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!

3.3 Generating Functional

To compute correlation functions, we introduce a source term J(x):

$$\boxed{Z[J] = \int \mathcal{D}\phi \, e^{i\int d^4x[\mathcal{L}(\phi) + J(x)\phi(x)]}}$$

Z[J] is called the generating functional because:

$$\langle 0|T\{\phi(x_1)\cdots\phi(x_n)\}|0\rangle = \frac{1}{Z[0]}\frac{1}{i^n}\frac{\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)}\Bigg|_{J=0}$$

Each functional derivative "pulls down" one field operator! This is the key formulaconnecting path integrals to correlation functions.

Free Field Result

For the free field, we can complete the square:

$$Z_0[J] = Z_0[0] \exp\left[\frac{i}{2}\int d^4x d^4y \, J(x)D_F(x-y)J(y)\right]$$

where D_F(x-y) is the Feynman propagator:

$$D_F(x-y) = \int \frac{d^4k}{(2\pi)^4} \frac{ie^{-ik \cdot (x-y)}}{k^2 - m^2 + i\epsilon}$$

The 2-point function is:

$$\boxed{\langle 0|T\{\phi(x)\phi(y)\}|0\rangle = D_F(x-y)}$$

This reproduces our result from canonical quantization!

3.4 Wick Rotation (Euclidean Path Integral)

Minkowski spacetime (real time t) has an oscillating integrand e^iS. We can rotate to imaginary time τ = it (Wick rotation):

$$t \to -i\tau \quad \Rightarrow \quad S \to -S_E \quad \Rightarrow \quad e^{iS} \to e^{-S_E}$$

The Euclidean action is:

$$S_E[\phi] = \int d^4x_E \left[\frac{1}{2}(\partial_\mu\phi)^2 + \frac{1}{2}m^2\phi^2\right]$$

Now the path integral is:

$$Z_E = \int \mathcal{D}\phi \, e^{-S_E[\phi]}$$

This is a convergent integral! e^-S_E is a damping factor. Field configurations with large action are suppressed, making calculations easier.

💡 Why Wick Rotation?

Makes path integral well-defined mathematically
Connection to statistical mechanics (partition function Z = Tr e^-βH)
Easier to evaluate integrals (no oscillations)
Can analytically continue back to Minkowski spacetime

3.5 Interacting Fields

For interacting theories like φ⁴:

$$S[\phi] = S_0[\phi] + S_{\text{int}}[\phi], \quad S_{\text{int}} = -\frac{\lambda}{4!}\int d^4x \, \phi^4$$

We can't evaluate the path integral exactly! But we can expand in powers of λ:

$$Z[J] = \int \mathcal{D}\phi \, e^{iS_0[\phi]} e^{iS_{\text{int}}[\phi]} e^{i\int J\phi}$$

Expand e^iS_int in a Taylor series:

$$e^{iS_{\text{int}}[\phi]} = 1 + iS_{\text{int}}[\phi] + \frac{(iS_{\text{int}}[\phi])^2}{2!} + \cdots$$

Each term gives Feynman diagrams at a given order! This is perturbation theory.

🎯 Key Takeaways

Functional integral: Z = ∫𝒟φ e^iS[φ] (sum over field configurations)
Free field: Gaussian integral gives propagator D_F
Generating functional Z[J]: Derivatives give correlation functions
Wick rotation: t → -iτ makes integral convergent (Euclidean QFT)
Interacting theories: Expand in coupling constant → Feynman diagrams
Path integral ↔ Canonical quantization (same physics, different formulation)
Next: Computing correlation functions systematically!

Quantum Field Theory Course

Path Integrals in QFT

🔗Course Connections

📚Prerequisites

Video Lecture

3.1 From Particles to Fields