Think of a vibrating drumhead. At each point (x,y) and time t, the height is φ(x,y,t). A "path" in field theory is a complete history of the entire drumhead vibration!

We sum over all possible vibration patterns, each weighted by e^iS[φ].

The functional integral (path integral for fields) is:

$$\boxed{Z = \int \mathcal{D}\phi \, e^{iS[\phi]}}$$

where ∫𝒟φ means "integrate over all field configurations" and the action is:

$$S[\phi] = \int d^4x \, \mathcal{L}(\phi, \partial_\mu\phi)$$

3.2 Free Scalar Field

For the Klein-Gordon field:

$$S[\phi] = \int d^4x \left[\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - \frac{1}{2}m^2\phi^2\right]$$

After integration by parts:

$$S[\phi] = -\frac{1}{2}\int d^4x \, \phi(x)(\Box + m^2)\phi(x)$$

This is a Gaussian functional integral! The result is:

$$\boxed{Z_0 = \int \mathcal{D}\phi \, e^{iS_0[\phi]} = \mathcal{N} \cdot \det^{-1/2}(-i(\Box + m^2))}$$

𝒩 is an infinite normalization constant (we'll divide it out). The determinant is also infinite but well-defined!

Key Concepts (This Page)

QFT path integral sums over field configurations: $Z = \int\mathcal{D}\phi\,e^{iS[\phi]}$
Free scalar field action is quadratic, yielding a Gaussian functional integral
Result involves the functional determinant $\det^{-1/2}(-i(\Box + m^2))$
Integration by parts casts the action in the form $-\frac{1}{2}\phi\,(\Box+m^2)\,\phi$
This sets the stage for introducing sources and computing correlation functions

Next: Generating Functional Z[J]

Quantum Field Theory Course

Path Integrals in QFT

🔗Course Connections

📚Prerequisites

Video Lecture

3.1 From Particles to Fields

💡Field Configuration Space

3.2 Free Scalar Field

Key Concepts (This Page)