In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-symmetry is an abbreviation of the phrase "charge conjugation symmetry", and is used in discussions of the symmetry of physical laws under charge-conjugation. Other important discrete symmetries are P-symmetry (parity) and T-symmetry (time reversal).

These discrete symmetries, C, P and T, are symmetries of the equations that describe the known fundamental forces of nature: electromagnetism, gravity, the strong and the weak interactions. Verifying whether some given mathematical equation correctly models nature requires giving physical interpretation not only to continuous symmetries, such as motion in time, but also to its discrete symmetries, and then determining whether nature adheres to these symmetries. Unlike the continuous symmetries, the interpretation of the discrete symmetries is a bit more intellectually demanding and confusing. An early surprise appeared in the 1950s, when Chien Shiung Wu demonstrated that the weak interaction violated P-symmetry. For several decades, it appeared that the combined symmetry CP was preserved, until CP-violating interactions were discovered. Both discoveries led to Nobel Prizes.

The C-symmetry is particularly troublesome, physically, as the universe is primarily filled with matter, not anti-matter, whereas the naive C-symmetry of the physical laws suggests that there should be equal amounts of both. It is currently believed that CP-violation during the early universe can account for the "excess" matter, although the debate is not settled. Earlier textbooks on cosmology, predating the 1970s,^[which?] routinely suggested that perhaps distant galaxies were made entirely of anti-matter, thus maintaining a net balance of zero in the universe.

This article focuses on exposing and articulating the C-symmetry of various important equations and theoretical systems, including the Dirac equation and the structure of quantum field theory. The various fundamental particles can be classified according to behavior under charge conjugation; this is described in the article on C-parity.

Charge conjugation occurs as a symmetry in three different but closely related settings: a symmetry of the (classical, non-quantized) solutions of several notable differential equations, including the Klein–Gordon equation and the Dirac equation, a symmetry of the corresponding quantum fields, and in a general setting, a symmetry in (pseudo-)Riemannian geometry. In all three cases, the symmetry is ultimately revealed to be a symmetry under complex conjugation, although exactly what is being conjugated where can be at times obfuscated, depending on notation, coordinate choices and other factors.

The charge conjugation symmetry is interpreted as that of electrical charge, because in all three cases (classical, quantum and geometry), one can construct Noether currents that resemble those of classical electrodynamics. This arises because electrodynamics itself, via Maxwell's equations, can be interpreted as a structure on a U(1) fiber bundle, the so-called circle bundle. This provides a geometric interpretation of electromagnetism: the electromagnetic potential ${\displaystyle A_{\mu }}$ is interpreted as the gauge connection (the Ehresmann connection) on the circle bundle. This geometric interpretation then allows (literally almost) anything possessing a complex-number-valued structure to be coupled to the electromagnetic field, provided that this coupling is done in a gauge-invariant way. Gauge symmetry, in this geometric setting, is a statement that, as one moves around on the circle, the coupled object must also transform in a "circular way", tracking in a corresponding fashion. More formally, one says that the equations must be gauge invariant under a change of local coordinate frames on the circle. For U(1), this is just the statement that the system is invariant under multiplication by a phase factor ${\displaystyle e^{i\phi (x)}}$ that depends on the (space-time) coordinate ${\displaystyle x.}$ In this geometric setting, charge conjugation can be understood as the discrete symmetry ${\displaystyle z=(x+iy)\mapsto {\overline {z}}=(x-iy)}$ that performs complex conjugation, that reverses the sense of direction around the circle.

In quantum field theory, charge conjugation can be understood as the exchange of particles with anti-particles. To understand this statement, one must have a minimal understanding of what quantum field theory is. In (vastly) simplified terms, it is a technique for performing calculations to obtain solutions for a system of coupled differential equations via perturbation theory. A key ingredient to this process is the quantum field, one for each of the (free, uncoupled) differential equations in the system. A quantum field is conventionally written as

where ${\displaystyle {\vec {p}}}$ is the momentum, ${\displaystyle \sigma }$ is a spin label, ${\displaystyle n}$ is an auxiliary label for other states in the system. The ${\displaystyle a}$ and ${\displaystyle a^{\dagger }}$ are creation and annihilation operators (ladder operators) and ${\displaystyle u,v}$ are solutions to the (free, non-interacting, uncoupled) differential equation in question. The quantum field plays a central role because, in general, it is not known how to obtain exact solutions to the system of coupled differential questions. However, via perturbation theory, approximate solutions can be constructed as combinations of the free-field solutions. To perform this construction, one has to be able to extract and work with any one given free-field solution, on-demand, when required. The quantum field provides exactly this: it enumerates all possible free-field solutions in a vector space such that any one of them can be singled out at any given time, via the creation and annihilation operators.