TThe Kauffman Constraint Coefficients K (n , n ,…, n -1 ) Kenneth A. Griggs
Phoenix project
Washington, DC 20009
I. Introduction
In Reference [1], Louis H. Kauffman presents an
Algebra of Constraints . Its purpose is to explicitly define the algebraic conditions that best align non-commutative operators with their corresponding classical variables in the usual context of continuum calculus.
In creating this classical variable (CV) to non-commutating operator (NCO) correspondence, two assumptions are made: (1) there exists an exact mapping between CVs and NCOs of the form (I-1)
OperatorgCommutatinNoniableclassical var nn nn nn Xx Xx Xx
22 11 where n is a positive integer, and (2) there exists an exact mapping between linear combinations of CVs and linear combinations of NCOs of the form (I-2) OperatorsgCommutatinNoniablesclassical var nn Xx These two requirements establish constraint conditions on the NCOs that take the form of
Symmetrizers ; namely, (I-3) n n S XXXnXXX !1 where we are summing over all n -permutations of the permutation group S n . a a Kauffman uses {} brackets to designate
Symmetrizers as opposed to the [] which designate
Commutators . We will only use his Symmetrization convention in equation (I-3). In all cases that follow, curly brackets {} designate simple parenthesis. he Kauffman Constraint Coefficients K Ken n eth A. Gri ggs | 2 l
Phoe ni x proj e ct
These mappings enable Kauffman to align classical variable differential formulas with their corresponding non-commutative operator differential forms. In fact, by defining the first temporal derivative of a variable with respect to time (I-4) )0(1 hh Kauffman provides a general algorithm for obtaining a -order time-derivative in a recursive fashion. Namely, the next derivative is obtained from the previous by applying the product rule for differentiation and using the (I-4) identity. By programming this algorithm into Mathematica™, Kauffman is able to calculate the first nine levels of derivatives. However, Kauffman notes that “ The structure of the coefficients in this recursion is unknown territory…To penetrate the full algebra of constraints we need to understand the structure of these derivatives and their corresponding non-commutative symmetrizations. ” b While Kauffman employs computational differentiation using Mathematica™, we present below a closed-form expression for the
Kauffman Constraint Coefficient ,,,, nnnnK both as a solution to his challenge and to foster a greater understanding of the full Algebra of Constraints . II. Solution
The
Algebra of Constraints can be described combinatorically in terms of its structures. We begin by defining a -order time-derivative in terms of three structures: the first is the original variable ; the second is a set of -derivatives ,,,,,, HHHHH ; and the third is a set defining the multiplicity of those -derivatives ,,,,,, nnnnn . From these structures, two generalized functions are defined: firstly, there is the -Elemental E , which is a product of the -derivatives with their n multiplicities; secondly, there is the Kauffman Constraint Coefficient K , which is a count of the -Elementals. With these two functions, Kauffman’s derivative series takes the general form (II-1) N HnHnHnEnnnK ,,,,,,,,, where every Elemental E of the derivative series is explicitly given by c b Ibid., p. 31. c Because the Coefficient K is independent of the existence of the time functional = (0) =T (where T is used in Kauffman’s paper) for every Elemental of the derivative ( ) expansion, it is not displayed in the E Elementals. he Kauffman Constraint Coefficients K Ken n eth A. Gri ggs | 3 l
Phoe ni x proj e ct (II-2) ,,,,,, nnnnn n HHHHH HHnHnHnE with the number of derivatives of with respect to the Hamiltonian T , ; N the total number of distinct Elementals E , with N , of the derivative series; n the exponent and number of -derivatives, n H ; ,,, nnnK the -Kauffman Constraint Coefficient for the -Elemental of The key to this construction is that all -derivatives H , where , are present for every Elemental E of the series . As such, the form of the Elemental is encoded in the set of exponents n . Determining the correct -sets ,,, nnn of exponents generates the correct E elementals and their corresponding Kauffman Constraint Coefficients K . d Because of this, the
Kauffman Constraint Coefficient for each E Elemental is purely a function of the n exponents instead of the -derivatives H . As such, the Kauffman Constraint Coefficient is defined as (II-3) !1! !,,, n nnnnK III. Redefining the Kauffman Recursion Relation for ( ) As explained in the
Introduction , Kauffman generates each time derivative ( ) via a commutative recursion relation with all smaller derivatives (III-1) ,,, We have derived a new recursion relation that does not depend on using the product rule for differentiation but instead involves only summations and substitutions. e Namely, (III-2) !!1!1 H d Calculating the E Elementals can be performed as an iterative optimization problem through recursive differentiation and application of the product rule of differentiation or alternatively by
Combinatoric methods which will be explored in an upcoming paper. e This technique may provide a cleaner, more efficient algorithm for Mathematica™. he Kauffman Constraint Coefficients K Ken n eth A. Gri ggs | 4 l
Phoe ni x proj e ct
As a demonstration, we explicitly generate the first 4 such derivatives in the following
Table III-1 . Table III-1: New Recursive Calculation of !1 H Recursive Calculation Result !0 H !0!0 H HH !1 H HHH HH HH HH
31 !3!1 !2 H HHHHHH
HHH HHH HHHH
41 !4!1 !3 H
32 121 213 6226 !0!3!1!2!2!1!3!0
33 3!3 !3
HHH HHH HHHHH
HHHH HHHH
34 6
H HHH HHH
IV. Generating K for (9) As discussed in the
Introduction , Kauffman calculates the first nine levels of derivatives via Mathematica™. f We now present the calculation for those same coefficients using Equation (II-3). In so doing, we demonstrate how properly to use the Elementals ,,,,
EEEE corresponding to the derivative with the Kauffman Constraint Coefficient function to obtain the 30 coefficients ,,,,
KKKK . Table IV-1: Kauffman Constraint Coefficient for Mathematica™ E ,,,,, nnnnnK Calculation TH H K !1!9 !9!1! ! n n THH HH K nn nn THH HH K nn nn f Ibid., p. 32. he Kauffman Constraint Coefficients K Ken n eth A. Gri ggs | 5 l
Phoe ni x proj e ct Mathematica™ E ,,,,, nnnnnK Calculation THH HH K nn nn THH HH K nn nn THH HH K nn nn HTHH HHH K nnn nnn HTHH HHH K nnn nnn HTH HH K nn nn THH HH K nn nn HTHH HHH K nnn nnn TH H K n n THH HH K nn nn HTHH HHH K nnn nnn HTHH HHH K nnn nnn HTHH HHH K nnn nnn HHTH HHH K nnn nnn he Kauffman Constraint Coefficients K Ken n eth A. Gri ggs | 6 l
Phoe ni x proj e ct Mathematica™ E ,,,,, nnnnnK Calculation THH HH K nn nn THH HH K nn nn HTHH HHH K nnn nnn HTH HH K nn nn HTHH HHH K nnn nnn HTH HH K nn nn THH HH K nn nn HTHH HHH K nnn nnn HTH HH K nn nn THH HH K nn nn HTH HH K nn nn THH HH K nn nn TH H K !9!9 !9! ! n n he Kauffman Constraint Coefficients K Ken n eth A. Gri ggs | 7 l
Phoe ni x proj e ct
V. Generating K and K for (12) We extend our application of the
Kauffman Constraint Coefficient K to the slightly more difficult derivative. In so doing, we will only calculate the coefficients for two Elementals, namely, (V-1) HHHE HHE The corresponding
Kauffman Constraint Coefficients are (V-2) nnn nn nnn nn KK The reader should find these calculations using Mathematica™, or many other software programs, computationally reasonable.
VI. The (40) Challenge
To further demonstrate the usefulness of the
Kauffman Constraint Coefficient , a quick calculation can be made that would otherwise push the computational limits of Mathematica™ on many non-supercomputers. Namely, the reader may find it challenging to find the coefficient of the Elemental (VI-1)
HHHE of the fortieth time derivative . The predicted Kauffman Constraint Coefficient is (VI-2) nnn nnn K VII. Conclusion
Because this paper is not designed to recount the full scope of
Non-Commutative Worlds g as explicitly described by Louis H. Kauffman, we invite the reader to become more fully engaged in his revolutionary work. As such, we have presented only a summary of Kauffman’s mathematics and expressed motivations for creating an Algebra of Constraints . From this, we answered an outstanding question of the algebra:
What is the structure of the coefficients in the Kauffman Recursion Relation for ( ) ? g Ibid., pps. 1-34. he Kauffman Constraint Coefficients K Ken n eth A. Gri ggs | 8 l
Phoe ni x proj e ct
The
Kauffman Constraint Coefficients K and their corresponding Elementals E are presented as solutions to the construction of the derivative. Additionally, a new recursion relation is provided that requires only operational substitutions and summations; this algorithmically simplifies Kauffman’s original technique. To demonstrate K , we generate the 30 K Coefficients from the corresponding
Elementals E for (9) and find that our results are in complete agreement with Kauffman’s Mathematica™ solutions. We further present a calculation of two coefficients for the (12) derivative and invite readers to use Mathematica™ or any other means to calculate and verify our results. Finally, we present a challenging calculation for a coefficient of the (40) derivative series; owing to the vast numbers of permutations involved, a Mathematica™ approach may require substantial computer resources to obtain the solution in a reasonable time. These formula, calculations and techniques for the Kauffman Constraint Coefficients K and their corresponding Elementals E are presented to enable the further development and discussion of Kauffman’s emerging Algebra of Constraints . VII. References1