3.4 Spinor field theory

So far, we have focused on scalar fields, which live in the trivial representation of the Lorentz group and correspond to spin-$0$ bosons. In this section, we discuss the field theory for spin-$\frac{1}{2}$ particles, or fermions, which constitute all matter in the universe.

3.4.1 The Dirac equation

Like the Klein-Gordon equation, the Dirac equation was also an attempt at a relativistic version of the Schrödinger equation. Before the development of QFT, the quantized KG equation was thought to produce negative probabilities due to its second derivative in time.¹² Dirac thus sought a relativistic first-order differential equation in space and time.

Legend has it he was staring into a fire in Cambridge when he came up with an equation of the form

where ${\gamma }^{\mu }$ are constants that will be defined in a moment, and $\psi$ is a complex field. It is difficult to make this equation Lorentz covariant; indeed, it is impossible if $\psi$ is a scalar and each ${\gamma }^{\mu }$ is simply a number.¹³ Dirac’s brilliant insight, however, was that it can be covariant if ${\gamma }_{\mu }$ are $4\times 4$ complex matrices and $\psi$ a four component field.

The key is that ${\gamma }^{\mu }{\partial }_{\mu }$ is essentially the “square-root” of the d’Alembertian $\text{□}$ from the KG-equation:

if (and only if) ${\gamma }^{\mu }$ and ${\gamma }^{\nu }$ satisfy the Clifford algebra:

where $\{A,B\}=\mathit{AB}+\mathit{BA}$ is the anticommutator. Dirac found this is possible with $4\times 4$ matrices such as

where ${\sigma }^i$ are the Pauli matrices (Chapter 2.2). These are called the gamma, or Dirac, matrices, and plugging them into Eq. 3.101 yields the Dirac equation, which can be written even more compactly by defining $\cancel{\partial }\equiv {\gamma }^{\mu }{\partial }_{\mu }$:

This equation is considered one of the most significant breakthroughs in theoretical physics, “on par with the works of Newton, Maxwell, and Einstein before him” [36]. The insights that followed, as we will outline in this section, provided a theoretical basis for fermion spin, implied the existence of antiparticles, and overall were foundational to the development of the SM.¹⁴

3.4.2 Spinors

Before discussing solutions and quantization of the Dirac equation, let us examine what kind of object $\psi$ is. A related property of the Clifford algebra is that

satisfies the Lorentz algebra (Eq. 2.32). This means ${\mathrm{\Sigma }}_{\mathit{\mu \nu }}$ are generators of Lorentz transformations

where $\mathrm{\Lambda }$ is a Lorentz transformation with parameters ${\omega }^{\mathit{\mu \nu }}$, and $S[\mathrm{\Lambda }]$ is a particular 4D representation.

It can be shown¹⁵ that the Dirac equation is only Lorentz covariant if the components of $\psi$, ${\psi }_{\alpha }$, transform under this exact representation:

It is important to note here that $S[\mathrm{\Lambda }]$ is acting on the $\psi$ components — also called the spinor indices – and not on the spacetime coordinates $x^{\mu }$, which transform under the vector representation (Eq. 2.22). Explicitly, including the spacetime coordinates, $\psi (x)$ transforms as:

where both $S[\mathrm{\Lambda }]$ and $\mathrm{\Lambda }$ share the same transformation parameters ${\omega }^{\mathit{\mu \nu }}$ and thus correspond to the same Lorentz transformation.¹⁶

Dirac and Weyl spinors

What is this representation? Let’s look at the rotation and boost generators individually:

Comparing this with Eqs. 2.28 and 2.29, we see that the top left and bottom right blocks are exactly the left- and right-handed Weyl spinor irreps of the generators. The handedness of a spinor is called its chirality, and its physical significance will be discussed in a moment. Thus, we identify $S[\mathrm{\Lambda }]$ with the $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$, or Dirac spinor, representation.

This also means that, in this basis of the gamma matrices (called the Weyl, or chiral, basis), the Dirac spinor $\psi$ can be decomposed into two Weyl spinors:

which transform under their respective representations. The two components can be isolated if we consider a fifth gamma matrix:

${\gamma }^5$ is similar to our main four matrices in that $\{{\gamma }^5,{\gamma }^{\mu }\}=0$ and ${({\gamma }^5)}^2=𝟙$ . Importantly, we see from its form in the Chiral basis that projection operators $P_L$ and $P_R$ can be defined as:

which satisfy the projection property $P_{L∕R}^2=P_{L∕R}$ and project out the left- and right-handed components of a Dirac spinor:

Note that while the specific form depends on the basis, the definitions in Eq. 3.113 are basis-independent and can be considered to define chirality.

Chirality

The two Weyl spinor representations are related by a complex conjugation, meaning ${\psi }_L^{\ast }$ is a right-handed Weyl spinor, and vice versa. For a complex scalar field, we interpreted the conjugate as the antiparticle. The same interpretation applies here; hence, if a left-handed spinor describes a particle, its antiparticle is described by its conjugate, right-handed spinor.

The Dirac equation can be rewritten in the Weyl basis as two coupled equations of the Weyl spinors. Let us define ${\sigma }^{\mu }=(𝟙,\sigma )$ and ${\overline{\sigma }}^{\mu }=(𝟙,-\sigma )$, so that

Hence, we see the mass term couples the left- and right-handed components. This is why all massive fermions must exist in pairs of particles and antiparticles. An important special case, however, is for a neutral Majorana fermion, where $\psi$ equals its charge conjugate ${\psi }^c$ (to be defined below). Such a particle is its own antiparticle and can have a left-handed- or right-handed-only mass term. As discussed in Chapter 2.3, the only Majorana candidate in the SM is the right-handed neutrino.

For $m=0$, the Dirac equation decouples and leaves us with the Weyl equations describing massless fermions:

In Fourier space, these are:

where we used $E=\left|p\right|$ for massless particles. You may recall $\frac{\sigma \cdot p}{\left|p\right|}$ is the helicity operator, projecting the particle spin along its momentum. Thus, in the massless limit, we see that the left- and right-handed Weyl spinors are the $+1$ and $-1$ helicity eigenstates, respectively. This is not the case for massive particles, as helicity is no longer Lorentz invariant: one can always boost into a frame where the momentum is inverted while the spin remains the same, changing the sign of the helicity. Chirality is thus a more abstract concept for massive particles, related only to how they transform under Lorentz transformations.

Theories not symmetric under exchange of left- and right-handed components are called chiral, and symmetric theories vector. QED and QCD are both vector theories, but weak interactions are, surprisingly, chiral. This necessarily means it violates parity and charge conjugation symmetries ($P$ and $C$), which we will discuss soon in Section 3.4.6.

3.4.3 The Dirac Lagrangian

Recall that to quantize the scalar theory, we first needed the Lagrangian and the classical solutions of the K-G equation, to then obtain Hamiltonian and canonical fields and Poisson brackets before finally promoting them to quantum commutatation relations. We will proceed in similar (though condensed) fashion for the spinor theory, and first derive the Lagrangian corresponding to the Dirac equation.

Since we are no longer dealing with trivial representation of the Lorentz group, we have to be more careful with the types of terms we put into the Lagrangian; it must be composed of good Lorentz-invariant objects. A first guess at a Lorentz scalar formed of spinors may be ${\psi }^{†}\psi$. This is indeed a scalar, but it is not Lorentz invariant: $\psi$ and ${\psi }^{†}$ transform as $\psi \to S[\mathrm{\Lambda }]\psi$, ${\psi }^{†}\to {\psi }^{†}S{[\mathrm{\Lambda }]}^{†}$ and, hence

However, recall from Chapter 2.3 that (finite-dimensional) representations of Lorentz transformations are not unitary. (We can see this as well from the fact that the generators of $S[\mathrm{\Lambda }]$ in Eq. 3.110 are not anti-Hermitian.) Thus, $S{[\mathrm{\Lambda }]}^{†}S[\mathrm{\Lambda }]\ne 1$ in general and ${\psi }^{†}\psi$ is not a Lorentz scalar.

Instead, with a bit of matrix algebra¹⁷, one can show that

and hence

is a Lorentz scalar. Thus, we define $\overline{\psi }\equiv {\psi }^{†}{\gamma }^0$ as the “natural” conjugate to $\psi$, and end up with a nice Lorentz scalar $\overline{\psi }\psi$ for our Lagrangian.

Similarly, one can show that $\overline{\psi }{\gamma }^{\mu }\psi$ transforms as a Lorentz $4$-vector and, hence, contracting it with ${\partial }_{\mu }$ as $\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$ yields another scalar. These two terms, which are analogous to the mass and derivative terms a free complex scalar field (Eq. 3.26), are enough to build the Dirac lagrangian:

One can check that the EL equations reproduce the Dirac equation for $\psi$ and $\overline{\psi }$.

The U(1) conserved current

As with the complex scalar field, observe that the Dirac Lagrangian is invariant under global $U(1)$ symmetry $\psi \to e^{\mathit{i\alpha }}\psi$. Using Noether’s theorem, we can derive the conserved current and charge associated with this symmetry:

As for the complex scalar field, these represent the electromagnetic $4$-current and charge, respectively — a connection we will explore further in Section 3.5.2.

3.4.4 Quantizing the Dirac field

Solutions to the Dirac equation

Before quantizing, we first need the classical solutions to the Dirac equation. Multiplying both sides of it by $-(i{\gamma }^{\mu }{\partial }_{\mu }+m)$ gives us:

which means each component of $\psi$ individually satisfies the KG-equation. Thus, we can assume similar plane wave solutions:

where $u(p)$ and $v(p)$ are now spinors, and again we have positive and negative frequency solutions that correspond to particles and antiparticles, respectively, after quantization.

One can check using Fourier space, as we did for the Weyl equations, that

are general solutions to the Dirac equation, where $\xi$ and $\eta$ are the familiar two-component spinors from QM for spin-$\frac{1}{2}$ particles (although technically they do not have this interpretation before quantization). As is conventional, we will use a basis of ${\sigma }_z$ eigenstates ${\xi }_1={\eta }_1={(1,0)}^T$ and ${\xi }_2={\eta }_2={(0,1)}^T$, corresponding to spin-up and spin-down, respectively.

For example, in the rest frame $p_{\mu }=(m,0,0,0)$, we have:

More generally, we can always orient a particle’s 3-momentum along the $z$-axis, in which case:

Quantization

Now that we have a sensible Lagrangian and the classical solutions to the Dirac equation, the remaining steps to quantization follow closely that for our complex scalar field in Section 3.2.4, but with two notable differences. The first is that we now must sum over the two spin components of $u_s(p)$ and $v_s(p)$, in addition to integrating over the momentum:

As before, we have positive and negative frequency solutions, with the $b∕b^{†}$ and $c∕c^{†}$ operators associated with particles of the same mass and opposite charge. For spinors, we find that the ${\widehat{b}}^{s†}|0⟩$ and ${ĉ}^{s†}|0⟩$ also have opposite spins, i.e. for the $z$-axis angular momentum operator $J_z$ (which can be derived through Noether’s theorem as we did for the momentum operator in Section 3.1.3):

By convention, we take $b^{s†}$ and $b^s$ to be the creation and annihilation operators for the electron, and $c^{s†}$ and $c^s$ for its antiparticle, the positron. Thus, ${\overline{\psi }}_s(x)|0⟩$ corresponds to an electron at $x$ with spin state $s$, and ${\psi }_s(x)|0⟩$ to a positron at $x$ with the opposite spin state to $s$.

Through his equation, Dirac was the first to predict the existence of antimatter in 1930 [38] (although he initially thought the electron’s antiparticle was the proton). This prediction was soon confirmed by the discovery of a particle with the same mass as the electron but opposite charge by Carl Anderson in a bubble chamber in 1932 [39]. Both were awarded the Nobel prize.

The spin-statistics connection

The second, extremely important difference from scalar quantization is that, because spinors are spin-$\frac{1}{2}$ particles, they must obey anticommutation relations:

which also means the creation and annihilation operators satisfy:

Thus, unlike bosons, exchanging two particles yields a minus sign: ${\widehat{b}}_{p_1}^{r†}{\widehat{b}}_{p_1}^{s†}|0⟩=-{\widehat{b}}_{p_2}^{s†}{\widehat{b}}_{p_1}^{r†}|0⟩$, confirming that spinors obey Fermi-Dirac statistics and obey the Paul-Exclusion principle. Were we to try and impose our earlier commutation relations for spinors (or indeed, any half-integer-spin field), we would run into several issues. These include the time-ordered product in the $S$-matrix not being Lorentz invariant, and antiparticles contributing arbitrarily negative energies, making the theory unstable. They are all related to the deep connection between spin and statistics: the requirement of Lorentz invariance, stability, and causality in a QFT necessitates that half-integer-spin particles obey Fermi-Dirac, and integer-spin particles Bose-Einstein statistics.¹⁸

3.4.5 Interactions and Feynman rules

Having quantized the free Dirac field, we now discuss interactions, again focusing on small (and renormalizable) perturbations to the free theory. We start by presenting the propagators for the Dirac field and then extending our scalar Yukawa theory from Section 3.3 to spinor “nucleons”.

Propagators

We define the propagator for the Dirac field the same as for scalar fields in Section 3.2.5:

where $\alpha$ and $\beta$ index the spinor components. Again, we have an extra sum over the spin states. With some more matrix algebra one can show that these kinds of sums simplify nicely to

so that we end up with, in momentum space, the Feynman propagator:

Note that we have now suppressed the spinor indices; ${\mathrm{\Delta }}_F$ is still a $4\times 4$ matrix in spinor space. Note as well the relative minus sign in the time-ordering operator for fermions, due to exchanging the fields:

External lines

For scalars, external line terms such as $\varphi |p⟩$ simply contributed a factor of $1$ to the matrix element, where $|p⟩$ is again a one-particle meson state with momentum $p$:

(The $e^{-\mathit{ip}\cdot x}$ factor contributes only to the momentum conservation delta function in the $S$-matrix element.) For spinors, we instead end up with a spinor factor. For example, for an incoming fermion with momentum $q$ and spin $s$:

We can see looking at the form of the quantized fields (Eq. 3.128), and which terms will contribute something non-zero, that incoming (outgoing) external fermions will be associated with a $u$ $(ū$) and antifermions with a $\overline{v}$ $(v$) factor.¹⁹

Yukawa theory reloaded

We now revisit Yukawa theory, the simplest possible theory of interactions for spinors. The Lagrangian is the same as in Eq. 3.69, but now with $\psi$ a spinor:

Note that through dimensional analysis, since $[M\overline{\psi }\psi ]=[\overline{\psi }\cancel{\partial }\psi ]\stackrel{!}{=}4$ we can deduce that $[\psi ]=\frac{3}{2}$. This means that (1) the Yukawa interaction is marginal, with $[\varphi \overline{\psi }\psi ]=4$ and $[g]=0$, and (2) importantly, there are no other renormalizable, Lorentz-invariant interactions we can write down for spinors with the fields at our disposal (modulo some ${\gamma }^5$’s thrown in, as we’ll discuss in Section 3.4.6). Terms like $\psi {\varphi }^2$, $\cancel{\partial }\mathit{\psi \varphi }$, or $\overline{\psi }\psi {\varphi }^2$ are all either not Lorentz-scalars or of dimension $\ge 5$. In this sense, because their possible interactions are so heavily constrained by their $\frac{3}{2}$-dimensionality, spinors in QFT are quite simple! There is only one other spinor interaction in the SM, which we will see in Section 3.5, with gauge bosons.

We again refer to $\varphi$ and $\psi$ as the “meson” and “nucleon” fields, which is slightly more accurate now since nucleons are in reality fermions. The two main features missing from this theory are that the relevant mesons, the pions, are pseudoscalars (to be discussed in the next section) and are a strong isospin triplet (to be described briefly in Chapter. 4.1).

Meson decay and the Higgs decay width

The matrix element for meson decay into a fermion-antifermion pair with spin and momentum $s_1,q_1$ and $s_2,q_2$, respectively, to first-order can be read off from the Feynman diagram in Figure 3.7:

We can calculate the decay rate as in Section 3.3.4, except now we have to sum over the spins of the fermions:

In the COM frame, we can choose $q_1=(\frac{m}{2},0,0,q)$ and $q_2=(\frac{m}{2},0,0,-q)$, with $q^2=\frac{m^2}{4}-M^2$ by energy conservation. Using the forms of ${ū}_s$ and $v_s$ we found in Eq. 3.127, we see that the sum over spin states simplifies nicely:

Since this is independent of the final state kinematics, the integral of $d{\mathrm{\Pi }}_{\mathrm{LIPS}}$ is the same as for the scalar meson decay, and we obtain an overall decay rate of:

As we hinted at in Section 3.3.4, this is in fact the decay width of the Higgs boson to fermions at tree level, if we plug in the Higgs Yukawa coupling constant $g_f=\frac{\sqrt{2}{m}_f}{v}$. Here $m_f$ is the fermion mass and $v$ is the Higgs vacuum expectation value, $246\text{GeV}$. For example, for the $H\to {\mu }^{+}{\mu }^{-}$ decay, with $M=m_{\mu }=105.7\text{MeV}$ and $m=m_H=125\text{GeV}$, we get $\mathrm{\Gamma }\approx 900\text{eV}$, exactly in line with the predicted value [40]!

One can similarly update our nucleon scattering amplitudes from Section 3.3.3, which simply gain some inner products between the incoming and outgoing spin states (see e.g. Tong QFT [3] Chapter 5.7). Notably, however, the $t$-channel and $u$-channel diagrams (Figure 3.3) now have a relative minus sign, in accordance with Fermi-Dirac statistics.

3.4.6 CPT Symmetries

In this section, we discuss three important discrete symmetries in QFT. As discussed in Chapter 2.3, the full Lorentz group includes the parity $P$ and time reversal $T$ operators. In the $4$-vector representation, they have the simple forms $P=\mathrm{diag}(1,-1,-1,-1)$ and $T=\mathrm{diag}(-1,1,1,1)$, meaning

However, their forms in other representations, such as spinors, are not as straightforward.

Observe also that all our complex Lagrangians so far have been invariant under some form of complex conjugation $\psi \leftrightarrow {\psi }^{\ast }$. This represents another discrete symmetry, and since we know from Eq. 3.28 that complex conjugation inverts “charge”, we call this charge conjugation, or $C$, symmetry.

All local, relativistic QFTs are necessarily invariant under the combined $\mathit{CPT}$ symmetry; this is known as the CPT theorem [41, 42].²⁰ Whether a theory is individually $C$, $P$, or $T$ invariant, however, must be determined by experiment,²¹ as we give examples of below. If it is, we must impose the symmetries in our mathematical formulation by carefully defining the actions of the relevant operators; i.e., we have to consider how $\psi$ must transform under $P$ to maintain $P$-invariance of the Lagrangian, etc.

Such symmetries are crucial handles for understanding QFTs, particularly in the case of the weak and strong interactions for which we have otherwise little classical intuition. By studying them, we often glean important insights into the theory, such as why certain processes are forbidden: for example, we now understand that the pion cannot decay into three photons because this would violate the $C$-invariance of QED.

$P$- and $\mathit{CP}$-violation

Historically, it was thought that parity individually is a universal symmetry of nature. Indeed, this was verified experimentally for electromagnetism and the strong interaction, but, surprisingly, in 1956 an experiment measuring the isotropy of the beta decay of cobalt-60 to nickel-60 by Chien-Shiung Wu showed that the weak interaction in fact violates parity- (and $C$-) invariance [43]. The two theorists, Yang Chen-Ning and Lee Tsung-Dao, who proposed this experiment won the Nobel prize the year after but, controversially, Wu did not.

It was then proposed by Lev Landau [44] and others that perhaps the combined $\mathit{CP}$-symmetry is the true symmetry of nature. As we define below, the $\mathit{CP}$ operation transforms a particle into its antiparticle, hence, $\mathit{CP}$-invariance can be thought of as saying the laws of physics are the same for particles and antiparticles. This indeed appeared to be the case until 1964, when the Fitch-Cronin experiment discovered small, indirect $\mathit{CP}$-violation by the weak interaction by measuring decays of neutral kaons [45], for which another Nobel prize was awarded to James Cronin and Val Fitch. Since then, several experiments have observed both direct and indirect $\mathit{CP}$-violation, and quantifying the magnitude of $\mathit{CP}$-violation in different sectors of the SM remains an active area of research in HEP (see Ref. [46] Chapters 13-14 for a nice comprehensive review).

Interestingly, $\mathit{CP}$-violation is only possible through the weak interaction if there exist $\ge 3$ generations of fermions, whereas it is expected for the strong interaction but not observed (the so-called “strong $\mathit{CP}$ problem” [47, 48].²² Furthermore, the experimentally determined magnitude of $\mathit{CP}$-violation in the weak interaction is about $1000\times$ smaller than what is allowed [46, 48]. These mysterious “coincidences” — Why did nature “choose” exactly the minimum number of generations needed for $\mathit{CP}$-violation? Why is there no strong $\mathit{CP}$-violation? etc. — suggest deeper underlying physics, such as “axions” [49].

Scalar fields

We see from our complex scalar Lagrangian in Eq. 3.26 that it can only be invariant under $C$, $P$, or $T$ if they transform the field $\varphi$ by at most a complex phase: $\varphi \to e^{\mathit{i\alpha }}\varphi$. A further physical requirement, however, is that applying any of the operators twice should return the original field, which thus constrains the possible transformations to:

The time-reversal operation is a bit subtle, as it must be anti-unitary. We will not discuss it much further, although its implications can be fun to think about.

Nomenclature Whether a field transforms with a $+$ or $-$ sign under $P$ is called its intrinsic parity, and similarly under $C$ its intrinsic $C$-parity. We also refer to them as “even” or “odd” under the transformation, respectively. In particular, an odd-parity scalar, i.e. one which transforms with a minus sign under parity, is called a pseudoscalar. The Higgs field, for example, is a scalar, while the pion is a pseudoscalar (as was determined based on nuclear interactions).

Vector fields

Though we introduce vector fields in detail in the next section, their transformation properties are analogous to scalars and simple enough to describe here:

where ${\eta }_{\mathit{\mu \nu }}$ is the Minkowski metric (i.e. $P$ and $T$ flip the sign of the first and the last three components of $A^{\mu }$, respectively). We use similar “odd” and “even” nomenclature for vectors, with an odd-parity vector called a pseudovector. Recall for example that the electric and magnetic $3$-vector fields are vectors and pseudovectors, respectively. Notably, the photon is odd under $C$ while the neutral pion; this explains why the pion can decay into two photons (since the two photons have a combined parity of $(-1)(-1)=+1$), but not to three, even though either would be allowed kinematically.

Spinors: parity

Spinors live in a more complicated representation of the Lorentz group, so it takes more work to derive their transformations. On the other hand, this also means their properties and the physical consequences are more interesting.

If $P$ is a true symmetry of the theory, after a parity transformation ${\psi }^{\prime }(x^{\prime })=\mathit{P\psi }(x)P^{†}$ must satisfy the parity-transformed Dirac equation:

where $x^{\mu }\to {x{}^{\prime }}^{\mu }=(x^0,-x)$ and ${\partial }_{\mu }^{\prime }\equiv \partial ∕\partial {x{}^{\prime }}^{\mu }$ under parity. One can see, by multiplying the original Dirac equation by ${\gamma }^0$, that this is satisfied if ${\psi }^{\prime }(x^{\prime })=\pm {\gamma }^0\psi (x)$:

Again, the sign in the transformation indicates the intrinsic parity of the field.

Looking at the form of ${\gamma }^0$ and $\psi$ in the Weyl basis (Eqs. 3.104 and 3.111), we see that the parity transformation swaps around left- and right-handed spinors:

Chirality being inverted makes sense given its (loose) connection to helicity, which is flipped under parity. Similarly, remembering from Section 3.4.4 that particle and anti-particle solutions to the Dirac equation have the form $u(p)\propto {(\xi ,\xi )}^T$ and $v(p)\propto {(\eta ,-\eta )}^T$, respectively, we see that fermions and antifermions have even and odd parity, respectively. The weak interaction breaks parity symmetry by interacting only with left-chiral fermions and right-chiral antifermions.

We can also check that the Lorentz scalars and vectors we constructed, $\overline{\psi }\psi$ and $\overline{\psi }{\gamma }^{\mu }\psi$, are indeed invariant under parity, e.g.:

However, we can also construct pseudoscalars and pseudovectors by throwing in a ${\gamma }^5$ matrix: $\overline{\psi }{\gamma }^5\psi$ and $\overline{\psi }{\gamma }^5{\gamma }^{\mu }\psi$. One can confirm this by grinding it out as above, or by simply looking at their form in the Weyl basis, e.g.:

We thus see that this will pick up an overall minus sign under ${\psi }_L\leftrightarrow {\psi }_R$.

Spinors: charge conjugation and $\mathit{CP}$

Under charge conjugation, $\psi \to {\psi }_c=C{\psi }^{\ast }$, where $C$ is a matrix that can mix up the spinor components. We can follow similar reasoning as for parity to show that ${\psi }_c$ satisfies the Dirac equation only if:

In the Weyl basis, this means $C=\pm i{\gamma }^2$ and thus

where as always the sign in the transformation indicates the intrinsic $C$-parity of the field. Looking at the individual components:

${\gamma }^2$ and complex conjugation both flip chirality, so combined we see that charge conjugation retains it, transforming left-(right-)chiral fermions into left-(right-)chiral antifermions. Thus, the weak interaction violates $C$-symmetry as well by coupling only to opposite-chirality fermions and antifermions.

Combining parity and charge conjugation gives us, in the Weyl basis:

or, in terms of the Weyl spinors:

The combination thus transforms fermions into their opposite-chirality antifermions, and vice versa. Often, this transformation is considered to define the relation between particles and antiparticles, and is a better symmetry of the weak interaction (and, hence, the SM) than $C$ or $P$ individually. However, as discussed above, it is violated as well, to a lesser extent, through the mixing of the three generations of fermions.

Spinors: time reversal and CPT

The time reversal operation is more subtle, as it is anti-unitary. We will forego a detailed discussion of these subtleties (see e.g. Schwartz  [16] Chapter 11.6), and note that the time reversal operator $T$ is defined to transform a Dirac spinor in the Weyl basis as:

It flips both the spin and momenta of the fermions, and is violated as well by the weak interaction (as it must be to ensure $\mathit{CPT}$-invariance, given $\mathit{CP}$-violation).

Finally, we can combine all these operations to obtain the $\mathit{CPT}$-transformation of the Dirac spinor:

This transforms a particle into an antiparticle reversed in space and time.

One interesting way of testing $\mathit{CPT}$-invariance is to measure the rates of a process’ $\mathit{CP}$- and $T$-conjugates, and confirm that they are equal. All experimental tests to this date have confirmed $\mathit{CPT}$-invariance [46].