6. Second Quantization and Quantum Field Theory 6.0. Preliminary 6.1. The Occupation Number Representation 6.2. Field Operators and Observables 6.3. Equation of Motion and Lagrangian Formalism for Field Operators

6.0. 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.

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.

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.

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

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

A,C real → For bosons, nα = 0, 1, 2, 3, … For fermions, nα = 0, 1 → 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).

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

→ →

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 :

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

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

Summary 1-P operator: 2-P operator:

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

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

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

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

Equation of Motion

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

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