An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 3.4

The Dirac Field — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · majorana-fermions

Problem

Majorana fermions. Recall from Eq. (3.40) that one can write a relativistic equation for a massless 2-component fermion field that transforms as the upper two components of a Dirac spinor (ψL\psi_L). Call such a 2-component field χa(x)\chi_a(x), a=1,2a = 1, 2. (a) Show that it is possible to write an equation for χ(x)\chi(x) as a massive field in the following way: iσˉ⋅∂χ−imσ2χ∗=0.i\bar{\sigma}\cdot\partial\chi - im\sigma^2\chi^* = 0. That is, show, first, that this equation is relativistically invariant and, second, that it implies the Klein-Gordon equation, (∂2+m2)χ=0(\partial^2 + m^2)\chi = 0. This form of the fermion mass is called a Majorana mass term. (b) Does the Majorana equation follow from a Lagrangian? The mass term would seem to be the variation of (σ2)abχa∗χb∗(\sigma^2)_{ab}\chi_a^*\chi_b^*; however, since σ2\sigma^2 is antisymmetric, this expression would vanish if χ(x)\chi(x) were an ordinary c-number field. When we go to quantum field theory, we know that χ(x)\chi(x) will become an anticommuting quantum field. Therefore, it makes sense to develop its classical theory by considering χ(x)\chi(x) as a classical anticommuting field, that is, as a field that takes as values *Grassmann numbers* which satisfy αβ=−βαfor any α,β.\alpha\beta = -\beta\alpha \qquad \text{for any } \alpha, \beta. Note that this relation implies that α2=0\alpha^2 = 0. A Grassmann field ξ(x)\xi(x) can be expanded in a basis of functions as ξ(x)=∑nαnϕn(x),\xi(x) = \sum_n \alpha_n \phi_n(x), where the ϕn(x)\phi_n(x) are orthogonal c-number functions and the αn\alpha_n are a set of independent Grassmann numbers. Define the complex conjugate of a product of Grassmann numbers to reverse the order: (αβ)∗≡β∗α∗=−α∗β∗.(\alpha\beta)^* \equiv \beta^*\alpha^* = -\alpha^*\beta^*. This rule imitates the Hermitian conjugation of quantum fields. Show that the classical action, S=∫d4x[χ†iσˉ⋅∂χ+im2(χTσ2χ−χ†σ2χ∗)],S = \int d^4x \left[\chi^\dagger i\bar{\sigma}\cdot\partial\chi + \frac{im}{2}(\chi^T\sigma^2\chi - \chi^\dagger\sigma^2\chi^*)\right], (where χ†=(χ∗)T\chi^\dagger = (\chi^*)^T) is real (S∗=SS^* = S), and that varying this SS with respect to χ\chi and χ∗\chi^* yields the Majorana equation. (c) Let us write a 4-component Dirac field as ψ(x)=(ψLψR),\psi(x) = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}, and recall that the lower components of ψ\psi transform in a way equivalent by a unitary transformation to the complex conjugate of the representation ψL\psi_L. In this way, we can rewrite the 4-component Dirac field in terms of two 2-component spinors: ψL(x)=χ1(x),ψR(x)=iσ2χ2∗(x).\psi_L(x) = \chi_1(x), \qquad \psi_R(x) = i\sigma^2\chi_2^*(x). Rewrite the Dirac Lagrangian in terms of χ1\chi_1 and χ2\chi_2 and note the form of the mass term. (d) Show that the action of part (c) has a global symmetry. Compute the divergences of the currents Jμ=χ†σˉμχ,Jμ=χ1†σˉμχ1−χ2†σˉμχ2,J^\mu = \chi^\dagger\bar{\sigma}^\mu\chi, \qquad J^\mu = \chi_1^\dagger\bar{\sigma}^\mu\chi_1 - \chi_2^\dagger\bar{\sigma}^\mu\chi_2, for the theories of parts (b) and (c), respectively, and relate your results to the symmetries of these theories. Construct a theory of NN free massive 2-component fermion fields with O(N)O(N) symmetry (that is, the symmetry of rotations in an NN-dimensional space). (e) Quantize the Majorana theory of parts (a) and (b). That is, promote χ(x)\chi(x) to a quantum field satisfying the canonical anticommutation relation {χa(x),χb†(y)}=δabδ(3)(x−y),\{\chi_a(\mathbf{x}), \chi_b^\dagger(\mathbf{y})\} = \delta_{ab}\delta^{(3)}(\mathbf{x} - \mathbf{y}), construct a Hermitian Hamiltonian, and find a representation of the canonical commutation relations that diagonalizes the Hamiltonian in terms of a set of creation and annihilation operators. (Hint: Compare χ(x)\chi(x) to the top two components of the quantized Dirac field.)

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