Path Integrals in Scalar Field Theory

Hello! Welcome back to our exploration of the path integral.

In our last lesson, we successfully reformulated single-particle quantum mechanics, replacing the operator formalism with Feynman's intuitive "sum over all histories." We found that the amplitude for a particle to travel between two points, the propagator, could be written as an integral over all possible paths, with each path weighted by a phase $e^{i S /ℏ}$ determined by its classical action.

Today, we make the crucial leap from a single particle to a quantum field. Our goal is to extend the path integral formalism to a scalar field theory and define the generating functional for Green's functions. This object, denoted $Z [J]$ , is a cornerstone of modern QFT. It acts as a master formula from which we can, in principle, extract all the physical information about the theory, particularly the correlation functions that are essential for calculating scattering amplitudes.

1. From a Particle to a Field: The Functional Integral

The transition from quantum mechanics to quantum field theory in the path integral language is conceptually straightforward. A quantum mechanical system with one degree of freedom is described by a path $q (t)$ . A system with $N$ degrees of freedom would be described by a set of paths ${q_{i} (t)}_{i = 1}^{N}$ .

A scalar field, $ϕ (x, t)$ , can be viewed as a system with an infinite number of degrees of freedom, one for each point $x$ in space. The path integral for a single particle, $\int D q (t)$ , thus becomes a functional integral over all possible field configurations, written as $\int D ϕ$ .

The following video provides an excellent and clear explanation of this transition.

QFT2 Lecture 1b: path integrals for free field theory

Dr. Nabil Iqbal from Durham University explains how the path integral for quantum mechanics is generalized to quantum field theory by thinking of a field as a collection of degrees of freedom on a spatial lattice.

Please watch the first 4 minutes and 40 seconds of this video. Focus on the analogy between a set of N particles, q_a(t), and a field \phi(x,t) where the discrete index 'a' becomes the continuous spatial coordinate 'x'.

Just as the action for a particle was a functional of the path $q (t)$ , the action for a field is a functional of the field configuration $ϕ (x)$ , where $x$ denotes the spacetime 4-vector $x^{μ}$ . The action is the integral of the Lagrangian density $L$ over all spacetime:

S [ϕ] = \int d^{4} x L (ϕ (x), \partial_{μ} ϕ (x))

The basic path integral in QFT, which represents the vacuum-to-vacuum transition amplitude, is then formally written as:

Z = \int D ϕ e^{i S [ϕ]}

This integral sums over all possible functions $ϕ (x)$ that can exist throughout spacetime.

2. The Generating Functional for Green's Functions

In the operator formalism, the fundamental objects we calculated were the time-ordered vacuum expectation values of products of field operators, known as Green's functions or correlation functions:

G^{(n)} (x_{1}, \dots, x_{n}) = ⟨ 0∣ T {ϕ^(x_{1}) \dots ϕ^(x_{n})} ∣0 ⟩

In the last lesson, we saw that for a single particle, this corresponded to inserting the classical variable $q (t)$ into the path integral. The same principle applies here. We could calculate $G^{(n)}$ by evaluating:

G^{(n)} (x_{1}, \dots, x_{n}) = \frac{\int D ϕ ϕ ( x _{1} ) \dots ϕ ( x _{n} ) e ^{i S [ϕ]}}{\int D ϕ e ^{i S [ϕ]}}

However, calculating a new integral for each $n$ -point function is cumbersome. The path integral formalism provides a more elegant method through the use of a source field.

We introduce a classical, non-dynamical source field $J (x)$ that couples linearly to our quantum field $ϕ (x)$ . The Lagrangian density is modified to:
$L \to L^{'} = L + J (x) ϕ (x)$
We then define the generating functional $Z [J]$ as the path integral in the presence of this source:
$Z [J] = \int D ϕ e^{i S [ϕ] + i \int d^{4} x J (x) ϕ (x)}$
The remarkable property of $Z [J]$ is that it contains information about all correlation functions. We can extract them by taking functional derivatives with respect to the source $J (x)$ and then setting $J (x) = 0$ . The functional derivative acts like a regular derivative, but for functionals:
$\frac{δ J ( y )}{δ J ( x )} = δ^{(4)} (x - y)$
Using this, you can see that each derivative with respect to $J (x_{k})$ brings down a factor of $i ϕ (x_{k})$ inside the integral. This leads to the master formula for Green's functions:

G^{(n)} (x_{1}, \dots, x_{n}) = \frac{1}{Z [ 0 ]} (\frac{1}{i})^{n} \frac{δ ^{n} Z [ J ]}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0}

This is a powerful idea: if we can calculate the single object $Z [J]$ , we can find any correlation function through differentiation.

The following reading from notes by Prof. Markus Luty provides a clear, text-based summary of this concept.

Correlation Functions and Diagrams

This resource provides a concise introduction to the source method and the definition of the generating functional Z[J].

Please read sections 1 'Sources' and the part of section 1.1 'Correlation Functions' up to equation 1.22 (pages 1-4). Focus on how adding the source term Jφ allows for the definition of Z[J] and how functional differentiation of Z[J] generates the correlation functions.

3. Calculating the Generating Functional for a Free Scalar Field

Let's now apply this formalism to the simplest case: the free real scalar field (the Klein-Gordon field). The action is purely quadratic:

S_{0} [ϕ] = \int d^{4} x \frac{1}{2} (\partial_{μ} ϕ \partial^{μ} ϕ - m^{2} ϕ^{2}) = \int d^{4} x \frac{1}{2} ϕ (- \partial^{2} - m^{2}) ϕ

The generating functional $Z_{0} [J]$ is therefore:

Z_{0} [J] = \int D ϕ exp (i \int d^{4} x [\frac{1}{2} ϕ (- \partial^{2} - m^{2}) ϕ + J ϕ])

This is a Gaussian functional integral. You are familiar with finite-dimensional Gaussian integrals of the form:

\int_{- \infty}^{\infty} d^{n} x exp (- \frac{1}{2} x^{T} A x + b^{T} x) = \frac{( 2 π ) ^{n}}{det A} exp (\frac{1}{2} b^{T} A^{- 1} b)

By drawing an analogy where the vector $x$ becomes the field $ϕ$ , the matrix $A$ becomes the differential operator $i (\partial^{2} + m^{2})$ , and the source term $b$ becomes $i J$ , we can solve this functional integral. This is a common strategy in physics: solve a finite-dimensional version and boldly generalize to the infinite-dimensional case.

The following video walks through this exact calculation.

QFT2 Lecture 1b: path integrals for free field theory

We'll now evaluate Z_0[J] by treating it as an infinite-dimensional Gaussian integral. Pay close attention to how the inverse of the Klein-Gordon operator appears in the final result.

Please watch from 07:51 to 17:36. The first part introduces the goal of calculating the two-point function via functional derivatives. The second part (starting at 10:05) recalls the formula for a Gaussian integral and applies it to the functional integral for Z[J].

The result of the integration is one of the most important formulas in QFT:

Z_{0} [J] = Z_{0} [0] exp (- \frac{i}{2} \int d^{4} x d^{4} y J (x) Δ_{F} (x - y) J (y))

Here, $Z_{0} [0]$ is the (divergent) result with the source turned off, which will cancel in all physical calculations. The crucial piece is $Δ_{F} (x - y)$ . It is the inverse of the Klein-Gordon operator, satisfying:

(- \partial^{2} - m^{2} + i ϵ) Δ_{F} (x - y) = - i δ^{(4)} (x - y)

This is precisely the equation for the Feynman propagator for the scalar field! The $i ϵ$ prescription arises naturally from the need to ensure the path integral converges, just as we discussed in the last lesson.

This result is profound. The path integral for the free theory automatically generates the Feynman propagator as the kernel in the exponent of the generating functional.

Test your understanding!

Using the formula for $Z_{0} [J]$ we just found, and the master formula for Green's functions, explicitly calculate the two-point function $G^{(2)} (x_{1}, x_{2}) = ⟨ 0∣ T {ϕ^(x_{1}) ϕ^(x_{2})} ∣0 ⟩$ .

Show answer

We need to calculate:

G^{(2)} (x_{1}, x_{2}) = \frac{1}{Z _{0} [ 0 ]} (\frac{1}{i})^{2} \frac{δ ^{2} Z _{0} [ J ]}{δ J ( x _{1} ) δ J ( x _{2} )}_{J = 0}

Let's first take one derivative. Let $K [J] = - \frac{i}{2} \int d^{4} x d^{4} y J (x) Δ_{F} (x - y) J (y)$ . Then $Z_{0} [J] = Z_{0} [0] e^{K [J]}$ .

\frac{δ Z _{0} [ J ]}{δ J ( x _{1} )} = Z_{0} [0] \cdot e^{K [J]} \cdot \frac{δ K [ J ]}{δ J ( x _{1} )}

Using the product rule for functional derivatives and the symmetry of $Δ_{F}$ , we get:

\frac{δ K [ J ]}{δ J ( x _{1} )} = - \frac{i}{2} \int d^{4} y Δ_{F} (x_{1} - y) J (y) - \frac{i}{2} \int d^{4} x J (x) Δ_{F} (x - x_{1}) = - i \int d^{4} y Δ_{F} (x_{1} - y) J (y)

Now, we take the second derivative:

\frac{δ ^{2} Z _{0} [ J ]}{δ J ( x _{1} ) δ J ( x _{2} )} = Z_{0} [0] [e^{K [J]} \frac{δ ^{2} K [ J ]}{δ J ( x _{2} ) δ J ( x _{1} )} + \frac{δ e ^{K [J]}}{δ J ( x _{2} )} \frac{δ K [ J ]}{δ J ( x _{1} )}]

Let's evaluate the pieces:

\frac{δ ^{2} K [ J ]}{δ J ( x _{2} ) δ J ( x _{1} )} = \frac{δ}{δ J ( x _{2} )} (- i \int d^{4} y Δ_{F} (x_{1} - y) J (y)) = - i Δ_{F} (x_{1} - x_{2})

Now, we set $J = 0$ . When $J = 0$ , $K [J] = 0$ and $e^{K [J]} = 1$ . The second term in the square brackets has a factor of $J (y)$ from $δ K [J] / δ J (x_{1})$ , so it vanishes. We are left with:

\frac{δ ^{2} Z _{0} [ J ]}{δ J ( x _{1} ) δ J ( x _{2} )}_{J = 0} = Z_{0} [0] \cdot (1) \cdot (- i Δ_{F} (x_{1} - x_{2}))

Plugging this into the formula for $G^{(2)}$ :

G^{(2)} (x_{1}, x_{2}) = \frac{1}{Z _{0} [ 0 ]} (- \frac{1}{1}) (Z_{0} [0] (- i Δ_{F} (x_{1} - x_{2}))) = i Δ_{F} (x_{1} - x_{2})

This confirms that the two-point function is, up to a factor of $i$ , the Feynman propagator, which is exactly what we found using canonical quantization.

Conclusion

In this lesson, we have taken the path integral from the realm of single particles to the universe of quantum fields. This generalization provided us with an elegant and powerful new tool: the generating functional.

Key Takeaways:

The path integral for a field theory, known as a functional integral, is a sum over all possible field configurations $ϕ (x)$ in spacetime.
By introducing a classical source field $J (x)$ , we defined the generating functional $Z [J]$ .
All time-ordered Green's functions (correlation functions) of the theory can be obtained by taking functional derivatives of $Z [J]$ with respect to the source $J$ .
For the free scalar field, $Z_{0} [J]$ is a Gaussian functional. Its explicit calculation shows that it is the exponential of a quadratic in $J$ , where the kernel is the Feynman propagator, $Δ_{F} (x - y)$ .

Preview of the Next Lesson:
We have defined the machinery and shown how it works for a free theory. In the next lesson, we will formalize the result of our exercise and derive the Feynman propagator for a scalar field from the generating functional. We will then explore how to compute higher-point correlation functions and see how Wick's theorem emerges naturally from this formalism, setting the stage for tackling interacting theories.

Recommended resources

1. From a Particle to a Field: The Functional Integral

2. The Generating Functional for Green's Functions

3. Calculating the Generating Functional for a Free Scalar Field

Conclusion