Charge Conjugation and the Positron

The Dirac equation solved the crisis of the negative-energy states in one way — the filled sea of The Dirac Sea and Antiparticles — but that picture is baroque: an infinite charge density in the vacuum, a construction that leans on Pauli exclusion and so has no analogue for bosons, and a manifest asymmetry between a particle and its antiparticle. Charge conjugation replaces all of that with a single matrix identity. It is a definite operation $\psi \mapsto \psi^{c}$ that takes any solution of the Dirac equation coupled to the electromagnetic field with charge $q$ and returns a genuine solution with charge $-q$ — the positron as a partner of the electron, not a hole. The whole construction hangs on one algebraic object, the charge-conjugation matrix $C$ , fixed by how it acts on the gamma matrices: $C\,\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ . That single relation, applied to the complex conjugate of the minimally coupled equation, flips the sign of the coupling and turns a charge- $q$ solution into a charge- $(-q)$ one. In the Dirac representation it is solved by $C=i\gamma^{2}\gamma^{0}$ , a matrix that is antisymmetric, $C^{T}=-C$ , and unitary, $C^{\dagger}=C^{-1}$ . This article owns the operation itself: the derivation of the defining relation from charge reversal, the explicit $C$ and its properties, the map $u\leftrightarrow v$ between electron and positron spinors developed in Dirac Spinors and Plane-Wave Solutions, and one clean physical consequence — the charge parity of the photon and the selection rule it enforces on neutral-pion decay. What charge conjugation is not is the full discrete-symmetry story: $C$ is violated by the weak interaction, and only the combined $CPT$ survives. We flag that boundary; we do not cross it.

Key takeaways

Charge conjugation is the operation $\psi^{c}=C\bar\psi^{T}=C\gamma^{0T}\psi^{*}$ that maps a charge- $q$ Dirac solution to a charge- $(-q)$ one — the positron as a positive-energy partner of the electron.
Defining relation (owned): $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ . It is the transpose, not the naive conjugate, that $C$ intertwines — a genuine trap we verify explicitly.
Explicit matrix (owned): in the Dirac representation $C=i\gamma^{2}\gamma^{0}$ , with $C^{\dagger}=C^{-1}=C^{T}=-C$ (unitary and antisymmetric).
$C$ maps the electron spinor to the positron spinor, $v^{s}(\mathbf p)=C\bar u^{s}(\mathbf p)^{T}$ ; it is the modern, one-particle alternative to the filled sea.
Punchline: the electromagnetic current is $C$ -odd $\Rightarrow$ the photon has charge parity $C=-1$ , so $\pi^{0}\to\gamma\gamma$ is allowed but $\pi^{0}\to 3\gamma$ is forbidden — a selection rule confirmed to one part in $10^{8}$ .
Scope: $C$ is exact for QED but violated by the weak interaction; the full $CPT$ story belongs elsewhere.
Key idea: one matrix identity $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ turns $\psi^{c}=C\bar\psi^{T}$ into a charge-reversing map, making the positron a real particle rather than a hole — and forcing the photon's $C=-1$ .

The core insight

Charge conjugation $\psi^{c}=C\bar\psi^{T}$ , built from the single relation $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ (solved by $C=i\gamma^{2}\gamma^{0}$ in the Dirac representation), sends every charge- $q$ solution of the Dirac equation to a charge- $(-q)$ solution. It replaces the infinite Dirac sea with a symmetric particle/antiparticle bookkeeping, maps the electron spinor onto the positron spinor, and — because the electromagnetic current is $C$ -odd — fixes the photon's charge parity at $-1$ .

Where this lives today

Charge conjugation is a permanent fixture of every quantum field theory. In modern collider physics the $C$ -parity selection rules it enforces are working tools: the two-photon versus three-photon distinction for $C=\pm1$ states, the classification of quarkonium and positronium decays, and the constraints on which final states a given resonance can reach. In antimatter experiments at facilities like CERN's Antihydrogen program, charge conjugation is the theoretical statement being tested to extraordinary precision — comparing the spectrum of antihydrogen with hydrogen is a direct $CPT$ test, and any measured difference would overturn the deepest symmetry of relativistic quantum theory. And the one place $C$ is not a good symmetry — the weak interaction — is where the frontier sits: the observed matter–antimatter asymmetry of the universe requires $C$ and $CP$ violation, and whether the neutrino is its own antiparticle (a Majorana fermion) is the question driving a generation of neutrinoless double-beta-decay searches.

Figure

The photon is C-odd: pi-zero to two photons is allowed, to three is forbidden

The physical picture

Picture the Dirac equation as a machine with a dial set to charge $q$ . Charge conjugation reads off any valid solution, applies one fixed matrix recipe, and hands back a guaranteed solution of the same machine with the dial flipped to $-q$ . Nothing about the spacetime motion is invented; the recipe re-encodes the four spinor components so that the electromagnetic coupling comes out with the opposite sign. The positron's positive energy and positive charge are then not emergent properties of a vacancy in an infinite reservoir — they are intrinsic quantum numbers of a real particle. Why does the recipe involve complex conjugation? Because minimal coupling enters the Dirac equation as $i\partial_{\mu}-qA_{\mu}$ . Take the complex conjugate and the $i$ flips sign, which is the same as flipping the sign of $q$ — provided you can simultaneously undo the damage conjugation does to the gamma matrices. The matrix $C$ is precisely the object that repairs the gammas, and the demand that it do so is what pins down the relation $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ . Everything else follows.

Figure

Charge conjugation flips the dial: a charge-q solution becomes a charge-minus-q solution

From charge reversal to the defining relation

The entire content of charge conjugation is forced by a single demand: that the conjugate of a charge- $q$ solution be a charge- $(-q)$ solution. Working out what matrix accomplishes this yields the relation $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ , and there is a genuine subtlety in whether it is the transpose or the conjugate of the gammas that $C$ must intertwine.

Derivation ACharge conjugation reverses the electromagnetic coupling

We start from the minimally coupled Dirac equation, take its complex conjugate, and force the result back into Dirac form with the charge reversed. The route is: conjugate (which flips $i$ and hits the gammas with a star), then repackage the starred gammas as transposes, then apply the matrix $C$ that turns each $\gamma^{\mu T}$ back into $-\gamma^{\mu}$ . What emerges is that $\psi^{c}\equiv C\bar\psi^{T}$ solves the equation with $q\to -q$ . Work in natural units $\hbar=c=1$ ; the coupling is $q$ (electron $q=-e$ ).

Step 1. Write the equation for charge $q$ . The minimally coupled Dirac equation is

\bigl[\gamma^{\mu}(i\partial_{\mu}-qA_{\mu})-m\bigr]\psi=0 \qquad (1)

where: $A_{\mu}$ is the real electromagnetic potential and $m$ the real mass. The coupling appears as the single combination $i\partial_{\mu}-qA_{\mu}$ .

Step 2. Take the complex conjugate. Since $A_{\mu}$ and $m$ are real and $i^{*}=-i$ ,

\bigl[-\gamma^{\mu *}(i\partial_{\mu}+qA_{\mu})-m\bigr]\psi^{*}=0 \qquad (2)

Why the sign flips: conjugating $i\partial_{\mu}$ gives $-i\partial_{\mu}$ , and factoring an overall minus out of the bracket turns $-qA_{\mu}$ into $+qA_{\mu}$ . The gammas are hit with a star, $\gamma^{\mu}\to\gamma^{\mu *}$ — the damage we must repair.

Step 3. Repackage as a transpose. Using $\gamma^{0\dagger}=\gamma^{0}$ , $\gamma^{i\dagger}=-\gamma^{i}$ (Hermiticity properties), the conjugate is rewritten $\gamma^{\mu *}=(\gamma^{\mu\dagger})^{T}$ , and the Dirac adjoint developed in Bilinear Covariants and Dirac Adjoint lets us pass to $\bar\psi^{T}$ . The upshot is that the object we must transform is the transpose $\gamma^{\mu T}$ , not the bare conjugate.

Step 4. Introduce $C$ . Multiply (2) from the left by a matrix $C$ and demand

C\,\gamma^{\mu T}C^{-1}=-\gamma^{\mu} \qquad (3)

where: this is the defining relation for $C$ . Substituting and identifying $\psi^{c}=C\bar\psi^{T}=C\gamma^{0T}\psi^{*}$ turns (2) into

\bigl[\gamma^{\mu}(i\partial_{\mu}+qA_{\mu})-m\bigr]\psi^{c}=0 \qquad (4)

— the Dirac equation for charge $-q$ . Why this is the whole game: (4) is identical to (1) with $q\to-q$ , so $\psi^{c}$ is a genuine solution describing the antiparticle. The mass and dynamics are untouched; only the charge is reversed. *Verified: the sign bookkeeping $q\to -q$ is confirmed symbolically, and the relation (3) is checked to hold for all four $\mu$ with $C=i\gamma^{2}\gamma^{0}$ (next derivation). Crucially, the analogous relation with the bare conjugate, $C\gamma^{\mu *}C^{-1}=-\gamma^{\mu}$ , is verified to be FALSE for $\mu=1,2,3$ — it is the transpose that $C$ intertwines.*

Go deeper · Derivation B · 1 minThe explicit matrix C in the Dirac representation

Derivation BThe explicit matrix C in the Dirac representation

We solve the defining relation (3) explicitly, constructing $C=i\gamma^{2}\gamma^{0}$ and confirming its unitarity and antisymmetry. The strategy uses the transposition pattern of the Dirac-representation gammas — two symmetric, two antisymmetric — and builds $C$ from the two symmetric ones so that conjugation produces exactly the required sign flips.

Step 1. The transposition pattern. In the Dirac representation,

\gamma^{0T}=\gamma^{0},\quad \gamma^{1T}=-\gamma^{1},\quad \gamma^{2T}=\gamma^{2},\quad \gamma^{3T}=-\gamma^{3} \qquad (5)

where: $\gamma^{0}$ and $\gamma^{2}$ are symmetric, $\gamma^{1}$ and $\gamma^{3}$ antisymmetric. This asymmetry is what a single matrix $C$ must reconcile.

Step 2. The ansatz. Take

C=i\gamma^{2}\gamma^{0} \qquad (6)

built from the two symmetric gammas, with the factor $i$ chosen to make $C$ unitary. Why this combination: it is the unique (up to phase) product that, conjugating each $\gamma^{\mu T}$ , yields $-\gamma^{\mu}$ using only the anticommutator $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$ recalled from Dirac Gamma Matrices and Clifford Algebra.

Step 3. Verify (3). Using $(\gamma^{2})^{2}=-1$ , $(\gamma^{0})^{2}=1$ and anticommutation, one checks $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ for every $\mu$ directly. The symmetric gammas ( $\mu=0,2$ ) and antisymmetric ones ( $\mu=1,3$ ) both come out with the correct sign.

Step 4. Read off the properties. From (6),

C^{\dagger}=C^{-1}=C^{T}=-C \qquad (7)

where: $C$ is antisymmetric ( $C^{T}=-C$ ) and unitary ( $C^{\dagger}=C^{-1}$ ). Why antisymmetry matters: it is what makes $\psi^{c}$ a genuinely independent solution rather than a rephasing of $\psi$ , and it governs the C-parities of the bilinear covariants. *Verified: $C=i\gamma^{2}\gamma^{0}$ satisfies (3) for all $\mu$ ; the properties (7) all hold; and acting on a rest-frame electron spinor $u$ , $C$ lands the result purely in the lower (positron) sector — the electron $\to$ positron map.*

Limiting cases

regime	statement	reading
free particle	$\psi^{c}_{(-E,\mathbf p,s)}\to$ positron $(+E,-\mathbf p,s)$	$C$ turns a negative-energy plane wave into a positive-energy positron
electromagnetic coupling	$[i\gamma^{\mu}(\partial_{\mu}+iqA_{\mu})-m]\psi^{c}=0$	$C$ exactly reverses the charge; QED conserves $C$
bosons (Klein–Gordon)	$\phi\to\phi^{*}$	charge conjugation exists without a sea — the sea has no bosonic analogue
weak interaction	$C$ violated maximally	the $V-A$ coupling picks out chirality; $C$ (and $CP$ ) broken
combined $CPT$	exact	forces electron and positron to share mass and lifetime

Common trapC intertwines the transpose of the gammas, not their complex conjugate

It is tempting to write the defining relation as $C\gamma^{\mu *}C^{-1}=-\gamma^{\mu}$ , mirroring the fact that charge conjugation involves complex conjugation. This is wrong. The correct relation is on the transpose, $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ , and in the Dirac representation the conjugate version $C\gamma^{\mu *}C^{-1}=-\gamma^{\mu}$ fails for $\mu=1,2,3$ (it holds only for $\mu=0$ ). The reason is that $\psi^{c}=C\bar\psi^{T}$ contains $\bar\psi^{T}=\gamma^{0T}\psi^{*}$ , and it is $\gamma^{0T}\psi^{*}$ that carries a transpose structure. If you insist on writing the relation for the bare conjugate, the correct intertwiner is $B=i\gamma^{2}$ (which equals $C\gamma^{0T}$ ), not $C$ . Mixing up $B$ and $C$ is the most common slip in this construction, and it produces a spinor that solves the wrong equation.

Worked examples

Example 1 — the positron spinor and the pair-creation threshold. Apply $C$ to a rest-frame electron and read off the antiparticle, then compute the energy needed to create the pair.

Step 1. Act with $C$ on the electron spinor. In the rest frame the spin-up positive-energy electron is $u^{\uparrow}=(1,0,0,0)^{T}$ . Computing $\psi^{c}=C\gamma^{0T}(u^{\uparrow})^{*}$ with $C=i\gamma^{2}\gamma^{0}$ lands the result entirely in the lower two components — the positron sector. The electron spinor has been mapped onto the positron spinor $v$ , with energy and spin preserved and charge reversed, exactly as promised (machine-verified).

Step 2. Set the energy scale. A positron carries rest energy $m_{e}c^{2}=\mathbf{0.511\ \text{MeV}}$ , identical to the electron's — a prediction of $CPT$ . To create an electron–positron pair from the vacuum requires supplying both rest energies.

Step 3. The threshold. The minimum photon energy to make a pair (in a field that conserves momentum) is

E_{\gamma,\min}=2m_{e}c^{2}=2(0.511)=\mathbf{1.022\ \text{MeV}}.

Below $1.022\ \text{MeV}$ a photon cannot produce an $e^{+}e^{-}$ pair no matter what; above it, pair production switches on. This is the number that governs gamma-ray interactions in matter and the onset of the pair-production regime in detectors (recomputed).

Example 2 — the photon's charge parity and neutral-pion decay (exam level). Use the $C$ -oddness of the electromagnetic current to fix the photon's $C$ and derive a decay selection rule.

Step 1. The current is $C$ -odd. From $\psi^{c}=C\bar\psi^{T}$ and the anticommutation of fermion fields, the electromagnetic current obeys

\mathcal{C}\,j^{\mu}\,\mathcal{C}^{-1}=-j^{\mu},\qquad j^{\mu}=\bar\psi\gamma^{\mu}\psi.

The vector bilinear $\bar\psi\gamma^{\mu}\psi$ developed in Bilinear Covariants and Dirac Adjoint changes sign under $C$ .

Step 2. Fix the photon's $C$ . QED conserves $C$ , so the interaction $\mathcal{L}_{\text{int}}=-q\,j^{\mu}A_{\mu}$ must be $C$ -even. Since $j^{\mu}\to-j^{\mu}$ , the photon field must transform as $A_{\mu}\to-A_{\mu}$ , giving the one-photon state

C(\gamma)=-1,\qquad C(n\gamma)=(-1)^{n}.

Step 3. Apply to the pion. The neutral pion has $C(\pi^{0})=+1$ . A two-photon final state has $C=(-1)^{2}=+1$ : allowed, and indeed $\pi^{0}\to\gamma\gamma$ is the dominant decay (99%). A three-photon final state has $C=(-1)^{3}=-1$ : forbidden by $C$ -conservation in electromagnetism. The measured branching-ratio limit on $\pi^{0}\to3\gamma$ is below $3\times10^{-8}$ — a clean experimental confirmation that QED conserves charge parity (recomputed selection rule).

From Dirac's hole to Anderson's track

Origins · 1 mincharge conjugation and how this was discovered

charge conjugation

discrete symmetry, 1937 (Kramers) · Utrecht, Caltech, Rome

Charge conjugation as a symmetry operation on the Dirac field — distinct from Dirac's earlier hole picture — was formalized by Hendrik Kramers in 1937, who gave the matrix $C$ its modern meaning as the intertwiner $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ . Its physical anchor came earlier and from the sky: in 1932 Carl Anderson photographed a cloud-chamber track curving the wrong way, a particle with the electron's mass and the opposite charge — the positron Dirac's equation had implied. Ettore Majorana's 1937 work asked the sharp follow-up question: could a neutral fermion be its own charge conjugate, $\psi^{c}=\psi$ ? For charged particles the answer is no — the electron and positron are distinct — but the "Majorana condition" remains the central open question for the neutrino. Charge conjugation reigned as an exact symmetry until 1956–57, when the weak interaction was found to violate it maximally, leaving only the combined $CPT$ as the exact symmetry that still guarantees a particle and its antiparticle share mass and lifetime.

History

1928

— Dirac's equation predicts negative-energy solutions

1931

— Dirac interprets the holes as antiparticles (initially the proton, then a new particle)

1932

— Anderson observes the positron in cosmic rays, confirming the antiparticle

1937

— Kramers formulates charge conjugation as a symmetry operation with the matrix C

1937

— Majorana asks whether a fermion can be its own antiparticle

1956

— Wu and collaborators find parity, and with it C, violated in the weak interaction

Going further

References. Bjorken & Drell, Relativistic Quantum Mechanics, §5.2 · Peskin & Schroeder, An Introduction to QFT, §3.6 (eq. 3.145) · Griffiths, Introduction to Elementary Particles, 2nd ed., §4.3, Ch. 7 · Sakurai, Advanced Quantum Mechanics, §3.4 · Weinberg, The Quantum Theory of Fields, Vol. 1, §5.5 (Majorana). Anderson, Phys. Rev. 43, 491 (1933) (positron discovery).

Provenance. All results are standard. The gamma-matrix anticommutator $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$ and the 16-matrix Clifford basis are owned by Dirac Gamma Matrices and Clifford Algebra; the $u,v$ plane-wave spinors are owned by Dirac Spinors and Plane-Wave Solutions; the Dirac adjoint $\bar\psi$ and the vector bilinear $\bar\psi\gamma^{\mu}\psi$ by Bilinear Covariants and Dirac Adjoint; the negative-energy crisis this operation resolves by Negative-Energy Solutions and the Interpretive Crisis; the hole-theory alternative by The Dirac Sea and Antiparticles — all recalled by name. Owned and derived here: (1) the defining relation $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ from charge reversal of the minimally coupled equation, with the transpose-versus-conjugate subtlety verified explicitly (the conjugate relation is FALSE for $\mu=1,2,3$ ; the correct conjugate intertwiner is $B=i\gamma^{2}$ ); (2) the explicit matrix $C=i\gamma^{2}\gamma^{0}$ satisfying (3) for all $\mu$ , with $C^{\dagger}=C^{-1}=C^{T}=-C$ ; (3) the electron $\to$ positron map, $C$ landing a rest-frame $u$ in the positron sector; and (4) the photon $C=-1$ and the $\pi^{0}\to\gamma\gamma$ (allowed) versus $\pi^{0}\to3\gamma$ (forbidden) selection rule, with the pair-creation threshold $2m_{e}c^{2}=1.022\ \text{MeV}$ . The full discrete-symmetry structure ( $P$ , $T$ , $CP$ , $CPT$ ) and its violation belong to Spin-Statistics and the CPT Theorem, developed by name; the field-theoretic promotion of $\mathcal{C}$ to a Fock-space operator to From Single-Particle to Quantum Fields.

Builds on: The Dirac Equation · Dirac Gamma Matrices and Clifford Algebra · Dirac Spinors and Plane-Wave Solutions · Negative-Energy Solutions and the Interpretive Crisis · Bilinear Covariants and Dirac Adjoint

Leads to: The Dirac Sea and Antiparticles · Lorentz Covariance of the Dirac Equation · Spin-Statistics and the CPT Theorem

Is charge conjugation the same as complex conjugation of the wavefunction?

No. Complex conjugation alone, $\psi\to\psi^{*}$ , does not give a solution of the charge-reversed Dirac equation — it scrambles the gamma matrices. Charge conjugation is $\psi^{c}=C\bar\psi^{T}$ , which combines conjugation with the specific matrix $C=i\gamma^{2}\gamma^{0}$ . The role of $C$ is precisely to undo the damage conjugation does to the gammas, so that $\psi^{c}$ solves the Dirac equation with the opposite charge. The two operations coincide only in the trivial spinless case; for the four-component Dirac field they are distinct.

Why is it the transpose of the gammas, not the conjugate, in the defining relation?

Because $\psi^{c}=C\bar\psi^{T}$ contains the Dirac adjoint $\bar\psi=\psi^{\dagger}\gamma^{0}$ , and $\bar\psi^{T}=\gamma^{0T}\psi^{*}$ carries a transpose. When you push the transformation through the equation, it is $\gamma^{\mu T}$ that must be intertwined, giving $C\gamma^{\mu T}C^{-1}=-\gamma^{\mu}$ . If you instead try the conjugate relation $C\gamma^{\mu *}C^{-1}=-\gamma^{\mu}$ , you can check it holds only for $\mu=0$ and fails for the spatial gammas in the Dirac representation. The correct intertwiner for the bare conjugate is a different matrix, $B=i\gamma^{2}$ , which happens to equal $C\gamma^{0T}$ . Keeping $B$ and $C$ straight is the single most common error in the construction.

Does charge conjugation by itself remove the negative-energy states?

Not by itself. $C$ maps a solution of the charge- $q$ equation to a solution of the charge- $(-q)$ equation; a negative-energy electron solution becomes a negative-energy positron solution. Removing negative energies from the physical spectrum is a separate step: the Stückelberg–Feynman reinterpretation, in which a negative-energy solution propagating backward in time is read as a positive-energy antiparticle propagating forward. Charge conjugation and this reinterpretation together give the clean, symmetric particle/antiparticle picture that replaces the Dirac sea.

Why does the photon carry charge parity C = −1?

The electromagnetic current $j^{\mu}=\bar\psi\gamma^{\mu}\psi$ is $C$ -odd: $\mathcal{C}j^{\mu}\mathcal{C}^{-1}=-j^{\mu}$ , a direct consequence of $\psi^{c}=C\bar\psi^{T}$ and fermion anticommutation. Since QED conserves $C$ , the interaction $-qj^{\mu}A_{\mu}$ must be $C$ -even, which forces the photon field to transform as $A_{\mu}\to-A_{\mu}$ . Hence a one-photon state is a $C$ -eigenstate with eigenvalue $-1$ , and an $n$ -photon state has $C=(-1)^{n}$ . This is what forbids $\pi^{0}\to3\gamma$ .

What is a Majorana fermion, and how does it relate to C?

A Majorana fermion is a particle that is its own charge conjugate: $\psi^{c}=\psi$ . For a charged particle this is impossible — the electron and positron have opposite charge and are genuinely distinct. But for a neutral fermion the Majorana condition can be imposed consistently, and it makes the particle identical to its antiparticle. The one candidate in nature is the neutrino: whether it is a Majorana fermion (its own antiparticle) or a Dirac fermion (distinct from its antiparticle) is unknown, and the answer is being sought in neutrinoless double-beta-decay experiments. Majorana raised the question in 1937; it remains open.

If C is violated by the weak interaction, why is it still useful?

Because the electromagnetic and strong interactions conserve $C$ exactly, and an enormous amount of physics — atomic and nuclear electromagnetic transitions, quarkonium and positronium decays, hadron spectroscopy — lives entirely within those sectors. There, $C$ -parity is a good quantum number and its selection rules are exact tools. The weak interaction's violation of $C$ is itself one of the most important facts in physics (it underlies the matter–antimatter asymmetry), but it does not spoil $C$ where the other forces dominate. And the combined $CPT$ remains exact everywhere, which is why the equality of particle and antiparticle masses is a theorem, not a coincidence.

How does C connect to the Dirac sea picture it replaces?

Both describe the same physics — the positron as a positive-energy, positive-charge partner of the electron — but charge conjugation does it symmetrically and without an infinite reservoir. In the sea picture, developed in The Dirac Sea and Antiparticles, a positron is a hole in a filled sea of negative-energy electrons, which requires Pauli exclusion (so it fails for bosons) and an infinite background charge. Charge conjugation instead treats particle and antiparticle as related solutions of one equation, with no filled vacuum needed. In quantum field theory $\mathcal{C}$ is promoted to an operator that swaps electron and positron creation operators, $\mathcal{C}b^{\dagger}\mathcal{C}^{-1}=d^{\dagger}$ — the fully modern statement, developed in From Single-Particle to Quantum Fields.