Graph model of the Heisenberg-Weyl algebra
P. Blasiak, A. Horzela, G.H.E. Duchamp, K.A. Penson, A.I. Solomon
GGraph model of the Heisenberg-Weyl algebra
P Blasiak , A Horzela , G H E Duchamp , K A Penson and A I Solomon , H. Niewodnicza´nski Institute of Nuclear Physics, Polish Academy of Sciencesul. Radzikowskiego 152, PL 31342 Krak´ow, Poland Universit´e Paris-Nord, LIPN, CNRS UMR 703099 Av. J.-B. Clement, F-93430 Villetaneuse, France Universit´e Pierre et Marie Curie, LPTMC, CNRS UMR 7600Tour 24 - 2i`eme ´et., 4 pl. Jussieu, F 75252 Paris Cedex 05, France The Open University, Physics and Astronomy DepartmentMilton Keynes MK7 6AA, United KingdomE-mail: [email protected], [email protected],[email protected], [email protected], [email protected]
Abstract.
We consider an algebraic formulation of Quantum Theory and develop acombinatorial model of the Heisenberg–Weyl algebra structure. It is shown that by liftingthis structure to the richer algebra of graph operator calculus, we gain a simple interpretationinvolving, for example, the natural composition of graphs. This provides a deeper insightinto the algebraic structure of Quantum Theory and sheds light on the intrinsic combinatorialunderpinning of its abstract formalism.
1. Introduction
Quantum Theory seen in action is an interplay of mathematical ideas and physical concepts.From a present-day perspective its formalism and structure is founded on the theory of Hilbertspace [1, 2, 3]. According to a few basic postulates, the physical notions of transformationsand measurements on a system are described in terms of operators. In this way the algebraof operators constitutes the proper mathematical framework within which quantum theoriesare built. The structure of this algebra is determined by two operations, the addition andmultiplication of operators; this lies at the root of all fundamental aspects of Quantum Theory[4].However, the physical content of Quantum Theory transcends the abstract mathematicalformalism. It is provided by the correspondence rules assigning operators to physical quantities.This is always an ad hoc procedure invoking concrete representations of the operator algebrachosen to best reflect the physical concepts related to the phenomena under investigation. Themost common structure in Quantum Theory is the Heisenberg–Weyl algebra. This describesthe algebraic relation between the position and momentum operators, equally the creation and annihilation operators, which provide our link to the most fundamental physical concepts.Accordingly, we take the Heisenberg–Weyl algebra as the central point of our study. a r X i v : . [ qu a n t - ph ] J un nterest in combinatorial representations of mathematical entities stems from a wealth ofconcrete models they provide. Their convenience comes from simplicity, which, being based onthe elementary notion of enumeration, directly appeals to intuition, often rendering invaluableinterpretations illustrating abstract mathematical constructions [5, 6, 7]. This makes thecombinatorial perspective particularly attractive in quantum physics, given the latter’s activepursuit of a proper understanding of fundamental phenomena.In this paper we develop a combinatorial representation of the operator algebra of QuantumTheory which is based on the Heisenberg–Weyl algebra. We recast it in the language of graphswith a simple composition rule and show how, from this perspective, abstract algebraic structuresgain an intuitive meaning. In some respects this draws on the Feynman idea of representingphysical processes as diagrams, familiar as a bookkeeping tool in the perturbation expansionsof quantum field theory [8, 9]. The combinatorial approach, however, has much more to offer ifapplied to the overall structure of Quantum Theory seen from the algebraic point of view. Wewill show that the process of lifting to a more structured algebra of graphs gives the abstractoperator calculus a straightforward interpretation, reflecting natural operations on graphs. Thisprovides an interesting insight into the algebraic counterpart of the theory and sheds light onthe intrinsic combinatorial structures which lie behind its abstract formalism.
2. Quantum Theory as an Algebra of Operators
The usual setting for Quantum Theory consists of specifying a Hilbert space H , whosevectors are the states of the system, and identifying operators on those states with physicallyrelevant quantities. Operators acting on H naturally form an algebra with addition andmultiplication, which we denote by O . The most interesting structures in O are, of course,those generated by operators having a physical interpretation. They usually originate fromconsidering some observables of interest along with operations causing changes in the state ofa system. Accordingly, one takes a hermitian operator, say N , representing some observableand defines a basis in H related to states with definite values of the corresponding physicalquantity. The eigenvectors of N are given by N | n (cid:105) = n | n (cid:105) , numbering the chosen eigenbasis | n (cid:105) in H ( n = 0 , , , ... ). One is then interested in describing processes which change the state ofthe system, e.g. time evolution, interactions and other transformations. For that purpose it isconvenient to introduce annihilation a and creation a † operators which shift the basis vectors byone, i.e. [ a, N ] = a and [ a † , N ] = − a † . Conventionally, these operators are required to satisfythe canonical commutation relation [ a, a † ] = 1 , (1)constituting the Heisenberg–Weyl algebra structure [10, 11] which has became the hallmarkof noncommutativity in Quantum Theory [4]. The operators defined above play the role ofelementary processes altering the system by changing its state with respect to the chosenphysical characteristic, i.e. they cause a jump between the eigenstates | n (cid:105) according to therule a | n (cid:105) = √ n | n − (cid:105) and a † | n (cid:105) = √ n + 1 | n + 1 (cid:105) . We shall assume that any change ofstate can be obtained by the action of some combination of such creation and annihilation acts,making the operators a and a † convenient building blocks describing the transformations of asystem.The creation and annihilation operators can be used to represent elements of the algebra O . Indeed, each operator can be seen as an element of the free algebra generated by a and a † , i.e. written as a linear combination of words in generators. This procedure is, however,ambiguous due to the commutation relation Eq. (1) which yields different representations of thesame operator [12]. To solve this problem, the order of a and a † has to be fixed. Conventionallythis is done by choosing the normally ordered form in which all annihilators stand to the rightf creators [13, 14]. Consequently, each operator A ∈ O can be uniquely written in the normallyordered form such as A = (cid:88) r,s α rs a † r a s . (2)In this way elements of the operator algebra O are represented in terms of the ladder operators a and a † , and interpreted as combinations of the elementary acts of annihilation and creation.Eq. (2) will be the starting point of our combinatorial representation of the algebra O .
3. Graphs and their Algebra
Considering combinatorial realizations of operator algebras, we shall specify two classes of graphs g and g , the latter being the shadow of the former under a suitable forgetful procedure. We shallemploy the convenient notion of graph composition to show how these structures are naturallymade into algebras, providing the representation of the algebra O . & Composition
A graph is a collection of vertices connected by lines with internal structure determined bysome construction rules. For our purposes, we consider a specific class of graphs defined in thefollowing way. & Lines.
The basic building blocks of the graphs are vertices • attached to twosorts of lines, those coming into, and those going out of, the vertex, having loose ends markedwith grey (cid:78)(cid:77) and white (cid:77) arrows respectively. A generic one-vertex graph Γ ( r,s ) is characterizedby two numbers r and s counting incoming and outgoing lines respectively, see Fig. 1. We shalldenote by g the class comprising all such one-vertex graphs, and by Ø the empty, or void,graph (no vertices, no lines). In a further construction we shall assume that all lines attachedto vertices are distinguishable. r outgoing lines s ingoing lines Figure 1.
A generic one-vertex graph Γ ( r,s ) ∈ g . We do not specify limits of summation and constraints on coefficients since it does not affect the algebraicconsiderations and can be introduced at each step if needed. .1.2. Construction Rules.
A multi-vertex graph Γ is a set of vertices with additional structureintroduced by joining some of the outgoing lines to the incoming ones. The requirement thatthe original direction of lines is preserved results in a directed structure of graphs indicated byblack arrows (cid:78) on the inner lines. We further restrict the class of graphs we consider to thosewithout cycles, i.e. we exclude graphs with closed paths. An example of a multi-vertex graphis shown in Fig. 2. Figure 2.
Example of a multi-vertex graph (8 vertices and 9 outgoing lines, 6 incoming lines,5 inner lines) built of two kinds of vertices: Γ (2 , ( X shape) and Γ (2 , ( Y shape).The rules specified above define the class of graphs denoted by g . In a less formal manner,we can describe these graphs as having an inner structure determined by directed connectionsbetween vertices and a characteristic set of outer lines marked with grey (cid:78)(cid:77) and white (cid:77) arrowsat the loose ends. These graphs can be seen as a kind of process, with the vertices beingintermediate steps. This observation can be developed further with the help of the convenientnotion of graph composition . Two graphs can be composed by joining some of the incoming lines (greyarrows) of the first one with some of the outgoing lines (white arrows) of the second one. Thisoperation is inner in g since it preserves the direction of the lines and does not introduce cycles.Observe that two graphs can be composed in many ways, i.e. as many as there are possiblechoices of pairs of lines (grey arrows from the first one and white arrows from the second one)which are joined, see Fig. 2. Note also that composing graphs in reverse order yields differentresults.The notion of graph composition allows for an iterative definition, i.e. any element of g canbe constructed starting from the void graph by successive composition with one-vertex graphs.Consequently, the class of one-vertex graphs g ⊂ g can be seen as representing basic processesor events happening one after another and constituting a composite process – a multi-vertexgraph. & ,, ,, , , ,., . Figure 3.
Two one-vertex graphs composed in different order. Note that distinguishability ofthe lines is taken into account.
In many cases one is not interested in the inner structure of a graph and needs only to focus onthe outer lines. This is equivalent to considering the graph’s one-vertex equivalents obtained byreplacing all inner vertices and lines by a single vertex and keeping all the outer lines untouched, i.e. Γ ∼ −→ Γ ( r,s ) where Γ is the graph with r outgoing and s incoming lines. For example,for the graph in Fig. 2 this gives Γ ∼ −→ Γ (9 , . The mapping g ∼ −→ g reduces to forgettingabout the inner structure of graphs and introduces an equivalence relation in g . Accordingly,two graphs are equivalent Γ ∼ Γ if and only if both have the same number of incoming andoutgoing lines respectively. The simplest choice of representatives of equivalence classes are theone-vertex graphs and so the quotient set g / ∼ is isomorphic to the set of one-vertex graphs g .There are two characteristic mappings between g and g : the canonical projection mapdescribed above and the inclusion map g ⊂ g , i.e. g ∼ (cid:43) (cid:43) g ⊃ (cid:107) (cid:107) (3)n this sense g is a shadow of the more structured class g . Observe that the arrows in Diag. (3)can not be reversed, i.e. once the inner structure of a graph is forgotten it cannot be restored. Both classes of graphs g and g can be endowed with the structure of a noncommutative algebrabased on the natural concept of graph composition. An algebra requires two operations, additionand multiplication, which are constructed as follows. We define the vector space G (over C )generated by the basis set g , by G = (cid:110) (cid:88) i α i Γ i : α i ∈ C , Γ i ∈ g (cid:111) . (4)Addition in G has the usual form (cid:88) i α i Γ i + (cid:88) i β i Γ i = (cid:88) i ( α i + β i ) Γ i . (5)The less trivial part in the definition of an algebra G concerns multiplication, which by bilinearity (cid:88) i α i Γ i ∗ (cid:88) j β j Γ j = (cid:88) i,j α i β j Γ i ∗ Γ j , (6)only requires definition on the basis set g . Recalling the notion of graph composition, thedefinition which suggests itself is (compare with Fig. 3) Γ i ∗ Γ j = (cid:88) all compositions of Γ i with Γ j . (7)Note that all the terms in the sum are distinct with coefficient equal to one. The multiplicationthus defined is noncommutative and produces G an associative algebra with unit (void graph).Imposing an algebraic structure on g roughly follows the above scheme. Accordingly, onedefines the vector space G = (cid:110) (cid:88) i,j α i,j Γ ( i,j ) : α i,j ∈ C , Γ ( i,j ) ∈ g (cid:111) , (8)with addition defined analogously to Eq. (5). Multiplication again reduces to defining it on thebasis set g , but the obstacle here is that the composition rule is not closed within the class g , i.e. it produces two-vertex graphs which belong to g . This however can be overcome by applyingthe forgetful mapping Γ ∼ −→ Γ ( r,s ) to the result. Specifically, multiplication of two graphs in g follows Diag. (3) and consists of:1) treating graphs as elements of g ,2) multiplying them according to Eq. (7),3) forgetting the inner structure of the resulting two-vertex graphs in the sum.Note that, contrary to Eq. (7), some of the resulting terms may be equal and their sum mayinvolve nontrivial integer coefficients. Grouping terms with respect to the number of joined linesyields the explicit formula Γ ( r,s ) ∗ Γ ( k,l ) = min { k,s } (cid:88) i =0 (cid:18) si (cid:19)(cid:18) ki (cid:19) i ! Γ ( r + k − i,s + l − i ) . (9)For example: Γ (2 , ∗ Γ (2 , = Γ (4 , + 2 Γ (3 , and Γ (2 , ∗ Γ (2 , = Γ (4 , + 4 Γ (3 , + 2 Γ (2 , ,see Fig. 3. In this way, multiplication in the richer structure G is naturally projected onto G .he resulting combinatorial algebra G is associative and noncommutative. Again, similarly to g and g in Diag. (3), both algebras are related by G ∼ (cid:43) (cid:43) G ⊃ (cid:107) (cid:107) (10)Hence, the algebra G is an image of the more structured algebra of graphs G .
4. Graph representation of operator algebra
The structures described above are examples of algebras having concrete representations basedon the natural concept of graph composition. It appears that both are intimately related to thealgebra of operators O . As suggested by the similarity of elements in O and G , see Eqs. (2)and (8), we make a correspondence of the basis sets a † r a s ←→ Γ ( r,s ) (11)establishing the isomorphism of the vector spaces. This would not be surprising if it were not forthe fact that this mapping also preserves multiplication in both algebras. Indeed, multiplicationof the basis elements in O gives a † r a s a † k a l = min { k,s } (cid:88) i =0 (cid:18) si (cid:19)(cid:18) ki (cid:19) i ! a † r + k − i a s + l − i , (12)which is the result of transforming a s a † k to the normally ordered form using the commutator[ a s , a † k ] = (cid:80) min { k,s } i =1 (cid:0) si (cid:1)(cid:0) ki (cid:1) i ! a † k − i a s − i , see e.g. [15]. It suffices to compare Eqs. (9) and (12) toshow that Eq. (11) establishes an isomorphism between the algebras O and G . In other words,both algebras are essentially the same, i.e. they have all elements and operations equivalent. Wecan thus enjoy the advantages of concrete realization of the abstract operator algebra in termsof graphs. For example, instead of multiplying operators in O one can do it in G simply bycomposing graphs. The crucial role in this procedure is played by the more structured algebraof graphs G where all these operations have a simple interpretation. Accordingly, Diag. (10) canbe complemented to G ∼ (cid:44) (cid:44) G ⊃ (cid:107) (cid:107) (cid:111) (cid:111) : (cid:47) (cid:47) O (13)In this way, the algebra O gains a combinatorial representation via G and can be seen asreflecting the natural processes taking place in G .
5. Discussion
We have considered the Heisenberg–Weyl algebra within the operator structure of QuantumTheory and exploited a convenient representation of operators by normally ordered expressionsin ladder operators a and a † . This allowed us to identify the operator algebra O withthe combinatorial algebra G constructed as the projection of the algebra of graphs G .The main result of the paper shows that these combinatorial structures provide us with asimple interpretation of the operator calculus in terms of the natural composition of graphs.Accordingly, the operator algebra can be seen as an image (via G ) of the algebra G which is acategorical version of the algebra O [16, 17]. This may be illustrated by redrawing Diag. (13) as ∼ (cid:32) (cid:32) (cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65) Graph Algebra (cid:111) (cid:111) (cid:111)(cid:47) (cid:111)(cid:47) (cid:111)(cid:47) O ⊂ (cid:62) (cid:62) (cid:125)(cid:125)(cid:125)(cid:125)(cid:125)(cid:125)(cid:125)(cid:125)(cid:125) (cid:111) (cid:111) : (cid:47) (cid:47) G Quantum Theory
Heisenberg − W eyl algebra (cid:111) (cid:111) (cid:111)(cid:47) (cid:111)(cid:47) (cid:111)(cid:47)
The diagram indicates the existence of the fundamental graph structure G of which the algebraicstructure of Quantum Theory is just the reflection. That is, all objects in the theory, as well asits calculus, can be seen, via simple inclusion, as elements and operations present in the richeralgebra of graphs described by natural composition rules. At any time it is possible to returnto the coarser level of an algebra of operators, simply by forgetting about the inner structure ofgraphs. As a result, the algebra of graphs suggests itself as a fundamental combinatorial levelof Quantum Theory. The process described here, of lifting the theory to the richer structure G , is motivated by the natural interpretation of graphs as processes transforming quantities orobjects, which is an attractive concept from the physical point of view. Moreover, the majoradvantage of the combinatorial representation of the algebra O , presented above, is that theabstract operator calculus can be seen intuitively as a straightforward composition of graphs. Acknowledgments
We wish to thank Philippe Flajolet for important discussions on the subject. Most of thisresearch was carried out in the Mathematisches Forschungsinstitut Oberwolfach (Germany) andthe Laboratoire d’Informatique de l’Universit´e Paris-Nord in Villetaneuse (France) whose warmhospitality is greatly appreciated. The authors acknowledge support from the Polish Ministry ofScience and Higher Education grant no. N202 061434 and the Agence Nationale de la Rechercheunder programme no. ANR-08-BLAN-0243-2.
References [1] Isham C J 1995
Lectures on Quantum Theory: Mathematical and Structural Foundations (London: ImperialCollege Press)[2] Peres A 2002
Quantum Theory: Concepts and Methods (New York: Kluwer Academic Publishers)[3] Ballentine L E 1998
Quantum Mechanics: A Modern Development (Singapore: World Scientific)[4] Dirac P A M 1982
The Principles of Quantum Mechanics
Analytic Combinatorics (Cambridge: Cambridge University Press)[6] Bergeron F, Labelle G and Leroux P 1998
Combinatorial Species and Tree-like Structures (Cambridge:Cambridge University Press)[7] Aigner M 2007
A Course in Enumeration
Graduate Texts in Mathematics (Berlin: Springer-Verlag)[8] Bjorken J D and Drell S D 1993
Relativistic Quantum Fields (New York: McGraw-Hill)[9] Mattuck R D 1992
A Guide to Feynman Diagrams in the Many-Body Problem
Algebra (Prentica Hall)[11] Bourbaki N 1998
Algebra vol I (Springer)[12] Cahill K E and Glauber R J 1969
Phys.Rev.
Coherent States. Application in Physics and Mathematical Physics (Singapore: World Scientific)[14] Blasiak P, Horzela A, Penson K A, Solomon A I and Duchamp G H E 2007
Am. J. Phys. J. Phys. A: Math. Theor. Mathematics Unlimited - 2001 and Beyond (Berlin: Springer Verlag) pp 29–50arXiv:math.QA/0004133 [math.QA][17] Morton J 2006
Theory and Applications of Categories16