How does QFT differ from Quantum Mechanics?
Short Answer
Quantum Mechanics (QM) describes a fixed number of particles as quantum objects. Quantum Field Theory (QFT) promotes fields to quantum objects, allowing particle creation and annihilation.
Key Differences
| Aspect | Quantum Mechanics | Quantum Field Theory |
|---|---|---|
| Basic object | Particle wave function \Psi(x,t) | Field operator \hat{\phi}(x,t) |
| Particle number | Fixed | Variable (creation/annihilation) |
| Relativity | Non-relativistic | Relativistic |
| Position | Observable \hat{x} | Label (parameter) |
| Hilbert space | L^2(\mathbb{R}^3) | Fock space |
The Conceptual Shift
In Quantum Mechanics
- Particles are fundamental objects
- Wave function gives probability amplitude for finding particle at position x
- Number of particles is conserved
In Quantum Field Theory
- Fields are fundamental objects
- Particles are excitations of the underlying field
- Particle number can change (pair creation, annihilation)
Mathematical Structure
QM: Wave Functions
States are vectors in L^2(\mathbb{R}^3):
\Psi(x) \in \mathcal{H} = L^2(\mathbb{R}^3)
QFT: Fock Space
States live in Fock space, a direct sum over particle number sectors:
\mathcal{F} = \bigoplus_{n=0}^{\infty} \mathcal{H}_n
where \mathcal{H}_n is the n-particle Hilbert space.
Field Operators
In QFT, we have operator-valued distributions:
\hat{\phi}(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \left( \hat{a}_p e^{-ipx} + \hat{a}_p^\dagger e^{ipx} \right)
where:
- \hat{a}_p^\dagger creates a particle with momentum p
- \hat{a}_p annihilates a particle with momentum p
Why QFT is Necessary
Relativistic Consistency
Special relativity + quantum mechanics leads to:
- Negative energy solutions (antiparticles)
- Pair creation when E > 2mc^2
- Non-conservation of particle number
The Locality Principle
In QFT, causality is built in:
[\hat{\phi}(x), \hat{\phi}(y)] = 0 \quad \text{when } (x-y)^2 < 0
Operators at spacelike-separated points commute (or anti-commute for fermions).
Example: From QM to QFT
Harmonic Oscillator (QM)
\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2 = \hbar\omega\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)
Free Scalar Field (QFT)
\hat{H} = \int \frac{d^3p}{(2\pi)^3} E_p \left(\hat{a}_p^\dagger\hat{a}_p + \frac{1}{2}\right)
QFT is like an infinite collection of harmonic oscillators, one for each momentum mode!
References
- M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995), Chapters 1–2
- S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, 1995)
- A. Zee, Quantum Field Theory in a Nutshell, 2nd ed. (Princeton University Press, 2010)
- L. H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge University Press, 1996)
- D. Tong, Lectures on Quantum Field Theory — Free online notes