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

a^†_i = (τ_i /2\(\hbar\))^1/2 q_i − i (1/2\(\hbar\)τ_i)^1/2 p_i ;

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 (a^†a + 1/2); In the holomorphic representation,

a:= (1/\(\sqrt2\)) (ξ + d/dξ) ,   a^†:= (1/\(\sqrt2\)) (ξ − d/dξ) ,   where   ξ:= (mω/\(\hbar\))^1/2 x ;

Other relationships are that q = (\(\hbar\)/2mω)^1/2 (a + a^†); L₃ = i\(\hbar\) (a₂^† a₁ − a₁^† 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 a_k^† corresponding to the coefficient of a negative-frequency mode in a field expansion

φ(x) = ∑_k (a_k u_k(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, \(\sqrt2\) α₁ ⊗ σ, \(\sqrt3\) α₂ ⊗ σ, ...) ;   basically,   a^† |n\(\rangle\) = (n+1)^1/2 |n+1\(\rangle\) .

* Properties: Bosonic ones satisfy the commutation relations [a_k^†, a_k'^†] = 0, [a_k , a_k'^†] = δ_kk' and, for general powers (with n ≤ m)

a^†n aⁿ = N (N−1) ··· (N−n+1) ,   [aⁿ, a^†m] = ∑_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 {b_k, b_k'^†} = δ_kk'.
@ 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].

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