Annihilation Operator

In Quantum Mechanics > s.a. creation operator; Lowering Operator; Normal Order.
* Idea: An operator that takes a state in a family labeled by a discrete parameter to a lower one; E.g., it lowers the energy of an oscillator by one level.
$ Def: A lowering operator for the i-th degree of freedom, which depends on the choice of value for a parameter τ_i, 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 := 2^−1/2 (ξ + d/dξ) ,   a^† := 2^−1/2 (ξ − d/dξ) ,  where  ξ:= (mω/\(\hbar\))^1/2 x .

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

φ = ∑_k (a_k u_k + a*_k u*_k) ;

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

a(σ) ψ:= (σ · α₁, 2^1/2 σ · α₂, 3^1/2 σ · α₃, ...) ;   basically,   a |n\(\rangle\) = n^1/2 |n−1\(\rangle\) .

* Properties: Bosonic ones satisfy the commutation relations [a_k, a_{k '}] = 0 and [a_k, a_{k '}^†] = δ_kk', while fermionic ones satify {b_k, b_{k '}^†} = δ_kk' (Note that composite bosons satisfy non-standard commutation relations, > see particle statistics); > s.a. creation operator.
* On a Riemannian M: The Dirac operator d + δ on the Hodge complex of M.

References > s.a. Normal Order.
@ General: Bueler mp/01 [on Riemannian manifolds]; Mizrahi & Dodonov JPA(02)qp [paradoxical example]; Odake & Sasaki JMP(06)qp [solvable systems]; Guadagnini JPA(13)-a1212 [representation].
@ Fermionic: Derrick JMP(63) [representation on the space of periodic functions on a real interval].
@ Generalized / modified: Ghosh JMP(98); Bagarello JMP(07)-a0903 [bounded version]; Trifonov JPA(12) [non-linear, and coherent states]; > s.a. Ladder Operators; relativistic quantum particle [3D, deformed algebra].
@ Related topics: Petrović a1001 [analytic functions]; Gupta & Kumar a1105 [canonical brackets from continuous symmetries, BRST formalism]; Kumar et al PRL(13) + Zavatta & Bellini Phy(13) [bosonic, experimental characterization by quantum process tomography].

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