Anti Particles

Prerequisites: Dirac Equation, Special Relativity. Full path (24 pages)

• Intuitive
• Concrete
• Abstract
• FAQ

Intuitive

For every kind of particle there is a partner with the same mass and the same spin but with all of its charges reversed. The partner of the electron is the positron: equally heavy, spin 1/2, but with positive electric charge. The partner of the proton is the antiproton, with negative charge and negative baryon number. When a particle meets its antiparticle, all the charges add up to zero, so nothing forbids the pair from disappearing and leaving behind only energy, for example two photons. The reverse also happens: a photon with enough energy near a nucleus can turn into an electron and a positron.

Antiparticles became necessary when quantum mechanics was combined with special relativity. The relativistic wave equations for the electron have solutions with negative energy, and these solutions cannot simply be thrown away. Dirac proposed that all negative-energy states are already filled by a sea of electrons. A missing negative-energy electron behaves like a particle with positive energy and positive charge. The positron was found in cosmic rays a few years later.

Today we read the same mathematics differently. In quantum field theory one field describes both the particle and its antiparticle. The field can create a particle or destroy an antiparticle, and both processes change the total charge by the same amount. Some particles carry no charges at all, like the photon. They are their own antiparticles.

• https://paul.physicsnotes.org/doku.php?id=distillery:cpv

Concrete

We use natural units $\hbar = c = 1$ and the metric signature $(+,-,-,-)$.

Negative energy solutions. The relativistic energy relation $E^2 = \vec p^{\,2} + m^2$ has two roots, $E = \pm\sqrt{\vec p^{\,2}+m^2}$. Both the Klein-Gordon equation and the Dirac equation therefore have plane wave solutions $e^{-ip\cdot x}$ with $p^0 = +E_p$ and $e^{+ip\cdot x}$ with $p^0 = +E_p$, where $E_p = \sqrt{\vec p^{\,2}+m^2}$. The second kind oscillates in time as $e^{+iE_p t}$, which in single-particle quantum mechanics means energy $-E_p$. Since these states form part of a complete set, an electron coupled to the electromagnetic field could decay into them without limit.

Charge conjugation of the Dirac equation. Take an electron of charge $q$ in an external field, $(i\gamma^\mu(\partial_\mu + iqA_\mu) - m)\psi = 0$. In the Dirac or Weyl representation $\gamma^2$ is the only imaginary gamma matrix, and one checks $\gamma^2 (\gamma^\mu)^* = -\gamma^\mu \gamma^2$. Taking the complex conjugate of the equation and multiplying from the left by $\gamma^2$ gives $$\left(i\gamma^\mu(\partial_\mu - iqA_\mu) - m\right)\psi^c = 0, \qquad \psi^c \equiv i\gamma^2 \psi^* .$$ So $\psi^c$ obeys the Dirac equation for a particle of charge $-q$. If $\psi \propto e^{+iE_p t}$ is a negative-energy solution, then $\psi^c \propto e^{-iE_p t}$ is a positive-energy solution. The negative-energy electron solution is equivalent to a positive-energy solution with opposite charge: the positron.

Field theory. In quantum field theory the coefficients of the two kinds of solutions become operators. For the Dirac field $$\psi(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}} \sum_s \left( a^s_{\vec p}\, u^s(p)\, e^{-ip\cdot x} + b^{s\dagger}_{\vec p}\, v^s(p)\, e^{+ip\cdot x}\right),$$ where $a^s_{\vec p}$ annihilates an electron and $b^{s\dagger}_{\vec p}$ creates a positron, and both obey anticommutation relations. All excitations now have positive energy. The conserved charge of the $U(1)$ symmetry $\psi \to e^{-i\alpha}\psi$ is, after normal ordering, $$Q = \int d^3x\, :\psi^\dagger \psi: \; = \int \frac{d^3p}{(2\pi)^3}\sum_s \left(a^{s\dagger}_{\vec p} a^s_{\vec p} - b^{s\dagger}_{\vec p} b^s_{\vec p}\right).$$ Electrons and positrons carry opposite values of $Q$, while both have positive energy. For a complex scalar field the same structure appears with $\phi(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}(a_{\vec p}e^{-ip\cdot x} + b^\dagger_{\vec p} e^{+ip\cdot x})$. For a real field, $\phi = \phi^\dagger$ forces $b = a$, and the particle is its own antiparticle.

• Particle/Anti-Particle Annihilation | Of Particular Significance
• http://physics.stackexchange.com/questions/391/is-anti-matter-matter-going-backwards-in-time?rq=1

Abstract

In relativistic quantum field theory a free particle species corresponds to a unitary irreducible representation of the Poincaré group, labelled by mass and spin (or helicity), together with a representation $R$ of the internal symmetry group $G$ that carries its charges. The antiparticle carries the same Poincaré labels and the complex conjugate representation $\bar R$. For $G = U(1)$ this means charge $-q$ instead of $q$; for $SU(3)$ color a quark in the $\mathbf 3$ has an antiquark in the $\bar{\mathbf 3}$.

The need for antiparticles comes from locality. To build a field $\phi(x)$ that transforms covariantly and whose commutator (or anticommutator, for fermions) with $\phi^\dagger(y)$ vanishes at spacelike separation, the positive-frequency part, which annihilates particles, must be combined with a negative-frequency part, which creates particles. If the field carries a charge, so that $[Q,\phi] = -q\,\phi$, both parts must lower the charge by $q$. The negative-frequency part therefore creates particles of charge $-q$, which are a second species with the same mass. Only when $R$ is a real representation can these be identified with the original particles. This happens for the photon and the $\pi^0$, and for neutral fermions described by Majorana fields.

The relation between particle and antiparticle is implemented by the CPT theorem: every local, Lorentz invariant quantum field theory with a Hermitian Hamiltonian has an antiunitary symmetry $\Theta = CPT$. Applied to a one-particle state it maps a particle to its antiparticle. It follows that particle and antiparticle have exactly equal masses and equal total lifetimes, and opposite values of all additive quantum numbers. Charge conjugation $C$ by itself is a symmetry of QED and QCD but is violated by the weak interaction, so a process and its charge-conjugate process need not have the same rate.

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1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Lie Groups
10. Newtonian Mechanics
11. Variational Calculus
12. Configuration Space
13. Euler-Lagrange Equations
14. SO(3)
15. Special Relativity
16. Lagrangian Formalism
17. Lorentz Group
18. Phase Space
19. SU(2)
20. Hamiltonian Formalism
21. Spinors
22. Schrödinger Equation
23. Klein-Gordon Equation
24. Dirac Equation