Namespaces
Variants
Actions

Difference between revisions of "Wodzicki residue"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (link)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
''non-commutative residue''
 
''non-commutative residue''
  
In algebraic quantum field theory (cf. also [[Quantum field theory|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|Yang–Mills field]]) this is the Dixmier trace, which is the unique extension of the usual [[Trace|trace]] to the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301601.png" /> of the compact operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301602.png" /> such that the partial sums of its 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 operators (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301603.png" />DOs; cf. [[Pseudo-differential operator|Pseudo-differential operator]]) which are not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301604.png" />. It is the only trace one can define in the algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301605.png" />DOs (up to a multiplicative constant), its definition being: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301606.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301607.png" /> the [[Laplace operator|Laplace operator]]. It satisfies the trace condition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301608.png" />. A very important property is that it can be expressed as an integral (local form):
+
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).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w1301609.png" /></td> </tr></table>
+
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
 +
$$
 +
\mathrm{res}_{s=1} \zeta_A(s) = \frac{1}{n} \mathrm{res} A^{-1} \ .
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016010.png" /> the co-sphere bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016011.png" /> (some authors put a coefficient in front of the integral, this gives the Adler–Manin residue).
+
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]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016013.png" /> a compact [[Riemannian manifold|Riemannian manifold]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016014.png" /> an elliptic operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016015.png" />), it coincides with the Dixmier trace, and one has
+
====References====
 
+
<table>
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016016.png" /></td> </tr></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>
The Wodzicki residue continues to make sense for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016017.png" />DOs of arbitrary order and, even if the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016019.png" />, are not invariant under coordinate choice, their integral is, and defines a trace. All residues at poles of the zeta-function of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130160/w13016020.png" />DO can be easily obtained from the Wodzicki residue [[#References|[a2]]].
+
<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>
  
====References====
+
{{TEX|done}}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Connes,  "Noncommutative geometry" , Acad. Press  (1994)</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</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</TD></TR></table>
 

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
How to Cite This Entry:
Wodzicki residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wodzicki_residue&oldid=18226
This article was adapted from an original article by E. Elizalde (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article