Quantum field theory 1, lecture 20
Charge conjugation
Consider a Dirac spinor and its conjugate as in eq. \(\eqref{eq:DiracSpinor}\). The charge conjugate spinors are defined as \[\Psi_\mathsf{C} = \begin{pmatrix} \xi_a \\ \bar \psi^{\dot a} \end{pmatrix}, \quad \quad \bar\Psi_\mathsf{C} = \left( \psi^a, \bar \xi_{\dot a} \right).\] A Majorna spinor is a Dirac spinor which obeys \(\psi_\mathsf{C} = \psi\) or, in terms of Weyl spinors, \(\xi_a = \psi_a\) and \(\bar\psi^{\dot a} = \bar \xi^{\dot a}\). In other words, we can write the four component Majorana spinor as \[\Psi = \Psi_\mathsf{C}= \begin{pmatrix} \psi_a \\ \bar \psi^{\dot a} \end{pmatrix}.\] The relation between Dirac spinors and Majorana spinors is as the relation between a complex and a real scalar field. As one can construct complex scalar fields out of two real fields, one can construct Dirac spinors out of Majorana spinors (exercise). Note that a Majorana spinor has formally as many real degrees of freedom as a Weyl spinor.
The charge conjugate fields can be written as \[\Psi_\mathsf{C} = \mathscr{C} \bar \Psi^T, \quad\quad\quad \bar \Psi_\mathsf{C} =- \Psi^T \mathscr{C}^{-1}, \label{eq:DefChargeConjugateDiracField}\] with the transpose spinors \[\bar \Psi^T = \begin{pmatrix} \xi^a \\ \bar \psi_{\dot a} \end{pmatrix}, \quad\quad\quad \Psi^T=\left(\psi_a, \bar\xi^{\dot a}\right)\] and the charge conjugation matrix1 \[\mathscr{C} = \begin{pmatrix} \varepsilon_{ab} && \\ && \varepsilon^{\dot a\dot b} \end{pmatrix}, \quad \quad \quad \mathscr{C}^{-1}= \begin{pmatrix} \varepsilon^{ab} && \\ && \varepsilon_{\dot a\dot b} \end{pmatrix}.\] As a matrix, \(\mathscr{C}\) obeys \[\mathscr{C} = - \mathscr{C}^{-1} = - \mathscr{C}^\dagger = - \mathscr{C}^T = \mathscr{C}^*= \begin{pmatrix} & -1 & & \\ 1 & & & \\ & & & 1\\ & & -1 & \end{pmatrix} .\] One has also \[\begin{split} \mathscr{C}^{-1} \gamma^\mu \mathscr{C} = & \begin{pmatrix} \varepsilon^{ab} && \\ && \varepsilon_{\dot a \dot b} \end{pmatrix} \begin{pmatrix} && -i (\sigma^\mu)_{b \dot c} \\ -i (\bar \sigma^\mu)^{\dot b c} && \end{pmatrix} \begin{pmatrix} \varepsilon_{cd} && \\ && \varepsilon^{\dot c \dot d} \end{pmatrix}\\ = & \begin{pmatrix} && -i \varepsilon^{ab} (\sigma^\mu)_{b \dot c} \varepsilon^{\dot c \dot d} \\ -i \varepsilon_{\dot a \dot b} (\bar \sigma^\mu)^{\dot b c} \varepsilon_{cd} \end{pmatrix} \\ = & \begin{pmatrix} && i (\bar \sigma^\mu)^{\dot d a} \\ i (\sigma^\mu)_{d \dot a} \end{pmatrix}, \end{split}\] or, in matrix notation \[\mathscr{C}^{-1} \gamma^\mu \mathscr{C} = - (\gamma^\mu)^T. \label{eq:C1}\] Similarly, \[\begin{split} \mathscr{C}^{-1} \sigma^{\mu\nu} \mathscr{C} & = - (\sigma^{\mu\nu})^T, \\ \mathscr{C}^{-1} \gamma_5 \mathscr{C} & = (\gamma_5)^T,\\ \mathscr{C}^{-1} \gamma^5 \gamma^\mu \mathscr{C} & = (\gamma^5 \gamma^\mu)^T. \end{split} \label{eq:C2}\] The index structure in \(\eqref{eq:C1}\) and \(\eqref{eq:C2}\) is appropriate for transposed spinors.
Parity
Parity transforms the coordinates as \((t,\vec x) \to (t, -\vec x)\). For a Dirac spinor and its conjugate as in \(\eqref{eq:DiracSpinor}\) one defines the parity transformed spinors \[\Psi_\mathsf{P} = \begin{pmatrix} i \bar\xi^{\dot a} \\ i \psi_a \end{pmatrix}, \quad \quad \quad \bar \Psi_\mathsf{P} = \left( i \bar \psi_{\dot a}, i\xi^{a} \right). \label{eq:ParityTransformedSpinors}\] Note that the role of right-handed and left-handed spinors (and their corresponding indices) is interchanged for the parity-transformed field. Eq. \(\eqref{eq:ParityTransformedSpinors}\) can also be written as \[\Psi_\mathsf{P} = i \beta \Psi \quad\quad\quad \bar\Psi_\mathsf{P} = - i \bar \Psi \beta^{-1},\] with the matrix \[\beta = \begin{pmatrix} && \delta^{\dot a}_{\;\;\dot b} \\ \delta^{\;\;b}_a && \end{pmatrix}, \quad\quad\quad \beta^{-1} = \begin{pmatrix} && \delta_a^{\;\;b} \\ \delta^{\dot a}_{\;\;\dot b} && \end{pmatrix}.\] As a matrix, \(\beta\) obeys \[\beta=\beta^{-1} = \beta^\dagger = \beta^T = \beta^*= \begin{pmatrix} && \mathbb{1} \\ \mathbb{1} && \end{pmatrix}.\] Parity transformations of the gamma matrices are given by \[\begin{split} \beta \gamma^j \beta^{-1} & = - \gamma^j,\\ \beta \gamma^0 \beta^{-1} & = \gamma^0. \end{split} \label{eq:ParityTransformedGamma}\] The gamma matrices on the right hand side of \(\eqref{eq:ParityTransformedGamma}\) agree with \(\eqref{eq:GammaMatrices}\) as matrices but have a different index structure such that they fit to the spinors in \(\eqref{eq:ParityTransformedSpinors}\), \[\beta \gamma^\mu \beta^{-1} = \begin{pmatrix} && -i (\bar \sigma^\mu)^{\dot a b} \\ -i (\sigma^\mu)_{a\dot b} && \end{pmatrix}.\] When doing a parity transform of an expression (e.g. a Lagrangian) and replacing spinors with parity transformed spinors, one should also replace gamma matrices by the expressions in \(\eqref{eq:ParityTransformedGamma}\). Similarly for the antisymmetric matrices \[\begin{split} \beta \sigma^{ij} \beta^{-1} & = \sigma^{ij},\\ \beta \sigma^{j0} \beta^{-1} & = - \sigma^{j0}. \end{split}\] The matrix \(\gamma_5\) is a pseudoscalar in the sense \[\beta\gamma_5 \beta^{-1} = - \gamma_5.\]
Time reversal
Time reversal changes the time direction, \((t,\vec x) \to (-t,\vec x)\). It is also a anti-unitary transformation that transforms all complex numbers to there complex conjugates. The time-reversed version of the Dirac spinor and its conjugate as in \(\eqref{eq:DiracSpinor}\) is given by \[\Psi_\mathsf{T} = \begin{pmatrix} \psi^a \\ - \bar\xi_{\dot a} \end{pmatrix}, \quad\quad\quad \bar \Psi_\mathsf{T} = \left( - \xi_a, \bar \psi^{\dot a} \right). \label{eq:TimeTransformedSpinors}\] Note that the role of upper and lower indices has been interchanged. With the matrices \(\mathscr{C}\) and \(\gamma_5\) one can write this as \[\Psi_\mathsf{T} = \mathscr{C}^{-1} \gamma_5 \Psi, \quad\quad\quad \bar \Psi_\mathsf{T} = \bar \Psi \gamma_5 \mathscr{C}.\] When considering time-reversal transformations, the following identities are useful \[\begin{split} \mathscr{C}^{-1} \gamma^0 \mathscr{C}^{-1} & = (\gamma^0)^*\\ \mathscr{C}^{-1} \gamma^j \mathscr{C}^{-1} & = - (\gamma^j)^*\\ \mathscr{C}^{-1} \gamma_5 \mathscr{C}^{-1} & = (\gamma_5)^*\\ \mathscr{C}^{-1} \sigma^{ij} \mathscr{C}^{-1} & = - (\sigma^{ij})^*\\ \mathscr{C}^{-1} \sigma^{j0} \mathscr{C}^{-1} & = (\sigma^{j0})^*. \end{split}\] Similarly as in the case of parity, the index structure of these expressions is such that it fits to eq. \(\eqref{eq:TimeTransformedSpinors}\). In other words, one should use these expressions for \((\gamma^\mu)^*\) ect. in connection with time-reversed spinors.
Grassmann property of fields and index-free notation
Define products of left-handed two-component Weyl spinors as \[\chi \psi = \chi^a \psi_a,\] and similarly for right handed spinors, \[\bar\chi \bar\psi = \bar\chi_{\dot a} \bar\psi^{\dot a}.\] In other words, contracted indices that are not written should be interpreted as \(^c_{\,\,c}\) for left-handed spinors and \(_{\dot c}^{\;\;\dot c}\) for right-handed spinors. Since \(\chi\) and \(\psi\) are Grassmann valued their components anti-commute, e. g. \[\chi \psi = \chi^a \psi_a = - \psi_a \chi^a = \psi^a \chi_a = \psi \chi.\] So with this notation one has \(\chi \psi = \psi \chi\) and similarly \(\bar\chi \bar\psi = \bar\psi \bar\chi\) etc.
Hermitian conjugation includes also a commutation of Grassmann numbers, e. g. \[(\chi \psi)^\dagger = \left( \chi^a \psi_a \right)^\dagger = \left[ \psi_a \right]^\dagger \left[ \chi^a \right]^\dagger = (\psi^\dagger)_{\dot a} (\chi^\dagger)^{\dot a} = \psi^\dagger \chi^\dagger.\] Another example for manipulating spinor indices is \[\begin{split} \bar \psi \bar \sigma^\mu \chi & = \bar \psi_{\dot a} (\bar \sigma^\mu)^{\dot a b} \chi_b = - \chi_{b} \left[ \epsilon^{\dot a \dot c} \epsilon^{b d}(\sigma^\mu)_{d \dot c} \right] \bar \psi_{\dot a}\\ & = - \chi^{d} (\sigma^\mu)_{d \dot c} \bar \psi^{\dot c} = - \chi \sigma^\mu \bar \psi. \end{split} \label{eq:spionrIndexManipulation}\] The minus sign in the second equation is due to the Grassmann property and the interchange of spinors.
Lagrangian for Weyl fermions
A Lagrangian for a left-handed, two-component Weyl fermion \(\psi_a\) and its right-handed hermitean conjugate \(\bar \psi_{\dot a}=\psi^\dagger_{\dot a}\) in Minkowski space can be written as \[\begin{split} \mathscr{L} = & i \bar\psi_{\dot a} (\bar \sigma^\mu)^{\dot a b} \partial_\mu \psi_b - \frac{1}{2}m \psi^a \psi_a - \frac{1}{2} m \bar\psi_{\dot a} \bar\psi^{\dot a} = i \bar\psi \bar \sigma^\mu \partial_\mu \psi - \frac{1}{2} m \psi \psi - \frac{1}{2}m \bar\psi \bar \psi. \end{split} \label{eq:LagrangianLeftWeyl}\] In the second equation we used the short hand notation introduced above, keeping in mind that we deal here with two-component spinors.
The two mass terms go into each other under hermitian conjugation assuming real \(m\). More general one could allow them to have complex conjugate masses \(m\) and \(m^*\) respectively, but the complex phase of \(m=e^{i\beta} |m|\) can be absorbed into a redefinition \(\psi\to e^{-i\beta/2}\psi\), \(\bar\psi\to e^{i\beta/2}\bar\psi\), so that real \(m>0\) can be assumed without loss of generality.
On the other side, if the theory is supposed to be invariant under the U(1) symmetry \(\psi\to e^{i\alpha}\psi\), \(\bar\psi \to e^{-i \alpha} \bar\psi\), a mass term as in \(\eqref{eq:LagrangianLeftWeyl}\) is actually excluded. In other words, such a symmetry would only be unbroken for \(m=0\).
Let us also consider the hermitian conjugate of the first, kinetic term in the Lagrangian, \[\begin{split} \left[ i \bar\psi_{\dot a} (\bar\sigma^\mu)^{\dot a b} \partial_\mu \psi_b \right]^\dagger & = -i \partial_\mu [\psi_b]^\dagger \left[ (\bar\sigma^\mu)^{\dot a b} \right]^* [\bar \psi_{\dot a}]^\dagger = -i \partial_\mu [\psi_b]^\dagger (\bar\sigma^\mu)^{b\dot a} [\bar \psi_{\dot a}]^\dagger = -i \partial_\mu \bar\psi_{\dot b} (\bar\sigma^\mu)^{\dot b a} \psi_{a} \\ & = i \bar\psi_{\dot b} (\bar\sigma^\mu)^{\dot b a} \partial_\mu \psi_{a} - \partial_\mu \left[ i \bar\psi_{\dot b} (\bar\sigma^\mu)^{\dot b a} \psi_{a} \right]. \end{split}\] The last term on the right is a total derivative and contributes only an irrelvant boundary term in the action. This shows that the kinetic term in also hermitian. Note also that using \(\eqref{eq:spionrIndexManipulation}\) with \(\chi = \partial_\mu \psi\) and droping another boundary term from partial integration, one can also bring the kinetic term into the form \[i \psi \sigma^\mu \partial_\mu \bar \psi.\] This shows that \(\eqref{eq:LagrangianLeftWeyl}\) is as well a Lagrangian for the left-handed field \(\psi_a\) as for the right-handed field \(\bar \psi^{\dot a}\).
Lagrangian for Majorana fermions
We can now also write down directly a Lagrangian for four-component Majorana spionors, \[\Psi = \Psi_\mathsf{C}= \begin{pmatrix} \psi_a \\ \bar \psi^{\dot a} \end{pmatrix}, \quad\quad\quad \bar\Psi = \bar\Psi_\mathsf{C} = \begin{pmatrix} \psi^a, \bar \psi_{\dot a} \end{pmatrix}.\] In fact, the Lagrangian \(\eqref{eq:LagrangianLeftWeyl}\) can be rewritten in terms of a Majorana spinor \(\Psi\) as \[\mathscr{L} = - \frac{1}{2}\bar \Psi_\mathsf{C} \gamma^\mu \partial_\mu \Psi - \frac{1}{2}m \bar \Psi_\mathsf{C} \Psi = \frac{1}{2} \Psi^T \mathscr{C}^{-1} \gamma^\mu \partial_\mu \Psi + \frac{1}{2}m \Psi^T \mathscr{C}^{-1} \Psi. \label{eq:MajoranaLagrangian}\] In the second equation we wrote everything in terms of the spinor \(\Psi\) only, using \(\eqref{eq:DefChargeConjugateDiracField}\), to make explicit that there is only one independant spionor field here.
Lagrangian for Dirac fermions
Dirac fermions represented by the spinors \[\Psi = \begin{pmatrix} \psi_b \\ \bar\xi^{\dot b} \end{pmatrix}, \quad\quad\quad \bar \Psi = \begin{pmatrix} \xi^b, \bar \psi_{\dot b} \end{pmatrix},\] are charged fermions. This means one can do a U(1) transformation \[\Psi \to e^{i \alpha} \Psi, \quad\quad\quad \bar\Psi \to \bar \Psi e^{-i\alpha},\] or \[\psi_b \to e^{i\alpha} \psi_b, \quad\quad\quad \bar \xi^{\dot b} \to e^{i\alpha}\bar \xi^{\dot b}, \quad\quad\quad \xi^b \to e^{-i\alpha} \xi^b, \quad\quad\quad \bar \psi_{\dot b} \to e^{-i\alpha} \bar \psi_{\dot b}. \label{eq:transformElectricU1DiracCompo}\] This implies that a mass term as for the Majorana fermions is not allowed. However, for Dirac spinors, which have twice as many degrees of freedom as the Weyl or Majorana fermions, it is possible to include a differnt kind of mass term, involving the combination \[- m\bar \Psi \Psi = - m \left[\xi^a \psi_a + \bar \psi_{\dot a} \bar \xi^{\dot a} \right]. \label{eq:DiracMassTerm}\] Note that this mixes the spinors \(\psi\) and \(\xi\) and is only allowed because they have opposite charges under the U(1) transformation in eq. \(\eqref{eq:transformElectricU1DiracCompo}\). In turn a mass term as in \(\eqref{eq:DiracMassTerm}\) actually breaks another U(1) symmetry, the so-called chiral or axial symmetry \[\psi_b \to e^{i\beta} \psi_b, \quad\quad\quad \bar \xi^{\dot b} \to e^{-i\beta}\bar \xi^{\dot b}, \quad\quad\quad \xi^b \to e^{i\beta} \xi^b, \quad\quad\quad \bar \psi_{\dot b} \to e^{-i\beta} \bar \psi_{\dot b}.\] In terms of gamma five this can be written as \[\Psi \to e^{i \beta \gamma_5} \Psi, \quad\quad\quad \bar\Psi \to \bar \Psi e^{i\beta \gamma_5}.\] Indeed, the typical Lagrangian for charged massive Dirac fermions is given by \[\begin{split} \mathscr{L} & = - \bar \Psi \gamma^\mu \partial_\mu \Psi - m \bar\Psi \Psi \\ & = i \bar\psi_{\dot a} (\bar \sigma^\mu)^{\dot a b} \partial_\mu \psi_b + i \xi^a (\sigma^\mu)_{a\dot b} \partial_\mu \bar \xi^{\dot b} - m [\xi^a \psi_a + \bar \psi_{\dot a} \bar \xi^{\dot a}]. \end{split}\]
Dirac equation
Variation with respect to \(\bar\Psi\) yields the Dirac equation, \[(\gamma^\mu \partial_\mu +m) \Psi =0.\] The equation of motion following from the Majorana Lagrangian \(\eqref{eq:MajoranaLagrangian}\) would actually be of the same form, but it would be for a constrained or “real” Majorana spinor and not for an unconstrained or “complex” Dirac spinor.
Relation to Klein-Gordon equation
It is interesting to apply another derivative operator to the Dirac equation, \[(-\gamma^\mu \partial_\mu +m)(\gamma^\nu \partial_\nu +m) \Psi =0.\] Here one can replace \[\gamma^\mu \gamma^\nu \partial_\mu \partial_\nu = \frac{1}{2}\{ \gamma^\mu, \gamma^\nu \} \partial_\mu \partial_\nu = \eta^{\mu\nu} \partial_\mu\partial_\nu,\] because the partial derivatives commute. This leads to \[(- \eta^{\mu\nu}\partial_\mu\partial_\nu+m^2) \Psi = 0,\] which shows that all components of the Dirac spionor that solves the free Dirac equation are also solutions to the Klein-Gordon equation. This implies also that one can solve the free Dirac equation in terms of plane waves.
This is the convention of Srednicki↩︎