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/d*ξ*) , *a*^{†} :=
2^{−1/2} (*ξ* −
d/d*ξ*) , 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* = (*α*_{0},
*α*_{1}, *α*_{2},
*α*_{3},...)}, the annihilation operator
*a*(*σ*) associated with any *σ* ∈ \(\cal H\) is

*a*(*σ*) *ψ*:=
(*σ* · *α*_{1},
2^{1/2} *σ* · *α*_{2},
3^{1/2} *σ* · *α*_{3},
...) ; 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

**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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