Abstract
The fundamental building blocks of matter, i.e. quarks and leptons, carry spin 1/2. There are two formally different but in essence equivalent methods of describing particles with spin: The representation theory of the Poincaré group, in the framework of Wigner’s classification hypothesis of particles (see e.g. [QP07], Chap. 6), and the Van der Waerden spinor calculus based on SL(2, \(\mathbb{C}\)).
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Notes
- 1.
We have inserted a vertical and a horizontal line in order to emphasize the space–space components in the lower right 3 ×3 block as opposed to the time–time, time–space, and space–time components. We shall keep these auxiliary lines occasionally for the sake of clarity, but we shall drop them later in the text.
- 2.
Note that the rotations (1.6) are orthogonal matrices, i.e. \({\mathcal{R}}^{-1} = {\mathcal{R}}^{T}\), while the boosts (1.7) are symmetric matrices. The theorem writes an arbitrary Λ ∈ L + ↑ as the product of a symmetric and an orthogonal matrix. This is analogous to the decomposition of a complex number in its (real) modulus and a (unitary) phase. This analogy will become even more striking for SL(2, \(\mathbb{C}\)) representations below.
- 3.
- 4.
- 5.
This construction can be found for example in [RUE70].
- 6.
See e.g. [HAM62].
- 7.
The term contragredience is a general term in linear algebra which helps to distinguish covariant and contravariant objects. For example, in the decomposition v = ∑ v a v e v of a vector in terms of the set of base vectors e = {e v }, this set is covariant, while the set of expansion coefficients a = {a v} is contravariant, i.e. in the example
$$\begin{array}{l} {\mathrm{\mathbf{e}}}^{{\prime}} = \mathrm{\mathbf{Ae}},\quad {\mathrm{\mathbf{a}}}^{{\prime}} = {({\mathrm{\mathbf{A}}}^{T})}^{-1}\mathrm{\mathbf{a}},\quad \mathrm{so\ that} \\ \mathrm{\mathbf{v}} ={ \mathrm{\mathbf{a}}}^{{\prime}}\,\cdot \,{\mathrm{\mathbf{e}}}^{{\prime}} = \left ({({\mathrm{\mathbf{A}}}^{T})}^{-1}\mathrm{\mathbf{a}}\right ) \cdot (\mathrm{\mathbf{Ae}}) = \mathrm{\mathbf{a}}({\mathrm{\mathbf{A}}}^{-1}\mathrm{\mathbf{A}})\mathrm{\mathbf{e}} = \mathrm{\mathbf{a}} \cdot \mathrm{\mathbf{e}}\\ \end{array}$$stays invariant. Expansion coefficients and base vectors are said to be contragredient to each other.
- 8.
We recall: A T is the transposed, A ∗ the complex conjugate of A, and A † = A T ∗ is the hermitean conjugate.
- 9.
Se e.g. [FAR59] where this is called the “U-transformation”.
- 10.
Summation over repeated, contragredient indices is implied.
- 11.
SL(2, \(\mathbb{C}\)) is characterized by the invariant skew-symmetric scalar product \({b}_{a}{\epsilon }^{\mathit{ab}}{c}_{b}\) and hence is isomorphic to Sp(2, \(\mathbb{C}\)), the symplectic group in two complex dimensions.
- 12.
A spinor field is a spinor with respect to SL (2, \(\mathbb{C}\)) whose entries are complex functions instead of complex numbers.
- 13.
Note, however, that whilst \({\sigma }^{0} = \nVdash \) belongs to SL (2, \(\mathbb{C}\)) the Pauli matrices σ do not.
- 14.
Note the minus sign in front of σ ⋅ ∇ which is due to ∂ μ = (∂ 0, ∇ ) being the covariant derivative (C.24).
- 15.
More on representations of Dirac matrices can be found in [QP07], Sect. 9.1. For instance, in the description of neutrinos the class of Majorana representations is particularly relevant, i.e. the representations in which all γ-matrices are pure imaginary.
- 16.
We note, however, that quantum electrodynamics and other field theories can equivalently be formulated in the two-component formalism discussed above. This has been worked out, for the case of QED, by L. M. Brown, Proc. of Colorado Theor. Physics Institute, Colorado (1961). In some applications this formalism is simpler than the standard one, see also (Kersch et al., 1986).
- 17.
The theorem says that for any two sets of Dirac matrices which fulfill the anticommutation relations (1.75) there is a nonsingular matrix S which transforms one set into the other.
- 18.
See the discussion in Sect.3.2.1.
- 19.
In some textbooks the normalization of one-fermion states is taken to be p 0 ∕ m instead of 2p 0. Our normalization is the same for fermions and for bosons.
- 20.
This limiting procedure is only applicable for spin 1/2. For higher spin J > 1 ∕ 2 the limit m → 0 is discontinuous: Whilst a massive particle with spin J has (2J + 1) magnetic substates, a massless particle can have only two helicity states \(h = (\mathbf{\mathit{J}} \cdot \mathbf{\mathit{p}})/\vert \mathbf{\mathit{p}}\vert = \pm J\).
- 21.
Recall that complex conjugation converts small into capital, capital into small indices, (χB) ∗ = χ ∗ b, (ϕ a ) ∗ = ϕ A ∗ , and that indices are lowered and raised by means of ε-matrices.
- 22.
Note that here we apply charge conjugation C only to ψ(x), but not to the sources of the external fields. Had we done so, we would have found (1.100) to be invariant under C. The electromagnetic interaction is invariant under C.
- 23.
In a system of a finite number of fermions the hole theory is a prefectly consistent and useful approach. This is not so in a field theory with infinitely many degrees of freedom and in which genuine antiparticles occur.
- 24.
See Appendix A.
- 25.
Recall that C − 1 = − C, so that
$$\begin{array}{l} C{\phi }_{a}{C}^{-1} = \chi {{_\ast}}_{a},\quad \quad C{\chi }^{B}{C}^{-1} = -{\phi }^{{_\ast}B}, \\ {C}^{-1}\chi {{_\ast}}_{a}C = {\phi }_{a},\quad \quad {C}^{-1}{\phi }^{{_\ast}B}C = {\chi }^{B}\end{array}$$ - 26.
This is a special case of the famous spin-statistics theorem (Fierz 1938, Pauli 1940).
- 27.
See e.g. [SAK67].
- 28.
The most general case is treated below in Sect. 1.8.4.
- 29.
In fact the fields (1.122) are not well-behaved and, strictly speaking, we should smooth them out with appropriate weight functions.
- 30.
See e.g. [OMN70]
- 31.
This sphere is originally defined in the description of polarized or partially polarized light. The formalism describing polarized electromagnetic waves (or photons) is the same as for spin-1/2 particles. The real quantities ζ1, ζ2, ζ3 are called Stokes parameters in electrodynamics.
- 32.
We use the “slash” notation, \(\not \!\!{p} \equiv {a}^{\mu }{\gamma }_{\mu }\).
- 33.
- 34.
Normalization and sign in agreement with (1.93).
- 35.
It is interesting to remark that these considerations can be generalized to an arbitrary number of fields.
- 36.
- 37.
See e.g. [SCH68].
- 38.
Obviously, γ0 and γi cannot be even simultaneously.
- 39.
This is worked out in [GAS66].
- 40.
If G has several branches the formula (1.209) generates those elements which can be deformed continuously into the identity element.
- 41.
In fact, this ansatz for L int conserves parity and is not in accord with experiment. A more realistic description of β-decay contains γα(1 − γ5) instead of γα in both factors of (1.213), and \(\kappa = {G}_{\mathrm{F}}/\sqrt{2}\) with G F the Fermi constant, cf. Chap. 3.
- 42.
The nucleonic factor contains the matrix γ5 because the pion field is pseudoscalar, not scalar, with respect to Lorentz transformations and because the interaction is known to conserve parity which means that (1.214) must be even under P.
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Scheck, F. (2012). Fermion Fields and Their Properties. In: Electroweak and Strong Interactions. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20241-4_1
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