Creation Operator

In Quantum Mechanics > s.a. annihilation operator [including modified versions]; Ladder Operators; Normal Order; Raising Operator.
* Idea: An operator that takes a state in a family labeled by a discrete parameter to a higher one; E.g., it raises the energy of an oscillator by one level.
\$ Def: The raising operator for the i-th degree of freedom for a system, which depends on the choice of value for a parameter τ, and can be expressed as

ai = (τi /2$$\hbar$$)1/2 qi – i (1/2$$\hbar$$τi)1/2 pi ;

A choice of value for τ is equivalent to a choice of complex structure on phase space.
* Harmonic oscillator: One normally chooses τ = mω, so H = $$\hbar$$ω (aa + 1/2); In the holomorphic representation,

a:= (1/√2) (ξ + d/dξ) ,   a:= (1/√2) (ξ – d/dξ) ,   where   ξ:= (/$$\hbar$$)1/2 x ;

Other relationships are that q = ($$\hbar$$/2)1/2 (a + a); L3 = i$$\hbar$$ (a2 a1a1 a2).
> Specific theories: see relativistic quantum particle [3D, deformed algebra].

In Quantum Field Theory > s.a. approaches to quantum field theory [covariant]; fock space [number operator, generalizations].
* Idea: An operator that adds a quantum of given momentum k to a (free) field.
* In quantum field theory: The operator ak corresponding to the coefficient of a negative-frequency mode in a field expansion

φ(x) = ∑k (ak uk(x) + a*k u*k(x)) ;

In a Fock space $$\cal F$$($$\cal H$$):= {ψ = (α0, α1, α2, α3, ...)}, the creation operator a(σ) associated with any σ ∈ $$\cal H$$ is the adjoint of the corresponding annihilation operator a(σ),

a(σ) ψ:= (0, α0 σ, √2 α1 ⊗ σ, √3 α2 ⊗ σ, ...) ;   basically,   a |n$$\rangle$$ = (n+1)1/2 |n+1$$\rangle$$ .

* Properties: Bosonic ones satisfy the commutation relations [ak, ak '] = 0, [ak, ak '] = δkk' and, for general powers (with nm)

an an = N (N–1) ··· (Nn+1) ,   [an, am] = ∑k=1n k! $${m \choose k}{n \choose k}$$ a(m–k) a(n–k) ;

* Different commutation relations: Note that composite bosons satisfy non-standard commutation relations (> see particle statistics), and fermionic operators satisfy the anticommutation relations {bk, bk'} = δkk'.
@ References: Szafraniec RPMP(07) [characterizations]; Kim et al PRL(08)-a0901 [commutation relations, for photons, proposed experiment]; Gupta & Kumar a1105 [canonical brackets from continuous symmetries, BRST formalism]; Guadagnini JPA(13)-a1212 [representation]; Kumar et al PRL(13) + Zavatta & Bellini Phy(13) [characterization by quantum process tomography].
> Specific theories: see quantum dirac fields.

References > s.a. Normal Order.
@ References: Odake & Sasaki JMP(06)qp [solvable systems].