# Difference between revisions of "Wodzicki residue"

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''non-commutative residue'' | ''non-commutative residue'' | ||

− | In algebraic | + | In algebraic [[quantum field theory]], in order to write down an action in operator language one needs a functional that replaces integration [[#References|[a1]]]. For the Yang–Mills theory (cf. [[Yang–Mills field]]) this is the Dixmier trace, which is the unique extension of the usual [[trace]] to the ideal $\mathcal{L}^{(1,\infty)}$ of the compact operators $T$ such that the partial sums of the spectrum diverge logarithmically as the number of terms in the sum. The Wodzicki (or non-commutative) residue [[#References|[a3]]] is the only extension of the Dixmier trace to the class of [[pseudo-differential operator]]s (ΨDOs) which are not in $\mathcal{L}^{(1,\infty)}$. It is the only trace one can define in the algebra of ΨDOs (up to a multiplicative constant), its definition being: $\mathrm{res} A = 2\mathrm{res}_{s=0} \tr(A\Delta^{-s})$, with $\Delta$ the [[Laplace operator]]. It satisfies the trace condition: $\mathrm{res}(AB) = \mathrm{res}(BA)$. A very important property is that it can be expressed as an integral (local form): |

+ | $$ | ||

+ | \mathrm{res} A = \int_{S^*M} \tr a_{-n}(x,\xi) d\xi | ||

+ | $$ | ||

+ | with $S^*M \subset T^*M$ the co-sphere bundle on $M$ (some authors put a coefficient in front of the integral, this gives the Adler–Manin residue). | ||

− | + | If $\dim M = n = -\ord A$ ($M$ a compact [[Riemannian manifold]], $A$ an [[elliptic operator]], $n \in \mathbf{N}$), it coincides with the Dixmier trace, and one has | |

+ | $$ | ||

+ | \mathrm{res}_{s=1} \zeta_A(s) = \frac{1}{n} \mathrm{res} A^{-1} \ . | ||

+ | $$ | ||

− | + | The Wodzicki residue continues to make sense for ΨDOs of arbitrary order and, even if the symbols $a_j(x,\xi)$, $j < n$, are not invariant under coordinate choice, their integral is, and defines a trace. All residues at poles of the zeta-function of a ΨDO can be easily obtained from the Wodzicki residue [[#References|[a2]]]. | |

− | + | ====References==== | |

− | + | <table> | |

− | < | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Connes, "Noncommutative geometry" , Acad. Press (1994) {{MR|1303779}} {{ZBL|0818.46076}} </TD></TR> |

− | + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Elizalde, "Complete determination of the singularity structure of zeta functions" ''J. Phys.'' , '''A30''' (1997) pp. 2735 {{MR|1450345}} {{ZBL|0919.58065}} </TD></TR> | |

− | + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Wodzicki, "Noncommutative residue I" Yu.I. Manin (ed.) , ''K-Theory, Arithmetic and Geometry'' , ''Lecture Notes in Mathematics'' , '''1289''' , Springer (1987) pp. 320–399 {{MR|0923140}} {{ZBL|0649.58033}} </TD></TR> | |

+ | </table> | ||

− | + | {{TEX|done}} | |

− |

## Latest revision as of 18:13, 25 September 2017

*non-commutative residue*

In algebraic quantum field theory, in order to write down an action in operator language one needs a functional that replaces integration [a1]. For the Yang–Mills theory (cf. Yang–Mills field) this is the Dixmier trace, which is the unique extension of the usual trace to the ideal $\mathcal{L}^{(1,\infty)}$ of the compact operators $T$ such that the partial sums of the spectrum diverge logarithmically as the number of terms in the sum. The Wodzicki (or non-commutative) residue [a3] is the only extension of the Dixmier trace to the class of pseudo-differential operators (ΨDOs) which are not in $\mathcal{L}^{(1,\infty)}$. It is the only trace one can define in the algebra of ΨDOs (up to a multiplicative constant), its definition being: $\mathrm{res} A = 2\mathrm{res}_{s=0} \tr(A\Delta^{-s})$, with $\Delta$ the Laplace operator. It satisfies the trace condition: $\mathrm{res}(AB) = \mathrm{res}(BA)$. A very important property is that it can be expressed as an integral (local form): $$ \mathrm{res} A = \int_{S^*M} \tr a_{-n}(x,\xi) d\xi $$ with $S^*M \subset T^*M$ the co-sphere bundle on $M$ (some authors put a coefficient in front of the integral, this gives the Adler–Manin residue).

If $\dim M = n = -\ord A$ ($M$ a compact Riemannian manifold, $A$ an elliptic operator, $n \in \mathbf{N}$), it coincides with the Dixmier trace, and one has $$ \mathrm{res}_{s=1} \zeta_A(s) = \frac{1}{n} \mathrm{res} A^{-1} \ . $$

The Wodzicki residue continues to make sense for ΨDOs of arbitrary order and, even if the symbols $a_j(x,\xi)$, $j < n$, are not invariant under coordinate choice, their integral is, and defines a trace. All residues at poles of the zeta-function of a ΨDO can be easily obtained from the Wodzicki residue [a2].

#### References

[a1] | A. Connes, "Noncommutative geometry" , Acad. Press (1994) MR1303779 Zbl 0818.46076 |

[a2] | E. Elizalde, "Complete determination of the singularity structure of zeta functions" J. Phys. , A30 (1997) pp. 2735 MR1450345 Zbl 0919.58065 |

[a3] | M. Wodzicki, "Noncommutative residue I" Yu.I. Manin (ed.) , K-Theory, Arithmetic and Geometry , Lecture Notes in Mathematics , 1289 , Springer (1987) pp. 320–399 MR0923140 Zbl 0649.58033 |

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Wodzicki residue.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Wodzicki_residue&oldid=18226