Permutation relationships

permutation relations

Rules for permuting the product of two creation or annihilation operators. That is, for the annihilation operators $ \{ {a( f ) } : {f \in H } \} $ and the adjoint creation operators $ \{ {a ^ \star ( f ) } : {f \in H } \} $, where $ H $ is some Hilbert space, acting in the symmetric Fock space $ F( H) $ over $ H $, these relationships take the form

$$ \tag{1 } a( f _ {1} ) a ( f _ {2} ) - a( f _ {2} ) a( f _ {1} ) = $$

$$ = \ a ^ \star ( f _ {1} ) a ^ \star ( f _ {2} ) - a ^ \star ( f _ {2} ) a ^ \star ( f _ {1} ) = 0, $$

$$ a( f _ {1} ) a ^ \star ( f _ {2} ) - a ^ \star ( f _ {2} ) a( f _ {1} ) = ( f _ {1} , f _ {2} ) E ,\ f _ {1} , f _ {2} \in H, $$

where $ ( \cdot , \cdot ) $ is the inner product in $ H $ and $ E $ is the identity operator acting in $ F( H) $. The relations (1) are also called the commutation relations. In the case of an anti-symmetric Fock space, the creation and annihilation operators permute in accordance with the rules

$$ \tag{2 } a( f _ {1} ) a( f _ {2} ) + a( f _ {2} ) a( f _ {1} ) = $$

$$ = \ a ^ \star ( f _ {1} ) a ^ \star ( f _ {2} ) + a ^ \star ( f _ {2} ) a ^ \star ( f _ {1} ) = 0, $$

$$ a( f _ {1} ) a ^ \star ( f _ {2} ) + a ^ \star ( f _ {2} ) a( f _ {1} ) = ( f _ {1} , f _ {2} ) E ,\ f _ {1} , f _ {2} \in H, $$

which are called the anti-commutation relations.

In the case of an infinite-dimensional space $ H $, besides the creation and annihilation operators acting in Fock spaces over $ H $ there exist other irreducible representations not equivalent to them for the commutation and anti-commutation relations, i.e. other families of operators acting in some Hilbert space and satisfying the permutation rules (1) or (2) [1], . In the case of a finite-dimensional Hilbert space $ H $, all the irreducible representations of (1) or (2) are unitarily equivalent.

References

[1]	F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989)
[2a]	L. Gårding, A. Wightman, "Representations of the anticommutation relations" Proc. Nat. Acad. Sci. USA , 40 : 7 (1954) pp. 617–621
[2b]	L. Gårding, A. Wightman, "Representations of the commutation relations" Proc. Nat. Acad. Sci. USA , 40 : 7 (1954) pp. 622–626

Comments

The abbreviations CCR and CAR, which stand for canonical commutation relations and canonical anti-commutation relations are often used for relations (1) and (2). One also speaks of CCR algebras and CAR algebras.

References

[a1]	N.N. Bogolyubov, A.A. Logunov, A.I. Oksak, I.T. Todorov, "General principles of quantum field theory" , Kluwer (1990) pp. 265ff; 295 (Translated from Russian)
[a2]	G.G. Emch, "Algebraic methods in statistical mechanics and quantum field theory" , Wiley (Interscience) (1972)
[a3]	S.S. [S.S. Khorozhii] Horuzhy, "Introduction to algebraic quantum field theory" , Kluwer (1990) pp. 256ff (Translated from Russian)
[a4]	O.I. [O.I. Zav'yalov] Zavialov, "Renormalized quantum field theory" , Kluwer (1990) pp. 3ff (Translated from Russian)