Canonical Quantization

Table of Contents

1. Introduction
2. What is Quantization?
3. Classical vs Quantum Descriptions
4. Canonical Quantization: Overview
5. From Classical Mechanics to Field Theory
6. Canonical Variables and Poisson Brackets
7. Field Commutation Relations
8. Scalar Field Quantization
9. Mode Expansion and Creation/Annihilation Operators
10. Commutators in Momentum Space
11. Quantization of Complex Scalar Fields
12. Fermionic Quantization: Anticommutators
13. Hamiltonian and Energy Spectrum
14. Fock Space and Vacuum
15. Normal Ordering and Divergences
16. Physical Interpretation and Observables
17. Limitations and Beyond
18. Conclusion

1. Introduction

Canonical quantization is one of the foundational procedures for constructing quantum theories from classical systems. In the context of quantum field theory (QFT), it provides a way to describe fields quantum mechanically, treating them similarly to position and momentum in quantum mechanics.

2. What is Quantization?

Quantization refers to promoting classical observables and equations into quantum operators that obey non-commuting algebra. There are multiple approaches to quantization, including:

• Canonical quantization
• Path integral quantization
• Geometric quantization

This article focuses on the canonical approach.

3. Classical vs Quantum Descriptions

In classical field theory, fields \( \phi(x) \) are real- or complex-valued functions. Upon quantization:

• Fields become operators.
• Classical Poisson brackets become commutators.
• Classical observables become operator-valued observables on a Hilbert space.

4. Canonical Quantization: Overview

Canonical quantization follows these steps:

1. Define a classical Lagrangian and derive the equations of motion.
2. Determine canonical conjugate momenta.
3. Define Poisson brackets for fields.
4. Promote fields and momenta to operators.
5. Replace Poisson brackets with commutators.

5. From Classical Mechanics to Field Theory

In particle mechanics, for coordinate \( q \) and momentum \( p \):

\[
\{q_i, p_j\} = \delta_{ij}
\quad \Rightarrow \quad
[\hat{q}i, \hat{p}_j] = i\hbar \delta{ij}
\]

In field theory, the generalization is:

\[
\{\phi(x), \pi(y)\} = \delta^3(x – y)
\quad \Rightarrow \quad
[\hat{\phi}(x), \hat{\pi}(y)] = i\hbar \delta^3(x – y)
\]

6. Canonical Variables and Poisson Brackets

Given a Lagrangian \( \mathcal{L} \), the conjugate momentum is:

\[
\pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}(x)}
\]

Define the equal-time Poisson bracket:

\[
\{\phi(\vec{x}, t), \pi(\vec{y}, t)\} = \delta^3(\vec{x} – \vec{y})
\]

7. Field Commutation Relations

Upon quantization, we impose:

\[
[\hat{\phi}(\vec{x}, t), \hat{\pi}(\vec{y}, t)] = i\delta^3(\vec{x} – \vec{y})
\]

\[
[\hat{\phi}(\vec{x}, t), \hat{\phi}(\vec{y}, t)] = [\hat{\pi}(\vec{x}, t), \hat{\pi}(\vec{y}, t)] = 0
\]

These are the canonical commutation relations.

8. Scalar Field Quantization

Consider the Klein-Gordon field with Lagrangian:

\[
\mathcal{L} = \frac{1}{2} \partial^\mu \phi \partial_\mu \phi – \frac{1}{2} m^2 \phi^2
\]

The conjugate momentum is \( \pi(x) = \dot{\phi}(x) \). Canonical quantization requires:

\[
[\phi(\vec{x}, t), \pi(\vec{y}, t)] = i \delta^3(\vec{x} – \vec{y})
\]

9. Mode Expansion and Creation/Annihilation Operators

The field is expanded as:

\[
\phi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \left( a_p e^{-ipx} + a_p^\dagger e^{ipx} \right)
\]

Here:

• \( a_p^\dagger \): creation operator
• \( a_p \): annihilation operator

These satisfy:

\[
[a_p, a_{p’}^\dagger] = (2\pi)^3 \delta^3(p – p’)
\]

10. Commutators in Momentum Space

All other commutators vanish:

\[
[a_p, a_{p’}] = [a_p^\dagger, a_{p’}^\dagger] = 0
\]

This structure underlies the definition of Fock space for multiparticle states.

11. Quantization of Complex Scalar Fields

For complex fields \( \phi(x) \), introduce separate operators \( a_p \) and \( b_p \) for particles and antiparticles:

\[
\phi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}} \left( a_p e^{-ipx} + b_p^\dagger e^{ipx} \right)
\]

\[
[\phi(x), \phi^\dagger(y)] = \text{non-zero}, \quad [\phi(x), \phi(y)] = 0
\]

12. Fermionic Quantization: Anticommutators

For spin-½ fields like Dirac fermions, canonical quantization involves anticommutation relations:

\[
\{\psi_\alpha(\vec{x}), \psi_\beta^\dagger(\vec{y})\} = \delta_{\alpha\beta} \delta^3(\vec{x} – \vec{y})
\]

Fermionic Fock space arises from these anticommutators.

13. Hamiltonian and Energy Spectrum

The Hamiltonian operator is derived from:

\[
H = \int d^3x\, \left[ \frac{1}{2} \pi^2 + \frac{1}{2} (\nabla \phi)^2 + \frac{1}{2} m^2 \phi^2 \right]
\]

In terms of creation/annihilation operators:

\[
H = \int \frac{d^3p}{(2\pi)^3} E_p \left( a_p^\dagger a_p + \frac{1}{2} \delta^3(0) \right)
\]

14. Fock Space and Vacuum

The vacuum state \( |0\rangle \) satisfies:

\[
a_p |0\rangle = 0
\]

States with particles are constructed as:

\[
|p\rangle = a_p^\dagger |0\rangle
\]

Fock space spans all possible multi-particle states.

15. Normal Ordering and Divergences

To remove infinite vacuum energy \( \delta^3(0) \), we apply normal ordering:

\[
:H: = H – \langle 0 | H | 0 \rangle
\]

This sets the vacuum energy to zero.

16. Physical Interpretation and Observables

• The number operator \( N_p = a_p^\dagger a_p \)
• Momentum operator: derived from \( T^{0i} \)
• Particle interactions: expressed through interaction terms in the Hamiltonian

17. Limitations and Beyond

• Not manifestly Lorentz invariant (compared to path integrals)
• Difficult to apply in curved spacetime
• Functional methods and path integrals often preferred for advanced formulations

18. Conclusion

Canonical quantization is a powerful and intuitive procedure that bridges classical fields with quantum theory. By promoting fields to operators and enforcing commutation relations, it enables a consistent description of particle creation, annihilation, and interactions in quantum field theory.