# 量子電動力學

## 費曼的量子電動力學觀

### 簡介

• 光子從一時間地點，移動到另一時間地點
• 電子從一時間地點，移動到另一時間地點
• 一電子在某一時間地點，發射或吸收一光子

1. 若一事件能以多種不同的方式發生，那麼其機率幅為各發生方式機率幅的和；
2. 若一過程中包括多種獨立的子過程，則其機率幅為各子過程機率幅的積。

### 機率幅

$P=|\mathbf{v}+\mathbf{w}|^2$

$P=|\mathbf{v}\times\mathbf{w}|^2.$

$E(A\rightarrow D)\times E(B\rightarrow C) - E(A\rightarrow C) \times E(b\rightarrow D),$

### 傳播子

$P(A \rightarrow B) \rightarrow D_F(x_B-x_A),\quad E(C \rightarrow D) \rightarrow S_F(x_D-x_C)$

## 數學

 $\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,$

$\gamma_\mu \,\!$狄拉克矩陣
$\psi$自旋1/2粒子（例如電子正電子場）的
$\bar\psi\equiv\psi^\dagger\gamma^0$
$D_\mu = \partial_\mu+ieA_\mu \,\!$規範協變導數，而$\ e$為耦合強度（等同於基本電荷），
$\ A_\mu$協變電磁場向量勢
$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!$電磁場張量

### 運動方程式

$\mathcal{L} = i \bar{\psi} \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma_\mu A^\mu \psi -m \bar{\psi} \psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} \quad \quad \quad(1) \,$

$\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (2) \,$

$\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) = \partial_\mu \left( i \bar{\psi} \gamma^\mu \right) \,$
$\frac{\partial \mathcal{L}}{\partial \psi} = -e\bar{\psi}\gamma_\mu A^\mu - m \bar{\psi} \,$

$i \partial_\mu \bar{\psi} \gamma^\mu + e\bar{\psi}\gamma_\mu A^\mu + m \bar{\psi} = 0 \,$

$i \gamma^\mu \partial_\mu \psi - e \gamma_\mu A^\mu \psi - m \psi = 0 \,$

 $i \gamma^\mu \partial_\mu \psi - m \psi = e \gamma_\mu A^\mu \psi \,$

$\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (3) \,$

$\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) \,$
$\frac{\partial \mathcal{L}}{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi \,$

 $\partial_\nu F^{\nu \mu} = e \bar{\psi} \gamma^\mu \psi \,$

$\partial_{\mu} A^\mu = 0$

$\Box A^{\mu}=e\bar{\psi} \gamma^{\mu} \psi\,,$

### 相互作用繪景

$M_{fi}=\langle f|U|i\rangle.$

$V=e\int d^3x\bar\psi\gamma^\mu\psi A_\mu$

$U=T\exp\left[-\frac{i}{\hbar}\int_{t_0}^tdt'V(t')\right]$

### 費曼圖

$M_{fi}=(ie)^{2}\overline{u}(\vec{p}\,',s')\epsilon\!\!\!/\,'(\vec{k}\,',\lambda')^{*}{p\!\!\!/+k\!\!\!/+m_{e} \over (p+k)^{2}-m^{2}_{e}}\epsilon\!\!\!/(\vec{k},\lambda)u(\vec{p},s)+(ie)^{2}\overline{u}(\vec{p}\,',s')\epsilon\!\!\!/(\vec{k},\lambda){p\!\!\!/-k\!\!\!/'+m_{e}\over (p-k')^{2}-m^{2}_{e}}\epsilon\!\!\!/\,'(\vec{k}\,',\lambda')^{*}u(\vec{p},s)$

## 参考文献

1. ^ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 Feynman, Richard. QED: The Strange Theory of Light and Matter. Princeton University Press. 1985. ISBN 978-0-691-12575-6.
2. ^ P.A.M. Dirac. The Quantum Theory of the Emission and Absorption of Radiation. Proceedings of the Royal Society of London A. 1927, 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039.
3. ^ E. Fermi. Quantum Theory of Radiation. Reviews of Modern Physics. 1932, 4: 87–132. Bibcode:1932RvMP....4...87F. doi:10.1103/RevModPhys.4.87.
4. ^ F. Bloch; A. Nordsieck. Note on the Radiation Field of the Electron. Physical Review. 1937, 52 (2): 54–59. Bibcode:1937PhRv...52...54B. doi:10.1103/PhysRev.52.54.
5. ^ V. F. Weisskopf. On the Self-Energy and the Electromagnetic Field of the Electron. Physical Review. 1939, 56: 72–85. Bibcode:1939PhRv...56...72W. doi:10.1103/PhysRev.56.72.
6. ^ R. Oppenheimer. Note on the Theory of the Interaction of Field and Matter. Physical Review. 1930, 35 (5): 461–477. Bibcode:1930PhRv...35..461O. doi:10.1103/PhysRev.35.461.
7. ^ W. E. Lamb; R. C. Retherford. Fine Structure of the Hydrogen Atom by a Microwave Method,. Physical Review. 1947, 72 (3): 241–243. Bibcode:1947PhRv...72..241L. doi:10.1103/PhysRev.72.241.
8. ^ P. Kusch; H. M. Foley. On the Intrinsic Moment of the Electron. Physical Review. 1948, 73 (3): 412. Bibcode:1948PhRv...73..412F. doi:10.1103/PhysRev.73.412.
9. ^ H. Bethe. The Electromagnetic Shift of Energy Levels. Physical Review. 1947, 72 (4): 339–341. Bibcode:1947PhRv...72..339B. doi:10.1103/PhysRev.72.339.
10. ^ S. Tomonaga. On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields. Progress of Theoretical Physics. 1946, 1 (2): 27–42. doi:10.1143/PTP.1.27.
11. ^ J. Schwinger. On Quantum-Electrodynamics and the Magnetic Moment of the Electron. Physical Review. 1948, 73 (4): 416–417. Bibcode:1948PhRv...73..416S. doi:10.1103/PhysRev.73.416.
12. ^ J. Schwinger. Quantum Electrodynamics. I. A Covariant Formulation. Physical Review. 1948, 74 (10): 1439–1461. Bibcode:1948PhRv...74.1439S. doi:10.1103/PhysRev.74.1439.
13. ^ R. P. Feynman. Space–Time Approach to Quantum Electrodynamics. Physical Review. 1949, 76 (6): 769–789. Bibcode:1949PhRv...76..769F. doi:10.1103/PhysRev.76.769.
14. ^ R. P. Feynman. The Theory of Positrons. Physical Review. 1949, 76 (6): 749–759. Bibcode:1949PhRv...76..749F. doi:10.1103/PhysRev.76.749.
15. ^ R. P. Feynman. Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction. Physical Review. 1950, 80 (3): 440–457. Bibcode:1950PhRv...80..440F. doi:10.1103/PhysRev.80.440.
16. ^ 16.0 16.1 F. Dyson. The Radiation Theories of Tomonaga, Schwinger, and Feynman. Physical Review. 1949, 75 (3): 486–502. Bibcode:1949PhRv...75..486D. doi:10.1103/PhysRev.75.486.
17. ^ F. Dyson. The S Matrix in Quantum Electrodynamics. Physical Review. 1949, 75 (11): 1736–1755. Bibcode:1949PhRv...75.1736D. doi:10.1103/PhysRev.75.1736.
18. ^ The Nobel Prize in Physics 1965. Nobel Foundation. [2008-10-09].
19. ^ G.S. Guralnik, C.R. Hagen, T.W.B. Kibble. Global Conservation Laws and Massless Particles. Physical Review Letters. 1964, 13 (20): 585–587. Bibcode:1964PhRvL..13..585G. doi:10.1103/PhysRevLett.13.585.
20. ^ G.S. Guralnik. The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles. International Journal of Modern Physics A. 2009, 24 (14): 2601–2627. arXiv:0907.3466. Bibcode:2009IJMPA..24.2601G. doi:10.1142/S0217751X09045431.
21. ^ Griffiths, David J., Introduction to Elementary Particles 2nd revised, WILEY-VCH, 2008, ISBN 978-3-527-40601-2
22. ^ 22.0 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 Peskin, Michael; Schroeder, Daniel. An introduction to quantum field theory Reprint. Westview Press. 1995. ISBN 978-0201503975.
23. ^ Schwarz, Matthew D. Quantum Field Theory and Standard Model. Cambridge University Press. 2014. ISBN 978-1-107-034730.
24. ^ Dyson, Freeman, Divergence of Perturbation Theory in Quantum Electrodynamics, Physical Review, 1951, 85: 631–632
25. ^ Kinoshita, Toichiro. Quantum Electrodynamics has Zero Radius of Convergence Summarized from Toichiro Kinoshita. [06-10-2010].

## 延伸閱讀

### 學術期刊

• Dudley, J.M.; Kwan, A.M. Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University. American Journal of Physics. 1996, 64 (6): 694–698. Bibcode:1996AmJPh..64..694D. doi:10.1119/1.18234.