47.2 Majorana spinors and the Majorana field

47.2 Majorana spinors and the Majorana field

Concept links · terms present in this machine draft; source roles are unverified: orthogonal group · group action

The analog for spin $\begin{array}{l}\frac{1}{2}\end{array}$ of the real scalar field is known as the Majorana spinor field, and can be constructed using a choice of real-valued matrices for the generators ${\gamma }_0,{\gamma }_j$ , acting on a four-component real-valued field $\psi .$ . Such a choice was given explicitly in section 41.2, and can be rewritten in terms of 2 by 2 block matrices, using the real matrices

$${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),i{\sigma }_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

as follows

$${\gamma }_0^M=\left(\begin{array}{cc}0& -i{\sigma }_2\\ -i{\sigma }_2& 0\end{array}\right),{\gamma }_1^M=\left(\begin{array}{cc}{\sigma }_3& 0\\ 0& {\sigma }_3\end{array}\right)$$ $${\gamma }_2^M=\left(\begin{array}{cc}0& i{\sigma }_2\\ -i{\sigma }_2& 0\end{array}\right),{\gamma }_3^M=\left(\begin{array}{cc}-{\sigma }_1& 0\\ 0& -{\sigma }_1\end{array}\right)$$

Quadratic combinations of Cliford generators have a basis

$${\gamma }_0^M{\gamma }_1^M=\left(\begin{array}{cc}0& {\sigma }_1\\ {\sigma }_1& 0\end{array}\right),{\gamma }_0^M{\gamma }_2^M=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right),{\gamma }_0^M{\gamma }_3^M=\left(\begin{array}{cc}0& {\sigma }_3\\ {\sigma }_3& 0\end{array}\right)$$ $${\gamma }_1^M{\gamma }_2^M=\left(\begin{array}{cc}0& {\sigma }_1\\ -{\sigma }_1& 0\end{array}\right),{\gamma }_2^M{\gamma }_3^M=\left(\begin{array}{cc}0& -{\sigma }_3\\ {\sigma }_3& 0\end{array}\right),{\gamma }_1^M{\gamma }_3^M=\left(\begin{array}{cc}-i{\sigma }_2& 0\\ 0& -i{\sigma }_2\end{array}\right)$$

and one has

$${\gamma }_5^M=i{\gamma }_0^M{\gamma }_1^M{\gamma }_2^M{\gamma }_3^M=\left(\begin{array}{cc}{\sigma }_2& 0\\ 0& -{\sigma }_2\end{array}\right)$$

The quantized Majorana field can be understood as an example of a quantization of a pseudo-classical fermionic oscillator system (as described in section 30.3.2), by the fermionic analog of the Bargmann-Fock quantization method (as described in section 31.3). We take as dual pseudo-classical phase space $V$ the real-valued solutions of the Dirac equation in the Majorana representation. Using values of the solutions at $t=0$ , continuous basis elements of V are given by the four component distributional field $\Psi (\mathbf{x})$ , with components ${\Psi }_a(\mathbf{x})$ for $a=1,2,3,4$

This space $V$ comes with an inner product

$$(\psi ,\varphi )={\int }_{{\mathbf{R}}^3}{\psi }^T(\mathbf{x})\varphi (\mathbf{x})d^3\mathbf{x}\text{\hspace{1em}(47.11)}$$

In this form the invariance under translations and under spatial rotations is manifest, with the $S(\Lambda )$ acting by orthogonal transformations on the Majorana spinors when $\Lambda$ is a rotation. One way to see this is to note that the $S(\Lambda )$ in this case are exponentials of linear combinations of the antisymmetric matrices ${\gamma }_1^M{\gamma }_2^M,{\gamma }_2^M{\gamma }_3^M,{\gamma }_1^M{\gamma }_3^M$ , and thus are orthogonal matrices.

As we have seen in chapter 30, the fermionic Poisson bracket of a pseudoclassical system is determined by an inner product, with the one above giving in this case

$$\{{\Psi }_a(\mathbf{x}),{\Psi }_b({\mathbf{x}}^{\prime }){\}}_{+}={\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime }){\delta }_{ab}$$

The Hamiltonian that will give a pseudo-classical system evolving according to the Dirac equation is

$$h=\frac{1}{2}{\int }_{{\mathbf{R}}^3}{\Psi }^T(\mathbf{x}){\gamma }_0(\gamma \cdot \nabla -m)\Psi (\mathbf{x})d^3\mathbf{x}$$

One can see this by noting that the operator ${\gamma }_0(\gamma \cdot \nabla -m)$ is minus its adjoint with respect to the inner product 47.11, since ${\gamma }_0$ is an antisymmetric matrix, the ${\gamma }_0\gamma$ are symmetric, and the derivative is antisymmetric. Applying the finite dimensional theorem 30.1 in this infinite dimensional context, one finds that the pseudo-classical equation of motion is

$$\frac{\partial }{\partial t}\Psi (\mathbf{x})=\{\Psi (\mathbf{x}),h{\}}_{+}={\gamma }_0(\gamma \cdot \nabla -m)\Psi (\mathbf{x})$$

which is the Dirac equation in Hamiltonian form (see equations 47.4 and 47.5). The antisymmetry of the operator ${\gamma }_0(\gamma \cdot \nabla -m)$ that generates time evolution corresponds to the fact that time evolution gives for each t an (infinite dimensional) orthogonal group action on the space of solutions $\nu ,$ preserving the inner product 47.11.

Corresponding to the Poincar´e group action 47.9 on solutions to the Dirac equation, at least for translations by a and rotations $\Lambda ,$ one has a corresponding action on the fields, written

$$\Psi (\mathbf{x})\to u(\mathbf{a},\Lambda )\Psi (\mathbf{x})=S(\Lambda {)}^{-1}\Psi (\Lambda \cdot \mathbf{x}+\mathbf{a})$$

The quadratic pseudo-classical moment map that generates the action of spatial translations on solutions is the momentum

$$\mathbf{P}=-\frac{1}{2}{\int }_{{\mathbf{R}}^3}{\Psi }^T(\mathbf{x})\nabla \Psi (\mathbf{x})d^3\mathbf{x}$$

since it satisfies (generalizing equation 30.1)

$$\{\mathbf{P},\Psi (\mathbf{x}){\}}_{+}=\nabla \Psi (\mathbf{x})$$

For rotations, the moment map is the angular momentum

$$\mathbf{J}=-\frac{1}{2}{\int }_{{\mathbf{R}}^3}{\Psi }^T(\mathbf{x})(\mathbf{x}\times \nabla -\mathbf{s})\Psi (\mathbf{x})d^3\mathbf{x}$$

which satisfies

$$\{\mathbf{J},\mathbf{\Psi }(\mathbf{x}){\}}_{+}=(\mathbf{x}\times \nabla -\mathbf{s})\mathbf{\Psi }(\mathbf{x})$$

Here the components $s_j$ of s are the matrices

$$s_j=\frac{1}{2}{ϵ}_{jkl}{\gamma }_k{\gamma }_l$$

Our use here of the fixed-time fields $\Psi (\mathbf{x})$ as continuous basis elements on the phase space $\nu$ comes with two problematic features:

• One cannot easily implement Lorentz transformations that are boosts, since these change the fixed-time hypersurface used to define the $\Psi (\mathbf{x})$

• The relativistic complex structure on $V$ needed for a consistent quantization is defined by a splitting of ${\nu }_{\otimes }\mathbf{C}$ into positive and negative energy solutions, but this decomposition is only easily made in momentum space, not position space.

47.2.1 Majorana spinor fields in momentum space

Concept links · terms present in this machine draft; source roles are unverified: complexification

Recall that in the case of the real relativistic scalar field studied in chapter 43 we had the following expression (equation 43.8) for a solution to the Klein-Gordon equation

$$\varphi (t,\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}(\alpha (\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+\overline{\alpha }(\mathbf{p})e^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}$$

with $\alpha (\mathbf{p})$ and $\overline{\alpha }(\mathbf{p})$ parametrizing positive and negative energy subspaces of the complexified phase space $\mathcal{M}\otimes \mathbf{C}$ . This was quantized by an infinite dimensional version of the Bargmann-Fock quantization described in chapter 26, with dual phase space $\mathcal{M}$ the space of real-valued solutions of the Klein-Gordon equation, and the complex structure $J_r$ the relativistic one discussed in section 43.2.

For the Majorana theory, one can write four-component Majorana spinor solutions to the Dirac equation as

$$\psi (t,\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}(\alpha (\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+\overline{\alpha }(\mathbf{p})e^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}\text{\hspace{1em}(47.12)}$$

where $\alpha (\mathbf{p})$ is a four-component complex vector, satisfying

$$-{\gamma }_0^M({\gamma }^M\cdot \mathbf{p}+im)\alpha (\mathbf{p})={\omega }_{\mathbf{p}}\alpha (\mathbf{p})\text{\hspace{1em}(47.13)}$$

These $\alpha (\mathbf{p})$ are the positive energy solutions ${\tilde{\psi }}_{+}(\mathbf{p})$ of 47.6, with the conjugate equation for $\overline{\alpha }(\mathbf{p})$ giving the negative energy solutions of 47.7 (with the sign of $\mathbf{p}$ interchanged, $\overline{\alpha }(\mathbf{p})={\tilde{\psi }}_{-}(-\mathbf{p}))$ .

For each $\mathbf{p},$ there is a two dimensional space of solutions to equation 47.13. One way to choose a basis $u_{+}(\mathbf{p}),u_{-}(\mathbf{p})$ of this space is by first considering the case $\mathbf{p}=0$ . The equation 47.13 becomes

$${\gamma }_0^M\alpha (0)=i\alpha (0)$$

which will have a basis of solutions

$$u_{+}(0)=\frac{1}{2}\left(\begin{array}{c}1\\ 0\\ 0\\ i\end{array}\right),u_{-}(0)=\frac{1}{2}\left(\begin{array}{c}0\\ 1\\ -i\\ 0\end{array}\right)$$

These will satisfy

$$-\frac{i}{2}{\gamma }_1^M{\gamma }_2^M{u}_{+}(0)=\frac{1}{2}{u}_{+}(0),-\frac{i}{2}{\gamma }_1^M{\gamma }_2^M{u}_{-}(0)=-\frac{1}{2}{u}_{-}(0)$$

The two solutions

$$u_{+}(0)e^{-imt}+\overline{u_{+}(0)}{e}^{imt}=\left(\begin{array}{c}\cos (mt)\\ 0\\ 0\\ \sin (mt)\end{array}\right)$$ $$u_{-}(0)e^{-imt}+\overline{u_{-}(0)}{e}^{imt}=\left(\begin{array}{c}0\\ \cos (mt)\\ -\sin (mt)\\ 0\end{array}\right)$$

correspond physically to a relativistic spin $\begin{array}{l}\frac{1}{2}\end{array}$ particle of mass m at rest, with the first having spin ${}^{\bullet }{\mathrm{up}}^{\bullet }$ in the 3-direction, the second spin “down”.

The Majorana spinor field theory comes with a significant complication with respect to the case of scalar fields. The complex four-component $\alpha (\mathbf{p})$ provide twice as many basis elements as one needs to describe the solutions of the Dirac equation (put diferently, they are not independent, but satisfy the relation 47.13). Quantizing using four sets of annihilation and creation operators (one for each component of $\alpha$) would produce a quantum field theory with too many degrees of freedom by a factor of two. The standard solution to this problem is to make a choice of basis elements of the space of solutions for each value of $\mathbf{p}$ by defining polarization vectors

$$u_{+}(\mathbf{p})=L(\mathbf{p})u_{+}(0),u_{-}(\mathbf{p})=L(\mathbf{p})u_{-}(0)$$

Here $L(\mathbf{p})$ is an element of $SL(2,\mathbf{C})$ chosen so that, acting by a Lorentz transformation on energy-momentum vectors it takes $(m,0)$ to $({\omega }_{\mathbf{p}},\mathbf{p})$ . More explicitly, using equation 40.4, one has

$$L(\mathbf{p})\left(\begin{array}{cc}m& 0\\ 0& m\end{array}\right)L^{†}(\mathbf{p})=\left(\begin{array}{cc}{\omega }_{\mathbf{p}}+p_3& p_1-i{p}_2\\ p_1+i{p}_2& {\omega }_{\mathbf{p}}-p_3\end{array}\right)$$

Such a choice is not unique and is a matter of convention. Explicit choices are discussed in most quantum field theory textbooks (although in a diferent representation of the $\gamma$-matrices), see for instance chapter 3.3 of [67]. Note that these polarization vectors are not the same as the Bloch sphere polarization vectors used in earlier chapters. They are defined on the positive mass hyperboloid, not on the sphere, and for these there is no topological obstruction to a continuous definition.

Solutions are then written as

$$\psi (t,\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}\sum _{s=\pm }({\alpha }_s(\mathbf{p})u_s(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+{\overline{\alpha }}_s(\mathbf{p})\overline{u_s(\mathbf{p})}{e}^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}\text{\hspace{1em}(47.14)}$$

One now has the correct number of functions to parametrize pseudo-classical complexified dual phase space ${\nu }_{\otimes }\mathbf{C}$ . These are the single-component complex functions ${\alpha }_{+}(\mathbf{p}),{\alpha }_{-}(\mathbf{p})$ , providing elements of ${\mathcal{V}}_{J_r}^{+}={\mathcal{H}}_1$ and their conjugates ${\overline{\alpha }}_{+}(\mathbf{p}),{\overline{\alpha }}_{-}(\mathbf{p})$ , which provide elements of ${\gamma }_{J_r}^{-}$

To quantize the Majorana field in a way that allows a simple understanding of the action of the full Poincar´e group, we need a positive-definite Poincar´e invariant inner product on the space of solutions of the Dirac equation. We have already seen what the right inner product is (see equation 47.11), but unfortunately this is not written in a way that makes Lorentz invariance manifest. Unlike the case of the Klein-Gordon equation, working in momentum space does not completely resolve the problem. Using the ${\alpha }_{+}(\mathbf{p}),{\alpha }_{-}(\mathbf{p})$ allows for an explicitly positive-definite inner product, which is just two copies of the Klein-Gordon one for scalars, see equation 43.18. In the next section we will quantize the theory using these. This inner product is not however manifestly Lorentz invariant (due to the dependence on the choice of polarization vectors $u_{\pm }(\mathbf{p}))$ ).

47.2.2 Quantization of the Majorana field

Concept links · terms present in this machine draft; source roles are unverified: charge operator · symmetry group

Quantization of the dual pseudo-classical phase space $M$, using the fermionic Bargmann-Fock method, the relativistic complex structure, and the functions ${\alpha }_{\pm }(\mathbf{p})$ from equation 47.14 is given by annihilation and creation operators $a_{\pm }(\mathbf{p}),a_{\pm }^{†}(\mathbf{p})$ that anticommute, except for the relations

$$[a_{+}(\mathbf{p}),a_{+}^{†}({\mathbf{p}}^{\prime }){]}_{+}={\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime }),[a_{-}(\mathbf{p}),a_{-}^{†}({\mathbf{p}}^{\prime }){]}_{+}={\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$$

The field operator is then constructed using these, giving

Definition (Majorana field operator)

The Majorana field operator is given by

$$\hat{\Psi }(\mathbf{x},t)=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}\sum _{s=\pm }(a_s(\mathbf{p})u_s(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+a_s^{†}(\mathbf{p})\overline{u_s(\mathbf{p})}{e}^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}$$

If one uses commutation instead of anticommutation relations, the Hamiltonian operator will have eigenstates with arbitrarily negative energy, and there will be problems with causality due to observable operators at space-like separated points not commuting. These two problems are resolved by the use of anticommutation instead of commutation relations. The multi-particle state space for the theory has occupation numbers 0 or 1 for each value of p and for each value of $s=\pm$ . Like the case of the real scalar field, the particles described by these states are their own antiparticles. Unlike the case of the real scalar field, each particle state has a ${\mathbf{C}}^2$ degree of freedom corresponding to its spin $\begin{array}{l}\frac{1}{2}\end{array}$ nature.

One can show that the Hamiltonian and momentum operators are given by

$$\hat{H}={\int }_{{\mathbf{R}}^3}{\omega }_{\mathbf{p}}(a_{+}^{†}(\mathbf{p})a_{+}(\mathbf{p})+a_{-}^{†}(\mathbf{p})a_{-}(\mathbf{p}))d^3\mathbf{p}$$

and

$$\hat{\mathbf{P}}={\int }_{{\mathbf{R}}^3}\mathbf{p}(a_{+}^{†}(\mathbf{p})a_{+}(\mathbf{p})+a_{-}^{†}(\mathbf{p})a_{-}(\mathbf{p}))d^3\mathbf{p}$$

The angular momentum and boost operators are much more complicated to describe, again due to the dependence of the ${\alpha }_{\pm }(\mathbf{p})$ on a choice of polarization vectors $u_{\pm }(\mathbf{p})$

Note that, as in the case of the real scalar field, the theory of a single Majorana field has no internal symmetry group acting, so no way to introduce a charge operator and couple the theory to electromagnetic fields.

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