Step by Step Derivation of the Propagator

1        The Approach

The first part of QFT is a free particle theory (no interactions, see Unifying Chart 3, Part 1).  After this, interactions are introduced (U.C. 3, Part 2).  In the course of deriving the interaction theory, a mathematical relationship arises that is called the Feynman propagator.  Physically, it can be visualized as representing a virtual particle that exists fleetingly and carries energy, momentum, and in some cases, charge from one real particle to another.  Thus, it is the carrier or mediator of force (interaction.)

Heuristically, it may help to consider the virtual particle as created at a particular spacetime point and destroyed at a later spacetime point, and this is how Feynman diagrams portray it.  From this (heuristic) perspective the field operator   (y) can be considered to create a virtual scalar particle at y, and the field operator  (x) destroys that virtual particle at x.  The scalar propagator incorporates these two field operators in a sort of “short-hand” way.

Note that the above “creation/destruction at a point” perspective can help initially in understanding the derivation of the propagator, but we caution that it will have to be modified and refined.  We will save that to the end when, after digesting the derivation to follow, this modification will be easier to understand.

We will now derive a relationship for the propagator using the field operators acting on the vacuum, and will later see (U.C. 3, Part 2) that this derived relationship arises naturally in the full mathematical development of the interaction theory.

2        Milestones in the Derivation

We start with the coefficient commutation relations and proceed in five distinct steps.  The entire derivation is for continuous (rather than discrete) eigenstate solutions of the field equation (Klein-Gordon here), since the propagator represents a virtual particle in the vacuum and the vacuum is unbounded.  We represent the scalar Feynman propagator with the symbol .

      As noted, our starting point is the coefficient commutators



and from there we follow five steps.  The first two are purely mathematical, and will serve as background that will help us with the remaining steps.  The third step comprises a physical interpretation of the propagator, and the last two steps are mathematical manipulations to get that interpretation into convenient form (the form that is used in QFT interaction analysis.)

1.      First Step:  Use (1) to find commutation relations for positive and negative frequency solutions, i.e., determine what  is in


where Δ± is different from the propagator ΔF, and (2) is found in terms of an integral over 3 momentum k (as the ± (x) and   ± (y) are integrals over 3-momentum).

2.      Second Step:  Express  in terms of a contour integral in the complex plane (in the space of 4-momenta).

3.      Third Step:  Express the Feynman propagator ΔF as a time ordered operator (actually the vacuum expectation value of this operator), which represents a particle or antiparticle created at one point in space and time in the vacuum and destroyed at another place and time.

4.      Fourth Step:  Express ΔF as a contour integral in the space of 4-momenta (with the help of the second step above.)

5.      Fifth Step:  Re-express ΔF as an integral over real (rather than complex) energy and 3-momenta.  This is the form most suitable for analysis.


3        The Derivation

3.1    Step 1: Commutation Relations for Positive/Negative Frequency Solutions




Then, substituting the integral (boundary free, continuous) form for the Klein-Gordon solutions gives


and thus,





leads to






3.2    Step 2: Expressing iΔ± as Contour Integrals

Text Box: Figure 1. Contour Integral for Real, Positive FrequencyConsider the complex plane for a function f of the complex variable k0, i.e., f (k0).  Here, the symbol k0 is not a pole (poles are usually designated with null subscript), but represents a complex number generalization of the zeroth component (the energy) of 4-momentum k.  We concern ourselves with the particular case where k0 takes on the real value ωk. From complex variable theory,




Now, re-express (5) as



where we take the bracketed quantity as equal to f (ωk), and where



We can then use (10) in (8) to re-express f (ωk) in terms of a contour integral.  Using this for the bracket in (9), we find (9) becomes


where the integral notation now implies integration over four dimensions of the 4-momentum, with the 3-momentum part from   to + in real space and the energy part a contour integral in complex space.  Note that the integral does not “blow up” because k0  ωk over the contour integral.  We are using a mathematical trick that works, though it jars our usual understanding that, for real particles, the zeroth component of 4-momentum equals energy.  k0 has at this point become, for us, a variable that generally does not equal energy ωk.

We modify (11) a little by noting what is always true mathematically for any four vector, and thus true for 4-momentum components,


and what is physically true for rest mass, energy, and 3-momentum,



Solve these last two relations for k0 and ωk and substitute in the last line of (11) to get



Text Box: Figure 2. Contour Integral for Real, Negative Frequency

For iΔ (x  y), we carry out similar steps except that the contour integral (still c.c.w.) is now about  ωk.  When all is said and done, we find the only difference from (14) to be the contour, which is now about the negative frequency value and designed by C .





3.3    Step 3:  The Feynman Propagator as the VEV of a Time Order Operator

Text Box: Figure 3. Creation & Destruction of Virtual Particle/AntiparticleFigure 3a) represents creation of a particle, which will be virtual, at y and destruction of it at x.  Figure 3b) represents creation of an antiparticle at x and destruction of it at y.  Virtual particles are never detected when real particles interact, so the same effect on the real particles could be realized by either of the processes in Figure 3.  For example, a virtual particle carrying charge from y to x would represent the same charge exchanges as an antiparticle carrying opposite charge from x to y.  Thus, we need a relationship for the propagator that includes both scenarios as possibilities.

To this end, we need an operator that will create a particle first if ty < tx, but create an antiparticle first if tx < ty.    Consider the time ordered operator T, defined as follows.

If ty < tx,   (y) operates first (creates a particle) and is placed on the right, with  (x) operating second (destroys the particle) and placed on the left.


If tx < ty,  (x) operates first (creates an antiparticle) and is placed on the right with   (y) operating second (destroying the antiparticle) and placed on the left.


We now define what is called the transition amplitude density, which equals the vacuum expectation value (VEV) of the above time ordered operator.  It is an amplitude, similar to the amplitude of a wave function in NRQM, because, as we will shortly see, the square of its magnitude equals the probability density of it being observed.  (As the square of the magnitude of the amplitude for a component of the wave function equals the probability of it being observed.)

This transition amplitude density is



and this represents both possible scenarios of Figure 3.  In wave mechanics, the bracket above is an integration over all space.  This is still true, but note carefully that the integration variable is over the space variable of the bra and ket (think x), but not the time ordered variables x and y.  In matrix mechanics, we merely think of a bracket as equaling zero unless the bra and ket represent the same particle(s).

To gain insight into (18), consider the transition amplitude density operating on the vacuum when a virtual particle is propagated.  Then,


where G, F, and H are numeric factors that result from the creation and destruction operations (such as the normalization coefficients that are part of the field operators), which we will not express explicitly here.  Thus, we have a general ket left, which in this case is part vacuum state, with the amplitude of the vacuum state part being GF, and the amplitude of the multiparticle state (scalar plus anti-scalar) part being HF.  For appropriate normalization, the square of the magnitude of GF is the probability of observing the vacuum (no particles left after the transition.)  To find the amplitude GF, we need only form an inner product of the last line of (19) with 0, i.e.,



Thus, the VEV of the time ordered operator is an amplitude, the square of whose magnitude is the probability of the transition from vacuum initially to vacuum finally.  Actually, |G(x)F(y)|2 is a probability density (to be precise, a double density), because it is a function of x and y.  We would need to integrate this over all x and y to get the actual probability, and this is what one does in interaction theory to calculate probabilities and cross sections.

In a similar way, the same time ordered operator can be used for antiparticle propagation (with time for x and y reversed.)

Given all of this, we can define the Feynman propagator ΔF (where the factor of i is introduced with foresight) by




3.4    Expressing ΔF as a Contour Integral

For ty < tx,


where the last line follows from the prior one because Δ+(x  y) is numeric and includes no operators.

For tx < ty, in similar fashion, we find


In summary,


where in section 3.2, we expressed Δ±(x  y) as contour integrals, and thus we have here expressed the Feynman propagator in terms of contour integrals.  Note also, that the Feynman propagator, although encompassing operators in its initial definition, turns out to be simply a numeric quantity without operators.


3.5    Step 5: Re-express ΔF in Most Convenient Form

We would like two things more: 1) express the propagator as a single function so we don’t have to keep track (while we are integrating over spacetime and doing other things) of whether the virtual field is a particle or antiparticle (i.e., whether to use the Δ+ or Δ function), and 2) have all our integrations over real numbers rather than deal with contour integrals.

To do this, consider Figure 4 where we have shown two contour integrals.  The top loop represents Δ and encloses ωk with a ccw path.  The lower loop encloses +ωk, but since it has a cw integration path, represents  Δ+.

Thus, we can define the Feynman propagator of (24) as


ΔF    (integral of Fig 4 for either loop),


where the proportionality sign takes care of the concomitant integration over k, as well as the various constants involved.

Text Box: Figure 4.  Contour Integrals for D– and – D+So we can then re-write the propagator (24) with (14), (15), and Figure 4 as




where the CF on the integral defines the route we take in the plane of Figure 4.

Now, consider enlarging the outer hemispheric parts of the two loops in Figure 4 so they extend essentially to infinity.  The value of the contour integrals over them will remain unchanged.  But the k2 value in the denominator of (25) will become so large that any contribution to the integral over those parts of the path will become negligible.  This means that we can effectively take the integral of (25) as extending only along the real axis from minus to plus infinity as in Figure 5.

We can further simplify by moving the poles an infinitesimal distance η off the real axis as shown in Figure 6 and deform the contour so that it is all along the real axis.  In the limit as η  0, the integral will have the same value, though we must now include this slight pole shift in the propagator expression (25).  We do this by recalling from (12) and (13) that we used


to obtain the denominator of (25), so we must temporarily restate (25) using the right hand side of (26), then shift the poles.  Thus, (25) becomes


If we then use (26) again, ignore second order terms in η, and take ε = 2ηωk, we have our final result for the Feynman scalar propagator



Note the advantages of this form.  We now have a single mathematical relationship that automatically describes both a particle propagating from y to x and an antiparticle propagating from x to y.  We also have done away with the cumbersome contour integrals in favor of a simple 4D integral over the entire 4-momentum space.  In practice, we can evaluate this integral then take ε to zero after the integration is carried out.

4        Comments on the Propagator and Its Derivation

4.1    The Propagator and Interaction Theory

The derivation above was formulated with an eye to interaction theory.  In that theory, amplitudes are derived for various kinds of interactions between various particles.  The square of the magnitude of each amplitude turns out to be the probability of that particular interaction (transition) occurring.  These transition amplitudes each depend on the initial real particles, the final real particles, and the virtual particle(s) that mediate the transition.  It turns out that the factor in the amplitude representing the virtual particle contribution is identical to the Feynman propagator ΔF as we defined it in the VEV of the time ordered operator (21).  Thus, it is also equal to (28), so we can simply plug the RHS of (28) into the overall transition amplitude as part of our analysis.

This is one reason we started with the relation to create and destroy a virtual scalar particle, rather than what one might initially expect, the simpler creation and destruction operator relation a(k)a(k).  Our heuristic approach was tailored to match what we knew would be coming in the mathematical development of interaction theory.

4.2    Meaning of Spacetime Points y and x

In Figure 3a we imply the virtual particle is created at y and destroyed at x.  In Feynman diagrams virtual particles are depicted in this way, and the real incoming particle can be thought of a being destroyed at y, often with an outgoing real particle created at y.  The virtual particle is also thus created at y.  At x the virtual is destroyed, with the simultaneous creation of at least one outgoing real particle at x, often accompanied by the destruction of another real incoming particle at x.

To be precise, it is more correct to think of the incoming, outgoing, and virtual particles as moving waves spread out in space.  What we calculate for a given y and x is the probability density for the interaction as a function of the coordinates y and x.  If y and x are closer the probability density for the interaction to occur is greater; if farther away, the probability density is less.  Integrating over all x and y gives the total probability for observing the interaction.

4.3    Momentum Space Form of the Propagator

From (28), we can readily write down the 4-momentum space form of the propagator, which will be very useful,



4.4    Spinor and Vector Propagators

In parallel fashion to all of the above, we can derive Feynman propagators for Dirac particles and photons.

For spin  particles,




For photons,






Unifying Chart 3, Part 1  (From Field Eqs to Propagators and Observables)

Unifying Chart 3, Part 2  (Interaction Theory.  From Operators and Propagators to Feynman rules.)

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