Feynman–Stueckelberg Picture

The Feynman–Stueckelberg picture rewrites the negative-frequency part of a charged field’s propagation as a positive-energy antiparticle traveling between the oppositely ordered endpoints. The associated fermion-line arrow can point opposite to increasing coordinate time. That arrow records an orientation in the amplitude; it does not describe an observable particle carrying negative energy into the past. The kernel derivation and its normalization belong to Dirac Propagators. Here the task is to read its two time orderings correctly.

Required background. Dirac Propagators provides the time-ordered spin sums; Antiparticles fixes the positive energies and conjugate charges of physical states.

Helpful background. Time Reversal defines the symmetry operation; Locality and Causality Warnings explains why a Feynman correlation is not a signal kernel.

Two time orderings of the Dirac propagator

Use $\hbar=c=1$, metric $(+---)$, and future-directed on-shell labels $p=(E_{\mathbf p},\mathbf p)$, with $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}>0$. For the free vacuum define

$$(S_F)_{ab}(x-y) =\langle0|T\{\widehat\psi_a(x) \widehat{\bar\psi}_b(y)\}|0\rangle.$$

With $u^\dagger u=v^\dagger v=2E_{\mathbf p}$ and $d\Pi_p=d^3p/[(2\pi)^3\,2E_{\mathbf p}]$, the established propagator result, with $\Delta=x-y$, can be written

$$\begin{aligned} S_F(\Delta)={}&\theta(\Delta^0)S_+(\Delta)\\ &-\theta(-\Delta^0)S_-(\Delta), \end{aligned}$$

where the two mode sums are

$$S_+(\Delta)=\sum_s\int d\Pi_p\, u_s(p)\bar u_s(p)e^{-ip\cdot\Delta},$$ $$S_-(\Delta)=\sum_s\int d\Pi_p\, v_s(p)\bar v_s(p)e^{ip\cdot\Delta}.$$

For $x^0>y^0$, the particle is created at the earlier endpoint $y$ by the particle-creation part of $\bar\psi(y)$ and annihilated at $x$ by the particle-annihilation part of $\psi(x)$. Its propagation phase is $e^{-ip\cdot(x-y)}$.

For $x^0<y^0$, the antiparticle-creation part of $\psi(x)$ acts at the earlier endpoint $x$, and the antiparticle-annihilation part of $\bar\psi(y)$ acts at the later endpoint $y$. The phase can be written

$$e^{ip\cdot(x-y)}=e^{-ip\cdot(y-x)}.$$

On the right it has exactly the positive-energy time dependence for propagation from $x$ to $y$. The overall minus sign in the second term comes from exchanging fermionic fields in time ordering. It is not a negative transition probability or a negative antiparticle norm.

This interpretation uses a vacuum matrix element after field quantization. Reading an arbitrary classical negative-frequency spinor as a particle detector count would skip that necessary step.

Three directions that must be kept distinct

For a Dirac field whose particle has charge $q$, assign the fermion-line arrow along particle propagation and opposite antiparticle propagation. This is the conventional orientation of species number on the line. Multiplication by $q$ converts that orientation into electric-charge flow; for an electron field the arrow is therefore not an arrow of positive electric charge.

Contribution	Physical propagation	Fermion-line arrow
Particle, $y^0<x^0$	From $y$ to $x$, charge $q$, energy $E_{\mathbf p}>0$	From $y$ to $x$
Antiparticle, $x^0<y^0$	From $x$ to $y$, charge $-q$, energy $E_{\mathbf p}>0$	From $y$ to $x$

One can assign a bookkeeping momentum $k$ along the fermion arrow. For the second row it is $k=-p$, while the physical antiparticle momentum remains $p$. Thus $k^0<0$ says that the chosen line orientation opposes the physical propagation. It does not change the eigenvalue of the physical antiparticle Hamiltonian to a negative number.

The same distinction appears in the free-field expansion: the factor $v_s(p)e^{ip\cdot x}$ has negative-frequency Fourier labels, but its coefficient is the operator $b_s^\dagger(p)$ that creates a positive-energy antiparticle. The label on $b^\dagger$ is the future-directed physical momentum, not the Fourier momentum of that exponential.

The original picture is useful because a single oriented fermion line can organize electron scattering, positron scattering, and pair creation or annihilation in an amplitude. Feynman’s 1949 paper develops this external-potential organization and explicitly acknowledges Stueckelberg’s spacetime interpretation. An internal line in a perturbative diagram is nevertheless a propagator integrated over its internal variables, not a measured classical trajectory.

Endpoint relabeling is not time reversal

Changing $e^{ip\cdot(x-y)}$ into $e^{-ip\cdot(y-x)}$ is an identity in an amplitude. The physical time-reversal operation acts antiunitarily on states and transforms momenta, spins, and background fields according to its symmetry rules. It does not turn a particle into its antiparticle merely by reversing a line arrow.

Likewise, charge conjugation relates conjugate species and their field equations; it is not the same operation as choosing which endpoint of a time-ordered propagator is earlier. The particle and antiparticle terms already coexist in one charged field.

The diagrammatic observation also motivates the language of crossing: an outgoing antiparticle can be represented by an oppositely oriented incoming particle leg in the analytic description of an amplitude. A crossing relation additionally requires the correct external spinors, charge labels, fermionic signs, and analytic continuation between physical kinematic regions. Replacing $p$ by $-p$ inside a measured probability is not a derivation of that relation.

Correlations do not transmit information to the past

The Feynman propagator implements vacuum time ordering and generally has nonzero spacelike correlations. Its two time-ordering terms are therefore not alternative retarded signal paths. A causal source-response problem uses the retarded kernel; locality imposes the appropriate vanishing commutators or anticommutators at spacelike separation.

For spacelike-separated endpoints, different inertial frames can reverse their time order. The full covariant propagator and local field algebra accommodate that change; neither term separately defines a frame-independent history of a particle traveling between detectors. For timelike-separated endpoints the physical antiparticle interpretation still has positive energy and ordinary forward propagation.

Finally, this picture does not determine the number of pairs produced by a background. That calculation needs the initial state and the asymptotic mode relation developed in Pair Creation. The propagator organizes amplitudes; occupation numbers and vacuum persistence answer distinct statistical questions.

Exercises

A negative-time phase. Let $y=(\tau,\mathbf0)$, $x=(0,\mathbf0)$ with $\tau>0$, and take a rest-frame antiparticle. Rewrite the negative-time phase and state the physical energy and propagation direction.

Solution

$p=(m,\mathbf0)$ gives $e^{ip\cdot(x-y)}=e^{-im\tau}$. The physical antiparticle travels from the earlier $x$ to the later $y$ with energy $m$. A momentum assigned along the oppositely directed fermion arrow is $k=-p$. That bookkeeping variable does not alter the physical energy.

Charge flow at pair creation. A neutral source creates one particle and one antiparticle of a charged Dirac field. Which way do their fermion arrows point relative to the production event, and what is the outgoing charge?

Solution

Both physical particles propagate toward later detection. The particle’s arrow leaves the event; the antiparticle’s arrow enters it. Their outgoing charges are $q$ and $-q$, whose sum is zero. An entering arrow need not represent an incoming physical particle in this convention.

A tempting signaling argument. Why does the $\theta(y^0-x^0)$ term not let an experiment at $y$ send a message to $x$?

Solution

It is part of a vacuum time-ordered correlation, not the retarded change of an observable caused by a controlled source. The response kernel has the causal boundary condition. Reinterpreting an oriented line does not replace that response calculation or its support restrictions.

References

• Feynman, Richard P. “The Theory of Positrons.” Physical Review 76, 749–759 (1949). doi:10.1103/PhysRev.76.749. External-potential amplitudes and the spacetime interpretation of positron propagation.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press (2014). doi:10.1017/9781139540940. Fermion propagators, diagram orientation, and scattering amplitudes.
• Tong, David. Quantum Field Theory. University of Cambridge lecture notes (2006), sections 5.4–5.5. Lecture notes. Free Dirac-field quantization and the time-ordered propagator.