Abstract
In the previous chapter we have recalled the basic notions of non-relativistic quantum mechanics. We have seen that, in the Schroedinger representation, the physical state of a free particle of mass m is described by a wave function \(\psi (\mathbf{x},t)\) which is itself a classical field having a probabilistic interpretation.
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Notes
- 1.
The name second quantization is somewhat improper since, just as in the first quantization, only dynamical quantities are promoted to operators acting on states. However, while in the first quantization these quantities include the position and the momentum of a particle, in this new framework, the dynamical quantities to be quantized are fields, the space-coordinates being just a labels.
- 2.
Extension of the invariance to the full Poincaré group is obvious.
- 3.
Furthermore, erasing the negative energy solutions would spoil the completeness of the eigenstates of \(\hat{P}^\mu \) and the expansion in plane waves would be no longer correct.
- 4.
A possible interpretation of the negative-energy states as ‘holes’ in the sea of positive-energy ones was originally proposed by Dirac. For further reading on this we refer the reader to the references at the end of the chapter.
- 5.
The factor i has been inserted in order to have a real current.
- 6.
Actually this “charge” can be any conserved quantum number associated with invariance under \(\mathrm{U}(1)\) transformations, like baryon or lepton number etc. However we will always refer to the electric charge.
- 7.
For the sake of simplicity, we shall often omit the identity matrix when writing combinations of spinorial matrices. We shall for instance write the Dirac equation in the simpler form \( \left( i\hbar \gamma ^{\mu }\partial _{\mu }-mc\right) \,\psi (x)=0\).
- 8.
As mentioned in Chap. 7 the spinor representation cannot be obtained in terms of tensor representations of the Lorentz group.
- 9.
Note that, with respect to the last chapter, we have changed our convention for the standard momentum of a massless particle. Clearly the discussion in Chap. 9 equally applies to this new choice, upon replacing direction 1 with direction 3.
- 10.
This can be easily ascertained by multiplying both Eq. (10.148) to the left by \(S(\mathbf {\Lambda })\). We find that \(S(\mathbf {\Lambda })u(p,r)\) and \(S(\mathbf {\Lambda })v(p,r)\) satisfy the following equations: \((S(\mathbf {\Lambda })p\!\!/\,S(\mathbf {\Lambda })^{-1}-mc)\,S(\mathbf {\Lambda })u(p,r)=0\) and \((S(\mathbf {\Lambda })p\!\!/\,S(\mathbf {\Lambda })^{-1}+mc)\,S(\mathbf {\Lambda })v(p,r)=0\). Next we use property (10.88) and invariance of the Lorentzian scalar product \(\gamma \cdot p\equiv \gamma ^\mu p_\mu =p\!\!/\) to write \(S(\mathbf {\Lambda })p\!\!/\,S(\mathbf {\Lambda })^{-1}=p\!\!/'=\gamma ^\mu p'_\nu \), where \(p'=\mathbf {\Lambda }\,p\). Thus the transformed spinors satisfy Eq. (10.148) with the transformed momentum \(p'\), and consequently, should be a combination of \(u(p',s)\) and \(v(p',s)\), respectively.
- 11.
In the above derivation the time-dependent exponential \(e^{\frac{i}{\hbar }(p^{0}-p'^{0})x^{0}}\) equals one since the equality \(\mathbf{p} = \mathbf{p}'\) implemented by the delta-function implies \(p^{0}=p'^{0}\).
- 12.
We also observe that the Dirac equation is invariant under the transformations
- 13.
We recall that the Zeeman effect can only be explained if \(g=2\). We see that this value is correctly predicted by the Dirac relativistic equation in the non-relativistic limit.
- 14.
Recall that helicity is invariant under proper Lorentz transformations and labels irreducible representations of \(\mathrm{SO}(1,3)\) with \(m=0\).
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D’Auria, R., Trigiante, M. (2016). Relativistic Wave Equations. In: From Special Relativity to Feynman Diagrams. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22014-7_10
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DOI: https://doi.org/10.1007/978-3-319-22014-7_10
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