Antimatter

#1. Introduction to Antimatter

Antimatter consists of antiparticles — particles that have the same mass and spin as their corresponding matter particles, but with certain quantum numbers reversed (charge, lepton number, baryon number, magnetic moment direction). When a particle and its antiparticle meet, they annihilate, converting their combined rest-mass energy into photons (or other particle–antiparticle pairs).

Key historical note: The existence of antimatter was first predicted by Paul Dirac in 1928 through his relativistic wave equation, and the positron (antielectron) was experimentally discovered by Carl Anderson in 1932.

Anti Hydrogen model

#2. Properties of Antiparticles

For every particle, its antiparticle satisfies:

Annihilation

Conservation Laws in Particle–Antiparticle Processes

In any interaction involving particles and antiparticles, the following quantities are strictly conserved:

$\text{Charge: } \sum Q_i = \sum Q_f$

$\text{Linear Momentum: } \sum \vec{p}_i = \sum \vec{p}_f$

$\text{Energy: } \sum E_i = \sum E_f \quad \Rightarrow \quad E = \sqrt{(pc)^2 + (mc^2)^2}$

$\text{Angular Momentum: } \sum \vec{J}_i = \sum \vec{J}_f$

$\text{Lepton Number: } \sum L_i = \sum L_f$

$\text{Baryon Number: } \sum B_i = \sum B_f$

#3. Particle–Antiparticle Pairs

3.1 Electron ↔ Positron

$e^- \longleftrightarrow e^+$

Annihilation reaction:

$e^- + e^+ \longrightarrow 2\gamma$

Each photon carries energy $E = m_e c^2 = 0.511\ \text{MeV}$ (in the centre-of-mass frame), emitted back-to-back to conserve momentum.

Pair production (inverse process):

$\gamma \longrightarrow e^- + e^+$

Requires photon energy $E_\gamma \geq 2m_e c^2 = 1.022\ \text{MeV}$ (in the presence of a nucleus for momentum conservation).

3.2 Proton ↔ Antiproton

$p \longleftrightarrow \bar{p}$

Annihilation:

$p + \bar{p} \longrightarrow \text{mesons (e.g., pions)} \quad \text{or} \quad \gamma\text{-rays}$

$p + \bar{p} \longrightarrow \pi^+ + \pi^- + \pi^0 + \cdots$

Threshold energy for antiproton production:

$E_{\text{threshold}} = 4m_p c^2 \approx 3.76\ \text{GeV}$

The antiproton was discovered at the Bevatron accelerator (Berkeley) in 1955 by Segrè and Chamberlain.

3.3 Neutron ↔ Antineutron

$n \longleftrightarrow \bar{n}$

Although the neutron is electrically neutral, its magnetic moment is opposite for the antineutron, making them distinguishable. The antineutron was discovered in 1956.

Annihilation:

$n + \bar{n} \longrightarrow \pi^+ + \pi^- + \pi^0 + \cdots$

AntiProton AntiNeutron

3.4 Photon ↔ Itself (Self-Conjugate)

$\gamma \longleftrightarrow \gamma$

The photon is its own antiparticle. It has:

• Zero charge $Q = 0$
• Zero mass
• Spin $s = 1$
• Zero lepton number, zero baryon number

Since all additive quantum numbers are zero, the photon satisfies $\bar{\gamma} = \gamma$.

$C|\gamma\rangle = -|\gamma\rangle \quad (\text{charge conjugation eigenvalue} = -1)$

3.5 Muon and Antimuon

$\mu^- \longleftrightarrow \mu^+$

Decay:

$\mu^- \longrightarrow e^- + \bar{\nu}_e + \nu_\mu$

$\mu^+ \longrightarrow e^+ + \nu_e + \bar{\nu}_\mu$

Each decay conserves both electron and muon lepton numbers separately.

3.6 Pion (Pi Meson) ↔ Antipion

Pions are the lightest mesons, composed of quark–antiquark pairs:

$\pi^+ \longleftrightarrow \pi^-$

The $\pi^0$ is self-conjugate (its own antiparticle), similar to the photon.

Pion decay:

$\pi^+ \longrightarrow \mu^+ + \nu_\mu \quad (\tau \approx 26\ \text{ns})$

$\pi^0 \longrightarrow 2\gamma \quad (\tau \approx 8.5 \times 10^{-17}\ \text{s})$

#4. Quark–Antiquark Annihilation

At the fundamental level, particle–antiparticle annihilation is quark–antiquark annihilation mediated by gauge bosons. For example, in proton–antiproton annihilation:

$u + \bar{u} \longrightarrow g \longrightarrow q + \bar{q} \quad \text{(via gluon } g\text{)}$

$u + \bar{u} \longrightarrow \gamma \text{ or } Z^0 \quad \text{(electroweak)}$

The general quark annihilation vertex (QCD):

$q_i + \bar{q}_i \longrightarrow g^a \quad \text{with amplitude} \propto g_s\, T^a_{ij}$

where $g_s$ is the strong coupling constant and $T^a_{ij}$ are the $SU(3)$ colour generators.

Cross-section for $q\bar{q}$ annihilation into lepton pair (Drell–Yan):

$\hat{\sigma}(q\bar{q} \to \ell^+\ell^-) = \frac{4\pi\alpha^2}{3\hat{s}}\, e_q^2 \cdot N_c^{-1}$

where $\hat{s}$ is the partonic centre-of-mass energy squared, $e_q$ is the quark charge, and $N_c = 3$ is the colour factor.

#5. Baryons and Antibaryons

What is a Baryon?

A baryon is a hadron composed of three quarks (qqq), bound together by the strong force (QCD). Baryons carry baryon number $B = +1$.

$B = \frac{1}{3}(n_q - n_{\bar{q}})$

Each quark contributes $B = +\frac{1}{3}$; each antiquark contributes $B = -\frac{1}{3}$.

The antidbaryon is the corresponding antiparticle with three antiquarks $(\bar{q}\bar{q}\bar{q})$ and $B = -1$.

Baryon AntiBaryon

Associated Baryon Particles

Each of these has a corresponding antibaryon with all quarks replaced by antiquarks and $B = -1$, e.g., $\bar{p}$, $\bar{n}$, $\bar{\Lambda}^0$, $\bar{\Omega}^+$, etc.

Baryon number conservation:

$\Delta B = 0 \quad \text{in all known interactions}$

(Possible violation only in hypothetical proton decay or baryogenesis scenarios.)

Dirac Hole Theory

The Dirac Equation

In 1928, Paul Dirac formulated a relativistic quantum mechanical equation for the electron:

$\left(i\hbar\gamma^\mu \partial_\mu - mc\right)\psi = 0$

where $\gamma^\mu$ are the 4×4 Dirac gamma matrices satisfying the Clifford algebra:

$\{\gamma^\mu, \gamma^\nu\} = \gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = 2g^{\mu\nu}\mathbf{I}_{4\times 4}$

The energy solutions of the Dirac equation are:

$E = \pm\sqrt{(pc)^2 + (mc^2)^2}$

This yields both positive and negative energy eigenvalues. The negative-energy solutions presented a conceptual crisis: classically, an electron could cascade down to $-\infty$ energy.

The Dirac Sea

Dirac resolved this using the Pauli Exclusion Principle (valid for fermions like electrons):

Dirac's Hole Hypothesis: The vacuum (ground state) consists of an infinite "sea" of negative-energy states, all completely filled. Since the Pauli principle forbids two fermions in the same state, real electrons cannot fall into these occupied negative-energy states.

$|0\rangle_{\text{Dirac}} = \prod_{E < 0} a^\dagger_{E,s}|0\rangle_{\text{bare}}$

Holes as Antiparticles

If a negative-energy electron (energy $-|E|$, charge $-e$, momentum $-\vec{p}$) is excited out of the Dirac sea into a positive-energy state, it leaves behind a hole.

The hole behaves as a particle with:

• Energy: $+|E|$ (absence of $-|E|$ energy raises the sea's energy by $+|E|$)
• Charge: $+e$ (absence of $-e$ charge gives net $+e$)
• Momentum: $+\vec{p}$
• Spin: $+\frac{1}{2}$ (same as electron)

$\text{Hole} \equiv \text{Positron } e^+$

Pair creation in hole theory:

$\gamma \longrightarrow e^- \text{ (excited from sea)} + \text{hole } (e^+)$

Pair annihilation:

$e^- \text{ falls into hole} \longrightarrow \gamma + \gamma$

Limitation of Hole Theory: It only works for fermions (Pauli exclusion). For bosons (integer spin), there is no exclusion principle to fill the sea. This inadequacy led to the development of Quantum Field Theory.

Dirac Hole Model

Modern Theory of Antiparticles — Quantum Field Theory

In Quantum Field Theory (QFT), particles and antiparticles are both excitations of the same underlying quantum field. There is no Dirac sea; instead, the field is quantised with creation and annihilation operators.

Quantum Field

Electron Proton Field

The Electron Field

The electron is described by a Dirac spinor field $\psi(x)$, which upon canonical quantisation is expanded as:

$\psi(x) = \sum_{s} \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \left[ b^s_{\vec{p}}\, u^s(p)\, e^{-ip\cdot x} + d^{s\dagger}_{\vec{p}}\, v^s(p)\, e^{+ip\cdot x} \right]$

where:

• $b^s_{\vec{p}}$ = annihilation operator for an electron (spin $s$, momentum $\vec{p}$)
• $b^{s\dagger}_{\vec{p}}$ = creation operator for an electron
• $d^{s\dagger}_{\vec{p}}$ = creation operator for a positron
• $d^s_{\vec{p}}$ = annihilation operator for a positron
• $u^s(p)$, $v^s(p)$ = positive and negative energy Dirac spinors

The anticommutation relations (ensuring Fermi–Dirac statistics):

$\{b^r_{\vec{p}},\, b^{s\dagger}_{\vec{q}}\} = (2\pi)^3\,\delta^{(3)}(\vec{p} - \vec{q})\,\delta^{rs}$

$\{d^r_{\vec{p}},\, d^{s\dagger}_{\vec{q}}\} = (2\pi)^3\,\delta^{(3)}(\vec{p} - \vec{q})\,\delta^{rs}$

The conjugate field $\bar{\psi} = \psi^\dagger \gamma^0$ creates electrons and annihilates positrons.

The Proton Field

The proton is a composite particle (three quarks), but at an effective field theory level, it is also described by a Dirac spinor field $\Psi_p(x)$:

$\Psi_p(x) = \sum_s \int \frac{d^3p}{(2\pi)^3\sqrt{2E_p}} \left[b^s_{\vec{p}}\, u^s(p)\, e^{-ip\cdot x} + d^{s\dagger}_{\vec{p}}\, v^s(p)\, e^{ip\cdot x}\right]$

Here:

• $b^{s\dagger}_{\vec{p}}$ creates a proton; $d^{s\dagger}_{\vec{p}}$ creates an antiproton
• The proton and antiproton are excitations of the same proton quantum field $\Psi_p$

At the fundamental quark level, the proton is built from quark fields $q_f(x)$ (where $f = u, d, s, \ldots$):

$u\text{-quark field: } q_u(x) = \int \frac{d^3p}{(2\pi)^3\sqrt{2E_p}} \left[b_{\vec{p}}\, u(p)\, e^{-ip\cdot x} + d^\dagger_{\vec{p}}\, v(p)\, e^{ip\cdot x}\right]$

The $d^\dagger$ operator here creates an anti-up quark $\bar{u}$.

Crossing Symmetry

Crossing symmetry is a fundamental property of scattering amplitudes in QFT. It states that the amplitude for a process involving a particle in the initial state is related to the amplitude for the same process with the corresponding antiparticle in the final state (with reversed 4-momentum), and vice versa.

Formally, for an S-matrix element:

$\mathcal{M}\bigl(A + B \to C + D\bigr) = \mathcal{M}\bigl(B + \bar{C} \to \bar{A} + D\bigr)$

by analytically continuing the external momenta.

The Three Related Channels

Consider the generic $2 \to 2$ scattering. Crossing symmetry connects three "channels":

The three Mandelstam variables satisfy:

$s + t + u = \sum_i m_i^2$

The same Feynman amplitude $\mathcal{M}(s, t, u)$, when analytically continued, describes all three channels.

Example: Compton Scattering vs. Pair Annihilation

s-channel (Compton scattering):

$e^- + \gamma \longrightarrow e^- + \gamma$

t-channel (related by crossing):

$e^- + e^+ \longrightarrow \gamma + \gamma \quad \text{(pair annihilation)}$

These two processes share the same Feynman diagrams — the difference is only which legs are "crossed" (moved from initial to final state with reversed momentum):

$\mathcal{M}_{e^-\gamma \to e^-\gamma}(s,t) = \mathcal{M}_{e^-e^+ \to \gamma\gamma}(t,s)$

Mathematical Statement

If a particle $A$ with momentum $p$ is in the initial state, crossing symmetry allows us to replace it with its antiparticle $\bar{A}$ with momentum $-p$ in the final state:

$\text{Initial state particle } A(p) \quad \longleftrightarrow \quad \text{Final state antiparticle } \bar{A}(-p)$

For spinor fields, this replacement is:

$u^s(p) \text{ (initial electron)} \quad \longleftrightarrow \quad \bar{v}^s(-p) \text{ (final positron)}$

$\bar{u}^s(p) \text{ (final electron)} \quad \longleftrightarrow \quad v^s(-p) \text{ (initial positron)}$

#Summary Table

#Sources

• Antimatter - Wikipedia
• Antimatter Explained (YouTube)