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* = *m**ω*,
so *H* = \(\hbar\)*ω* (*a*^{†}*a* +
1/2); In the holomorphic representation,

*a*:= (1/√2)
(*ξ* + d/d*ξ*) , *a*^{†}:=
(1/√2)
(*ξ* – d/d*ξ*)
, where *ξ*:=
(*mω*/\(\hbar\))^{1/2} *x* ;

Other relationships are that *q* = (\(\hbar\)/2*mω*)^{1/2} (*a* + *a*^{†}); *L*_{3} =
i\(\hbar\) (*a*_{2}^{†} *a*_{1} – *a*_{1}^{†} *a*_{2}).

> __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* = (*α*_{0}, *α*_{1}, *α*_{2}, *α*_{3}, ...)},
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, *α*_{0 }*σ*, √2 *α*_{1} ⊗ *σ*, √3 *α*_{2} ⊗ *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}* a*^{n}
= *N* (*N*–1) ··· (*N*–*n*+1)
, [*a*^{n}, *a*^{†}^{m}]
= ∑_{k=1}^{n} *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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