kT

k

Useful Relations

i

j

ij

ijk

k

i

j

ijk

k

(7)

0

µ

−1

1

µ

−1 0

5

0

1

µ

0

µ

1

µ

−1

5

0

1

µ

−1

0

µ1 0 0 −1

γ

µ 0 σ

−σ 0

5

0

1

2

3

µ0 1 1 0

α

µ0 σ σ 0

0

µ0 1

1 0

γ

µ0 −σ σ 0

5

0

1

2

3

µ1 0 0 −1

α

µσ 0 0 −σ

0

µ

1

µ

### Here, we summarize some useful relations of the γ-matrices.

Anti-commutation relations

µ

ν}

µν

5

ν}

Hermiticity

µ

0

µ

0

0

0

k

−γk

5

5

### . (A.6.5)

Complex Conjugate

0

0

1

1

2

−γ2

3

3

5

5

Transposed

Tµ

0

µ

0

T5

5

(8)

|ψi

0i

|ψi.

00i

hψ|U

|ψi

O

O0

OU−1

O

hψ|O|ψi

0|O00i.

0|O00i

hψ|UO0

|ψi

hψ|O|ψi

O0

O











'

5

' ·pm

µ

'

³

p

´

µ

5

'

³σ·p

σ

´











(9)







0

0

−1

0

5

0

−1

5

5

0

k

0

−1

k

k

0

k

5

0

−1

k

5

k

5













0

0

−1

0

5

0

−1

5

5

0

k

0

−1

k

k

0

k

5

0

−1

k

5

k

5







×

XN

i=1

ii

−1}

X

n=1

n

n

a

a

a

iTa

(10)

a

¡

{ln

a}¢

¡

{Ta}¢

{Ta}

a

b

abc

c

abc

{Ta

b}

ab

{1}

µ}

5}

µ

µ

µν

µ

ν

{p

5

{p

1

2

3

4}

n

1

2

3

4

1

3

2

4

1

4

2

3

o

5

1

2

3

4}

αβγδ

α1

β2

γ3

δ4

(11)

Z

Z

Z µ

Z µ

·

µ

¶¸

·

µ

¶¸

(12)

L

Z

L µ

k

3

∂x∂ψk

∂x∂ψk

Z

µ

k

3

Z Ã

∂x∂ψ

k

µ

k

¶!

3

Z Ã

k

∂x∂ψ

k

!

3

k

∂x∂ψ

k

µ

µ

µ

L

µ

µ

LI£

5

µ

¤

(13)

0

0†

−iα

|α| ¿

0

0†

−iαψ

### . (A.11.2b)

Invariance of Lagrangian Density

L(ψ0

0†

µ

0

µ

0†

− L(ψ, ψ

µ

µ

µ

µ

µ

³

µ

´

·µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

¸

µ

·

µ

µ

¸

### where the equation of motion for ψ is employed.

Current Conservation

µ

µ≡ −i

·

µ

µ

¸

µ

µ

µ

µ

(14)

0

iαγ5

5

5µ

−i

µ

5

µ

5

µ

5µ

H

### is constructed from the Lagrangian density

L. The field theory

µ

Tµν

T00

L

³

i

i

∂x∂ψi

k

´

µ

i

i

i0

i

i

i

ν

i

ν

i0

i

i

i

ν

i

ν

0i

µ

0i

− L(ψi

µ

i

X

i

"

i

i

µ

i

µ

i

i

i

µ

i

µ

i

#

(15)

X

i

·

i

ν

i

µ

i

µ

ν

i

µ

µ

µ

i

ν

i

¶¸

ν

X

i

"

i

ν

i

µ

i

µ

ν

i

µ Ã

µ

i

ν

i

!#

ν

µ

"

LgµνX

i

Ã

µ

i

ν

i

µ

i

ν

i

!#

ν

### = 0. (A.12.2)

Energy Momentum TensorTµν

Tµν

Tµν X

i

Ã

µ

i

ν

i

µ

i

ν

i

!

− Lgµν

Tµν

µTµν

H

T00 H ≡ T00

X

i

Ã

0

i

0

i

0

i

0

i

!

− L.

L(ψi

i

∂ψ∂xi

k

ψi

ψ

i

ψi

i

ψ i

i

H

X

i

³

ψi

i

ψ i

i

´

− L.

(16)

Hamiltonian

Z

H

3

Z "X

i

ψi

i

ψ

i

i

− L

#

3

L

i

i

i

0γ·∇−

0

ij

j

ψi

ψ

i

ψi

i

i

ψ i

H

X

i

³

ψi

i

ψ i

i

´

−L

i

k

k

ij

j

·∇+m]

Z

H

3

Z

·∇

3

(17)

### A.12. Hamiltonian Density 207

Classical Field Theory

### In classical field theory, the situation is just the same as the classical mechanics case. If one stays in the classical field theory, then one should derive the field equation from the Hamiltonian by the functional variational principle as will be discussed in the next section.

Quantized Field Theory

|Ψi

space.

|Ωi,

|Ωi

|Ωi

Y

p,s

(s)p |0ii,

|0ii

(18)

Z ·

2

¸

3

### with respect to ψ(r), then one can obtain the static Schr¨odinger equation.

Functional Derivative

i

i

0

j

ij

−r0

i

### (r).

Functional Variation of Hamiltonian

Z

3

Z ·

0

02

0

0

0

¸

3

0

−E µZ

0

0

3

0

Z

−r0

·

02

0

0

0

¸

3

0

2

### which is just the static Schr¨odinger equation.

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