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In the mode expansion of a Dirac field, $b^s(p)$ is the fermionic annihilation operator for a particle of four-momentum $p$ and spin angular momentum label $s$, while $d^{s†}(p)$ is the fermionic creation operator for the corresponding antiparticle. The Dirac spinor $u^s(p)$ is the positive-frequency particle wavefunction and $v^s(p)$ is the negative-frequency antiparticle wavefunction; they solve $(/p-m)u^s=0$ and $(/p+m)v^s=0$, respectively.

annihilation can produce gamma rays, charged antiparticles and neutrinos, directly or through decay and hadronization of the primary products. For self-conjugate dark matter of mass $m_{\chi }$, the annihilation rate per unit volume is $⟨\sigma v⟩{\rho }_{\chi }^2/(2m_{\chi }^2)$. For a photon yield $d{N}_{\gamma }/dE$, the optically thin flux from a chosen solid angle isUse consistent mass-energy units. The factor $1/2$ assumes identical self-conjugate particles; distinct particle-antiparticle populations require their separate number densities. A gamma-ray continuum with an endpoint near the particle mass is generic for many channels, whereas a narrow line from $\chi \chi \to \gamma \gamma$ or $\gamma Z$ would be especially distinctive but often loop suppressed. dwarf spheroidal galaxies offer high dark-matter content with limited conventional emission; the Galactic centre has a potentially larger signal and much more complicated foregrounds.

Charged Positrons and antiprotons are redistributed by magnetic fields and lose energy, so their radiation spectrum depend on propagation and cannot usually point directly to the source. annihilation electrons can also produce synchrotron and inverse-Compton radiation. dark matter captured in the Sun could produce escaping high-energy neutrinos, with the signal depending on capture, annihilation equilibrium and particle interactions. Early annihilation energy injection can alter recombination and cosmic microwave background anisotropies.

A persuasive interpretation joins particle-like spectral structure with a consistent density-dependent morphology and other channels. Pulsars, cosmic-ray interactions and unresolved sources can imitate continuum excesses. Density-profile uncertainties, substructure and velocity-dependent annihilation modify the predicted signal; a lack of a detectable line does not exclude all annihilating dark matter.

The baryon excess requires a mechanism that produces different net baryon numbers from an initially symmetric ensemble. In the usual local, CPT-invariant framework, the Sakharov conditions are baryon-number violation, C and CP violation, and departure from thermal equilibrium. Baryon conservation would preserve zero net baryon number; exact conjugation symmetries equate the conjugate production processes; equilibrium inverse reactions and detailed balance erase a would-be asymmetry. These necessary conditions do not determine its magnitude.^[ref]

For example, let equal numbers of a heavy particle $X$ and its antiparticle decay into channels $i$ with baryon numbers $B_i$. Their mean net yield per pair isIt can be nonzero even though CPT requires equal total widths: the branching fractions differ among channels carrying different baryon numbers. A nonzero decay-rate asymmetry requires interference with both a CP-odd weak phase and an absorptive phase. If the expansion rate overtakes the inverse-reaction rate, decays can occur out of equilibrium; later baryon-violating washout must then be weak enough for the asymmetry to survive.

In electroweak baryogenesis, CP-asymmetric transport near a first-order phase-transition boundary can bias baryon-violating sphaleron transitions. CP violation alone is insufficient: the transition and washout dynamics also matter. In leptogenesis, out-of-equilibrium decays of heavy Majorana neutrinos can generate lepton asymmetry, part of which sphalerons convert into baryon asymmetry while conserving $B-L$. The CKM phase demonstrates CP violation in the Standard Model but does not by itself supply a successful cosmological mechanism. The key quantity is a surviving net charge yield, not merely the existence of a CP-violating reaction.

A Majorana neutrino is its own antiparticle, with a mass term that can violate lepton number by two units. It is not thereby a state of both readily observable helicities with equal weak rates: charged-current vertices still select chirality. In light-neutrino exchange, the needed helicity connection between two beta vertices supplies a neutrino-mass factor.

Ordinary two-neutrino double beta decay is $(A,Z)\to (A,Z+2)+2e^{-}+2{\bar{\nu }}_e$ and conserves lepton number. It can occur when the direct single-beta route is energetically forbidden or highly suppressed but the double change is allowed. neutrinoless double beta decay, $(A,Z)\to (A,Z+2)+2e^{-}$, violates lepton number by two. For the conventional light-Majorana exchange in neutrinoless double beta decay mechanism,Here $U$ is the lepton mixing matrix, $G_{0\nu }$ the phase-space factor in this convention and ${\mathcal{M}}_{0\nu }$ a nuclear matrix element. Complex phases can cancel contributions to the effective Majorana mass; oscillation mass differences alone do not fix it. Other lepton-number-violating mechanisms can also cause the process, so a rate is not a model-independent measurement of $m_{\beta \beta }$.

The experimental signature is a peak in the summed electron energy at the nuclear Q-value, apart from small recoil and energy losses. Two-neutrino decay instead gives a continuous sum-energy radiation spectrum because neutrinos carry away variable energy. Isotope enrichment, large exposure, good energy resolution and radiopure shielding improve sensitivity; gamma backgrounds, radon, surface events and the two-neutrino tail require calibration and sideband controls. Coincidence or track information can test whether the topology is compatible with two electrons. A later concrete example is GERDA's 2013 germanium-76 source-detector search, which used pulse-shape discrimination and background controls rather than identifying any isolated energy deposit as a decay.^[ref]

The negative-kinetic-energy branch is interpreted as an antiparticle channel. In the overlap $m{c}^2<E<V_0-m{c}^2$, particles can escape to the left and oppositely charged antiparticles to the right. This interval exists only when $V_0>2m{c}^2$. The external electrostatic source supplies the energy for pair creation; the reflected particle flux can exceed the incident one because the outgoing state need not contain only the incident particle.

The conserved quantity is signed charge current, not a positive one-particle probability in both regions. Negative $\mathcal{T}$ denotes rightward transport of opposite charge, so it is consistent with $\mathcal{R}>1$ and $\mathcal{R}+\mathcal{T}=1$. A quantized charged field with particle and antiparticle creation operators gives the appropriate interpretation of this Klein paradox. This bosonic result must not be replaced by a fermionic reflection law or interpreted as a failure of charge conservation.^[ref]

A Klein-Gordon solution is second order in time: both $\Psi$ and ${\partial }_t\Psi$ are needed as initial data. Its free modes have both signs of frequency,This differs from the first-order Schrödinger evolution of a fixed one-particle wavefunction.

Moreover, $|\Psi {|}^2$ does not generally obey the required probability continuity equation. The conserved Klein-Gordon density is proportional toup to a positive conventional constant. A pure positive-frequency mode gives $+2\omega |\Psi {|}^2$ and a negative-frequency mode gives $-2\omega |\Psi {|}^2$. This density is not positive definite and cannot be a universal position probability density.

For a free field, one can choose a positive-frequency subspace with a positive integrated inner product. Time-dependent backgrounds generally mix it with the negative-frequency subspace, so simply deleting the latter is not dynamically consistent. In quantum field theory the complex Klein-Gordon field instead has particle and antiparticle sectors; its conserved current is interpreted as charge current, and creation or annihilation of pairs need not fit a fixed-particle-number wavefunction. These are limitations of the naive single-particle interpretation, not a failure of the relativistic field equation.

Distinguish in-operators and out-operators before identifying the symbols in the final vacuum condition. The full charged-field Fourier component before the pulse containsThe conjugate incoming mode has the conjugate matched solution. Collecting the coefficients of $e^{-i\Omega t}$ and $e^{i\Omega t}$ after the pulse gives the real Bogoliubov mapSince ${\alpha }^2-{\beta }^2=1$, the inverse map isThe Inverse Bogoliubov vacuum condition therefore yieldsHere the $a,b$ inside $(u_{\mathbf{k}}{a}_{\mathbf{k}}+v_{\mathbf{k}}{b}_{-\mathbf{k}}^{†})$ mean out-operators. Using the forward-map coefficient $\beta$ with the same plus sign would be inconsistent with this convention. Mode rephasings can change a displayed coefficient sign, but the map and squeezed-state exponent must change together.

The complete in-vacuum also obeys the antiparticle annihilation condition. In out-operators this is $(u_{\mathbf{k}}{b}_{-\mathbf{k}}^{\mathrm{out}}+v_{\mathbf{k}}{a}_{\mathbf{k}}^{\mathrm{out}†})|0_{\mathrm{in}}⟩=0$. The two conditions jointly select the vacuum sector used below; the particle condition alone would not exclude arbitrary incoming antiparticle excitations.

Suppress the mode label temporarily and let $a,b$ be out-operators. With $q=v/u$, the commuting-mode identityshows that $(a+q{b}^{†})e^{-q{a}^{†}{b}^{†}}|0_{\mathrm{out}}⟩=0$. The analogous relation for $b$ gives the antiparticle condition. Thus in a finite regulated collection of modes,For one particle-antiparticle mode pair, $(a^{†}{b}^{†}{)}^n|0,0⟩=n!|n,n⟩$. The exponential's factorial therefore cancels:The resulting particle-antiparticle two-mode squeezed vacuum has the geometric pair-number distributionHere one pair means one particle in canonical mode $\mathbf{k}$ and one antiparticle in mode $-\mathbf{k}$. They are distinct species, and the state has equal particle and antiparticle occupation. In particular,For a finite set, $|\mathcal{N}|={\prod }_{\mathbf{k}}\sqrt{1-r_{\mathbf{k}}}={\prod }_{\mathbf{k}}|u_{\mathbf{k}}{|}^{-1}$. An overall phase is arbitrary. With no pulse, $\Omega =\omega$, so $v=0$ and only the zero-pair state remains.

The ultraviolet divergence of an instantaneous vector-potential quench qualifies this formal product. For an exactly instantaneous vector-potential step in three spatial dimensions, at large $k$ the mixing behaves as $v_{\mathbf{k}}\sim e{\mathbf{A}}_0\cdot \hat{\mathbf{k}}/(2\hslash k)$. Thus $\int {\mathrm{d}}^{3}k|v_{\mathbf{k}}{|}^2$ diverges in the ultraviolet. In a finite quantization box, a finite momentum cutoff makes the product a normalized Fock-space vector; a sufficiently smooth finite-duration pulse removes this particular instantaneous-quench pathology. The per-mode distribution remains the result above, while the unregulated infinite product should not be asserted to have a finite nonzero global normalization.

A quantum field is an operator-valued distribution on spacetime: smearing $\Phi (x)$ against a smooth test function gives an operator, whereas pointwise products can require renormalization. For example, a free complex scalar field contains particle annihilation operators and antiparticle creation operators. Its microcausality condition makes field commutators vanish at spacelike separation. For fermions, the corresponding locality condition uses anticommutators. This combines quantum operators with the spacetime locality of a quantum field theory.

The antiparticle of the strange quark has electric charge $+1/3$ and baryon number $-1/3$.

The tau is the charged lepton of the third fermion generation. In the massless approximation its charged-current W boson partial decay width equals those of the electron and muon, reflecting lepton universality.

The tau lepton has electric charge $-1$ in units of the positive elementary electric charge; ${\tau }^{+}$ is its antiparticle. Its weak charged current decays make the tau lepton useful for testing lepton universality. They also illustrate how particle detection can establish a short-lived parent through visible daughters and a consistent invisible recoil.

For a charged spin-zero complex scalar field, time reversal preserves charge and can include an intrinsic phase. In its free-field mode expansion of a free field, antiunitarity conjugates the plane-wave coefficients. Relabel spatial momentum to its negative and compare the annihilation and creation terms: $\hat{T}a(\mathbf{p}){\hat{T}}^{-1}={\eta }_{T}a(-\mathbf{p})$ and $\hat{T}{b}^{†}(\mathbf{p}){\hat{T}}^{-1}={\eta }_T{b}^{†}(-\mathbf{p})$. Particle annihilation remains annihilation; unlike charge conjugation, this operation does not exchange particle and antiparticle operators.

For a charged Dirac particle, incoming momenta $p,k$ and outgoing momenta $p^{\prime },k^{\prime }$ obey $p+k=p^{\prime }+k^{\prime }$. The two QED diagrams have internal fermion momenta $p+k$ and $p-k^{\prime }$, giving $s$ and $u$ channels. The amplitude is proportional to $\bar{u}(p^{\prime })[/{ϵ}^{\prime \ast }R(p+k)/ϵ+/ϵR(p-k^{\prime })/{ϵ}^{\prime \ast }]u(p)$, where $R(q)=(/q+m)/(q^2-m^2+i0)$. It is related to particle-antiparticle two-photon annihilation by crossing symmetry, with appropriate changes to external wavefunctions and conjugations.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.