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/d*ξ*) ,
*a*^{†}:= (1/\(\sqrt2\)) (*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*,
\(\sqrt2\) *α*_{1} ⊗ *σ*,
\(\sqrt3\) *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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