Noncommutative spectral geometry and the deformed Hopf algebra
Hopf Algebra Structure of a Model Quantum Field Theory
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Hopf Algebra Structure of Hopf Algebra Structure of a Model Quantum Field a Model Quantum Field Theory Theory Allan Solomon Open University and University of Paris VI Collaborators: Gerard Duchamp, U. Paris 13 Karol Penson, U. Paris VI Pawel Blasiak Inst. of Nuclear Phys., Krakow Andrzej Horzela, Inst. of Nuclear Physics Krakow G26 International Conference on Group Theoretical Methods in Physics New York , June 2006.
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Hopf Algebra Structure of a Model Quantum Field Theory. Allan Solomon Open University and University of Paris VI Collaborators: Gerard Duchamp, U. Paris 13 Karol Penson, U. Paris VI Pawel Blasiak Inst. of Nuclear Phys., Krakow Andrzej Horzela, Inst. of Nuclear Physics Krakow - PowerPoint PPT Presentation
Transcript of Hopf Algebra Structure of a Model Quantum Field Theory
Hopf Algebra Structure of a Model Hopf Algebra Structure of a Model Quantum Field TheoryQuantum Field Theory
Allan Solomon
Open University and University of Paris VI
Collaborators:
Gerard Duchamp, U. Paris 13
Karol Penson, U. Paris VI
Pawel Blasiak Inst. of Nuclear Phys., Krakow
Andrzej Horzela, Inst. of Nuclear Physics Krakow
G26 International Conference on Group Theoretical Methods in Physics New York , June 2006.
AbstractAbstractRecent elegant work [1] on the structure of Perturbative Quantum
Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. quantum theory of non-commuting non-field-dependent operators. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs [2, 3], analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure[4]. We illustrate these ideas by means of simple solvable models, e.g. the partition function for a free boson gas and for a superfluid bose system. Finally, we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra.
ReferencesReferences[1] A readable account may be found in Dirk Kreimer's
"Knots and Feynman Diagrams", Cambridge Lecture Notes in Physics, CUP (2000).
[2] We use the graphical description of
Bender, Brody and Meister, J.Math.Phys. 40, 3239 (1999) and arXiv:quant-ph/0604164
[3] This approach is extended in
Blasiak, Penson, Solomon, Horzela and Duchamp, J.Math.Phys. 46, 052110 (2005) and arXiv:quant-ph/0405103
[4] A preliminary account of the more mathematical aspects of this work may be found in arXiv: cs.SC/0510041
Content of TalkContent of Talk1. Partition Functions and Generating Functionsi: Partition Function Integrand (PFI)ii. Exponential Generating Functioniii: Bell numbers and their Graphsvi: Connected Graph Theoremiv: Combinatorial properties of the calculationsv: PFI General CaseExamples: Free Boson Gas, Superfluid bosons.
3. Relation to other Models (QFT)
2. Model « Feynman » Graphs and the Hopf Algebra of these Graphs
QFT and Simplified ModelQFT and Simplified Model
QFTQFT ModelModel
AnalysisAnalysis YesYes NoNo
Graphs Graphs (Topology)(Topology)
YesYes YesYes
AlgebraAlgebra YesYes YesYes
SolvableSolvable NoNo YesYes
Partition FunctionPartition Function
Free Boson Gas (single mode)
)exp( HtrZ aaH
1
0
0
))exp(1(
)exp(
|)exp(|
)exp(
n
naan
aatrZ
Partition FunctionPartition Function
Free Boson Gas (single mode) Coherent state basis
)exp( HtrZ aaH
),(
,))1(exp(
||))exp(1(
))1)(exp(exp(
|:)1)(exp(exp(|:
|)exp(|
0
2
0
1
2
0 0
2
2
2
)/1(
)/1(
)/1(
yxBdy
xryeydyZ
zr
rrdrd
zaazzd
zaazzdZ
x
Izzzd ||1 2
Partition Function IntegrandPartition Function Integrand
)||,(!)(
),(2
0
0
zyxnxyBdy
yxBdyZn
n
The Partition Function is the integral of a function which has a combinatorial interpretation, as the generating Function of the Bell polynomials in the Free Boson Case.
)(),()( zdzxFxZ Generally
We shall now give a Combinatorial, Graphical interpretation of the Partition Function Integrand
in more General cases.
),( zxF
Generating FunctionsGenerating FunctionsIn general, for combinatorial numbers c(n),
DEFINITION( )
( )!
nc n xG x
n
Bell Numbers B(n) :( )
( ) exp(exp( ) 1)!
nB n xG x x
n
EXAMPLES
G(x) is , more precisely, the Exponential Generating Function (egf).
x
n
nn
n
x
n
en
xxGnc
xn
xnxGnnc
xen
xnxGnnc
2
!2)(,2)(
)1/(1!
!)(,!)(
!)(,)(
Generating FunctionsGenerating FunctionsFor 2-parameter combinatorial numbers c(n,k),
DEFINITION
EXAMPLES
xy
nn
k
k
k
n
nk
n en
xyCyxGCknc )1(
00 !)(),(,),(
0 0 !
)),((),(n
n
k
nk
n
xykncyxG
)1(
0 0 !),(),(
),,(),(
xey
n
n
k
n
k en
xyknSyxG
knSknc Stirling, Bell
Combinatorial numbersCombinatorial numbers
,....)(,2,!, nBnnnBell numbers B(n) count the number of ways
of putting n different objects in n identical containers
)!,...(203,52,15,5,2,1)}({ nnB
1-parameter
2-parameter
Stirling numbers (of 2nd kind) S(n,k) count the number of ways of putting n different objects in k identical (non-empty)
containers so
n
k
knSnB1
),()(
‘Choose’ nCk Stirling 2nd kind S(n,k)
Bell Numbers, Bell polynomialsBell Numbers, Bell polynomialsAs Bell Numbers are, in a sense which we shall show, generic for the description of Partition Functions, we recap their generating functions.
...)!3/!2/!1/exp(
!/)()(32
0
xxx
nxnBxG n
)1(
0
0 0
!)(
!),(),(
xey
n
n
n
n
k
n
k
en
xyB
n
xyknSyxG
Graphs for Bell numbers Graphs for Bell numbers B(n)B(n)
n=1
n=2
1
1
n=3
3 1
Total
1
2
5
B(n)
1
1
...)!3/!2/!1/exp()( 32 xxxxG
Connected Graph Theorem*Connected Graph Theorem*
nxn
ncxC
1 !
)()(
*GW Ford and GE Uhlenbeck,Proc.Nat.Acad.42(1956)122
If C(x) is the generating function of CONNECTED labelled graphs
then
is the generating function of ALL graphs!
)](exp[)( xCxA
Connected Graphs for Bell numbers Connected Graphs for Bell numbers B(n)B(n)
n=1 n=2 n=3
1 C(n)
...!3/!2/!1/)( 32 xxxxC
1 1
Hence Generating function for ALL graphs is
...)!3/!2/!1/exp()( 32 xxxxG
Partition Function IntegrandPartition Function IntegrandGeneral caseGeneral case
)exp( HtrZ wH
)!/exp(
!/)(
!/||
|)exp(|),(
1
0
0
nxzV
nxzW
nxzwz
xzxwzzxF
n
n
n
n
nn
)(),()( zdzxFxZ w is a ‘word’ in the a,a* operators
Limit =1 from normalization of |z> Evaluation depends on finding ‘normal order’ of wn
Origin emits single lines only, and vertex Origin emits single lines only, and vertex receives any number of lines receives any number of lines
Vertex Strengths Vn
n=1
n=2
1
1
n=3
3 1
1
1
General Generating Functions
...)!3/!2/!1/exp()( 3
3
2
21 xVxVxVxG
V1
V12+V2
V13+3V1V2+V3
Connected Graphs in GeneralConnected Graphs in General
n=1 n=2 n=3
V3 C(n)
...!3/!2/!1/)( 3
3
2
21 xVxVxVxC
V1 V2
Hence Generating function for ALL graphs is
...)!3/!2/!1/exp()( 3
3
2
21 xVxVxVxG
Partition Function Integrand: Partition Function Integrand: Expansion for Free Boson GasExpansion for Free Boson Gas
n=1
n=2
1
1
n=3
3 1
x=- y=|z|2
(y+y2)x2/2!
(y+3y2+y3 )x3/3!
1
1
...)!3/)(!2/)(!1/)(exp(),( 3
3
2
21 xyVxyVxyVyxF
yx
Vn(y)=y all n
Example: Superfluid Boson SystemExample: Superfluid Boson System
zHzzxF
aacacacH
|)exp(|),(
)}2/1(2/2/{ 2
2
1
2
1
This formula generates an integer sequence for c1, c2 even integers.
Choosing c1, c2 and z real to simplify expression, x=-
)/()sinh(
/)sinh()cosh(
)}11
(exp{1
),(
11
2
22
1
2
2
1
xcy
xc
zyccx
yyzxF
Example: Superfluid Boson SystemExample: Superfluid Boson System
This formula generates an integer sequence for c1=2, c2=4.
Choose c1=2, c2 =4 and y=z2 real
...)!3/)(!2/)(!1/)(exp(),( 3
3
2
21 xyVxyVxyVyxF
yV
yV
yV
yV
3024144
28816
362
62
4
3
2
1
This corresponds to the integer sequence 8,102,1728,35916,...
Partition Function Integrand: Partition Function Integrand: Expansion for Superfluid Boson GasExpansion for Superfluid Boson Gas
n=1
n=2
1
1
n=3
3 1
x=- ,y=z2
(V2(y)+V1(y)2 )x2/2!
(V3+3V1V2+V13 )x3/3!
1
1
...)!3/)(!2/)(!1/)(exp(),( 3
3
2
21 xyVxyVxyVyxF
V1(y)x
Model Feynman GraphsModel Feynman GraphsThese graphs are essentially like the Feynman Diagrams of a zero-dimensional (no space-time dependence, no integration) Model Field Theory. (In the case of a single-mode free boson gas, associated with H = a+ a, one can give an alternative more transparent graphical representation). Each graph with m components corresponds to a propagator H = a+m am
We now show how to find the Vn associated with general partition functions, thus getting a general graphical expansion for the Partition Function Integrand.
Hopf Algebras - the basicsHopf Algebras - the basicsAlgebra A x,y A x+y A xy A k K kx A
Bialgebra also hasCo-product : A A A
Co-unit : A K
Hopf Algebra also has Antipode S : A A
Generic ExampleGeneric ExampleAlgebra A generated by symbols x,y
xxyxyy A xxy+2 yx A
Unit e e x=x=x eCoproduct On uniteeXe On GeneratorxxXe + eXxABA) Balgebra mapAntipode S
On unitS(e)=e On GeneratorS(x)=-x S(AB)= S(B) S(A)(anti-algebra map)
Counit (e)=1 (x)=0
(AB)= (A) (B)
Hopf Algebra of Model DiagramsHopf Algebra of Model DiagramsGenerators are connected diagrams
...
Addition
+ =
Hopf Algebra of Model Diagrams (continued)Hopf Algebra of Model Diagrams (continued)
Multiplication
=
Algebra Identity e(=)
.
Hopf Algebra of Model Diagrams (continued)Hopf Algebra of Model Diagrams (continued)
Coproduct (Examples)on UnitXnGenerator
X=X
=
where A =
and B =
Hopf Algebra of Model Diagrams (continued)Hopf Algebra of Model Diagrams (continued)
Counit (Examples)
on Unit
otherwise
=
where A = and B =
= 1
Hopf Algebra of Model Diagrams (continued)Hopf Algebra of Model Diagrams (continued)
Antipode S (Examples)
UnitSGenerator S =-
S =SSS
where A =
and B =
Relation to other ModelsRelation to other ModelsHopf Algebra MorphismsHopf Algebra Morphisms
MQSym
LDiag FQSym
Diag Cocommutative CMM
Foissy Decorated Trees
Connes-Kreimer
SummarySummary
We have shown a simple model based on the evaluation of the Partition Function, which leads to a combinatorial description, based on Bell numbers; a graphical description and a related Hopf Algebra.
This is a very simple relative of the QFT model, but displays some of the latter’s features.
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