Field approach to QFT

Quantum Fields: a learner’s perspective

As someone still learning QFT, I believe there's a special advantage in explaining the ideas while they’re fresh. I still remember what was confusing to me a few months ago—and I can highlight the key insights that helped me move forward, without overwhelming the reader with technical detail. That’s what this paper aims to do.

 

In this blog post, I’d like to offer you an overview of the first part of my paper, where I lay out the conceptual foundations of QFT—not through abstract postulates, but through familiar terrain: wavefunctions, Hilbert spaces, and harmonic oscillators. If you’ve studied QM before, you’re already halfway there.

QFT is Quantum Mechanics—just with a different raw material

A common misconception is that QFT somehow replaces quantum mechanics. In truth, it’s the opposite: QFT is quantum mechanics, applied in a particular way. The shift lies not in the rules of quantum theory, but in what we apply them to. Instead of describing particles with definite positions or discrete spin states, we describe entire field configurations—functions that assign values to every point in space.

In QM, we work with wavefunctions \( \Psi(q) \) over a space of positions \( q \in \mathbb{R} \). In QFT, we step up a level: we deal with wavefunctionals \( \Psi[\phi] \) defined over fields \( \phi(x) \), where each field is itself a function from space to real numbers. It’s functions-of-functions now, but the rules of quantum theory stay the same. Our Hilbert space just becomes much larger.

This perspective is sometimes called the field approach to QFT, and it’s the path I take in my exposition. My goal isn’t to rigorously define every object from the start (we’ll leave that to the textbooks), but rather to build intuition—to help the reader see the forest before walking into the trees.

From classical fields to operator-valued fields

To see the transition from classical to quantum fields, we start with the standard single harmonic oscillator, and then we add new ones. And as you let the number of oscillators grow toward infinity, you’re left with a continuous field. But before giving this last leap, you can quantize the infinite but discrete collection of oscillators, and you have operators for each oscillator.


When you have a continuous amount of oscillators, you have a continuous amount of operators: an operator-valued field, i.e., a quantum field.

You can find more details here.

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