1. 前回まとめ
自由scalar場の量子化
Lagrangian 密度
運動方程式
Klein Gordon方程式
正準共役運動量
量子条件
Hamiltonian
一般解
Fock space
真空状態
生成演算子
消滅演算子
Hamiltonian

2. Dirac spinor =
Dirac行列
g 0 = =
Lagrangian密度
正準共役運動量
Dirac equation
quantization condition　
solution　
vacuum state
Fock space
particle
creation operator
annihilation operator
antiparticle
creation operator
annihilation operator

3. Electromagnetic field
electric field
magnetic field
Maxwell equation　　
4-dimensional description　　
4-current
4-field strength
4-current

4. Electromagnetic field electric field magnetic field
Maxwell equation　　
4-dimensional description　　 *
4-current * - - ( -( ) - -
4-field strength
dual of Fmn
4-current * *
F01=e0123F 23
F23
=(×B)23
=-B1
F01
= (-E)1
=-E1
=-B1 *
F23=e2301F 01
=-E1
= (×E)23
: totally anti-symmetric tensor

5. Electromagnetic field
electric field
magnetic field
Maxwell equation　　
4-dimensional description　　 *
scalar potential j, vector potential A,　　
4-dim. vector field Am　　
field strength
field strength

6. Electromagnetic field electric field magnetic field
Maxwell equation　　
4-dimensional description　　 *
scalar potential j, vector potential A,　　
4-dim. vector field Am　　
field strength
∂0An
-∂nA0 n = = n m = m = =
∂mA0
∂mAn-∂nAm
-∂0Am
( )23
×(∇×A)
= - (∇×A)1
=-(∂2A3-∂3A2)
=∂2A3-∂3A2
: totally anti-symmetric tensor

7. Electromagnetic field
electric field
magnetic field
Maxwell equation　　
4-dimensional description　　 *
scalar potential j, vector potential A,　　
4-dim. vector field Am　　
field strength
gauge transformation
L: function of xm
Fmn : invariant under gauge transformation
, E, B

8. *
gauge transformation
L: function of xm
Fmn : invariant under gauge transformation
, E, B

9. require
(i)
vector field
(dynamical variable)
Maxwell eq.　　
(ii) Lorentzian invariance, locality
Maxwell equation　　
(iii) gauge invariance *
(iv)
simple interaction with the current
Lagrangian density
gauge transformation
L: function of xm
Fmn : invariant under gauge transformation
, E, B

10. require
(i)
vector field
(dynamical variable)
Maxwell eq.　　
(ii) Lorentzian invariance, locality
(iii) gauge invariance *
(iv)
simple interaction with the current
Lagrangian density *
Euler equation
symmetric
symmetric = = = = =
anti-symmetric
anti-symmetric
Maxwell equation　　 *

11. Quantization of free electromagnetic field Am
free-field Lagrangian

12. positive frequency part
Quantization of free electromagnetic field Am
free-field Lagrangian
canonical conjugate momentum
quantization condition = ≠
???
gauge fixing　
positive frequency part
add 　
to　
and impose 　
the subsidiary condition
physical states
good!
canonical conjugate momentum
quantization condition

13. eq. of motion　
solution
polarization vectors
general solution
Fock space
vacuum state
subsidiary condition　
creation operator
annihilation operator

14. complex scalar
荷電場の記述　
Dirac spinor
自由場Lagrangian
matter 場のgauge変換　
は不変
は不変でない
covariant derivative
gauge場の変換　
gauge invariant Lagrangian density　

15. Schrödinger Representation & Heisenberg Representation
operators
depend on time
Heisenberg eq.
H: Hamiltonian
states
: do not depend on time.
Schrödinger representation
states
Schrödinger equation
Operators
do not depend on time.
in state
out state
scattering matrix (S-matrix)　　
scattering matrix elements　　
scattering provability ∝　

16. perturbation
Interaction representation　　
operators
states
solution
T : time ordered product (T-product)

17. The solution of is proof T-product ordered! differentiation n!tn t3 t2 n!
The n! terms with different orders become the same after T-ordering!
differentiation

18. def.
positsve frequency part
(annihilation operator)
negative frequency part
(creation operator)
def.
N : normal product
place
to the left of
Wick's theorem
def.
vacuum expatation value
e. g.

19. Wick's theorem
e. g.

20. の証明 -) =
よって、このとき、与式は成立する　　

21. -)
よって、このとき、与式は成立する　　
の証明 -) =
よって、このとき、与式は成立する　　

22. spinor yi
vector Am
locality,
Lorentz inv.
gauge inv.
perturbation
interaction reprensatation
probability
S-matrix
Wick's theorem
def.

23. locality,
Lorentz inv.
gauge inv.
spinor yi
vector Am
interaction reprensatation
perturbation
S-matrix
probability
Wick's theorem
def.

24. scalar field
If x0>y0
If x0<y0
2E(2p)3d (k-k' )
2E(2p)3d (k-k' ) [ [ ] ]
for =
for
にpole

25. Dirac field =
for
にpole
for

26. electromagnetic field=
for
にpole
for

27. locality,
Lorentz inv.
gauge inv.
spinor yi
vector Am
interaction reprensatation
perturbation
S-matrix
probability
Wick's theorem
def.