Virtual Particles Explained — Quantum Field Theory

Definition / Summary

Virtual particles are a mathematical device used in perturbation expansions of the S-operator (transition matrix) for interactions in quantum field theory.

No virtual particle physically appears in the interaction: all possible virtual particles and their antiparticles occur together in the mathematics and are removed by integration over their momenta.

In the coordinate-space representation of a Feynman diagram, the virtual particles are on-mass-shell (physically allowed): only 3-momentum is conserved at each vertex, not 4-momentum, so 4-momentum-conserving delta functions do not appear immediately. In the momentum-space representation, the virtual particles are generally off-mass-shell (unphysical), but 4-momentum is conserved at each vertex and around each loop (implemented by a delta function for each loop).

In the coordinate-space representation each virtual particle appears “as itself”; in the momentum-space representation it is represented by a propagator (a function of its 4-momentum).

Equations

The calculation for an “H”-shaped Feynman diagram describing the interaction between an electron and a photon with given incoming and outgoing 4-momenta, and with exchanged 4-momentum Q = (Q⃗,E):

The “centre part” of the transition probability is:

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

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

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

(the integral is over all virtual electrons with 3-momentum $\boldsymbol{q}$ created on the left of the “H”, and all virtual positrons with 3-momentum $\boldsymbol{q}$ created on the right, and $\theta(t)\ =\ 1\text{ if }t>0\text{ but }=0\text{ if }t<0$)

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

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

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

(where in the terms with $\theta(t_2-t_1)$ we have replaced $\boldsymbol{q}\text{ and }d^3\boldsymbol{q}$ by $-\boldsymbol{q}\text{ and }-d^3\boldsymbol{q}$)

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

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

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

(here a fictitious energy variable, $s$, has been introduced, enabling $\theta(t)$ to be replaced by $\lim_{\varepsilon\rightarrow 0+}(-1/2\pi i)\int e^{-ist}ds/(s+i\varepsilon)$)

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

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

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

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

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

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

where $k_0$ is the (non-zero) final energy of the photon measured in the reference frame in which the initial electron is stationary, and $Q\hspace{-1.0ex}/\ =\ \gamma_{\mu}Q^{\mu}\ =\ \gamma_i\boldsymbol{Q}^i\ +\ \gamma_0E$.

Extended explanation

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): $T\{H(x_1)\cdots H(x_N)\}$. “Time-ordered” means the copies are rearranged by their t-components, with the earliest on the right. For example, if $t_3>t_1>t_2$, then $T\{H(x_1)H(x_2)H(x_3)\}=H(x_3)H(x_1)H(x_2)$.

Although time-ordering is generally not Lorentz-invariant, it is invariant for non-spacelike pairs of events, and therefore for Hamiltonians provided they commute at spacelike separation: $H(x_1)H(x_2)=H(x_2)H(x_1)\text{ if }(x_1-x_2)^2>0$ (see Weinberg (3.5.13–14)).

Each copy of the Hamiltonian is a sum (integral) of products of (usually) three field operators: creation or annihilation operators for three particles of fixed types (for example, two fermions and a photon). These operators are evaluated at the same 4-position.

These sums run over every possible 3-momentum for each particle and its antiparticle. For a typical vertex,

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

These intermediate excitations are often called virtual particles: they are not created or destroyed in the physical process but appear in the mathematical expansion.

On-mass-shell virtual particles

In the coordinate-space representation the intermediate excitations are “realistic”: they obey the on-shell energy–momentum relation $E^2=\boldsymbol{p}^2+m^2$, since only particles that can exist have creation and annihilation operators.

Matching 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; a product $H(x_1)\cdots H(x_N)$ vanishes unless each creation operator is matched by a corresponding annihilation operator. Thus, if the 3-momentum is $\boldsymbol{q}$ on one side of an internal line, it must be $-\boldsymbol{q}$ on the other.

Phase factor at each vertex

Each particle at a vertex $x_n$ contributes a phase factor $e^{ip\cdot x_n}$, 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 $e^{i(p-k’-q)\cdot x_n}$).

Dirac delta functions and momentum conservation

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). Thus integration over vertex coordinates enforces 4-momentum conservation at each vertex.

In coordinate-space, however, time-ordering prevents integrating over all values of the time component simultaneously (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. Only after removing the time-ordering restriction (via introduction of an auxiliary energy variable) can the diagram be integrated over all coordinates and manifest 4-momentum conservation emerge.

Note: the Dirac delta 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 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, satisfying $E^2=\boldsymbol{p}^2+m^2$). Particles are created at x_i and annihilated later at x_j; antiparticles can be thought of as created at x_j and annihilated earlier at x_i. Which is called “particle” or “antiparticle” depends on convention and the chosen Hamiltonian. Photons are their own antiparticles.

All these modes are virtual in the sense that none corresponds to observable creation or annihilation in the scattering process: the diagram is a bookkeeping device and must be summed (integrated) over the 3-momenta of every allowed on-shell particle and antiparticle and over allowed time orderings 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 $e^{i(q-Q)\cdot(x_j-x_i)}$; integrating over the coordinate difference yields a delta distribution $\delta(q,Q)$ and a propagator (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 (hence they are off-mass-shell), which is why the term “off-shell virtual particle” is used. The propagator example in the equations above is $(-i\,q\hspace{-0.8ex}/\ +\ m)/(q^2\ +\ m^2\ -\ i\varepsilon)$. Momentum-space integrals sum over these 4-momenta while coordinates have been eliminated by delta distributions.

An “H” diagram was chosen in the example because it involves a virtual electron rather than a virtual photon; this avoids extra gauge-theory complications and highlights that the electromagnetic interaction is not mediated solely by virtual photons.

Casimir effect

Discussion of the Casimir effect could be added here. (Original: “Would someone like to contribute a comment on the Casimir effect?”)

Avoiding off-mass-shell virtual particles

In principle, one can avoid introducing off-mass-shell variables and perform all calculations in coordinate-space, but this is usually much more cumbersome. In the simple example above one can substitute

$t=(E-\sqrt{\boldsymbol{q}^2+m^2})(t_1-t_2)$ and $t=(E+\sqrt{\boldsymbol{q}^2+m^2})(t_1-t_2)$ at the start to obtain

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

and then using $\theta(t)+\theta(-t)=1$ and the identity $\int(Ae^{it}\theta(t)+Be^{it}\theta(-t))dt=(A+B)/2$ to reach the same final result.

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