Quantum field theory 1, lecture 18

Yukawa theory

Yukawa theory

Let us now investigate a theory for a non-relativistic fermion with spin \(1/2\) and a real, relativistic scalar boson \[S = \int dt d^3x \left\{-\bar{\psi}\left(-i\partial_t - \frac{\boldsymbol{\nabla}^2}{2m} +V_0 - i\epsilon\right) \psi -\tfrac{1}{2} \phi \left(\partial^2_{t} - \boldsymbol{\nabla}^2 +M^2 -i\epsilon\right)\phi -g\phi\bar{\psi}\psi\right\}.\]

Partition function for Yukawa theory

We will discuss this theory in terms of the partition function \[Z[\bar{\eta},\eta, J] = \int D\bar{\psi}D\psi D\phi \; e^{i S[\bar{\psi},\psi,\phi] + i\int_x\{\bar{\eta}\psi + \bar{\psi}\eta +J\phi \} } .\] As usual, by taking functional derivatives with respect to the source fields, one can obtain various correlation functions. Our strategy will be to perform a perturbation expansion in the cubic term \(\sim g\).

Quadratic action

Let us first concentrate on the quadratic theory and the corresponding partition function derived from the action \[S_2 = \int dt d^3 x \left\{-\bar{\psi}\left(-i\partial_t -\frac{\boldsymbol{\nabla}^2}{2m}+V_0-i\epsilon\right)\psi -\frac{1}{2}\phi(\partial^2_{t} - \boldsymbol{\nabla}^2+M^2 -i\epsilon)\phi\right\} .\] By doing the Gaussian integration one finds \[\begin{split} Z_2 [\bar{\eta},\eta, J] &= \int D\bar{\psi}D\psi D\phi \; \exp\left[iS_2 + i\int_x \left\{\bar{\eta}\psi +\bar{\psi}\eta +J\phi\right\}\right]\\ &= \exp \left[i\int d^4x d^4y \left\{\bar{\eta}(x) \Upsilon(x-y)\eta(y) +\frac{1}{2} J(x) \Delta(x-y) J(y)\right\} \right] \end{split}\] where \(\Upsilon(x-y)\) is the Greens function for fermions in eq. \(\eqref{eq:nonrelativisticfermionprop}\). For the scalar bosons, the Green function is \[\Delta(x-y) = \int \frac{d^4p}{(2\pi)^4}\frac{1}{-(p^0)^2 + \mathbf{p}^2 +M^2 - i\epsilon} e^{ip(x-y)},\] as discussed previously.

Propagator for non-relativistic fermions

For the non-relativistic fermion, the propagator integral over \(p^0\) has just a single pole at \(p^0 = \frac{\mathbf{p}^2}{2m}+ V_0 - i\epsilon,\) \[\Upsilon (x-y) = \mathbb{1} \int \frac{dp^0}{2\pi}\frac{d^3 p}{(2\pi)^3} \frac{1}{-p^0 +\frac{\mathbf{p}^2}{2m}+V_0 - i\epsilon}e^{-ip^0(x^0-y^0)+ i\mathbf{p}(\mathbf{x} - \vec{y})}\] When \(x^0-y^0 > 0\) the contour can be closed below the real \(p^0\)-axis, leading to \[\Upsilon (x-y) = i\; \mathbb{1} \int \frac{d^3p}{(2\pi)^3}\; e^{-i {\big (} \frac{\mathbf{p}^2}{2m}+V_0 {\big )}(x^0-y^0) + i\mathbf{p}(\mathbf{x} - \vec{y})} \quad\quad\quad (x^0 - y^0 > 0).\] In contrast, for \(x^0 - y^0 < 0,\) the contour can be closed above and there is no contribution at all. In summary \[\Upsilon (x-y) = i\;\theta(x^0-y^0)\; \mathbb{1} \int \frac{d^3p}{(2\pi)^3}\; e^{-i{\big (} \frac{\mathbf{p}^2}{2m}+V_0{\big )}(x^0-y^0) + i\mathbf{p}(\mathbf{x} - \vec{y})}.\] As a consequence of the absence of anti-particle-type excitations, the time-ordered and retarded propagators agree here.

Propagator and correlation functions

Let us also note the relation between propagators and correlation functions. For the free (quadratic) theory one has in the fermionic sector \[\begin{split} \left\langle\psi_a (x) \bar{\psi}_b (y)\right\rangle &= \left(\frac{1}{Z_2} \frac{\delta}{\delta \bar{\eta}_a (x)}\frac{\delta}{\delta \eta_b (y)} Z_2[\bar{\eta}, \eta, J]\right)_{\bar{\eta} = \eta = J = 0}\\ &= -i \Upsilon_{ab} (x-y), \end{split}\] Note that some care is needed with interchanges of Grassmann variables to obtain this expression. Similarly for the bosonic scalar field \[\begin{split} \left\langle\phi(x) \phi(y) \right\rangle &= \left(\frac{1}{Z_2} \frac{\delta}{\delta J(x)}\frac{\delta}{\delta J(y)} Z_2[\bar{\eta}, \eta, J]\right)_{\bar{\eta} = \eta = J = 0}\\ &= -i \Delta(x-y). \end{split}\]

Wick’s theorem

As discussed previously, one finds for the free theory \[\begin{split} \left\langle \phi(x_1) \ldots \phi(x_n) \right\rangle &= \left( \frac{1}{Z_2} \left(-i \frac{\delta}{\delta J(x_1)}\right)\cdots\left(-i\frac{\delta}{\delta J(x_n)}\right)Z_2[\bar\eta, \eta, J]\right)_{\bar{\eta}= \eta = J =0}\\ &= \sum_{\text{pairings}}\left[-i \Delta(x_{j_1} - x_{j_2})\right] \cdots \left[-i \Delta(x_{j_{n-1}}-x_{j_n})\right]. \end{split}\] The sum in the last line goes over all possible ways to distribute \(x_1, \ldots, x_n\) into pairs \((x_{j_1}, x_{j_2})\), \((x_{j_3}, x_{j_4})\), \(\ldots\), \((x_{j_{n-1}}, x_{j_n})\). This result is known as Wick’s theorem. It follows directly from the combinatorics of functional derivatives acting on \(Z_2\).

For example, \[\begin{split} \left\langle \phi(x_1)\; \phi(x_2)\; \phi(x_3) \phi(x_4) \right\rangle = & [-i \Delta(x_1 - x_2)][-i \Delta(x_3 - x_4)]\\ & + [-i \Delta(x_1 - x_3)][-i \Delta(x_2 - x_4)]\\ &+ [-i\Delta (x_1 - x_4)] [-i \Delta(x_2 - x_3)]. \end{split}\]

In a similar way correlation functions involving \(\bar{\psi}\) and \(\psi\) can be written as sums over the possible ways to pair \(\psi\) and \(\bar{\psi}\). For example \[\begin{split} \left\langle \psi_{a_1}(x_1) \psi_{a_2}(x_2) \bar{\psi}_{a_3}(x_3) \bar{\psi}_{a_4}(x_4) \right\rangle = & -\left\langle \psi_{a_1}(x_1) \bar{\psi}_{a_3}(x_3)\right\rangle \left\langle\psi_{a_2}(x_2) \bar{\psi}_{a_4}(x_4) \right\rangle\\ &+\left\langle \psi_{a_1}(x_1) \bar{\psi}_{a_4}(x_4)\right\rangle \left\langle\psi_{a_2}(x_2) \bar{\psi}_{a_3}(x_3) \right\rangle\\ = & -[-i \Upsilon_{a_1a_3}(x_1-x_3)][-i\Upsilon_{a_2a_4}(x_2-x_4)]\\ &+[-i \Upsilon_{a_1a_4}(x_1-x_4)][-i \Upsilon_{a_2a_3}(x_2-x_3)]. \end{split}\] Note that correlation functions at quadratic level (for the free theory) need to involve as many fields \(\psi\) as \(\bar{\psi}\), otherwise they vanish. Similarly, \(\phi\) must appear an even number of times. For mixed correlation functions one can easily separate \(\phi\) from \(\psi\) and \(\bar{\psi}\) at quadratic level, because \(Z_2[\bar \eta, \eta, J]\) factorizes. For example, \[\left\langle \phi(x_1)\;\psi_a(x_2)\;\phi(x_3) \bar{\psi}_b(x_4) \right\rangle = [-i\Delta(x_1 - x_3)][-i\Upsilon_{ab}(x_2 - x_4)]. \label{eq:fermion2boson2correlation}\]

Graphical representation

It is useful to introduce also a graphical representation. We will represent the scalar propagator by a dashed line,

The Feynman propagator for the fermions will be represented by a solid line with arrow,

We can then represent correlation functions graphically, for example, the mixed correlation function in eqn. \(\eqref{eq:fermion2boson2correlation}\) for the free theory would be

Perturbation theory in \(g\)

Let us now also consider the interaction terms in the action. In the functional integral it contributes according to \[e^{iS[\bar{\psi},\psi,\phi]}= e^{iS_2[\bar{\psi},\psi,\phi]} \; \exp \left[-i g \int d^4x \phi(x) \bar{\psi}_a (x) \psi_a (x)\right].\] We can assume that \(g\) is small and simply expand the exponential where it appears. This will add field factors \(\sim \phi(x) \bar{\psi}_a (x) \psi_a(x)\) to correlation functions with an integral over \(x\) and an implicit sum over the spinor index \(a\). The resulting expression involving correlation functions can then be evaluated as in the free theory. For example, \[\begin{split} \left\langle \phi(x_1) \psi_b(x_2) \bar{\psi}_c(x_3) \right\rangle &= \left\langle \phi(x_1) \psi_b(x_2) \bar{\psi}_c(x_3) \right\rangle_0\\ &\qquad +\left\langle \phi(x_1) \psi_b(x_2) \bar{\psi}_c(x_3) \left[-ig\int_y \phi(y) \bar{\psi}_a(y) \psi_a(y)\right]\right\rangle_0 + \ldots \end{split}\] The index \(0\) indicates that the correlation functions get evaluated in the free theory. Graphically, we can represent the interaction term as a vertex,

For each such vertex we need to include a factor \(-ig\) as well as an integral over the space-time variable \(y\) and the spinor index \(a\).

Three point function

To order \(g\), we find for the example above

The sign in the last line is due to an interchange of Grassmann fields. The last expression involves the fermion propagator for vanishing argument \[\Upsilon_{ab}(0) = \delta_{ab} \int \frac{d^4 p}{(2\pi)^4}\frac{1}{-p^0 +\tfrac{\mathbf{p}^2}{2m}+V_0-i\epsilon} = i\theta(0)\delta_{ab} \delta^{(3)}(0).\] We will set here \(\theta(0) = 0\) so that the corresponding contribution vanishes. In other words, we will interpret \[\Upsilon_{ab}(0) = \lim_{\Delta t\to 0} \Upsilon_{ab}(-\Delta t, \vec 0 ) = 0.\] Although this is a little ambiguous at this point, it turns out that this is the right way to proceed.

Feynmann rules in position space

To calculate a field correlation function in position space we need to

• have a scalar line ending on \(x\) for a factor \(\phi(x)\):

  

• have a fermion line ending on \(x\) for a factor \(\psi_a(x)\):

  

• have a fermion line starting on \(x\) for a factor \(\bar{\psi}_a(x)\):

  

• include a vertex \(-ig\int_y\) with integral over \(y\) for every order \(g\):

  

• connect lines with propagators \(-i\Delta(x-y)\) or \(-i\Upsilon_{ab}(x-y)\)

• determine the overall sign for interchanges of fermionic fields.

S-matrix elements from amputated correlation functions

To calculate S-matrix elements from correlation functions, we need to use the LSZ formula. For an outgoing fermion, we need to apply the operator \[i\left[-i\partial_t -\frac{\boldsymbol{\nabla}^2}{2m}+V_0\right]\left\langle\ldots\psi_a(x)\ldots \right\rangle\] and also go to momentum space by a Fourier transform \[\int_x e^{+i\omega_p x^0-i\mathbf{p}\mathbf{x}}.\] The operator simply removes the propagator leading to \(x\), because of \[\begin{split} i\left[-i\partial_{x^0} - \tfrac{\boldsymbol{\nabla}^2_x}{2m}+V_0\right]\; \left[-i\Upsilon_{ab}(x-y)\right] &=\delta_{ab}\int\frac{d^4p}{(2\pi)^4} e^{ip(x-y)} \frac{-p^0+\tfrac{\mathbf{p}^2}{2m}+V_0}{-p^0+\tfrac{\mathbf{p}^2}{2m}+V_0} = \delta_{ab} \delta^{(4)}(x-y). \end{split}\] One says that the correlation function is “amputated” because the external propagator has been removed.

Feynman rules for S-matrix elements in momentum space

Moreover, all expressions are brought back to momentum space. One can formulate Feynmann rules directly for contributions to \(i{\cal T}\) as follows.

• Incoming fermions are represented by an incoming line

  

  (to be read from right to left) associated with a momentum \(\mathbf{p}\) and energy \(\omega_{\mathbf{p}} = \tfrac{\mathbf{p}^2}{2m}+V_0\).

• Outgoing fermions are represented by an outgoing line

  

• Incoming or outgoing bosons are represented by

  

  and

  

  respectively.

• Vertices,

  

  contribute a factor \(-ig\).

• Internal lines that connect two vertices are represented by Feynmann propagators in momentum space, e. g. 

  

• Energy and momentum conservation are imposed on each vertex.

• For tree diagrams, all momenta are fixed by energy and momenta conservation. For loop diagrams one must include an integral over the loop momentum \(l_j\) with measure \(d^4 l_j / (2\pi)^4.\)

• Some care is needed to fix overall signs for fermions.

• Some care is needed to fix overall combinatoric factors from possible interchanges of lines or functional derivatives.

For the last two points it is often useful to go back to the algebraic expressions or to have some experience. We will later discuss very useful techniques based on generating functionals.

Fermion-fermion scattering

We will now discuss an example, the scattering of (spin polarized) fermions of each other. The tree-level diagram is

Because the interaction with the scalar field does not change the spin, the outgoing fermion with momentum \(\mathbf{q}_1\) will have spin \(\uparrow\), the one with momentum \(\mathbf{q}_2\) will have spin \(\downarrow\). By momentum conservation the scalar line carries the four momentum \[(\omega_{\mathbf{p}_1} - \omega_{\mathbf{q}_1},\; \mathbf{p}_1 - \mathbf{q}_1) = \left(\tfrac{\mathbf{p}^2_1}{2m}- \tfrac{\mathbf{q}^2_1}{2m}, \mathbf{p}_1 - \mathbf{q}_1\right) = (\omega_{\mathbf{q}_2} - \omega_{\mathbf{p}_2},\; \mathbf{q}_2 - \mathbf{p}_2).\] The last equality follows from overall momentum conservation, \(p_1 + p_2 = q_1 + q_2.\) The Feynmann rules give \[i{\cal T} = (-ig)^2 \frac{-i}{-(\omega_{\mathbf{p}_1}-\omega_{\mathbf{q}_1})^2 + (\mathbf{p}_1 - \mathbf{q}_1)^2 +M^2}.\] In the center-of-mass frame, one has \(\omega_{\mathbf{p}_1} = \omega_{\mathbf{p}_2} = \omega_{\mathbf{q}_1} = \omega_{\mathbf{q}_2}\) and thus \[{\cal T} = \frac{g^2}{(\mathbf{p}_1 - \mathbf{q}_1)^2 + M^2}.\]

Limits of large and small mass

Note that for \(g^2 \to \infty,\) \(M^2 \to \infty\) with \(g^2/M^2\) finite, \({\cal T}\) becomes independent of momenta. This resembles closely the \(\lambda(\phi^* \phi)^2\) interaction we discussed earlier for bosons. More, generally, one can write \[(\mathbf{p}_1 - \mathbf{q}_1)^2 = 2|\mathbf{p}_1|^2 (1-\cos(\vartheta)) = 4|\mathbf{p}_1|^2 \sin^2(\vartheta/2),\] where we used \(|\mathbf{p}_1| = |\mathbf{q}_1|\) in the center of mass frame and \(\vartheta\) is the angle between \(\mathbf{p}_1\) and \(\mathbf{q}_1\) (incoming and outgoing momentum of the spin \(\uparrow\) particle). For the differential cross-section \[\frac{d\sigma}{d \Omega_{q_1}} = \frac{|{\cal T}|^2 m^2}{16 (\pi)^2},\] we find \[\frac{d\sigma}{d\Omega_{q_1}} = \frac{g^4 m^2}{16 \pi^2}\left[\frac{1}{4 \mathbf{p}_{1}^{2} \sin^2(\vartheta/2) +M^2}\right]^2 .\] Another interesting limit is \(M^2 \to 0\). One has then \[\frac{d\sigma}{d\Omega_{q_1}} = \frac{g^4 m^2}{64 \pi^2 |\mathbf{p}_1|^4}\frac{1}{\sin^4(\vartheta/2)}.\] This is the differential cross-section form found experimentally by Rutherford. It results from the exchange of a massless particle or force carrier which is here the scalar boson \(\phi\) and in the case of Rutherford experiment (scattering of \(\alpha\)-particles on Gold nuclei) it is the photon. This cross section has a strong peak at forward scattering \(\vartheta \to 0,\) and for \(\mathbf{p}^2 \to 0\). These are known as colinear and soft singularities. Note that they are regulated by a small, nonvanishing mass \(M > 0\).

Relativistic fermions

To understand relativistic fermions we need to first understand the properties of the Lorentz group in more detail. Diracs description of relativistic fermions follows then very naturally.

Rotations and Lorentz transformations

We use here conventions where the metric in four dimensional Minkowski space is given by \[\eta_{\mu\nu} = \eta^{\mu\nu} = \text{diag}(-1,+1,+1,+1). \label{eq:metricMinkowski}\] Infinitesimal Lorentz transformations and rotations in Minkowski space are of the form \[\Lambda^\mu_{\;\;\nu} = \delta^\mu_{\;\;\nu} + \delta \omega^\mu_{\;\;\nu}, \label{eq:LorentzTransformInfinitesimal}\] with \(\Lambda^\mu_{\;\;\nu} \in \mathbb{R}\) such that the metric \(\eta_{\mu\nu}\) is invariant, \(\eta_{\mu\nu}\to \eta_{\rho\sigma}\Lambda^\rho_{\;\;\mu} \Lambda^\sigma_{\;\;\nu}=\eta_{\mu\nu}\). This implies \((\Lambda^{-1})^\mu_{\;\;\nu} = \Lambda_\nu^{\;\;\mu}\) and, for the inifinitesimal transformation, \[\delta\omega_{\mu\nu} = - \delta\omega_{\nu\mu}.\] The spatial-spatial components describe rotations the three dimensional subspace and the spatial-temporal components Lorentz boost in Minkowski space or rotations around a particular three-dimensional direction in Euclidean space.

Scalar, vector and tensor representations

Lorentz scalars are defined as objects that do not change at all under Lorentz transformations (including rotations). For scalar fields only the argument gets transformed, \[\phi(x) \to \phi^\prime(x) = \phi(\Lambda^{-1}x).\] Lorentz vectors are defined as quantities that get transformed by the matrix \(\Lambda\). For example, the momentum of a particle transforms as \[p^\mu \to p^{\prime\mu} = \Lambda^\mu_{\;\;\nu} p^\nu.\] A vector field like for examle the velocity field of a relativistic fluid transforms as \[u^\mu(x) \to u^{\prime\mu}(x) = \Lambda^\mu_{\;\;\nu} u^\nu(\Lambda^{-1} x).\] In addition to the transformation of the space-time argument there is now an exlicit transformation matrix acting on the index of the field. In a similar way, a covector field like the electromagnetic gauge field transforms according to \[A_\mu(x) \to A^\prime_\mu(x) = (\Lambda^{-1})^\nu_{\;\;\mu} A_\nu(\Lambda^{-1}x) = \Lambda^{\;\;\nu}_{\mu} A_\nu(\Lambda^{-1}x).\] One can go on in this way and define tensor field representations, for example a \((2,0)\)-tensor field transforms like \[T^{\rho\sigma}(x) \to T^{\prime\rho\sigma}(x) = \Lambda^\rho_{\;\;\mu} \Lambda^\sigma_{\;\;\nu} T^{\mu\nu}(\Lambda^{-1} x).\] In the next step we generalize this concept even further.

Lie groups and representaions

We consider representations of a group acting on a complex vector space. It can be seen as a map \(\rho\) \[\rho: G \to \text{GL}(n,\mathbb{C}),\] where \(\text{GL}(n,\mathbb{C})\) is the general linear group in \(N\) complex dimensions or group of complex \(n\times n\) matrices. The map must be such that \[\rho(g_1) \rho(g_2) = \rho(g_1 g_2),\] for all \(g_1, g_2 \in G\). We are specifically interested in Lie groups where finite transformations can be written in terms of infinitesimal transformations through the exponential map, \[g= \exp(i \xi^j T_j) = \lim_{N\to\infty} \left( \mathbb{1} + i \frac{\xi^j T_j}{N} \right)^N.\] Here, \(T_j\) are the generators of the Lie algebra. The Lie algebra, and indirectly the Lie group, are characterized by the Lie bracket or commutation relation \[[T_j, T_k] = i f_{jk}^{\;\;\,l} \, T_l,\] where \(f_{jk}^{\;\;\,l}\) are the structure constants.

A repesentation of a group element can similarly be written as an exponential map \[\rho(g) = \exp\left( i \xi^j T^{(R)}_j\right)\] where \(T^{(R)}_j\) are now representations of the Lie algebra generators acting in some vector space. They must have the same Lie bracket relation as the original generators or fundamental representation, \[\left[T^{(R)}_j, T^{(R)}_k \right] = i f_{jk}^{\;\;\,l} \, T^{(R)}_l.\] In this sense one can construct representations of a Lie group by finding representations of the associated Lie algebra.

Complex conjugate representations

For Lie groups where the structure constants are real one can find for representations \(T^{(R)}_j\) acting in a complex vector space also the complex conjugate representations \[T_j^{(C)} = (T_j^{(R)})^\dagger.\] Indeed this also fulfills the Lie bracket relation as follows by taking the hermitean conjugate on both sides. Sometimes the (representations of the) Lie algebra generators \(T_j^{(R)}\) are hermitean already, and in this case the complex conjugate representation is equivalent to the original one, but that is not always the case.

Lie algebra of Lorentz group

Representations of the Lorentz group with \[L(\Lambda^\prime \Lambda) = L(\Lambda^\prime) L(\Lambda),\] can be written in infinitesimal form as \[L(\Lambda) = \mathbb{1} + \frac{i}{2} \delta \omega_{\mu\nu} M^{\mu\nu}, \label{eq:defGeneratorM}\] where \(M^{\mu\nu} = - M^{\nu\mu}\) are the generators of the Lorentz algebra (or Lie algebra associated to the Lorentz group) acting in some representation space with the commutation relation or Lie bracket \[\left[ M^{\mu\nu}, M^{\rho\sigma} \right] = i \left( \eta^{\mu\rho} M^{\nu\sigma} - \eta^{\mu\sigma} M^{\nu\rho} - \eta^{\nu\rho} M^{\mu\sigma} + \eta^{\nu\sigma} M^{\mu\rho} \right). \label{eq:LorentzAlgebra}\]

Decomposition of Lie algebra

In general, one can decompose the generators into the spatial-spatial part \[J_i = \frac{1}{2} \epsilon_{ijk} M^{jk}, \label{eq:defJ}\] and a spatial-temporal part, \[K_j = M^{j0}. \label{eq:defK}\] Equation \(\eqref{eq:LorentzAlgebra}\) implies the commutation relations \[\begin{split} & \left[ J_i, J_j \right] = + \,i\, \epsilon_{ijk} J_k,\\ & \left[ J_i, K_j \right] = +\, i\, \epsilon_{ijk} K_k,\\ & \left[ K_i, K_j \right] = - i\, \epsilon_{ijk} J_k. \end{split}\] One can define the linear combinations of generators \[N_j = \frac{1}{2} (J_j - i K_j), \quad \quad \tilde N_j = \frac{1}{2} (J_j + i K_j), \label{eq:defNjGeneral}\] for which the commutation relations become \[\begin{split} & [ N_i, N_j ] = i \epsilon_{ijk} N_k,\\ & [ \tilde N_i, \tilde N_j ] = i \epsilon_{ijk} \tilde N_k,\\ & [ N_i, \tilde N_j ] = 0. \end{split}\] This shows that the representations of the Lorentz algebra can be decomposed into two representations of SU(2) with generators \(N_j\) and \(\tilde N_j\), respectively.

Fundamental representation

In the fundamental representation \(\eqref{eq:LorentzTransformInfinitesimal}\) one has the generators \[(M_F^{\mu\nu})^\alpha_{\;\;\beta} = - i (\eta^{\mu\alpha}\delta^{\nu}_\beta - \eta^{\nu\alpha} \delta^\mu_\beta). \label{eq:fundamentalGeneratorLorentz}\] It acts on the space of four-dimensional vectors \(p^\alpha\) and the infinitesimal transformation in \(\eqref{eq:LorentzTransformInfinitesimal}\) induces the infinitesimal change \[\delta p^\alpha = \frac{i}{2} \delta\omega_{\mu\nu} (M_F^{\mu\nu})^\alpha_{\;\;\beta} \; p^\beta = \delta\omega^\alpha_{\;\;\beta} p^\beta.\] The generator of rotations in the fundamental representation is \[(J_{i}^F)^j_{\;\;k} = - i \epsilon_{ijk},\] where \(j,k\) are spatial indices. All other components vanish, \((J^F_{i})^0_{\;\;0} = (J^F_{i})^0_{\;\;j} = (J^F_{i})^j_{\;\;0}=0\). Note that \(J^F_{i}\) is hermitian, \((J^F_{i})^\dagger = J^F_{i}\). The generator \(K_j\) has the fundamental representation \[(K^F_j)^0_{\;\;m} = - i \delta_{jm}, \quad\quad\quad (K^F_j)^m_{\;\;0} = - i \delta_{jm},\] and all other components vanish, \((K^F_j)^0_{\;\;0} = (K^F_j)^m_{\;\;n}=0\). From these expression one finds that the conjugate of the fundamental representation of the Lorentz algebra has the generators \[J^C_j = (J_j^F)^{\dagger} = J^F_j, \quad\quad \quad K^C_j = (K^F_j)^\dagger = - K^F_j. \label{eq:defconjugateRepGeneral}\] This implies that \(K^F_j\) is anti-hermitian, \[(K^F_j)^\dagger = - K^F_j. %\quad \quad \text{(Minkowski)} \label{eq:defconjugateRepMinkowski}\]

Note that \(N_j\) and \(\tilde N_j\) are hermitian and linearly independent in the fundamental reprsentation. There is however an interesting relation between them: Consider the hermitian conjugate representation of the Lorentz group as related to the fundamental one by eq. \(\eqref{eq:defconjugateRepGeneral}\). The representation of the generators \(N_j, \tilde N_j\) is \[\begin{split} N^C_j & = \frac{1}{2} (J^C_j - i K^C_j) = \frac{1}{2} (J_j^F + i K_j^F) = \tilde N_j^F,\\ \tilde N^C_j & = \frac{1}{2} (J^C_j + i K^C_j) = \frac{1}{2} (J_j^F - i K_j^F) = N_j^F. \end{split} \label{eq:NjC}\] This implies that the role of \(N_j\) and \(\tilde N_j\) is interchanged in the conjugate representation.