Table of Contents
- Introduction
- First Quantization Recap
- What is Second Quantization?
- Motivation and Physical Meaning
- Fock Space and Quantum States
- Creation and Annihilation Operators
- Commutation and Anticommutation Relations
- Field Operators
- Operator Algebra
- Example: Quantization of a Scalar Field
- Number Operators and Observables
- Antisymmetric States and Fermions
- Symmetric States and Bosons
- Hamiltonian in Second Quantized Form
- Second Quantization in Quantum Field Theory
- Dirac Field and Second Quantization
- Application in Many-Body Physics
- Path to Quantum Electrodynamics
- Interpretation and Conceptual Significance
- Conclusion
1. Introduction
Second quantization is a formalism in quantum mechanics and quantum field theory where the fields themselves are quantized. It provides a powerful framework for describing systems of identical particles, allowing for the creation and annihilation of particles, and is essential for understanding many-body systems and quantum field theory.
2. First Quantization Recap
In first quantization:
- Wavefunction \( \psi(x) \) describes a single particle.
- Operators like \( \hat{x}, \hat{p} \) act on the Hilbert space.
- Schrödinger equation governs time evolution.
However, this framework is inadequate for:
- Variable particle number
- Relativistic consistency
- Field-theoretic interactions
3. What is Second Quantization?
In second quantization:
- The wavefunction becomes an operator.
- Particles are excitations of quantum fields.
- The Hilbert space is extended to Fock space: a direct sum of n-particle Hilbert spaces.
4. Motivation and Physical Meaning
Second quantization:
- Describes creation and annihilation of particles.
- Automatically enforces quantum statistics (Bose-Einstein or Fermi-Dirac).
- Enables treatment of interacting fields and many-body systems.
5. Fock Space and Quantum States
Fock space \( \mathcal{F} \) is the Hilbert space that contains:
- The vacuum \( |0\rangle \)
- One-particle states \( a^\dagger_k |0\rangle \)
- Two-particle states \( a^\dagger_{k_1} a^\dagger_{k_2} |0\rangle \)
- And so on
\[
\mathcal{F} = \bigoplus_{n=0}^\infty \mathcal{H}^{(n)}
\]
6. Creation and Annihilation Operators
For bosons:
\[
[a_k, a_{k’}^\dagger] = \delta_{kk’}
\]
For fermions:
\[
\{a_k, a_{k’}^\dagger\} = \delta_{kk’}
\]
- \( a_k^\dagger \): creates a particle in state \( k \)
- \( a_k \): annihilates a particle in state \( k \)
7. Commutation and Anticommutation Relations
Bosonic fields (integer spin):
\[
[a_i, a_j] = 0, \quad [a_i, a_j^\dagger] = \delta_{ij}
\]
Fermionic fields (half-integer spin):
\[
\{a_i, a_j\} = 0, \quad \{a_i, a_j^\dagger\} = \delta_{ij}
\]
These ensure the proper symmetry or antisymmetry of the multi-particle wavefunction.
8. Field Operators
In position space:
\[
\hat{\psi}(x) = \sum_k a_k \phi_k(x), \quad \hat{\psi}^\dagger(x) = \sum_k a_k^\dagger \phi_k^*(x)
\]
Here \( \hat{\psi}(x) \) destroys a particle at position \( x \), and \( \hat{\psi}^\dagger(x) \) creates one.
9. Operator Algebra
Field operators satisfy:
\[
[\hat{\psi}(x), \hat{\psi}^\dagger(x’)] = \delta(x – x’)
\quad (\text{bosons})
\]
\[
\{\hat{\psi}(x), \hat{\psi}^\dagger(x’)\} = \delta(x – x’)
\quad (\text{fermions})
\]
10. Example: Quantization of a Scalar Field
Consider the real scalar field \( \phi(x) \) with expansion:
\[
\phi(x) = \int \frac{d^3p}{(2\pi)^3 \sqrt{2E_p}} \left( a_p e^{-ipx} + a_p^\dagger e^{ipx} \right)
\]
Canonical commutation relations:
\[
[\phi(x), \pi(y)] = i\delta^3(x – y)
\]
11. Number Operators and Observables
The number operator counts particles in a given mode:
\[
\hat{N}_k = a_k^\dagger a_k
\]
Total number operator:
\[
\hat{N} = \sum_k \hat{N}_k
\]
Expectation values give average particle numbers.
12. Antisymmetric States and Fermions
For fermions, states are antisymmetric under exchange:
\[
a_k^\dagger a_k^\dagger |0\rangle = 0
\]
This enforces the Pauli exclusion principle: no two fermions can occupy the same quantum state.
13. Symmetric States and Bosons
For bosons, creation operators commute:
\[
a_k^\dagger a_k^\dagger |0\rangle \ne 0
\]
Multiple bosons can occupy the same state, leading to phenomena like Bose-Einstein condensation.
14. Hamiltonian in Second Quantized Form
The single-particle Hamiltonian \( H = p^2/2m + V(x) \) becomes:
\[
\hat{H} = \int dx\, \hat{\psi}^\dagger(x) \left( -\frac{1}{2m} \nabla^2 + V(x) \right) \hat{\psi}(x)
\]
For interactions:
\[
\hat{H}_{\text{int}} = \frac{1}{2} \int dx\, dx’\, \hat{\psi}^\dagger(x) \hat{\psi}^\dagger(x’) V(x – x’) \hat{\psi}(x’) \hat{\psi}(x)
\]
15. Second Quantization in Quantum Field Theory
In QFT:
- Fields are operator-valued distributions
- Particles are excitations of these fields
- The vacuum is the lowest energy state
- Scattering amplitudes are calculated using Fock space
16. Dirac Field and Second Quantization
The quantized Dirac field:
\[
\psi(x) = \int \frac{d^3p}{(2\pi)^3} \sum_s \left[ a_{p,s} u_s(p) e^{-ipx} + b_{p,s}^\dagger v_s(p) e^{ipx} \right]
\]
Anticommutation relations for fermionic operators ensure correct quantum statistics.
17. Application in Many-Body Physics
Used extensively in:
- Superconductivity
- Quantum Hall effect
- Bose gases
- Atomic physics and condensed matter theory
Second quantization is essential for treating systems with variable particle numbers.
18. Path to Quantum Electrodynamics
In QED:
- Electromagnetic field is quantized using second quantization
- Interaction terms (e.g., \( \bar{\psi}\gamma^\mu A_\mu \psi \)) emerge naturally
- Feynman diagrams are derived from field interactions
19. Interpretation and Conceptual Significance
Second quantization shifts focus from particles to fields:
- The field is the primary object.
- The notion of particle becomes emergent.
- Creation and annihilation reflect fundamental quantum transitions.
20. Conclusion
Second quantization is a profound extension of quantum mechanics. It allows for the treatment of systems with indistinguishable particles, naturally incorporates quantum statistics, and serves as the foundation for all quantum field theories. Mastery of this formalism is essential for both theoretical and applied quantum physics.
