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Quantum electrodynamics

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This article is about the physical theory. For the Latin phrase, see quod erat demonstrandum. For other uses, see QED (disambiguation).











Feynman's view of quantum electrodynamics[edit]




Feynman diagram elements




QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks, and then using the probability amplitudes to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while making the assumption that the square of the total of the probability amplitudes mentioned above (P(A to B), E(A to B) and 'j') acts just like our everyday probability. (A simplification made in Feynman's book.) Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman.


Basic constructions[edit]





Probability amplitudes[edit]






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


Addition of probability amplitudes as complex numbers
Multiplication of probability amplitudes as complex numbers






P(\mbox{A to B}) \rightarrow D_F(x_B-x_A),\quad  E(\mbox{C to D}) \rightarrow S_F(x_D-x_C)

where a shorthand symbol such as x_A stands for the four real numbers which give the time and position in three dimensions of the point labeled A.

Mass renormalization[edit]

Main article: Self-energy








 \gamma^\mu are Dirac matrices;
\psi a bispinor field of spin-1/2 particles (e.g. electronpositron field);
\bar\psi\equiv\psi^\dagger\gamma^0, called "psi-bar", is sometimes referred to as the Dirac adjoint;
D_\mu \equiv \partial_\mu+ieA_\mu+ieB_\mu \,\! is the gauge covariant derivative;
Bμ is the external field imposed by external source;
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\! is the electromagnetic field tensor.

Equations of motion[edit]


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


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






to find the field equations for QED.

The two terms from this Lagrangian are then

\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+B^\mu) - m \bar{\psi}. \,

Substituting these two back into the Euler–Lagrange equation (2) results in

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

with complex conjugate

i \gamma^\mu \partial_\mu \psi - e \gamma_\mu (A^\mu+B^\mu) \psi - m \psi = 0. \,


i \gamma^\mu \partial_\mu \psi - m \psi = e \gamma_\mu (A^\mu+B^\mu) \psi \,


One further important equation can be found by substituting the above Lagrangian into another Euler–Lagrange equation, this time for the field, Aμ:

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






The two terms this time are

\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 \,

and these two terms, when substituted back into (3) give us

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


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

then we get

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


Interaction picture[edit]

This theory can be straightforwardly quantized by treating bosonic and fermionic sectors[clarification needed] as free. This permits us to build a set of asymptotic states which can be used to start a computation of the probability amplitudes for different processes. In order to do so, we have to compute an evolution operator that, for a given initial state |i\rangle, will give a final state \langle f| in such a way to have[22]:5

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

This technique is also known as the S-matrix. The evolution operator is obtained in the interaction picture where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:[22]:123

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

and so, one has[22]:86



Feynman diagrams[edit]



Qed rules.jpg

To these rules we must add a further one for closed loops that implies an integration on momenta \int d^4p/(2\pi)^4, since these internal ("virtual") particles are not constrained to any specific energy–momentum – even that usually required by special relativity (see this article for details). From them, computations of probability amplitudes are straightforwardly given. An example is Compton scattering, with an electron and a photon undergoing elastic scattering. Feynman diagrams are in this case[22]:158–159

Compton qed.jpg


\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)






Nonconvergence of series[edit]



See also[edit]


  1. ^ a b c d e f g h i j k Feynman, Richard (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-12575-6. 
  2. ^ P.A.M. Dirac (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proceedings of the Royal Society of London A 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039. 
  3. ^ E. Fermi (1932). "Quantum Theory of Radiation". Reviews of Modern Physics 4: 87–132. Bibcode:1932RvMP....4...87F. doi:10.1103/RevModPhys.4.87. 
  4. ^ Bloch, F.; Nordsieck, A. (1937). "Note on the Radiation Field of the Electron". Physical Review 52 (2): 54–59. Bibcode:1937PhRv...52...54B. doi:10.1103/PhysRev.52.54. 
  5. ^ V. F. Weisskopf (1939). "On the Self-Energy and the Electromagnetic Field of the Electron". Physical Review 56: 72–85. Bibcode:1939PhRv...56...72W. doi:10.1103/PhysRev.56.72. 
  6. ^ R. Oppenheimer (1930). "Note on the Theory of the Interaction of Field and Matter". Physical Review 35 (5): 461–477. Bibcode:1930PhRv...35..461O. doi:10.1103/PhysRev.35.461. 
  7. ^ Lamb, Willis; Retherford, Robert (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method,". Physical Review 72 (3): 241–243. Bibcode:1947PhRv...72..241L. doi:10.1103/PhysRev.72.241. 
  8. ^ Foley, H.; Kusch, P. (1948). "On the Intrinsic Moment of the Electron". Physical Review 73 (3): 412. Bibcode:1948PhRv...73..412F. doi:10.1103/PhysRev.73.412. 
  9. ^ Schweber, Silvan (1994). "Chapter 5". QED and the Men Who Did it: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press. p. 230. ISBN 978-0-691-03327-3. 
  10. ^ H. Bethe (1947). "The Electromagnetic Shift of Energy Levels". Physical Review 72 (4): 339–341. Bibcode:1947PhRv...72..339B. doi:10.1103/PhysRev.72.339. 
  11. ^ S. Tomonaga (1946). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields". Progress of Theoretical Physics 1 (2): 27–42. doi:10.1143/PTP.1.27. 
  12. ^ J. Schwinger (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron". Physical Review 73 (4): 416–417. Bibcode:1948PhRv...73..416S. doi:10.1103/PhysRev.73.416. 
  13. ^ J. Schwinger (1948). "Quantum Electrodynamics. I. A Covariant Formulation". Physical Review 74 (10): 1439–1461. Bibcode:1948PhRv...74.1439S. doi:10.1103/PhysRev.74.1439. 
  14. ^ R. P. Feynman (1949). "Space–Time Approach to Quantum Electrodynamics". Physical Review 76 (6): 769–789. Bibcode:1949PhRv...76..769F. doi:10.1103/PhysRev.76.769. 
  15. ^ R. P. Feynman (1949). "The Theory of Positrons". Physical Review 76 (6): 749–759. Bibcode:1949PhRv...76..749F. doi:10.1103/PhysRev.76.749. 
  16. ^ R. P. Feynman (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction". Physical Review 80 (3): 440–457. Bibcode:1950PhRv...80..440F. doi:10.1103/PhysRev.80.440. 
  17. ^ a b F. Dyson (1949). "The Radiation Theories of Tomonaga, Schwinger, and Feynman". Physical Review 75 (3): 486–502. Bibcode:1949PhRv...75..486D. doi:10.1103/PhysRev.75.486. 
  18. ^ F. Dyson (1949). "The S Matrix in Quantum Electrodynamics". Physical Review 75 (11): 1736–1755. Bibcode:1949PhRv...75.1736D. doi:10.1103/PhysRev.75.1736. 
  19. ^ "The Nobel Prize in Physics 1965". Nobel Foundation. Retrieved 2008-10-09. 
  22. ^ a b c d e f g Peskin, Michael; Schroeder, Daniel (1995). An introduction to quantum field theory (Reprint ed.). Westview Press. ISBN 978-0201503975. 
  23. ^ Kinoshita, Toichiro. "Quantum Electrodynamics has Zero Radius of Convergence Summarized from Toichiro Kinoshita". Retrieved 06-10-2010.  Check date values in: |accessdate= (help)
  24. ^ Espriu and Tarrach. "Ambiguities in QED: Renormalons versus Triviality". arXiv:hep-ph/9604431. 

Further reading[edit]



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

External links[edit]