Negative energy states in Relativistic QM, but not in QFT

This is one of the deepest conceptual shifts from Relativistic Quantum Mechanics (RQM) to Quantum Field Theory (QFT).

The short answer is:

In RQM, negative-energy solutions appear because we are trying to describe relativistic particles with a single-particle wavefunction. In QFT, those same solutions are reinterpreted as antiparticle creation operators, so physical states always have positive energy.

Let’s walk through it carefully.

1. The Problem Appears in the Dirac Equation

The free-particle Dirac equation is $(i\gamma^\mu \partial_\mu – m)\psi = 0$ (iγμ∂μ−m)ψ=0

Assume a plane-wave solution $\psi(x)=u(p)e^{-ip\cdot x}.$ ψ(x)=u(p)e−ip⋅x.

Substituting gives $(\gamma^\mu p_\mu-m)u(p)=0.$ (γμpμ−m)u(p)=0.

Nontrivial solutions require $\det(\gamma^\mu p_\mu-m)=0,$ det(γμpμ−m)=0,

which yields $E^2=p^2+m^2.$ E2=p2+m2.

Therefore $E=\pm \sqrt{p^2+m^2}.$ E=±p2+m2.

The Dirac equation naturally contains:

Positive-energy solutions
Negative-energy solutions

2. Why This Is a Disaster in Single-Particle Quantum Mechanics

In ordinary QM, energy eigenstates are physical particle states.

Suppose a particle occupies $E=+10\ \text{MeV}.$ E=+10 MeV.

Then there are states with $E=-10,\,-20,\,-100\ \text{MeV}$ E=−10,−20,−100 MeV

available.

The particle could continuously emit photons and fall to lower and lower energies.

There is no lowest energy state.

The vacuum would be unstable.

This is unacceptable.

3. Dirac’s Original Solution: The Dirac Sea

Paul Dirac proposed:

Every negative-energy state is already occupied.
Electrons obey the Pauli exclusion principle.
A normal electron cannot fall into those states.

The vacuum becomes $\text{Vacuum} = \text{all negative-energy states filled}.$ Vacuum=all negative-energy states filled.

A missing electron (a “hole”) behaves like a positively charged particle.

This predicted the positron.

Historically this was brilliant.

But it has problems:

Works only for fermions.
Requires an infinite sea of particles.
Doesn’t generalize well.

QFT replaces it with something much cleaner.

4. The Key Idea in QFT

QFT quantizes the field, not the particle.

Instead of a wavefunction $\psi(x),$ ψ(x),

we promote it to an operator field $\hat\psi(x).$ ψ^(x).

The field is expanded as $\hat\psi(x) = \sum_s \int \frac{d^3p}{(2\pi)^3} \left[ b_s(p)u_s(p)e^{-ipx} + d_s^\dagger(p)v_s(p)e^{ipx} \right].$ ψ^(x)=s∑∫(2π)3d3p[bs(p)us(p)e−ipx+ds†(p)vs(p)eipx].

Notice something important.

There is no negative-energy operator.

Instead: $e^{-iEt}$ e−iEt

appears with an electron annihilation operator $b.$ b.

while $e^{+iEt}$ e+iEt

appears with an antiparticle creation operator $d^\dagger.$ d†.

The “negative-energy solution” has been reinterpreted.

5. Where Did the Negative Energy Go?

In RQM we read $v(p)e^{+iEt}$ v(p)e+iEt

a particle with energy $-E$ −E.

In QFT we read exactly the same mathematical object as

creation of an antiparticle with energy $+E$ +E.

The sign has moved from the energy to the operator interpretation.

This is the crucial step.

6. Hamiltonian in QFT

After quantization, the Hamiltonian becomes $H= \sum_s \int d^3p\, E_p \left( b_s^\dagger b_s + d_s^\dagger d_s \right) + E_{\rm vac}.$ H=s∑∫d3pEp(bs†bs+ds†ds)+Evac.

Every excitation contributes $+E_p.$ +Ep.

Electron: $E_p>0.$ Ep>0.

Positron: $E_p>0.$ Ep>0.

No negative-energy particles remain.

The only leftover infinity is the vacuum energy $E_{\rm vac},$ Evac,

which is handled separately by normal ordering or renormalization.

7. Feynman’s Interpretation

Richard Feynman provided another viewpoint.

A negative-energy electron moving forward in time can be reinterpreted as

a positive-energy positron moving backward in time.

Mathematically, $e^{+iEt}$ e+iEt

can be viewed as either:

negative-energy particle forward in time
positive-energy antiparticle backward in time

Both descriptions are equivalent.

QFT adopts the positive-energy antiparticle interpretation.

8. Why the Klein-Gordon Equation Has the Same Issue

The Klein-Gordon equation also gives $E=\pm \sqrt{p^2+m^2}.$ E=±p2+m2.

So the problem is not unique to Dirac particles.

The issue arises whenever we combine:

quantum mechanics
special relativity

The cure is always the same:

Stop treating the object as a single particle and treat it as a quantum field.

The Deep Physical Lesson

The negative-energy states are not actually removed in QFT.

They are reinterpreted.

Relativistic QM	QFT
Negative-energy electron	Positron creation operator
Single-particle wavefunction	Quantum field
Vacuum unstable	Vacuum stable
Dirac sea required	No Dirac sea
Negative energies appear physical	All physical excitations have positive energy

In that sense, the existence of antiparticles is not an additional prediction of QFT. It is the mechanism by which QFT resolves the negative-energy catastrophe of relativistic quantum mechanics. The negative-energy solutions never disappear from the mathematics—they are simply understood as antiparticle degrees of freedom rather than physical states of negative energy.