Abstract
The quantized Maxwell field provided us already with an example of a relativistic quantum field theory. On the other hand, the description of relativistic charged particles requires Klein-Gordon fields for scalar particles and Dirac fields for fermions. Relativistic fields are apparently relevant for high energy physics. However, relativistic effects are also important in photon-matter interactions, spectroscopy, spin dynamics, and for the generation of brilliant photon beams from ultra-relativistic electrons in synchrotrons. Quasirelativistic effects from linear dispersion relations \(E \propto \boldsymbol{ p}\) in materials like Graphene and in Dirac semimetals have also reinvigorated the need to reconsider the role of Dirac and Weyl equations in materials science. In applications to materials with quasirelativistic dispersion relations c and m become effective velocity and mass parameters to describe cones or hyperboloids in regions of \((E,\boldsymbol{k})\) space.
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Notes
- 1.
However, we will see that in the second quantized formalism in the Heisenberg and Dirac pictures, the time evolution of the field operators is given by Heisenberg equations of motion, and the corresponding time evolution of states in the Schrödinger and Dirac pictures is given by corresponding Schrödinger equations with relativistic Hamiltonians.
- 2.
E. Schrödinger, Annalen Phys. 386, 109 (1926); W. Gordon, Z. Phys. 40, 117 (1926); O. Klein, Z. Phys. 41, 407 (1927).
- 3.
O. Klein, Z. Phys. 53, 157 (1929). Klein actually discussed reflection and transmission of relativistic spin 1/2 fermions which are described by the Dirac equation (21.38).
- 4.
We cannot try to discuss motion of particles of mass m in the presence of a potential by simply including a scalar potential term in the form \(\left (\hbar ^{2}\partial _{t}^{2} - \hbar ^{2}c^{2}\partial _{x}^{2} + m^{2}c^{4}\right )\phi = \Theta (x)V ^{2}\phi\) in the Klein-Gordon equation. This would correspond to a local mass \(M(x)c^{2} = \sqrt{m^{2 } c^{4 } - \Theta (x)V ^{2}}\) rather than to a local potential, and yield tachyons in x > 0 for \(V ^{2}> m^{2}c^{4}\).
- 5.
P.A.M. Dirac, Proc. Roy. Soc. London A 117, 610 (1928). Dirac’s relativistic wave equation was a great success, but like every relativistic wave equation, it also does not yield a single particle interpretation. It immediately proved itself by explaining the anomalous magnetic moment of the electron and the fine structure of spectral lines, and by predicting positrons.
- 6.
The commutators S μ ν provide the spinor representation of the generators of Lorentz transformations. Furthermore, equation (21.85) is the invariance of the γ matrices under Lorentz transformations, see Appendix H.
- 7.
W. Pauli, Z. Phys. 43, 601 (1927). Pauli actually only studied the time-independent Schrödinger equation with the Pauli term in the Hamiltonian, and although he mentions Schrödinger in the beginning, he seems to be more comfortable with Heisenberg’s matrix mechanics in the paper.
- 8.
L.L. Foldy, S.A. Wouthuysen, Phys. Rev. 78, 29 (1950).
- 9.
E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960); Yu.A. Bychkov, E.I. Rashba, JETP Lett. 39, 78 (1984); J. Phys. C 17, 6039 (1984).
- 10.
See e.g. J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phys. Rev. Lett. 78, 1335 (1997); D. Grundler, Phys. Rev. Lett. 84, 6074 (2000); J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004); E.Y. Sherman, D.J. Lockwood, Phys. Rev. B 72, 125340 (2005); K.C. Hall et al., Appl. Phys. Lett. 86, 202114 (2005); P. Pietiläinen, T. Chakraborty, Phys. Rev. B 73, 155315 (2006); E. Cappelluti, C. Grimaldi, F. Marsiglio, Phys. Rev. Lett. 98, 167002 (2007).
- 11.
Of course, this implies that one cannot naively invoke Hamiltonians with Coulomb interaction terms if we describe photons in Lorentz gauge. Otherwise we would overcount interactions. Remember that the Coulomb interaction terms came from the contributions to Hamiltonians from electromagnetic fields in Coulomb gauge, see Section 21.4.
- 12.
A current J μ is sometimes denoted as strongly conserved if the local conservation law ∂ μ J μ = 0 is an identity.
Bibliography
C. Itzykson, J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)
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Dick, R. (2016). Relativistic Quantum Fields. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_21
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DOI: https://doi.org/10.1007/978-3-319-25675-7_21
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