Feynman Rules from Path Integrals and the S-Matrix

From Correlation Functions to Scattering Amplitudes

The ultimate goal of much of particle physics is to predict the outcomes of scattering experiments. In these experiments, we prepare a set of particles in an initial state, let them interact, and then measure the final state.

The key conceptual steps are:

1. Asymptotic States: We imagine preparing an initial state $|\alpha {⟩}_{in}$ of well-separated, non-interacting particles at time $t\to -\infty$. These are our "in" states.
2. Evolution: These particles travel inwards, interact in a complex, localized region of spacetime, and then produce a set of outgoing particles.
3. Final State: The outgoing particles travel outwards and, at time $t\to +\infty$, form a final state $|\beta {⟩}_{out}$ of well-separated, non-interacting particles (the "out" states).

The fundamental object that connects these asymptotic states is the S-matrix (or scattering matrix). It is the operator that evolves "in" states to "out" states, and its matrix elements, $S_{\beta \alpha }=⟨{\beta }_{out}|{\alpha }_{in}⟩$, are the probability amplitudes for a given scattering process to occur. All the dynamics of the interaction are encoded within the S-matrix.

To get a clear picture of this setup, please watch the following segment from an MIT OpenCourseWare lecture.

The crucial question is: how do we calculate these S-matrix elements? The path integral gives us correlation functions $⟨\Omega |T(\varphi (x_1)\dots \varphi (x_n))|\Omega ⟩$, where $|\Omega ⟩$ is the true vacuum of the interacting theory. There must be a connection between these two objects.

The LSZ Reduction Formula: The Bridge

The connection is provided by one of the most important results in QFT: the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula.

In essence, the LSZ formula states that S-matrix elements are the residues of the poles of momentum-space correlation functions, evaluated when the external particle momenta go "on-shell" (i.e., satisfy $p^2=m^2$).

Let's unpack this. A full n-point correlation function, or Green's function $G^{(n)}(p_1,\dots ,p_n)$, can be represented by the sum of all Feynman diagrams with $n$ external legs. The LSZ formula provides a prescription:

1. Take the full n-point Green's function in momentum space.
2. For each external leg, "amputate" it. This means you remove the full propagator associated with that leg.
3. Put the momenta of all external legs on the mass shell ($p_k^2=m^2$).
4. Multiply by a factor of $\sqrt{Z}$ for each external leg, where $Z$ is the wavefunction renormalization constant (which we'll discuss more in the context of renormalization).

The result is the desired S-matrix element (more precisely, the scattering amplitude $\mathcal{M}$, which is the non-trivial part of the S-matrix).

The following reading provides a more formal statement of the LSZ formula and a clear diagrammatic interpretation of "amputation."

The LSZ formula is our bridge:

Now we need to figure out how to calculate those interacting correlation functions using the path integral.

Calculating Interacting Correlation Functions

We have the generating functional for a full interacting theory:
$Z[J]=\int \mathcal{D}\varphi e^{iS[\varphi ]+i\int d^{4}xJ(x)\varphi (x)}$
where the action $S[\varphi ]$ is split into a free part and an interaction part, $S[\varphi ]=S_0[\varphi ]+S_{int}[\varphi ]$. For example, in ${\varphi }^4$ theory, $S_{int}[\varphi ]=\int d^{4}x\frac{\lambda }{4!}\varphi (x{)}^4$.

We cannot solve this integral exactly. The key is to use a perturbative expansion in the coupling constant ($\lambda$ in this case). We can pull the interaction term out of the path integral by treating it as an operator.

1. Expand the exponential:
   $Z[J]=\int \mathcal{D}\varphi e^{i{S}_0[\varphi ]+i\int J\varphi }\left({\sum }_{k=0}^{\infty }\frac{(i{S}_{int}[\varphi ]{)}^k}{k!}\right)$

2. Replace fields with derivatives: Notice that inside the integral, any instance of $\varphi (x)$ can be generated by acting on the $e^{i\int J\varphi }$ term with a functional derivative: $\varphi (x)\leftrightarrow -i\frac{\delta }{\delta J(x)}$. We can therefore replace every $\varphi$ in $S_{int}$ with this derivative operator, which then acts outside the integral.
   $Z[J]=\left({\sum }_{k=0}^{\infty }\frac{i^k}{k!}{\left(S_{int}\left[-i\frac{\delta }{\delta J}\right]\right)}^k\right)\int \mathcal{D}\varphi e^{i{S}_0[\varphi ]+i\int J\varphi }$

3. Final Form: The remaining integral is just the generating functional for the free theory, $Z_0[J]$, which we know how to calculate!
   $Z[J]=\exp \left(i{S}_{int}\left[-i\frac{\delta }{\delta J}\right]\right)Z_0[J]$
   where $Z_0[J]=Z_0[0]\exp \left(-\frac{1}{2}\int d^{4}x{d}^{4}yJ(x)D_F(x-y)J(y)\right)$, and $D_F$ is the free Feynman propagator.

This remarkable formula is our "machine" for generating all correlation functions in the interacting theory, order by order in the coupling constant.

Deriving Feynman Rules

Let's see this machine in action for ${\varphi }^4$ theory, where ${\mathcal{L}}_{int}=-\frac{\lambda }{4!}{\varphi }^4$, so $S_{int}=\int d^{4}z\frac{\lambda }{4!}\varphi (z{)}^4$.

The formula becomes:
$Z[J]=\exp \left(i\int d^{4}z\frac{\lambda }{4!}{\left(-i\frac{\delta }{\delta J(z)}\right)}^4\right)Z_0[J]$

To derive the Feynman rule for the interaction vertex, we calculate the 4-point correlation function to first order in $\lambda$.
$⟨\Omega |T(\varphi (x_1)\varphi (x_2)\varphi (x_3)\varphi (x_4))|\Omega ⟩\approx \frac{1}{Z[0]}\left(-i\frac{\delta }{\delta J_1}\right)\dots \left(-i\frac{\delta }{\delta J_4}\right)Z[J]{|}_{J=0}$
We only need the term linear in $\lambda$ from the expansion of $Z[J]$:
$\dots \left(i\int d^{4}z\frac{\lambda }{4!}{\left(-i\frac{\delta }{\delta J(z)}\right)}^4\right)Z_0[J]{|}_{J=0}$

This looks daunting, but it has a beautiful graphical interpretation.

• The four derivatives $\frac{\delta }{\delta J(x_i)}$ open up the $Z_0[J]$ exponential.
• The four derivatives $\frac{\delta }{\delta J(z)}$ from the interaction term also act on $Z_0[J]$.
• The derivatives acting on $Z_0[J]$ "contract" in pairs, each contraction spitting out a propagator $D_F$.

The term that corresponds to a single interaction vertex will have the four external points $x_1,x_2,x_3,x_4$ all connecting to the single internal point $z$. This involves pairing each $\frac{\delta }{\delta J(x_i)}$ with one of the $\frac{\delta }{\delta J(z)}$. There are $4!$ ways to do this.

The final result for the amputated connected 4-point function is simply $-i\lambda$. This is the Feynman rule for the vertex! The $1/4!$ symmetry factor in the Lagrangian is precisely cancelled by the $4!$ ways of connecting the fields at the vertex.

The following resource provides a concise summary of this procedure.

This generalizes to any interaction. The Feynman rules for vertices are read directly from the interaction Lagrangian ${\mathcal{L}}_{int}$:

• For each interaction term in $i{\mathcal{L}}_{int}$, draw a vertex connecting the fields in that term.
• The value of the vertex is the coefficient of that term in $i{\mathcal{L}}_{int}$, stripping off the fields.
• A symmetry factor is included depending on how many identical fields meet at the vertex.

For example, in QED, the interaction Lagrangian is ${\mathcal{L}}_{int}=e\bar{\psi }{\gamma }^{\mu }{A}_{\mu }\psi$. The vertex rule is therefore $ie{\gamma }^{\mu }$, representing an electron, a positron (or electron moving backward in time), and a photon meeting at a point.

Conclusion

Today, we have connected all the dots, establishing the path integral as a complete framework for perturbative QFT.

Key Takeaways:

• Physical observables in scattering experiments are the elements of the S-matrix, which describe transitions between asymptotic "in" and "out" states.
• The LSZ Reduction Formula provides the crucial bridge, relating S-matrix elements to the on-shell, amputated time-ordered correlation functions of the interacting theory.
• The path integral's generating functional, $Z[J]$, can be expressed for an interacting theory as a series of derivative operators (built from $S_{int}$) acting on the free-theory generating functional, $Z_0[J]$.
• This formulation allows us to systematically calculate interacting correlation functions perturbatively and shows that Feynman rules for vertices are read directly from the interaction terms in the Lagrangian.

We have completed the full chain of logic: from the action $S[\varphi ]$ to the calculation of scattering amplitudes $\mathcal{M}$.

Preview of the Next Lesson:
Our use of the path integral has been firmly in Minkowski spacetime, geared towards calculating scattering amplitudes. However, the formalism reveals a deep connection to another area of physics. In our next lesson, we will explain the relationship between quantum field theory in Euclidean spacetime and statistical mechanics. This is achieved through a "Wick rotation" of the time coordinate, which transforms the path integral into a statistical partition function. This connection is not just a mathematical curiosity; it forms the foundation for non-perturbative computational techniques like Lattice QFT.