Virtual Particles Explained: On-Shell vs Off-Shell in QFT

What Is a Virtual Particle?

A virtual particle is a mathematical device, not a physical entity, that appears in perturbation expansions of the S-operator (the transition matrix used in quantum field theory to calculate interaction probabilities). Virtual particles never physically appear in an interaction. All possible virtual particles and their antiparticles occur together in the mathematics and are removed by integration over their momenta. Whether a virtual particle looks “physical” (on-mass-shell) or “unphysical” (off-mass-shell) depends entirely on whether the calculation is done in coordinate space or momentum space.

Key Takeaways

• Virtual particles are mathematical bookkeeping devices in the Dyson perturbation expansion of the S-operator, not physically observable objects.
• In coordinate-space Feynman diagrams, virtual particles satisfy the on-shell relation E² = p² + m², the same relation obeyed by real, detectable particles.
• In momentum-space Feynman diagrams, virtual particles are generally off-mass-shell, meaning their energy and momentum do not satisfy E² = p² + m².
• 4-momentum is conserved at every vertex and around every loop in the momentum-space representation, enforced by a Dirac delta function per loop.
• In coordinate space, only 3-momentum is conserved at each vertex; full 4-momentum conservation only emerges after an auxiliary energy variable is introduced and the time-ordering restriction is removed.
• An “H”-shaped Feynman diagram for electron-photon scattering, worked through step by step below, resolves to the propagator (-iQ̸ + m)/(2mk₀), where k₀ is the final photon energy measured in the electron’s rest frame.

Glossary of Key Terms

• S-operator (S-matrix): the transition matrix that encodes the probability amplitudes for particles to scatter from an initial to a final state.
• Perturbation expansion: a method of approximating a quantity as a series of successively smaller correction terms.
• On-mass-shell (on-shell): a particle whose energy and momentum satisfy the relativistic relation E² = p² + m², consistent with being a real, observable particle.
• Off-mass-shell (off-shell): a particle whose energy and momentum do not satisfy E² = p² + m², a state impossible for a real particle but permitted for virtual particles in momentum-space calculations.
• Propagator: a mathematical function of a particle’s 4-momentum that represents an internal line in a momentum-space Feynman diagram.
• Feynman diagram: a pictorial representation of the mathematical terms in a perturbation expansion, showing particles as lines meeting at interaction vertices.
• Dirac delta function: a distribution, not an ordinary function, that has meaning only inside an integral and is used to enforce conservation constraints such as momentum conservation.
• Time-ordered product: a product of operators rearranged so that the operator evaluated at the latest time stands leftmost.

How Is the “H”-Shaped Feynman Diagram Calculated?

The following derivation calculates the “centre part” of the transition probability for an “H”-shaped Feynman diagram describing an interaction between an electron and a photon, with incoming and outgoing 4-momenta and an exchanged 4-momentum Q = (Q⃗, E). An “H” diagram was chosen for this example because it involves a virtual electron rather than a virtual photon, which avoids extra gauge-theory complications and illustrates that the electromagnetic interaction is not mediated solely by virtual photons.

The integral runs over all virtual electrons with 3-momentum q created on the left side of the “H” diagram and all virtual positrons with 3-momentum q created on the right side, where θ(t) = 1 if t > 0 and θ(t) = 0 if t < 0:

$$\frac{1}{(2\pi)^3}\ \int\frac{d^3\boldsymbol{q}}{2\sqrt{\boldsymbol{q}^2\ +\ m^2}}\ \int\ d^3(\boldsymbol{x}_1-\boldsymbol{x}_2)\ \int\ d(t_1-t_2)\ \ e^{i(E(t_1-t_2)-\boldsymbol{Q}\cdot(\boldsymbol{x}_1-\boldsymbol{x}_2))}\ e^{i\boldsymbol{q}\cdot(\boldsymbol{x}_1-\boldsymbol{x}_2)}$$

$$\times\ \left(\theta(t_1-t_2)\ e^{-i\sqrt{\boldsymbol{q}^2\ +\ m^2}(t_1-t_2)}\ (\gamma_{i}\boldsymbol{q}^{i}+ \gamma_{0}\sqrt{\boldsymbol{q}^2\ +\ m^2}+im)\right.$$

$$\left. +\ \theta (t_2-t_1)\ e^{i\sqrt{\boldsymbol{q}^2 \ +\ m^2}(t_1-t_2)}\ (\gamma_i\boldsymbol{q}^i+ \gamma_{0}\sqrt{\boldsymbol{q}^2\ +\ m^2}-im)\right)$$

After substituting q and d³q with −q and −d³q in the terms containing θ(t₂−t₁), the expression becomes:

$$=\ \frac{1}{(2\pi)^3}\ \int\frac{d^3\boldsymbol{q}}{2\sqrt{\boldsymbol{q}^2\ +\ m^2}}\ \int\ d^3(\boldsymbol{x}_1-\boldsymbol{x}_2)\ e^{i((\boldsymbol{q}-\boldsymbol{Q})\cdot(\boldsymbol{x}_1-\boldsymbol{x}_2))}\ \int\ d(t_1-t_2)$$

$$\times\ \left(\theta(t_1-t_2)\ e^{i(E-\sqrt{\boldsymbol{q}^2\ +\ m^2})(t_1-t_2)}\ (\gamma_{i}\boldsymbol{q}^{i}+ \gamma_{0}\sqrt{\boldsymbol{q}^2\ +\ m^2}+im)\right.$$

$$\left.+\ \theta(t_2-t_1)\ e^{i(E+\sqrt{\boldsymbol{q}^2\ +\ m^2})(t_1-t_2)}\ (\gamma_{i}\boldsymbol{q}^{i}- \gamma_{0}\sqrt{\boldsymbol{q}^2\ +\ m^2}+im)\right)$$

A fictitious energy variable, s, is introduced to replace θ(t) using the identity θ(t) = lim (ε→0+) (−1/2πi) ∫ e^−ist ds/(s+iε), which enables integration over the full time range:

$$=\ \lim_{\varepsilon\rightarrow 0+}\frac{1}{(2\pi)^3}\ \int\frac{d^3\boldsymbol{q}}{2\sqrt{\boldsymbol{q}^2\ +\ m^2}}\ \delta^3(\boldsymbol{q},\boldsymbol{Q})\ \int\ d(t_1-t_2)$$

$$\times\ \frac{-1}{2\pi i}\ \left(\ (\gamma_i\boldsymbol{q}^i+ \gamma_0\sqrt{\boldsymbol{q}^2\ +\ m^2}+im)\ \int\frac{e^{i(E\ -\ \sqrt{\boldsymbol{q}^2\ +\ m^2}\ -\ s)(t_1-t_2)}}{s\ +\ i\varepsilon}\,ds\right.$$

$$\left.+\ \ (\gamma_i\boldsymbol{q}^i-\gamma_0\sqrt{\boldsymbol{q}^2\ +\ m^2}+im)\ \int\frac{e^{i(E\ +\ \sqrt{\boldsymbol{q}^2\ +\ m^2}\ +\ s)(t_1-t_2)}}{s\ +\ i\varepsilon}\,ds\right)$$

Integrating the 3-momentum delta function collapses q to Q, giving:

$$=\ \lim_{\varepsilon\rightarrow 0+}\frac{1}{(2\pi)^3}\ \frac{1}{2\sqrt{\boldsymbol{Q}^2\ +\ m^2}}$$

$$\times\ \frac{-1}{2\pi i}\ \left(\ (\gamma_i\boldsymbol{Q}^i+ \gamma_0\sqrt{\boldsymbol{Q}^2\ +\ m^2}\ +\ im)\ \int\frac{\delta(s,E- \sqrt{\boldsymbol{Q}^2\ +\ m^2})}{s\ +\ i\varepsilon}\,ds\right.$$

$$\left.+\ \ (\gamma_i\boldsymbol{Q}^i-\gamma_0\sqrt{\boldsymbol{Q}^2\ +\ m^2}\ +\ im)\ \int\frac{\delta(s,-E-\sqrt{\boldsymbol{Q}^2\ +\ m^2})}{s\ +\ i\varepsilon}\,ds\right)$$

Evaluating the s-integrals against the remaining delta functions yields:

$$=\ \lim_{\varepsilon\rightarrow 0+}\frac{-1}{(2\pi)^4\,i}\ \frac{\gamma_i\boldsymbol{Q}^i\ +\ im}{2\sqrt{\boldsymbol{Q}^2\ +\ m^2}}\left(\frac{1}{E-\sqrt{\boldsymbol{Q}^2\ +\ m^2}\ +\ i\varepsilon}\ +\ \frac{1}{-E-\sqrt{\boldsymbol{Q}^2\ +\ m^2}\ -\ i\varepsilon}\right)$$

$$+\ \frac{-1}{(2\pi)^4\,i}\ \frac{\gamma_0}{2}\left(\frac{1}{E-\sqrt{\boldsymbol{Q}^2\ +\ m^2}\ +\ i\varepsilon}\ -\ \frac{1}{-E-\sqrt{\boldsymbol{Q}^2\ +\ m^2}\ -\ i\varepsilon}\right)$$

Combining terms produces the final propagator result:

$$=\ \frac{1}{(2\pi)^4\,i}\ \frac{\gamma_i\boldsymbol{Q}^i\ +\ \gamma_0E\ +\ im}{\boldsymbol{Q}^2\ -\ E^2\ +\ m^2}\ =\ \frac{1}{(2\pi)^4}\ \frac{-i\,Q\hspace{-1.0ex}/\ +\ m}{2mk_0}$$

Here k₀ is the non-zero final energy of the photon measured in the reference frame in which the initial electron is stationary, and Q̸ = γ_μQ^μ = γ_iQ^i + γ₀E.

What Is the Dyson (Perturbation) Expansion?

The nth order of the Dyson expansion of the S-operator includes a time-ordered product of n copies of the Hamiltonian, evaluated at n different 4-positions (events): T{H(x₁)⋯H(x_N)}. “Time-ordered” means the copies are rearranged by their t-components, with the earliest term placed on the right. For example, if t₃ > t₁ > t₂, then T{H(x₁)H(x₂)H(x₃)} = H(x₃)H(x₁)H(x₂).

Time-ordering is generally not Lorentz-invariant, but it is invariant for non-spacelike pairs of events, and therefore holds for Hamiltonians provided they commute at spacelike separation: H(x₁)H(x₂) = H(x₂)H(x₁) if (x₁−x₂)² > 0 (see Weinberg, equations 3.5.13–14).

Each copy of the Hamiltonian is a sum (integral) of products of, usually, three field operators representing creation or annihilation operators for three particles of fixed types, for example two fermions and a photon, all evaluated at the same 4-position. These sums run over every possible 3-momentum for each particle and its antiparticle. A typical vertex takes the form:

$H(x_n)=\int \boldsymbol{a}(m,\boldsymbol{p},x_n)\ \boldsymbol{a}(m’,\boldsymbol{p}’,x_n)\ \boldsymbol{a}(m”,\boldsymbol{p}”,x_n)\,d\boldsymbol{p}\,d\boldsymbol{p}’\,d\boldsymbol{p}”$

These intermediate excitations are called virtual particles. They are not created or destroyed in the physical process itself but appear only in the mathematical expansion used to calculate the process.

Why Are Virtual Particles On-Shell in Coordinate Space?

In the coordinate-space representation, the intermediate excitations are “realistic”: they obey the on-shell energy-momentum relation E² = p² + m², because only particles that can physically exist have well-defined creation and annihilation operators.

Matching Operators at Each End of an Internal Line

Each internal line in a Feynman diagram represents a pair of operators, a creation operator at one end and an annihilation operator at the other. The two operators must correspond to the same particle species; the product H(x₁)⋯H(x_N) vanishes unless each creation operator is matched by a corresponding annihilation operator. As a result, if the 3-momentum is q on one side of an internal line, it must be −q on the other side.

Phase Factors and Momentum Conservation at Each Vertex

Each particle at a vertex x_n contributes a phase factor e^(ip·x_n), where p is its 4-momentum. The three fields connected to that vertex share the same x_n, so their phases combine, for example as e^(i(p−k′−q)·x_n).

When a combined phase is integrated over all values of the vertex coordinate x_n, it yields a Dirac delta distribution enforcing momentum conservation. The oscillatory integral vanishes unless the combined 4-momentum factor is zero, in which case the integral produces a delta function up to factors of 2π. Integration over vertex coordinates therefore enforces 4-momentum conservation at each vertex.

In coordinate space, however, time-ordering prevents integrating over all values of the time component simultaneously, because certain time orderings are restricted for virtual particles versus virtual antiparticles. For each allowed time-ordering, all spatial components are integrated and only 3-momentum is conserved at that stage. Full 4-momentum conservation only emerges after removing the time-ordering restriction, typically by introducing an auxiliary energy variable, and integrating over all coordinates.

The Dirac delta function is not an ordinary function but a distribution. It only has meaning within an integral and enforces constraints by reducing the number of integration variables.

Coordinate-Space Versus Momentum-Space Representations

Coordinate-Space Representation

Each internal line x_jx_i in coordinate space denotes the creation and annihilation operators for every allowed particle and antiparticle, all on-mass-shell and satisfying E² = p² + m². Particles are created at x_i and annihilated later at x_j; antiparticles can equivalently be thought of as created at x_j and annihilated earlier at x_i. Whether a given mode is labeled “particle” or “antiparticle” depends on convention and on the chosen Hamiltonian. Photons are their own antiparticles.

All of these modes are virtual in the sense that none corresponds to an observable creation or annihilation event in the scattering process. The diagram is a bookkeeping device that must be summed, or integrated, over the 3-momenta of every allowed on-shell particle and antiparticle and over every allowed time ordering of the vertices.

Momentum-Space Representation

The time-ordering restriction is removed by introducing an auxiliary energy variable, often called s, and combining it with the spatial momentum to form a 4-variable q = (q⃗,s). The phase becomes e^(i(q−Q)·(x_j−x_i)); integrating over the coordinate difference yields a delta distribution δ(q,Q) and a propagator, which is a function of q.

In momentum space, each internal line is assigned its own 4-momentum variable q. These variables need not satisfy the on-shell mass relation for that line, which is why such particles are called off-mass-shell virtual particles. The propagator in the “H”-diagram example above is (−iq̸ + m)/(q² + m² − iε). Momentum-space integrals sum over these 4-momenta, while the coordinates themselves have already been eliminated by delta distributions.

Can the Off-Shell Momentum-Space Calculation Be Avoided?

In principle, calculations can be performed entirely in coordinate space, avoiding off-mass-shell variables altogether, but this approach is usually far more cumbersome. In the “H”-diagram example above, substituting

t = (E − √(q² + m²))(t₁ − t₂) and t = (E + √(q² + m²))(t₁ − t₂)

at the start yields:

$$\int dt \left(\theta(t)\,e^{it}\,\frac{(\gamma_{i}\boldsymbol{q}^{i}+ \gamma_{0}\sqrt{\boldsymbol{q}^2+m^2}+im)}{E-\sqrt{\boldsymbol{q}^2+m^2}}+\theta(-t)\,e^{it}\,\frac{(\gamma_{i}\boldsymbol{q}^{i}- \gamma_{0}\sqrt{\boldsymbol{q}^2+m^2}+im)}{E+\sqrt{\boldsymbol{q}^2+m^2}}\right)$$

Using θ(t) + θ(−t) = 1 together with the identity ∫(Ae^(it)θ(t) + Be^(it)θ(−t))dt = (A+B)/2 leads to the same final result reached in momentum space.

Frequently Asked Questions

What is the difference between a virtual particle and a real particle?

A real particle satisfies the on-shell energy-momentum relation E² = p² + m² and can be detected directly. A virtual particle, in momentum-space calculations, is generally off-mass-shell and never appears as an observable outcome of an interaction. It exists only as a mathematical term within a perturbation expansion of the S-operator.

Why do virtual particles appear on-shell in coordinate space but off-shell in momentum space?

In coordinate space, only particles that can physically exist have creation and annihilation operators, so intermediate excitations there satisfy E² = p² + m². In momentum space, each internal line is assigned an independent 4-momentum variable that need not satisfy this relation, because the time-ordering restriction present in coordinate space has been removed.

Is 4-momentum conserved at every vertex in a Feynman diagram?

In the momentum-space representation, yes: 4-momentum is conserved at each vertex and around each loop, enforced by a Dirac delta function per loop. In the coordinate-space representation, only 3-momentum is conserved at each vertex, since time-ordering prevents full 4-momentum conservation from emerging until the time-ordering restriction is removed.

What is a propagator?

A propagator is a function of a virtual particle’s 4-momentum that represents an internal line in the momentum-space representation of a Feynman diagram. For the electron-photon “H”-diagram discussed above, the propagator takes the form (−iQ̸ + m)/(q² + m² − iε).

Why was an “H”-shaped diagram used instead of one with a virtual photon?

An “H”-shaped diagram was chosen because it involves a virtual electron rather than a virtual photon, which avoids extra complications from gauge theory and demonstrates that the electromagnetic interaction between an electron and a photon is not mediated solely by virtual photons.

What role does the Dirac delta function play in these calculations?

The Dirac delta function is a distribution, not an ordinary function, and has meaning only within an integral. It enforces conservation constraints, such as momentum conservation at a vertex, by reducing the number of integration variables in the calculation.