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**Definition/Summary**Virtual particles are a mathematical device used in perturbation expansions of the S-operator (transition matrix) of an interaction in quantum field theory.

No virtual particle physically appears in the interaction: all possible virtual particles, and their antiparticles, occur equally and together in the mathematics, and must be removed by integration over the values of their momenta.

In the coordinate-space representation of a Feynman diagram, the virtual particles are on-mass-shell (realistic), but only 3-momentum is conserved at each vertex, not 4-momentum, so there is no immediate way of obtaining 4-momentum-conserving delta functions.

In the momentum-space representation, the virtual particles are both on- and off-mass-shell (unrealistic), but 4-momentum is conserved at each vertex, and also round each loop (as shown by a delta function for each).

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

**Equations**Calculation for an "H"-shaped Feynman diagram for the interaction between an electron and a photon with given incoming and outgoing 4-momentums, and with "exchanged" 4-momentum [itex]Q = (\boldsymbol{Q},E)[/itex]:

The "centre part" of the transition probability is:

[tex]\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)}[/tex]

[tex]\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.[/tex]

[tex]\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)[/tex]

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

[tex]=\ \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)[/tex]

[tex]\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.[/tex]

[tex]\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)[/tex]

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

[tex]=\ \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)[/tex]

[tex]\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.[/tex]

[tex]\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)[/tex]

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

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

[tex]\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.[/tex]

[tex]\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)[/tex]

[tex]=\ \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)[/tex]

[tex]+\ \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)[/tex]

[tex]=\ \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}[/tex]

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

**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): [itex]T\{H(x_1)\cdots H(x_N)\}[/itex]

"Time-ordered" means that the copies are re-arranged in order of their t-components, with the earliest on the right. For example, if [itex]t_3>t_1>t_2[/itex], then [itex]T\{H(x_1)H(x_2)H(x_3)\}\ =\ H(x_3)H(x_1)H(x_2)[/itex]

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

Each copy is the sum (integral) of products of (usually) three operators: they are the creation or annihilation operator for three particles of fixed type (for example, two electrons or positrons, and a photon).

These types must be one particle which is

*not*its own anti-particle (with two operators), such as an electron, and one particle which

*is*(with one operator), such as a photon,

*or*three particles which

*are*.

This sum is over

*every*possible 3-momentum for each particle and its anti-particle, each evaluated at the same 4-position: [itex]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}''[/itex]

These particles are known as virtual particles. They are not created or destroyed in the actual interaction: they appear only in the mathematics.

**On-mass-shell virtual particles:**

These virtual particles are realistic: in other words, they

*can*exist: after all, only particles which

*can*exist can have creation or annihilation operators!

Being realistic, each such particle is on-mass-shell (or "on-shell"): the energy is fixed by the 3-momentum: [itex]E^2\ =\ \boldsymbol{p}^2\ +\ m^2[/itex], where [itex]m[/itex] is the standard mass for that particle.

**Same on-mass-shell virtual particle at each end of every line:**

The two virtual particles (on-mass-shell), one with a creation operator on one side of an internal line in a Feynman diagram, and one with an annihilation operator on the other side,

*must be the same*.

This is because a product [itex]H(x_1)\cdots H(x_N)[/itex] is zero unless every creation operator at one 4-position is "matched" by its own annihilation operator at another 4-position: these matches are represented by internal lines joining each such pair of points. By definition, therefore, a Feynman diagram must have each internal line "matched".

So if the 3-momentum is [itex]\boldsymbol{q}[/itex] on one side, it must be [itex]-\boldsymbol{q}[/itex] on the other side.

**"Phase" at each vertex:**

Each particle at a vertex [itex]x_n[/itex] is associated with a "phase" [itex]e^{ip\cdot x_n}[/itex], where [itex]p[/itex] is its 4-momentum.

Each vertex is associated with a different value of [itex]x_n[/itex] (a 4-vector), and the three particles (literally) connected to that vertex all share that value, so that all three produce a combined "phase" such as [itex]e^{i(p\ -\ k'\ -\ q)\cdot x_n}[/itex].

**Dirac Delta functions:**

The importance of the combined "phase" at each vertex is that it may be replaced by a Dirac delta function,

*provided that it is integrated over*

**all**possible values of [itex]x_n[/itex].This is because the "phase" oscillates if the factor [itex](p\ -\ k'\ -\ q)[/itex] is non-zero, but is a constant ([itex]1[/itex]) if the factor is zero (for all values of x). The oscillations make the integral zero if [itex](p\ -\ k'\ -\ q)[/itex] is non-zero, and the constant, [itex]1[/itex], makes the integral the same as if the "phase" was not there if [itex](p\ -\ k'\ -\ q)[/itex] is zero, apart from a factor of [itex]2\pi[/itex].

Therefore, the "phase", and the integral over [itex]x_n[/itex], can both be replaced by the symbol [itex]\delta(p\ -\ k'\ -\ q)[/itex] (a Dirac delta function), which has the effect of eliminating all versions of the diagram except those in which the total momentum at the vertex is zero.

*This is another way of saying that 4-momentum is conserved at the vertex.*

*Unfortunately*, in the

*coordinate-space*representation, the diagram

*cannot*be integrated over all possible values of x, because values of the t-component of x which are less than the t-component of y (the value for the other vertex) are not allowed for virtual particles (and values that are greater are not allowed for virtual anti-particles).

For each allowed value of the t-component, all values of the

*other*three ("spatial") components are allowed, and so the 3-momentum (only)

*is*conserved, but this is of no immediate use.

Only when this time-ordering restriction is removed (by the mathematical trick of including

*unrealistic*particles) can the diagram be integrated over

*all*possible values of x and y, thereby conserving

*4-momentum*.

**Dirac delta function is not a function**

The Dirac delta function is not a function but a distribution.

It only makes sense in the middle of an integral: it reduces the number of variables to be integrated over, while imposing constraints on the eliminated variables.

For details, see this thread.

**Coordinate-space representation:**

Each internal line [itex]x_jx_i[/itex]

*in the coordinate-space representation*represents the creation and annihilation operators of

*every possible*particle and anti-particle. These particles and anti-particles are realistic (on-mass-shell): each obeys the self-energy-momentum equation [itex]E^2\ =\ \boldsymbol{p}^2\ +\ m^2[/itex].

The particles are created at [itex]x_i[/itex] and annihilated, later, at [itex]x_j[/itex]. The anti-particles are created at [itex]x_j[/itex] and annihilated, earlier, at [itex]x_i[/itex].

It is arbitrary which we call a particle and which an anti-particle: we usually call the positron an anti-particle, but we can call the electron an anti-particle instead, if we adjust the Hamiltonian accordingly.

A photon, of course, is its

*own*anti-particle.

They are all virtual in the sense that none of them is actually created and annihilated: the diagram is a mathematical device, and must be integrated (summed) over the 3-momentum of every possible realistic particle and anti-particle, and also over all time-ordered values of [itex]x_i[/itex] and [itex]x_j[/itex].

**Momentum-space representation:**

The trick which removes the time-ordering restriction is the introduction of a new phase, combining the time coordinate with a new energy variable [itex]s[/itex]: together with the original 3-momentum variable(s), this gives a new four-component variable [itex]q \ =\ (\boldsymbol{q},s)[/itex] which behaves as a 4-momentum, and appears in a combined phase of the form [itex]e^{i(q-Q)\cdot(x_j-x_i)}[/itex], which disappears when integrated over [itex]x_j-x_i[/itex], to be replaced by a delta function [itex]\delta(q,Q)[/itex], multiplied by a propagator (a function of [itex]q[/itex]).

For a simple example, see Equations above, in which the propagator is [itex](-i\,q\hspace{-0.8ex}/\ +\ m)/(q^2\ +\ m^2\ -\ i\varepsilon)[/itex]. This case is simpler than usual, since the delta functions in this case eliminate the need to integrate over [itex]q[/itex].

An "H"-diagram has been chosen, rather than the similar "stick-man" diagram, since it involves a virtual electron rather than a virtual photon: this is partly to avoid the complications of gauge theory, and partly to emphasise that the electromagnetic interaction is not mediated

*solely*by virtual photons.

A different

*variable*4-momentum [itex]q[/itex] is assigned to each internal line. Since they do not have the standard mass appropriate to their line (they have every possible mass), they are called off-mass-shell, and may also be considered the 4-momentums of a virtual particle.

Again, these are virtual in the sense that none of them is actually created and annihilated: indeed, most of them (unlike the coordinate-space virtual particles)

*could not*exist, since they do not have the mass of an actual particle.

The diagram must be now integrated (summed) over the 4-momentum of every realistic

*and unrealistic*particle and anti-particle, but

*not*over the coordinate values, [itex]x_1,x_2,\cdots x_n[/itex], since these have been replaced by delta functions.

This elimination of the coordinates, and of the need to integrate over them, justifies the change of name from "coordinate-space representation" to "momentum-space representation".

**Casimir effect:**

Would someone like to contribute a comment on the Casimir effect?

**Avoiding**

*off-mass-shell*virtual particles:*In principle*, the mathematical device of

*off-mass-shell*virtual particles (and the whole

*momentum-space*representation) can always be

*avoided*, but in practice the calculations will usually be horrendous. In the simple example above, however, we could easily have avoided them by using the substitutions [itex]t = (E-\sqrt{\boldsymbol{q}^2\ +\ m^2})(t_1-t_2)[/itex] and [itex]t = (E+\sqrt{\boldsymbol{q}^2\ +\ m^2})(t_1-t_2)[/itex] at the beginning, giving:

[tex]\int\ dt\ \times\ \left(\theta(t)\ e^{it}\ (\gamma_{i}\boldsymbol{q}^{i}+ \gamma_{0}\sqrt{\boldsymbol{q}^2\ +\ m^2}+im)\left/(E-\sqrt{\boldsymbol{q}^2\ +\ m^2})\right.\right.[/tex]

[tex]\left.+\ \theta(-t)\ e^{it}\ (\gamma_{i}\boldsymbol{q}^{i}- \gamma_{0}\sqrt{\boldsymbol{q}^2\ +\ m^2}+im)\left/(E+\sqrt{\boldsymbol{q}^2\ +\ m^2})\right.\right)[/tex]

and then using [itex]\theta(t)+\theta(-t) = 1[/itex] and so [itex]∫(Ae^{it}\theta(t)+Be^{it}\theta(-t))dt = (A+B)/2[/itex], thus achieving the same final result.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!