1. 6. Second Quantization and Quantum Field Theory
Preliminary
The Occupation Number Representation
Field Operators and Observables
Equation of Motion and Lagrangian Formalism for Field Operators

2. Preliminary
Systems with variable numbers of particles ~ Second quantization
High energy scattering and decay processes.
Relativistic systems.
Many body systems (not necessarily relativistic).
1st quantization:
Dynamical variables become operators;
E, L, … take on only discrete values.
2nd quantization:
Wave functions become field operators.
Properties described by counting numbers of 1-particle states being occupied.
Processes described in terms of exchange of real or virtual particles.
For system near ground state:
→ Quasi-particles (fermions) or elementary excitations (bosons).
→ Perturbative approach.

3. 6.1. The Occupation Number Representation
Many body problem ~ System of N identical particles.
{ | k  } = set of complete, orthonormal, 1-particle states that satisfy the BCs.
is an orthonormal basis.
Uncertainty principle → identical particles are indistinguishable. →
bosons fermions
Bose-Einstein Fermi-Dirac
integral half-integral 
statistics
spin
Spin-statistics theorem: this association is due to causality.

4. Basis with built-in exchange symmetry:
bosons fermions
P denotes a permutation
even odd
if P consists of an
number of transpositions (exchanges)
With ,
is orthonormal: if
2 states with N  N  are always orthogonal.

5. Number Representation: States
Let { | α  } be a set of complete, orthonormal 1-P basis.
α = 0,1,2,3,… denotes a set of quantum numbers with increasing E.
E.g., one electron spinless states of H atom: | α  = | nlm 
| 0  = | 100 , | 1  = | 111 , | 2  = | 110 , | 3  = | 111 , …
Number (n-) representation:
Basis = (symmetrized) eigenstates of number operator
nα= number of particles in | α 
orthonormality

6. Creation and Annihilation Operators
Conjugate variables in n-rep:
annihilation operators
creation operators

7. A,C real →
For bosons, nα = 0, 1, 2, 3, …
For fermions, nα = 0, → A(0) = 0, C(1) = 0 and 1 = C(0) A(1).
Set: C(0) = A(1) = 1.
Completeness of this basis is with respect to the Fock space.
There exists many particle states that cannot be constructed in this manner. E.g., BCS states (Cooper pairs).

8. Commutation Relations
Exchange symmetries of states  Commutation relations between operators
Fock space
is the “vacuum”.
 α
For fermions, nα = 0, 1 →
Boson Fermion
Exchange symmetries are established by requiring
Commutator
Anti-commutator

9. → →

10. Number Representation: Operators
1-P operator :
= matrix elements →
The vacuum projector
confines A to the 1-particle subspace.
i.e.,
if the number of particles in either  or  is not one.
Many body version :

11. 2-Particle Potential Basis vector for the 2-P Hilbert space:
Completeness condition:

12. confines V to the 2-particle subspace.
Many body version :

13. Summary
1-P operator:
2-P operator:

14. 6.2. Field Operators and Observables
Momentum eigenstaes for spinless particles:
Orthonormality:
Completeness:
where

15. Field Operators
The field operators are defined in the Schrodinger picture by
Momentum basis:
Commutation relations :

16. = total number of particles
 ρ(x) is the number density operator at x.

17. 6.3. Equation of Motion & Lagrangian Formalism for Field Operators
Heisenberg picture:
Equal time commutation relations:

18. Equation of Motion

19. Lagrangian
 is complex → it represents 2 degrees of freedom ( Re , Im ) or ( ,  * ).
Variation on  * :
E-L eq: →
Schrodinger equation

20. Generalized momentum conjugate to  =
Variation on  :
integration by part →
Generalized momentum conjugate to  =
Hamiltonian density →
~ Classical field
Quantization rule:
Quantum field theory