Quantum field theory 1, lecture 21

Partition function for Dirac fermions

We can now also write down a partition function for free Dirac fermions in the form of a Grassmann functional integral, \[Z_2[\bar \eta, \eta] = \int D\bar\Psi D \Psi \exp\left( i \int d^4x \left\{ - \bar \Psi \gamma^\mu \partial_\mu \Psi - m \bar\Psi \Psi \right\} + i \int d^4 x \left\{ \bar \eta \Psi + \bar \Psi \eta \right\} \right).\] The fields \(\Psi\) and \(\bar\Psi\), as well as the sources \(\eta\) and \(\bar \eta\) are here Grassmann valued fields which also have the structure of four-component Dirac spinors. For example, \[\bar \eta \Psi = \sum_{\alpha=1}^4 \bar\eta_\alpha \Psi_\alpha.\] As usual one can now determine correlation functions by taking functional derivatives, for example \[\langle \Psi_\alpha(x) \bar \Psi_\beta(y) \rangle = \frac{1}{Z_2[\bar\eta, \eta]} \left( \frac{1}{i} \frac{\delta}{\delta \bar\eta_\alpha(x)} \right) \left( i \frac{\delta}{\delta \eta_\beta(y)} \right) Z_2[\bar\eta, \eta] = \frac{1}{i}S_{\alpha\beta}(x-y).\] The signs take here the Grassmann properties into account.

Feynmann propagator for Dirac fermions

As usual, it is possible to perform the Gaussian integral by completing the square, \[Z_2[\bar \eta, \eta] = \exp\left( i \int d^4x \left\{ \bar \eta_\alpha(x) S_{\alpha\beta}(x-y) \eta_\beta(y) \right\} \right),\] where the propagator or time-ordered Greens function \(S_{\alpha\beta}(x-y)\) is defined such that \[\left( \gamma^\mu \frac{\partial}{\partial x^\mu} + m \right)_{\alpha\kappa} S_{\kappa\beta}(x-y) = \delta_{\alpha\beta}\delta^{(4)}(x-y).\] This opertor inversion can be done conveniently in Fourier space, \[\begin{split} S_{\alpha\beta}(x-y) &= \int \frac{d^4 p}{(2\pi)^4} \;e^{ip(x-y)} (i\slashed{p} +m)^{-1}_{\alpha\beta}\\ &= \int \frac{d^4 p}{(2\pi)^4} \; e^{ip(x-y)} \frac{(-i\slashed{p} + m)_{\alpha\beta}}{p^2 + m^2 -i\epsilon}. \end{split}\] We have used here that \[(-i \slashed{p} + m) (i \slashed{p} + m) = \slashed{p}\slashed{p} + m^2 = \gamma^\mu \gamma^\nu p_\mu p_\nu + m^2 = \frac{1}{2} \{ \gamma^\mu, \gamma^\nu \} p_\mu p_\nu + m^2 = p^2 + m^2,\] and have inserted the usual \(i\epsilon\) term to ensure the right causality properties for a Feynmann propagator.

Coupling to gauge fields

We can now also write down the Lagrangian for Dirac fermions coupled to the electromagnetic gauge field \(A_\mu\), \[\mathscr{L} = - \bar \Psi \gamma^\mu (\partial_\mu - i e A_\mu) \Psi - m \bar \Psi \Psi.\] This is invariant under the local U(1) gauge transformation \[\Psi \to e^{i\alpha(x)} \Psi, \quad\quad\quad \bar \Psi \to \bar \Psi e^{-i\alpha(x)}, \quad\quad\quad A_\mu(x) \to A_\mu(x) + \frac{1}{e}\partial_\mu \alpha(x).\]

Lagrangian for Quantum electrodynamics

We can now also write down a Lagrangian for quantum electrodynamics in complete form after adding a kinetic term for the gauge fields, \[\mathscr{L} = - \bar \Psi \gamma^\mu (\partial_\mu - i e A_\mu) \Psi - m \bar \Psi \Psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}.\] We use here the electromagnetic field strength tensor \[F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.\] It is obviously anti-symmetric and invariant under the U(1) gauge transformations introduced above. Accordingly entire Lagrangian for quantum electrodynamics (QED) is gauge invariant.

Poincaré group, fields and particles

We have seen that quantum fields build representations of the Lorentz group. We have specifically investigated scalar and spinor fields, but will soon also turn to vector fields. On the other side, we have seen that excitations of fields in the asymptotic regimes, where they propagate over large distances, can be understood as particles. In this section we will investigate the relation further using spacetime symmetries.

Asymptotic states and the Poincaré group

Asymptotic states, that describe freely propagating particles, correspond to fields correlated over large distances in spacetime. Formally they can be associated to poles in propagators or correlation functions in momentum space. These asymptotic regions are governed by a set of symmetries, and it was shown by Eugene Wigner that one can use them to characterize the properties of particles.

Transformations of fields

So far we have discussed how the “internal” indices of a field transform under Lorentz transformations. However, a field depends on a space-time position \(x^\mu\) which also transforms. This is already the case for a scalar field, \[\phi (x) \to \phi^\prime (x) = \phi (\Lambda^{-1} x).\] (A maximum at \(x^\mu\) is moved to a maximum at \(\Lambda_{\;\; \nu}^\mu x^\nu\).) In infinitesimal form \[(\Lambda^{-1})_{\;\; \nu}^\mu = \delta_{\;\; \nu}^\mu - \delta \omega_{\;\; \nu}^\mu,\] and thus \[\phi (x) \to \phi^\prime (x) = \phi (x) - x^\nu \delta \omega_{\;\; \nu}^\mu \partial_\mu \phi (x) .\] This can also be written as \[\phi^\prime (x) = \left(1 + \frac{i}{2} \delta \omega^{\mu \nu} \mathcal{M}_{\mu \nu} \right) \phi(x),\] with generator \[\mathcal{M}_{\mu \nu} = - i (x_\mu \partial_\nu - x_\nu \partial_\mu) .\] Indeed, these generators form a representation of the Lie algebra \(\eqref{eq:LorentzAlgebra}\), i. e.  \[= i (\eta_{\mu \rho} \mathcal{M}_{\nu \sigma} - \eta_{\mu \sigma} \mathcal{M}_{\nu \rho} - \eta_{\nu \rho} \mathcal{M}_{\mu \sigma} + \eta_{\nu \sigma} \mathcal{M}_{\mu \rho}). \label{eq:defMmunu}\] For fields with non-vanishing spin, the complete generator contains \(\mathcal{M}_{\mu \nu}\) and the generator of “internal” transformations, for example for a left-handed spinor \[\psi_a (x) \to \psi^\prime_a (x) = \left(\delta_a^{\;\; b} + \frac{i}{2} \delta \omega^{\mu \nu} (M_{\mu \nu})_a^{\;\; b} \right) \psi_b (x),\] with \[(M_{\mu \nu})_a^{\;\; b} = (M^L_{\mu \nu})_a^{\;\; b} + \mathcal{M}_{\mu \nu} \, \delta_a^{\;\; b}.\] This can now be extended to fields in arbitrary representations of the Lorentz group.

Poincaré group

Poincaré transformations consist of Lorentz transformations plus translations, \[x^\mu \to \Lambda_{\;\; \nu}^\mu x^\nu - b^\mu .\] Translations only (without Lorentz transformations) form themselves an abelian Lie group, the additive group \(\mathbb{R}^4\). It is clear that Poincaré transformations form a group. The composition law is \[(\Lambda_2, b_2) \circ (\Lambda_1, b_1) = (\Lambda_2 \Lambda_1, b_2+ \Lambda_2 b_1 ).\] [Exercise: Show this.] The composition law is an example for a semi-direct product, namely of the Lorentz group O\((1,3)\) and the additive group \(\mathbb{R}^4\) of space and time translations, \[\text{Poincaré group} \cong \text{O}(1,3) \ltimes \mathbb{R}^4.\] Lorentz transformations can be parametrized by six parameters, which are supplemented by four parameters for translations. The entire symmetry group of Minkowski space has therefore ten parameters.

Lie algebra of Poincaré group

Let us now find the Lie algebra associated with the Poincaré group. As transformations of fields, translations are generated by the momentum operator \[P_\mu = -i \partial_\mu .\] For example, as an infinitesimal transformation, \[\begin{aligned} \phi (x) \to \phi^\prime (x) &= \phi (\Lambda^{-1} (x + b)) \\ &= \phi (x^\mu - \delta \omega_{\;\; \nu}^\mu x^\nu + b^\mu) \\ &= \left(1 + \frac{i}{2} \delta \omega^{\mu \nu} \mathcal{M}_{\mu \nu} + i b^\mu P_\mu \right) \phi (x) . \end{aligned}\] One finds easily \[\left[P_\mu, P_\nu \right] = 0, \label{eq:momentum_op_comm1}\] and \[\left[\mathcal{M}_{\mu \nu}, P_\rho\right] = i \left(\eta_{\mu \rho} P_\nu - \eta_{\nu \rho} P_\mu \right), \label{eq:momentum_op_comm2}\] which together with \(\eqref{eq:defMmunu}\) forms the Lie bracket relations of the Poincaré algebra. The commutator \(\eqref{eq:momentum_op_comm1}\) tells that the different components of the energy-momentum operator can be diagonalized simultaneously, while \(\eqref{eq:momentum_op_comm2}\) says that \(P_\rho\) transforms as a covector under Lorentz transformations.

Representations of the Poincaré group

Let us now discuss representations of the Poincaré algebra (and corresponding representations of the Poincaré group). We concentrate here on the part of the group that is connected to the identity transformations, i. e. SO\(^\uparrow(1,3) \ltimes \mathbb{R}^4\). It turns out that single-particle states can be understood as examples for such representations.

As we have done before, we will use a maximal number of commuting generators to label states. In particular, the different components of the momentum operator \(P_\mu=-i\partial_\mu\) commute and we can work with corresponding eigenstates, namely plane waves \(e^{ip_\mu x^\mu}\). The eigenvalues are then energy and momentum, \(p_\mu=(-E, \vec p)\).

Casimir operators

To classify representations, we first search for Casimir operators, i. e. operators that commute with all generators. One Casimir operator is \[P^2 = P_\mu P^\mu,\] which obviously commutes with \(\mathcal{M}_{\mu \nu}\) and \(P_\mu\). For single particle states of massive particles we have \(p_\mu p^\mu + M^2 = 0\) so that \(-P^2=M^2\) gives the particle mass. The other Casimir operator follows from the Pauli-Lubanski vector \[W^\mu = \frac{1}{2} \epsilon^{\mu \nu \rho \sigma} \mathcal{M}_{\nu\rho} P_\sigma.\] It is orthogonal to the momentum, \(W^\mu P_\mu=0\), and has the commutation relations \[[W^\mu, P_\nu]=0, \quad\quad\quad [\mathcal{M}_{\rho\sigma}, W^\mu]= i \left(\delta_\rho^\mu W_\sigma - \delta_\sigma^\mu W_\rho \right),\] as well as \[[W^\mu, W^\nu] = - i \epsilon^{\mu\nu\rho\sigma} W_\rho P_\sigma.\] The second Casimir of the Poincaré algebra is then given by \[W^2 = W_\mu W^\mu.\]

The little group

When discussing representations of the Poincaré group it is convenient to first make a case separation in terms of the quadratic Casimir \(P^2=P_\mu P^\mu\). In each of the cases one can then fix a reference choice for \(p^\mu_*\) and discuss remaining transformations that leave this reference invariant, \[(\delta^\mu_{\;\;\nu} + \delta\omega^\mu_{\;\;\nu}) p^\nu_* = p^\mu_*. \label{eq:defLittleGroup}\] This remaining symmetry group is then known as the little group. We will see this working in practise below.

Representations with vanishing momentum

The eigenvalue of the momentum operator \(P_\mu\) may actually simply vanish, \(p^\mu_*=(0,0,0,0)\). In that case the little group corresponds to the entire Lorentz group SO\((1,3)\). An example for such a state is the vacuum.

Representations with positive mass squared

Let us now first consider situations with \(-P^2=M^2 > 0\). Examples for such representations are single particle states with positive mass.

We can fix a reference momentum \(p_*^{\mu} = (M, 0, 0, 0)\) which corresponds to a particle momentum in its rest frame. The little group then consists of transformations that leave \(p_*^\mu\) invariant. These are just rotations so the little group is here SO\((3)\) or its double cover SU\((2)\) which has the same Lie algebra. More explicitly, this follows from searching solutions to \(\eqref{eq:defLittleGroup}\) which is here equivalent to \(\delta\omega^{\mu0}=0\). Lorentz boosts are excluded; what is left are rotations.

In the particles rest frame, the Pauli-Lubanski vector evaluates to \[\begin{aligned} W_0 = 0, \quad\quad\quad W_j = \frac{M}{2} \epsilon_{jkl} \mathcal{M}_{kl} = M J_j, \end{aligned}\] with angular momentum or spin operator \(J_j\). The second Casimir of the Poincaré algebra is accordingly \(W^2 / M^2= \vec J^2\). Single particle states \(| p, j, m \rangle\) can be labeled by momentum \(p^\mu\), total spin \(\vec J^2 = j(j+1)\) and eigenvalue \(m\) of the spin operator in \(z\)-direction \(J_3\).

Symmetric spinor-tensor representation of SU\((2)\)

Besides the standard representations of SU\((2)\) discussed in the context of quantum mechanics, an alternativ representation is in terms of symmetric spinor-tensors. As we have seen, the spin-1/2 representation is a Pauli spionor \(\psi_s\), where \(s=1,2\), which transforms under rotations according to \[\psi_s \to L_s^{\;\;t} \psi_t = \left(\delta_{s}^{\;\;t}+ \frac{i}{4} \delta\omega_{ij} \epsilon_{ijk} (\sigma_k)_s^{\;\;t}\right) \psi_t , \label{eq:littleGroupSU2Transformation}\] where \(L_s^{\;\;t}\) describes here the little group rotation matrix. One can now simply construct higher order representations as symmtric spinors with several SU\((2)\) indices \(\psi_{st\cdots u}\), and they transform accordingly under the little group transformations. In this way one obtains representations with higher spin, and a symmetric SU\((2)\) tensor with \(n\) indices describes particles with spin \(n/2\). The interesting feature about the little group is that every incoming or outgoing massive particle in a scattering experiment as its own little group and can be rotated independently in its respective rest frames.

Representations with negative mass squared

Here we have a situation with \(-P^2=M^2 < 0\). This corresponds to so-called tachyonic modes and if they appear they are usually associated to an instability.

We can fix a reference momentum as \(p_*^{\mu} = (0, 0, 0, M)\). The little group consists now of transformations that leave \(p_*^\mu\) invariant and these are Lorentz transformations in the remaining \(1+2\) dimensional space, SO\((1,2)\). We will not discuss these representations in more detail.

Representations with vanishing mass

Let us now consider representations with \(P^2= 0\). This is again a rather interesting case. Examples are here single particle states with vanishing mass \(M=0\).

Massless particles do not have a restframe, so to discuss the little group one must pick another reference momentum, for example \(p_*^\mu = (p, 0, 0, p) = p (\delta^{\mu}_0+\delta^{\mu}_3)\). The little group consists of transformations that leave this invariant. Specifically, eq. \(\eqref{eq:defLittleGroup}\) implies here \(\delta\omega^{\mu0}=\delta\omega^{\mu3}\). One can write this as \[\omega^{\mu\nu} = \begin{pmatrix} 0 & \alpha & \beta & 0 \\ -\alpha & 0 & \theta & -\alpha \\ -\beta & - \theta & 0 & -\beta \\ 0 & \alpha & \beta & 0 \end{pmatrix}.\] Here, if only \(\theta\) was non-vanishing, it would be the angle of a rotation in the \(1\)-\(2\)-plane, i. e. around the propagation direction of the massless particle. Instead non-vanishing \(\alpha\) would parametrize a combination of a boost in \(1\)-direction together with a rotation in the \(1\)-\(3\) plane. Finally, \(\beta\) parametrizes a combination of a boost in the \(2\)-direction with a rotation in \(2\)-\(3\)-plane. An infinitesimal group transformation out of the litte group can be written as \[\mathbb{1} + i \delta \theta J_3 + i \delta\alpha A + i \delta\beta B, \label{eq:transformationLittleGroupMassless}\] with \[A = K_1 + J_2 = M_{10}+M_{31}, \quad\quad\quad B=K_2-J_1=M_{20}+M_{32}.\] The Lie algebra of the little group is \[= i B, \quad\quad\quad [J_3, B] = -i A, \quad\quad\quad [A,B]=0. \label{eq:E2CommutationRelations}\] This is in fact the Lie algebra of the so-called special Euclidean group E\(^+(2)\) consisting of translations and rotations in the two-dimensional Euclidean plane. It contains an \(\text{SO}(2)\) subgroup of rotations, as well as a subgroup of translations \(\mathbb{R}^2\). The abelian subgroup of translations is in fact a normal subgroup. Similar to the Poincaré group itself, the Euclidean group E\(^+(2)\) has the structure of a direct product, E\(^+(2)=\text{SO}(2) \ltimes \mathbb{R}^2\). [Exercise: Check all this!]

In the fundamental representation of the Lorentz algebra, the operators \(A\) and \(B\) are actually nilpotent. In fact, one has \(A^3=B^3=AB=BA=0\). However, there are also representations of \(\eqref{eq:E2CommutationRelations}\) where \(A\) and \(B\) are hermitian such that the group has a unitary representation. However, as for any non-compact group, such unitary representations are necessarily infinite dimensional.

Physically, \(A\) and \(B\) can be related to gauge transformations. To see this consider polarization vectors for photons with momentum \(p^\mu_*\), \[\epsilon^\mu_\pm = \frac{1}{\sqrt{2}} (0, 1, \pm i, 0).\] These are eigenstates of \(J_3\), namely in the fundamental or vector representation of the Lorentz group, \[(J_3)^\mu_{\;\;\nu} \epsilon^\nu_\pm = \pm \epsilon^\mu_\pm.\] The two polarizations \(\epsilon^\mu_\pm\) describe therefore states with helicity \(\pm 1\), respectively. Now consider the action of \(\eqref{eq:transformationLittleGroupMassless}\) with \(\delta\theta =0\), \[\epsilon^\mu_\pm \to \epsilon^\mu_\pm + \frac{(\delta\alpha \pm i \delta\beta)}{\sqrt{2}p} p^\mu_*.\] Interestingly, the generators \(A\) and \(B\) are not realized trivially here, but they actually do change the polarization vector by a term proportional to the momentum. This is in fact a gauge transformation! To see this, consider the gauge transformation \(A_\mu(x) \to A_\mu(x) + \partial_\mu \alpha(x)\), which becomes in momentum space \[A_\mu(p) \to A_\mu(p) + i p_\mu \alpha(p).\] The physical photon states are supposed to be independent of this gauge choice, and one takes them to be gauge equivalence classes. In other words, all states that differ by a gauge transformation are getting identified. This works similarly for massless particles of spin two, where the gauge symmetry is then the one of general relativity.

For such gauge equilvalence classes, or in a gauge fixed description, physical states of single massless particles can be characterized as having vanishing eigenvalues with respect to the operators \(A\) and \(B\). We are then left with \(J_3\) which generates rotations around the direction of propagation. This is in fact helicity, \(J_3=h\). The little group for massless particles is then U\((1)\).

Fermionic massless particle states can change by a factor \(-1\) under rotations of \(2\pi\) around the propagation direction. This implies half-integer helicity \(h\). In contrast, bosonic massless particle states should be invariant under \(2\pi\) rotations, so that helicity \(h\) must be integer valued. These quantization conditions arise here from topological properties of the group, and not from properties of the Lie algebra.

Spinor helicity variables for massless momenta

Consider the \((2,2)\) representaion of a momentum \(p^\mu\), \[p_\mu (\sigma^\mu)_{a\dot a} = \begin{pmatrix} - p^0 + p^3 && p^1 -i p^2 \\ p^1 + i p^2 && -p^0 - p^3 \end{pmatrix}.\] The determinant of this matrix is \((p^0)^2 - \mathbf{p}^2 = - \eta_{\mu\nu} p^\mu p^\nu\). For momenta corresponding to massless particles this vanishes. This means that one eigenvalue is zero, or that the matrix has rank \(1\). As an example, take \(p^\mu = p_*^\mu\), the reference momentum we have chosen above for massless particles. In that case, \[p_\mu (\sigma^\mu)_{a \dot a} = \begin{pmatrix} 0 && 0 \\ 0 && -2 p^0 \end{pmatrix}.\] This shows that one can write for such momenta \[p_\mu (\sigma^\mu)_{a \dot a} = -\lambda_a \tilde \lambda_{\dot a}. \label{eq:defmasslesstwistors}\] The objects \(\lambda_a\) and \(\tilde \lambda_{\dot a}\) are known as spinor helicity variables or twistors. They are formally left-handed and right-handed spinors, respectively, but are taken to be commuting, i. e. they are composed out of ordinary complex numbers and not Grassmann variables. For real momenta \(p_\mu\sigma^\mu\) is hermiten and one has \[\tilde \lambda_{\dot a} = \pm (\lambda_{a})^*\] where the positive sign must be choosen for \(p^0>0\) and the negative sign for \(p^0<0\). More generally one may also consider complex momenta and then \(\lambda_a\) and \(\tilde \lambda_{\dot a}\) become independent. An explicit realization for \(p^0>0\) and massless on-shell momenta is \[\lambda_a = \frac{1}{\sqrt{p^0 + p^3}}\begin{pmatrix} -p^1 + i p^2 \\ p^0 + p^3 \end{pmatrix}, \quad\quad\quad \tilde \lambda_{\dot a} = \frac{1}{\sqrt{p^0 + p^3}}\begin{pmatrix} -p^1 - i p^2 \\ p^0 + p^3 \end{pmatrix}.\] This assignment is not unique, however. Specifically one could transform \[\lambda_a \to e^{i\theta} \lambda_a, \quad\quad\quad \tilde \lambda_{\dot a} \to e^{-i\theta} \tilde \lambda_{\dot a},\] and obtain new twistors that also fulfill \(\eqref{eq:defmasslesstwistors}\). One may use this to analytically continue for negative frequency \(p^0<0\) such that for \(p^\mu \to - p^\mu\) \[\lambda_a \to \lambda_a, \quad\quad\quad \tilde\lambda_{\dot a} \to - \tilde\lambda_{\dot a}.\]

Pulling indices up and down

As we have seen previously one can sensibly pull indices up and down with \(\varepsilon_{ab}\) and its inverse \(\varepsilon^{ab}\), for example a twistor with upper left-handed index is obtained by \[\lambda^a = \varepsilon^{ab} \lambda_b.\] Similarly this can be done for right-handed indices. One can then also write \[p_\mu (\bar \sigma^\mu)^{\dot a a} = \varepsilon^{\dot a \dot b} \varepsilon^{a b} p_\mu (\sigma^\mu)_{b \dot b} = - \tilde\lambda^{\dot a} \lambda^a.\]

Bracket notation and scalar products

It is customary to introduce the following notation for the left handed twistor associated with the momentum \(p\), \[| p \rangle_a = \lambda_a, \quad\quad\quad \langle p |^a = \lambda^a = \varepsilon^{ab} | p \rangle_a,\] and similarly for the right handed twistor, \[| p ]^{\dot a} = \tilde \lambda^{\dot a}, \quad\quad\quad [ p |_{\dot a} = \tilde \lambda_{\dot a} = \varepsilon_{\dot a \dot b} | p ]^{\dot b}.\] The analytic continuation properties can be choosen such that \[| -p \rangle_a = |p \rangle_a, \quad\quad\quad |-p ]^{\dot a} = - |p ]^{\dot a}.\] The bracket notation (not to be confused with the bra-ket notation of quantum mechanics) allows also to work nicely in situations where differnt momenta are involved, for example \(|p\rangle_a = \lambda_a\) and \(|q\rangle_a = \mu_a\). One defines the SU\((2)\) invariant scalar product between left-handed twistors \[\langle p q \rangle = \langle p |^a | q \rangle_a = \lambda^a \mu_a = \varepsilon^{ab} \lambda_b \mu_a = - \mu^a \lambda_b = - \langle p |^a | q \rangle_a = - \langle q p \rangle,\] and between right-handed twistors, \[[ p q ] = [ p |_{\dot a} q ]^{\dot a} = \tilde \lambda_{\dot a} \tilde \mu^{\dot a} = \varepsilon_{\dot a \dot b} \tilde \lambda^{\dot b} \tilde \mu^{\dot a} = - \tilde\mu_{\dot a} \tilde \lambda^{\dot a} = - [ q |^{\dot a}| p ]_{\dot a} = - [ q p ].\] Note that it is not possible to sensibly define a scalar product between a left-handed and a right-handed twistor. The anti-symmetry implies also that for massless momenta \[\langle pp \rangle =[pp]=0.\] This will be useful in the following. In the bracket notation we can also write \[p_\mu(\sigma^\mu)_{a \dot b} = - |p\rangle_a [ p |_{\dot b}, \quad\quad\quad p_\mu(\bar\sigma^\mu)^{\dot a b} = - | p ]^{\dot a} \langle p |^{ b}.\] This also implies \[2 p_\mu q^\mu = - \text{tr} \{ p_\mu \sigma^\mu q_\mu \bar \sigma^\nu \} = - | p \rangle_a [p |_{\dot b} | q ]^{\dot b} \langle q|^a = - \langle q p \rangle [pq] = \langle pq \rangle [pq].\]

Weyl equations

The equation of motion \(i(\bar\sigma^\mu)^{\dot a b} \partial_\mu \psi_b(x)\) for free massless left-handed Weyl spinor in plane wave form \(\psi_b(x) = \psi_b(p) e^{ipx}\) reads now \[- p_\mu(\bar\sigma^\mu)^{\dot a b} \psi_b(p) = | p ]^{\dot a} \langle p |^{ b} \psi_b(p) = 0.\] One can see that \(\psi_b(p) \sim | p \rangle_b\) is actually a solution, \[| p ]^{\dot a} \langle p |^{ b} | p \rangle_b = | p ]^{\dot a} \langle p p \rangle = 0.\] (Some care is needed here, however, because \(\psi_b(p)\) is a spinor with Grassmann property while \(| p \rangle_b\) is a twistor with no Grassmann property.) The complex conjugate \[\bar\psi^{\dot a}(p) = \varepsilon^{\dot a\dot b}(\psi_b(-p))^* = - | - p ]^{\dot a} = |p ]^{\dot a}\] is similarly a solution of the right-handed Weyl equation, \[- p_\mu(\sigma^\mu)_{a \dot b} |p ]^{\dot b} = |p\rangle_a [ p |_{\dot b} | p ]^{\dot b} = .|p \rangle_a [ p p ] = 0.\] In other words, we can identify \(\psi_a(p)\sim| p \rangle_a\) and \(\bar\psi^{\dot a}(p) \sim |p ]^{\dot a}\) is the corresponding conjugate spinor (of course some care is needed with Grassmann properties).

A solution of the free Majorana equation at vanishing mass, \(\gamma^\mu p_\mu \Psi(p)=0\), can accordingly be written in the form \[\Psi(p) = \begin{pmatrix} \psi_a(p) \\ \bar\psi^{\dot a}(p) \end{pmatrix} \sim \begin{pmatrix} |p \rangle_a \\ |p ]^{\dot a} \end{pmatrix}.\]

Little group representation for massless twistors

To construct \(| p \rangle_b = \lambda_b\) explicitely it is useful to start with a specific reference frame, for example the one where \(p^\mu = p_*^\mu = (p,0,0,p)\) points in the three-direction. One has then \[| p \rangle_a = \lambda_a = \begin{pmatrix} 0 \\ \sqrt{2p} \end{pmatrix}.\] One can then apply a Lorentz transformation in the left-handed representation \(L_a^{\;\;b}\) to go to other momenta.

At this point it is also interesting how the little group acts here. For the left-handed representation one finds \[\mathbb{1} + i \delta \theta J_3 + i \delta \alpha A + i \delta \beta B = \begin{pmatrix} 1 + \frac{i}{2}\delta\theta && 0 \\ -\delta\alpha - i \delta \beta && 1 - \frac{i}{2} \delta\theta \end{pmatrix}.\] Interestingly this implies that under such a transformation \[| p \rangle_a \to e^{-i\theta/2} |p \rangle_a.\] For the left handed twistor the generators \(A\) and \(B\) are represented trivially and have no effect. The little group is here just a complex phase \(e^{-i\theta/2}\). The negative sign and the factor \(1/2\) tell that we are dealing with states of helicity \(h=-1/2\) as appropriate for left-handed Weyl fermions. For right-handed twistors oe has instead \[|p]^{\dot a} \to e^{i\theta/2} |p]^{\dot a},\] which tells that they have helicity \(h=1/2\). For practical caluclations it is useful to know that the little group transformations can be done independently for each incoming or outgoing particle and a scattering amplitude must transform accordingly with the respective phase factors.

Extension to nonzero mass

Consider now momenta \(p^\mu\) such that \(\eta_{\mu\nu} p^\mu p^\nu + m^2=0\) with nonzero mass \(m\). In that case one can introduce two twistors \(\lambda_{sa}\) with \(s=1,2\) and write \[p^\mu (\sigma_\mu)_{a \dot b} = - \lambda_{sa} \tilde\lambda^s_{\dot b} = - \lambda_{sa} \varepsilon^{st} \tilde \lambda_{t \dot b}.\] By taking the determinant of this matrix one obtains \(-(p^0)^2+\mathbf{p}^2=-m^2\) and accordingly one may, as a matrix with indices \(a\) and \(s\), take \[\det(\lambda_{sa}) = \det(\tilde \lambda_{t\dot b})= m.\] The index \(s\) is here a kind of internal SU\((2)\) index associated with a particle of momentum \(p^\mu\) where the group SU\((2)\) corresponds to the appropriate little group of roations for massive particles with spin 1/2. Accordingly, these indices can also be pulled up and down with \(\varepsilon_{st}\) and its inverse \(\varepsilon^{st}\), for example \[\tilde \lambda^s_{\dot a} = \varepsilon^{st} \lambda_{t \dot a}.\]

Bracket notation for massive twistors

Also for the massive twistors one can work with the bracket notation \[| p_s \rangle_a = \lambda_{sa}, \quad\quad\quad \langle p_s|^a = \lambda_s^{a} = \varepsilon^{ab} | p_s \rangle_b.\] We can follow conventions where the little group SU\((2)\) index is taken in an upper position. Similarly for right-handed massive twistors (where \(\tilde \lambda_{s\dot a} = (\lambda_{sa})^*\) in Minkowski space for positive energy) \[| p_s ]^{\dot a} = \tilde \lambda_s^{\dot a}, \quad\quad\quad [ p_s |_{\dot a} = \tilde \lambda_{s\dot a} = \varepsilon_{\dot a \dot b} | p_s ]^{\dot b}.\] Sometimes one uses also a notation where the little group SU\((2)\) or spin indices are suppressed but then uses bold face letters to indicate that these are massive twistors, e. g. \[| \boldsymbol{p} \rangle_a = |p_s \rangle_a.\]

Little group transformations for massive twistors

The little group of SU\((2)\) rotations acts on these twistors according to \[| p_s \rangle_a \to L_s^{\;\;t} | p_t \rangle_a,\] which is precisely the transformation in \(\eqref{eq:littleGroupSU2Transformation}\). This is actually the same transformation for any twistor with an upper little group index.

Dirac equation for Majorana spionors

The Majorana spinor in momentum space can be written as \[\Psi_s(p) = \begin{pmatrix} \psi_{sa}(p) \\ \bar \psi_s^{\dot a}(p) \end{pmatrix} \sim \begin{pmatrix} | p_s \rangle_a \\ | p_s ]^{\dot a} \end{pmatrix}.\] The index \(s\) labels now the little group or, in other words, parametrizes the two independent spin states in the particles rest frame. The Dirac equation \((i\gamma^\mu p_\mu +m) \Psi_s(p)=0\) reads \[\begin{pmatrix} m \, \delta_a^{\;\;b} && p_\mu (\sigma^\mu)_{a\dot b} \\ p_\mu (\bar \sigma^\mu)^{\dot a b} && m \, \delta^{\dot a}_{\;\;\dot b} \end{pmatrix} \begin{pmatrix} | p_s \rangle_b \\ | p_s ]^{\dot b} \end{pmatrix} = \begin{pmatrix} m \, \delta_a^{\;\;b} && - |p_t \rangle_a [ p^t |_{\dot b} \\ - | p_t ]^{\dot a} \langle p^t |^b && m \, \delta^{\dot a}_{\;\;\dot b} \end{pmatrix} \begin{pmatrix} | p_s \rangle_b \\ | p_s ]^{\dot b} \end{pmatrix} = 0.\] One infers from this that \[m | p_s \rangle_a + p_\mu (\sigma^\mu)_{a \dot b} |p_s ]^{\dot b} = m | p_s \rangle_a - | p_t \rangle_a [ p^t p_s ] =0,\] and similarly \[m | p_s ]^{\dot a} + p_\mu(\bar\sigma^\mu)^{\dot a b} | p_s\rangle_b = m | p_s ]^{\dot a} - | p_s ]^{\dot a} \langle p^t p_s \rangle = 0.\] These are just other ways to write the free Dirac equation for free massive Majorana particles in Minkowski space. Here we are using the scalar products \(\langle p_s q_t \rangle = - \langle q_t p_s \rangle\) and \([ p_s q_t ] = - [ q_t p_s ]\), which generalize the corresponding definitions for massless twistors. Because of the additional index, such scalar products with the same momentum \(\langle p_s p_t \rangle\) and \([ p_s p_t ]\) are now non-zero, albeit they are antisymmetric and proportional to \(\varepsilon_{st}\).

We leave the generalization of the formalism from Majorana to Dirac spinors for the future.