Antiparticles

An antiparticle is a positive-energy excitation with the conjugate internal quantum numbers of a particle species. In a free charged scalar or Dirac field, particle and antiparticle have the same mass and spin but opposite additive charge. Negative-frequency modes help construct those excitations; they are not negative-energy particles recorded by a detector. This page identifies the species through the action of conserved generators and explains why electric neutrality alone does not imply self-conjugacy.

Required background. Dirac Negative-Energy Solutions and Interpreting the Klein–Gordon Equation provide the free-field energy and charge results. Charge Conjugation distinguishes a classical solution map from a field-theory operation. Helpful background. Majorana Spinors and CPT Preview explain self-conjugacy and the boundary of interacting symmetry claims.

Creation operators identify energy and charge

Use $\hbar=c=1$ and a finite normalization box so mode labels $r$ are discrete. Let $a_r^\dagger$ create a particle and $b_r^\dagger$ its antiparticle. Take the established normal-ordered free-field operators as inputs:

$$H=\sum_r E_r(N_{a,r}+N_{b,r}), \qquad Q=q\sum_r(N_{a,r}-N_{b,r}),$$

where $E_r>0$, $N_{a,r}=a_r^\dagger a_r$, and $N_{b,r}=b_r^\dagger b_r$. The scalar and Dirac energy and charge results are established on the two prerequisites above; this page uses their generator action rather than repeating field quantization.

Both bosonic and fermionic number operators obey $[N_{a,r},a_s^\dagger]=\delta_{rs}a_s^\dagger$. Number operators of the other species commute with $a_s^\dagger$. Consequently

$$\begin{aligned} [H,a_r^\dagger]&=E_r a_r^\dagger, &[Q,a_r^\dagger]&=q a_r^\dagger,\\ [H,b_r^\dagger]&=E_r b_r^\dagger, &[Q,b_r^\dagger]&=-q b_r^\dagger. \end{aligned}$$

With $H|0\rangle=Q|0\rangle=0$, these commutators give

State	Excitation energy	Charge
$a_r^\dagger\lvert0\rangle$	$E_r$	$q$
$b_r^\dagger\lvert0\rangle$	$E_r$	$-q$

The charge sign is a generator eigenvalue. It is not the sign of a Hilbert norm, a lower spinor component, or a classical Fourier frequency. The same logic works for bosons, whose occupation numbers are unbounded, and fermions, whose single-mode occupations are zero or one.

For example, a normalized antiparticle packet $|g\rangle=\sum_r g_r b_r^\dagger|0\rangle$ has $\sum_r|g_r|^2=1$ and

$$\langle g|H|g\rangle=\sum_r E_r|g_r|^2>0, \qquad Q|g\rangle=-q|g\rangle.$$

It has opposite charge without needing negative probability or energy.

Conjugate internal representations

An additive $U(1)$ charge is the simplest example of an internal representation. To make its phase convention explicit, choose a charge unit $q_0$ and write

$$U(\theta)=e^{i\theta Q/q_0}.$$

The commutators imply

$$\begin{aligned} U a_r^\dagger U^\dagger &=e^{i\theta q/q_0}a_r^\dagger,\\ U b_r^\dagger U^\dagger &=e^{-i\theta q/q_0}b_r^\dagger. \end{aligned}$$

The two one-particle species carry conjugate phases. Under this same adjoint action, a field containing an annihilator $a$ and creator $b^\dagger$ transforms with $e^{-i\theta q/q_0}$. Using $U^\dagger\widehat\psi U$ instead reverses that displayed phase. This convention choice does not change the charge eigenvalues of the created states or the already fixed classical gauge convention.

More generally, if particle states carry a unitary internal representation $R(g)$, antiparticle states carry its complex conjugate $R(g)^*$. If $R=\exp(i\theta^A T^A)$ with Hermitian generators, the conjugate generators are

$$T^A_{\rm anti}=-(T^A)^*.$$

For an Abelian generator this reverses the additive weight. For a non-Abelian representation, the statement is conjugation of the whole representation, not a rule that every species label, spin label, or quantum number is simply negated. This representation viewpoint is developed in Weinberg’s free-field construction.

Same mass and spin, with the correct scope

In the free scalar construction both species have $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$ and spin zero. In the free massive Dirac construction both have the same dispersion and a two-state spin-half multiplet. Rest-spin or helicity conventions can attach different labels to the mode columns; they do not make the antiparticle a negative-spin representation.

For interacting relativistic field theories satisfying the appropriate locality, Lorentz covariance, spectral, and vacuum hypotheses, CPT relates particle and antiparticle properties, including equal masses and corresponding spins. The theorem-level assumptions belong to CPT in the Symmetry volume and the scoped CPT Preview. A classical charge-conjugation matrix alone is not a proof of those interacting statements.

The existence of antiparticles does not require every interaction to be invariant under C separately. A theory can distinguish charge-conjugate processes while retaining particle and antiparticle species and satisfying CPT.

A fixed external environment can also distinguish the two species: opposite charges see opposite electrostatic potential energy. Different scattering or bound-state behavior in that environment does not by itself imply unequal vacuum masses. The transformed background must be included when testing charge-conjugation covariance.

Neutral does not always mean self-conjugate

A particle is electrically neutral if its electric-charge eigenvalue is zero. It is self-conjugate if particle and antiparticle are the same species, with the corresponding field constraints and state identifications. These are different conditions.

A simple theoretical example is a complex scalar with zero electromagnetic charge but an unbroken global $U(1)$ charge. Its particle and antiparticle are both electrically neutral, yet their opposite global charges distinguish them. Electrical neutrality alone therefore cannot identify them.

A real scalar field has a different mode construction, with one species rather than two independent charge-conjugate species. For spinors, an appropriate Majorana constraint relates the two mode sectors. A massive free Majorana field has two spin states; an unconstrained massive Dirac field has two particle and two antiparticle spin states. The matrix real structure is derived on Majorana Spinors.

A self-conjugate species cannot carry a nonzero conserved additive $U(1)$ charge whose sign reverses under conjugation: the same species could not have a definite charge both $q$ and $-q$ unless $q=0$. This statement does not forbid every interaction of a neutral field, nor does it classify all real non-Abelian representations.

Mode labels are not detector labels

The classical negative-frequency mode $v(p)e^{+ip\cdot x}$, with $p^0=E_{\mathbf p}>0$, has Fourier eigenvalues $(-E_{\mathbf p},-\mathbf p)$. In the quantized Dirac field it multiplies $b^\dagger(p)$, which creates an antiparticle state of physical energy $+E_{\mathbf p}$ and momentum $+\mathbf p$. This is the mode-to-operator assignment explained on the negative-energy owner.

Conjugating a classical wavefunction instead maps between charge-labelled equations and reverses its Fourier signs. It is a useful equation-level operation, but is not itself the statement that a detector registers a positive-energy antiparticle. That interpretation uses the quantum state, the generators, and the observable being measured.

Exercises

1. For $q\ne0$, a free state has three particle occupations and one antiparticle occupation in allowed modes. Find its total number and charge. Does charge determine total number?

Solution

$N_{\rm tot}=4$ and $Q=2q$. The same charge also occurs with two particles and no antiparticle, or with four particles and two antiparticles. Charge fixes a difference, not the total. Fermionic examples use distinct occupied modes where exclusion requires them.

2. Let a particle multiplet have a diagonal conserved generator $T=\operatorname{diag}(1,-1,0)$. What are the conjugate weights?

Solution

$T_{\rm anti}=-T^*=\operatorname{diag}(-1,1,0)$. The zero weight is unchanged, but this fact alone does not identify a state with its antiparticle. The other generators and species labels must also be considered.

3. Why can a real scalar have no independent antiparticle species even though its classical wave equation has two frequency signs?

Solution

The reality condition relates the two classical frequency sectors. After quantization they are the annihilation and creation parts of the same field species. Counting frequency signs alone would double count that species; an unconstrained complex field has additional independent internal degrees of freedom.

References

• Dreiner, Herbi K., Howard E. Haber, and Stephen P. Martin. “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry.” Physics Reports 494, 1–196 (2010). doi:10.1016/j.physrep.2010.05.002. Corrected manuscript. Dirac and Majorana state conventions.
• Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006, sections 2.5 and 5.2. Complex scalar fields and Dirac particle states.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, chapters 2 and 5. Particle representations, conjugate species, and relativistic fields.