Antiparticles and Charge-Conjugate Excitations

An antiparticle is a positive-energy state in the charge-conjugate sector. The phrase negative frequency describes a spacetime factor such as $e^{+ip\cdot x}$; it does not assign negative energy to the state created by the operator multiplying that factor. In a local complex scalar field, the positive-frequency term annihilates a particle and the negative-frequency term creates its antiparticle. Both particle and antiparticle states lie on the future mass shell.

Required background. One-Particle States: Mass, Spin, and Relativistic Normalization supplies relativistic state normalization, while Fields, Observables, and Interpolating Operators supplies the operator–state distinction.

The worked model is a free complex scalar. Detailed spinor charge-conjugation matrices, CPT theorems, and interacting scattering consequences are deferred.

All point-field formulas below are distributional identities on the common finite-particle domain. A genuine Hilbert-space vector is obtained only after smearing, as in $\Phi(f)\Omega_0$.

One local field, two positive-energy sectors

Let

$$p^\mu=(E_{\mathbf p},\mathbf p), \qquad E_{\mathbf p} = \sqrt{\mathbf p^2+m^2}>0,$$

and define

$$\mathrm d\mu_{\mathbf p} := \frac{\mathrm d^3\mathbf p} {(2\pi)^3\sqrt{2E_{\mathbf p}}}.$$

A free complex scalar and its adjoint have the expansions

$$\begin{aligned} \Phi(x) &= \int\mathrm d\mu_{\mathbf p} \Bigl[ a(\mathbf p)e^{-ip\cdot x} + b^\dagger(\mathbf p)e^{ip\cdot x} \Bigr],\\ \Phi^\dagger(x) &= \int\mathrm d\mu_{\mathbf p} \Bigl[ a^\dagger(\mathbf p)e^{ip\cdot x} + b(\mathbf p)e^{-ip\cdot x} \Bigr]. \end{aligned}$$

The two independent oscillator families obey

$$\begin{aligned} [a(\mathbf p),a^\dagger(\mathbf q)] &= (2\pi)^3\delta^{(3)} (\mathbf p-\mathbf q),\\ [b(\mathbf p),b^\dagger(\mathbf q)] &= (2\pi)^3\delta^{(3)} (\mathbf p-\mathbf q), \end{aligned}$$

with every mixed commutator zero. This exact complex-scalar decomposition, with positive $\omega_{\mathbf p}$ in both sectors, appears in Schwartz 2014, § 9.1, pp. 140–142.

The covariantly normalized one-particle states are

$$\begin{aligned} |\mathbf p,+\rangle &:= \sqrt{2E_{\mathbf p}}\, a^\dagger(\mathbf p)\Omega_0,\\ |\mathbf p,-\rangle &:= \sqrt{2E_{\mathbf p}}\, b^\dagger(\mathbf p)\Omega_0. \end{aligned}$$

The labels $+$ and $-$ denote opposite charges, not opposite energies. Both satisfy

$$P^0|\mathbf p,\pm\rangle = E_{\mathbf p}|\mathbf p,\pm\rangle, \qquad E_{\mathbf p}>0.$$

Energy and charge are operator tests

Choose

$$\begin{aligned} U(\alpha) &= e^{i\alpha Q},\\ U(\alpha)\Phi(x)U(\alpha)^{-1} &= e^{-i\alpha}\Phi(x). \end{aligned}$$

For the free Lagrangian

$$\mathcal L = \partial_\mu\Phi^\dagger\partial^\mu\Phi -m^2\Phi^\dagger\Phi,$$

the chosen phase transformation and translations have densities

$$\begin{aligned} j^\mu &= i\left[ \Phi^\dagger\partial^\mu\Phi - (\partial^\mu\Phi^\dagger)\Phi \right],\\ T^{\mu\nu} &= \partial^\mu\Phi^\dagger\partial^\nu\Phi + \partial^\mu\Phi\partial^\nu\Phi^\dagger\\ &\quad- \eta^{\mu\nu}\mathcal L. \end{aligned}$$

Substitute the mode expansions into $Q=\int\mathrm d^3\mathbf x\,j^0$ and $P^\mu=\int\mathrm d^3\mathbf x\,T^{0\mu}$. The spatial integral supplies momentum delta functions; after normal ordering, the cross terms cancel and the generators reduce to

$$\begin{aligned} {:}P^\mu{:} &= \int \frac{\mathrm d^3\mathbf p}{(2\pi)^3}\, p^\mu\\ &\quad\times \left[ a^\dagger(\mathbf p)a(\mathbf p) + b^\dagger(\mathbf p)b(\mathbf p) \right],\\ {:}Q{:} &= \int \frac{\mathrm d^3\mathbf p}{(2\pi)^3}\\ &\quad\times \left[ a^\dagger(\mathbf p)a(\mathbf p) - b^\dagger(\mathbf p)b(\mathbf p) \right]. \end{aligned}$$

Normal ordering removes the vacuum constants; it does not change the commutators that identify the one-particle quantum numbers. Henceforth denote these normal-ordered operators by $P^\mu$ and $Q$. Then

$$\begin{aligned} [P^\mu,a^\dagger(\mathbf p)] &= p^\mu a^\dagger(\mathbf p),\\ [P^\mu,b^\dagger(\mathbf p)] &= p^\mu b^\dagger(\mathbf p),\\ [Q,a^\dagger(\mathbf p)] &= +a^\dagger(\mathbf p),\\ [Q,b^\dagger(\mathbf p)] &= -b^\dagger(\mathbf p). \end{aligned}$$

Thus $a^\dagger$ and $b^\dagger$ both create future-mass-shell states, while their charges are opposite. Equivalently,

$$[Q,\Phi(x)]=-\Phi(x).$$

The field $\Phi$ lowers charge by one unit: it can annihilate the $+$ particle or create the $-$ antiparticle. Its adjoint raises charge.

Negative frequency is not negative state energy

The two c-number time dependences obey

$$\begin{aligned} i\partial_0 e^{-ip\cdot x} &= +E_{\mathbf p}e^{-ip\cdot x},\\ i\partial_0 e^{ip\cdot x} &= -E_{\mathbf p}e^{ip\cdot x}. \end{aligned}$$

This is why $e^{ip\cdot x}$ is called negative-frequency. But the coefficient of that term is $b^\dagger(\mathbf p)$, and

$$[H,b^\dagger(\mathbf p)] = +E_{\mathbf p}b^\dagger(\mathbf p).$$

The Hamiltonian commutator is the state-energy test. The Fourier sign of a field component is not. Treating the field as a one-particle wavefunction would conflate these two statements; Srednicki 2006, § 3, pp. 38–43 diagnoses that mistake and derives the positive free Hamiltonian.

The matrix elements display the two roles directly:

$$\begin{aligned} \langle\Omega_0| \Phi(x)|\mathbf p,+\rangle &= e^{-ip\cdot x},\\ \langle\mathbf p,-| \Phi(x)|\Omega_0\rangle &= e^{ip\cdot x}. \end{aligned}$$

Distributionally, the second line identifies the positive-energy antiparticle coefficient of $\Phi(x)\Omega_0$ even though its spacetime factor is negative-frequency. The adjoint field reverses the particle and antiparticle roles.

Why locality needs both frequency pieces

Write the positive-frequency Wightman kernel as

$$\begin{aligned} W_0(z) &:= \int\mathrm d\Pi_{\mathbf p}\, e^{-ip\cdot z},\\ \mathrm d\Pi_{\mathbf p} &:= \frac{\mathrm d^3\mathbf p} {(2\pi)^3\,2E_{\mathbf p}}. \end{aligned}$$

If one kept only the particle annihilation field

$$\Phi_a(x) := \int\mathrm d\mu_{\mathbf p}\, a(\mathbf p)e^{-ip\cdot x},$$

then

$$[\Phi_a(x),\Phi_a^\dagger(y)] = W_0(x-y)\mathbf 1.$$

This is generally nonzero at spacelike separation. The independent antiparticle-creation term supplies the second ordering. With $z:=x-y$,

$$\begin{aligned} [\Phi(x),\Phi^\dagger(y)] &= \left[ W_0(z)-W_0(-z) \right]\mathbf 1\\ &= i\Delta_m(z)\mathbf 1. \end{aligned}$$

For nonzero spacelike $z$, a proper orthochronous Lorentz transformation sends $z$ to $(0,\mathbf r)$. In that frame the invariant measure and $\mathbf p\mapsto-\mathbf p$ give $W_0(z)=W_0(-z)$, so $\Delta_m(z)=0$. This is the free-scalar cancellation proved more fully in Spacelike Compatibility and Local Observables. Also,

$$[\Phi(x),\Phi(y)]=0.$$

The two frequency pieces therefore turn a nonlocal one-sided kernel into the causal commutator of a local charged scalar. The equal mass of the two sectors is what permits the cancellation. Weinberg derives the charged scalar from covariance and spacelike commutation in Weinberg 1995, § 5.2, pp. 201–205.

This is a controlled free-field result: positive spectrum, covariance, independent conjugate-charge oscillators, the full local field–adjoint pair, and a Hermitian Hamiltonian are all being used. It is not a proof that the word “locality,” without those state, field, and spectrum assumptions, implies every antiparticle statement.

Charge conjugation exchanges the sectors

After a conventional phase choice, define the unitary free-model charge-conjugation operator by

$$\begin{aligned} C\Omega_0&=\Omega_0,\\ Ca(\mathbf p)C^{-1}&=b(\mathbf p),\\ Cb(\mathbf p)C^{-1}&=a(\mathbf p). \end{aligned}$$

It follows that

$$\begin{aligned} C\Phi(x)C^{-1} &= \Phi^\dagger(x),\\ CP^\mu C^{-1} &= P^\mu,\\ CQC^{-1} &= -Q,\\ C|\mathbf p,+\rangle &= |\mathbf p,-\rangle. \end{aligned}$$

Creation-operator phases can be redefined, so the displayed phase choice carries no physics. In this free complex-scalar model, $C$ is a symmetry. Notice, however, that the positive-energy $b^\dagger$ sector and the locality cancellation were established before $C$ was introduced; exact $C$ invariance was not a logical premise of either result. Whether an interacting dynamics has $C$ symmetry is a separate question. The scalar transformation and its phase freedom are discussed in Weinberg 1995, § 5.2, pp. 205–206.

Distinct and self-conjugate particles

For a real scalar,

$$\Phi=\Phi^\dagger,$$

so the creation and annihilation terms belong to one oscillator family. There is no independent antiparticle species: the scalar particle is self-conjugate.

This real-versus-complex scalar distinction is made explicitly in Weinberg 1995, § 5.2, p. 204. Electric neutrality alone does not imply self-conjugacy: another conserved quantum number may still distinguish a neutral state from its conjugate. Self-conjugacy requires agreement of all particle labels, not merely zero electric charge.

Limits and handoffs

The free mode expansion is not an exact expansion of a generic interacting Heisenberg field. In an interacting theory, stable particle and antiparticle content is identified through isolated spectral sectors and interpolating matrix elements. If no isolated one-particle subspace exists, the sharp ket language used here is unavailable. Coleman displays the one-particle overlap together with additional multiparticle content, and makes stability an explicit hypothesis, in Coleman 2019, § 13.5, pp. 279–281.

The full canonical complex-scalar derivation, including the Noether current, belongs to Complex Scalars and Conserved Charge. Fermionic implementations belong to Canonical Quantization of the Free Dirac Field and Majorana Fields and Reality Conditions. Equality statements that rely on the CPT theorem, rather than on an exact $C$ symmetry, belong to CPT: Hypotheses, Content, and Limits. Crossing relations belong to Analyticity and Crossing of Amplitudes.

Common pitfalls

“Negative frequency means negative particle energy.” Frequency labels the plane wave. State energy is fixed by the Hamiltonian action on the creation operator, and both $a^\dagger$ and $b^\dagger$ raise the energy by $E_{\mathbf p}>0$.

“The adjoint field is the charge-conjugation operator.” $\Phi^\dagger$ is another operator-valued distribution. $C$ is a unitary operator acting on the state space and mapping $\Phi$ to $\Phi^\dagger$ in the free model.

“The free-model $C$ operator is needed to obtain the antiparticle.” The positive-energy $b^\dagger$ sector and the locality cancellation were established first. The displayed $C$ operator then exhibits a symmetry that exchanges the two sectors in this free model.

“Every neutral particle is its own antiparticle.” Self-conjugacy requires all relevant quantum numbers and the representation content to agree, not merely zero electric charge.

Check your understanding

1. Use the displayed $P^\mu$ and $Q$ to find the energy and charge of $a^\dagger(\mathbf p)\Omega_0$ and $b^\dagger(\mathbf p)\Omega_0$.

   Answer

   The canonical oscillator brackets give $[P^0,a^\dagger]=E_{\mathbf p}a^\dagger$ and $[P^0,b^\dagger]=E_{\mathbf p}b^\dagger$, so both states have positive energy. The charge brackets give $[Q,a^\dagger]=+a^\dagger$ and $[Q,b^\dagger]=-b^\dagger$, so their charges are $+1$ and $-1$.

2. Why does the field containing only $a(\mathbf p)e^{-ip\cdot x}$ fail the spacelike-locality test, while the full complex scalar passes?

   Answer

   The one-sided field has commutator $W_0(x-y)$ with its adjoint, and $W_0$ is generally nonzero at spacelike separation. The $b^\dagger e^{ip\cdot x}$ term contributes $-W_0(y-x)$. Their difference is $i\Delta_m(x-y)$, whose support is causal, so the full commutator vanishes for spacelike separation.

3. Verify that charge conjugation flips the sign of $Q$ in the free model.

   Answer

   Conjugation exchanges $a^\dagger a$ with $b^\dagger b$. Therefore it sends $a^\dagger a-b^\dagger b$ to $b^\dagger b-a^\dagger a$, and hence $CQC^{-1}=-Q$.

References

• Coleman, Sidney. Lectures of Sidney Coleman on Quantum Field Theory. Edited by Bryan Gin-ge Chen, David Derbes, David Griffiths, Brian Hill, Richard Sohn, and Yuan-Sen Ting. Singapore: World Scientific, 2019. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Author-hosted manuscript. Santa Barbara: University of California, 2006. Author page.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.