Scalar Field Feynman Propagator Derivation

1. The Amplitude for Propagation

Let's start with a simple question: what is the quantum mechanical amplitude for a particle that is created at a spacetime point $y$ to be found later at a spacetime point $x$?

In our QFT framework, the operator $\hat{\varphi }(y)$ can be thought of as creating a particle from the vacuum at position $y$. More precisely, $\hat{\varphi }(y)|0⟩$ is a state containing a particle localized at $y$. The amplitude to find this particle at a later point $x$ is given by the inner product of the state $\hat{\varphi }(y)|0⟩$ with the state $⟨0|\hat{\varphi }(x)$, which represents the annihilation of a particle at $x$.

So, the amplitude is given by the vacuum expectation value (VEV) $⟨0|\hat{\varphi }(x)\hat{\varphi }(y)|0⟩$. Let's calculate this. Recall the mode expansion of the field operator:

When we substitute this into the VEV, many terms will be zero. For example, any term with an annihilation operator ${\hat{a}}_{\vec{p}}$ acting on the right vacuum $\left({\hat{a}}_{\vec{p}}|0⟩=0\right)$ or a creation operator ${\hat{a}}_{\vec{p}}^{†}$ acting on the left vacuum $\left(⟨0|{\hat{a}}_{\vec{p}}^{†}=0\right)$ will vanish. The only non-zero contribution comes from the term $⟨0|{\hat{a}}_{\vec{p}}{\hat{a}}_{\vec{q}}^{†}|0⟩$.

Using the commutation relation $[{\hat{a}}_{\vec{p}},{\hat{a}}_{\vec{q}}^{†}]=(2\pi {)}^3{\delta }^{(3)}(\vec{p}-\vec{q})$ (using the normalization from previous lessons, although conventions can vary), we find:

Here, the four-momentum in the exponent is on-shell, meaning $p^0={\omega }_{\vec{p}}$. This function, $D(x-y)$, gives the amplitude for a particle to travel from $y$ to $x$.

2. Time Ordering and the Feynman Propagator

There's a subtle but crucial point. The expression $⟨0|\hat{\varphi }(x)\hat{\varphi }(y)|0⟩$ is the amplitude for creating a particle at $y$ and annihilating it at $x$. For this to be a sensible physical process, causality suggests that the annihilation must happen after the creation, i.e., $x^0>y^0$.

What if we want to describe propagation from $x$ to $y$? That would require $y^0>x^0$, and the corresponding amplitude would be $⟨0|\hat{\varphi }(y)\hat{\varphi }(x)|0⟩=D(y-x)$.

It would be convenient to have a single mathematical object that handles both cases automatically. This is achieved by introducing the time-ordering operator, $T$. When acting on a product of fields, it arranges them so that the fields at later times are always to the left of fields at earlier times. For two fields, it's defined as:

This can be written compactly using the Heaviside step function, $\Theta (t)$:

We can now define the Feynman propagator, ${\Delta }_F(x-y)$, as the vacuum expectation value of this time-ordered product:

This single object contains the amplitude for propagation from $y$ to $x$ (if $x^0>y^0$) and from $x$ to $y$ (if $y^0>x^0$). This is the fundamental definition of the propagator in position space.

The following reading provides a concise derivation based on this definition.

3. The Propagator in Momentum Space

The expression we've derived,

is correct, but it's not manifestly Lorentz covariant because time and space are treated differently. For practical calculations, especially when we get to interactions, a momentum-space representation that is fully Lorentz covariant is indispensable.

The remarkable result is that the expression above is exactly equivalent to the following four-dimensional integral:

Here, $p$ is a general four-vector, and its component $p^0$ is an integration variable, no longer fixed to ${\omega }_{\vec{p}}$. This form is manifestly Lorentz invariant.

The mysterious "+i$ϵ$" term in the denominator is a small positive real number that we take to zero at the end of the calculation. It's not just a mathematical trick; it's the heart of the time-ordering aspect of the propagator. It tells us how to handle the poles in the complex $p^0$-plane that occur when $p^2-m^2=0$, i.e., when the momentum is on-shell. This is known as the $iϵ$ prescription.

4. Justification via Contour Integration

To prove that the momentum-space integral is equivalent to our time-ordered definition, we perform the $p^0$ integration using complex analysis and Cauchy's residue theorem. Your strong background in mathematical methods, including complex analysis, will be very useful here.

The integral over $p^0$ is:

The denominator has poles at $p^0=\pm \sqrt{{\omega }_{\vec{p}}^2-iϵ}$. A Taylor expansion for small $ϵ$ shows the poles are at $p^0\approx {\omega }_{\vec{p}}-i\frac{ϵ}{2{\omega }_{\vec{p}}}$ and $p^0\approx -{\omega }_{\vec{p}}+i\frac{ϵ}{2{\omega }_{\vec{p}}}$.
Notice that the pole corresponding to positive energy $({\omega }_{\vec{p}})$ is shifted slightly below the real axis, and the pole for negative energy $(-{\omega }_{\vec{p}})$ is shifted slightly above it.

The choice of integration contour depends on the sign of $x^0-y^0$:

• Case 1: $x^0>y^0$. The exponent is $e^{-i{p}^0(\text{positive})}$. For the integral to converge, we must close the contour in the lower half of the complex $p^0$-plane. This contour encloses the pole at $p^0={\omega }_{\vec{p}}$. Applying the residue theorem correctly reproduces the term $\Theta (x^0-y^0)D(x-y)$.
• Case 2: $y^0>x^0$. The exponent is $e^{-i{p}^0(\text{negative})}$. We must close the contour in the upper half-plane. This contour encloses the pole at $p^0=-{\omega }_{\vec{p}}$, and the residue theorem reproduces the term $\Theta (y^0-x^0)D(y-x)$.

This is a beautiful result, showing how the simple $iϵ$ prescription elegantly encodes the causal time ordering of propagation.

The following video gives an extremely clear and detailed walkthrough of this derivation, starting from the motivation and carefully explaining the contour integration.

5. The Propagator as a Green's Function

There is one more crucial property to understand. The Feynman propagator is a Green's function for the Klein-Gordon operator. This means it is a solution to the Klein-Gordon equation with a point-like source:

You can verify this by applying the operator $({\partial }_x^2+m^2)$ to the momentum-space integral form of the propagator. The derivatives bring down a factor of $-p^2$, which cancels the denominator (up to the $iϵ$), leaving an integral that is the definition of a delta function.

This property connects the propagator back to classical physics. It represents the response of the field at point $x$ to a unit impulse source at point $y$. This is the foundation for why the propagator is the building block of all interactions: particles are sources for the fields with which they interact, and the propagator describes how the influence of that source travels through spacetime.

The following reading provides a second, very detailed derivation that also demonstrates this Green's function property.

Conclusion

Today, we've derived one of the most important objects in quantum field theory, the Feynman propagator. It is the bridge between the free theory we've developed so far and the interacting theories we will study next.

Key Takeaways:

• The Feynman propagator, ${\Delta }_F(x-y)$, gives the amplitude for a scalar particle to propagate between spacetime points $x$ and $y$.
• It is formally defined as the vacuum expectation value of the time-ordered product of two field operators: ${\Delta }_F(x-y)=⟨0|T\{\hat{\varphi }(x)\hat{\varphi }(y)\}|0⟩$.
• The time-ordering operator $T$ ensures that the propagator describes propagation forward in time, regardless of whether $x^0>y^0$ or $y^0>x^0$.
• In momentum space, the propagator has a manifestly Lorentz-invariant form:
  ${\Delta }_F(x-y)=\int \frac{d^{4}p}{(2\pi {)}^4}\frac{i}{p^2-m^2+iϵ}{e}^{-ip\cdot (x-y)}$
• The $iϵ$ prescription correctly implements the time-ordering condition by shifting the poles in the complex $p^0$-plane, which is resolved using contour integration.
• The propagator is a Green's function for the Klein-Gordon equation, meaning it describes the field's response to a point source.

Preview of the Next Lesson:

Having defined the propagator, a natural question arises: does our theory respect causality? We've built in time-ordering, but can information travel faster than light? In the next lesson, we will demonstrate how imposing commutation relations ensures the locality of observables and preserves causality outside the light cone. This will involve calculating the commutator $[\hat{\varphi }(x),\hat{\varphi }(y)]$ for spacelike separations and showing that it vanishes, a non-trivial result with profound implications.