Graph model of the Heisenberg-Weyl algebra
We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg–Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple interpretation involving, for example, the natural composition of graphs. This provides a deeper insight into the algebraic structure of Quantum Theory and sheds light on the intrinsic combinatorial underpinning of its abstract formalism.
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 Hilbert space [IshamBook, PeresBook, BallentineBook]. According to a few basic postulates, the physical notions of transformations and measurements on a system are described in terms of operators. In this way the algebra of operators constitutes the proper mathematical framework within which quantum theories are built. The structure of this algebra is determined by two operations, the addition and multiplication of operators; this lies at the root of all fundamental aspects of Quantum Theory [DiracBook].
However, the physical content of Quantum Theory transcends the abstract mathematical formalism. 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 algebra chosen to best reflect the physical concepts related to the phenomena under investigation. The most common structure in Quantum Theory is the Heisenberg–Weyl algebra. This describes the 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.
Interest in combinatorial representations of mathematical entities stems from a wealth of concrete models they provide. Their convenience comes from simplicity, which, being based on the elementary notion of enumeration, directly appeals to intuition, often rendering invaluable interpretations illustrating abstract mathematical constructions [FlajoletBook, BergeronBook, AignerBook]. This makes the combinatorial perspective particularly attractive in quantum physics, given the latter’s active pursuit of a proper understanding of fundamental phenomena.
In this paper we develop a combinatorial representation of the operator algebra of Quantum Theory which is based on the Heisenberg–Weyl algebra. We recast it in the language of graphs with a simple composition rule and show how, from this perspective, abstract algebraic structures gain an intuitive meaning. In some respects this draws on the Feynman idea of representing physical processes as diagrams, familiar as a bookkeeping tool in the perturbation expansions of quantum field theory [BjorkenDrell, MattuckBook]. The combinatorial approach, however, has much more to offer if applied to the overall structure of Quantum Theory seen from the algebraic point of view. We will show that the process of lifting to a more structured algebra of graphs gives the abstract operator calculus a straightforward interpretation, reflecting natural operations on graphs. This provides an interesting insight into the algebraic counterpart of the theory and sheds light on the 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 , whose vectors are the states of the system, and identifying operators on those states with physically relevant quantities. Operators acting on naturally form an algebra with addition and multiplication, which we denote by . The most interesting structures in are, of course, those generated by operators having a physical interpretation. They usually originate from considering some observables of interest along with operations causing changes in the state of a system. Accordingly, one takes a hermitian operator, say , representing some observable and defines a basis in related to states with definite values of the corresponding physical quantity. The eigenvectors of are given by , numbering the chosen eigenbasis in (). One is then interested in describing processes which change the state of the system, e.g. time evolution, interactions and other transformations. For that purpose it is convenient to introduce annihilation and creation operators which shift the basis vectors by one, i.e. and . Conventionally, these operators are required to satisfy the canonical commutation relation
constituting the Heisenberg–Weyl algebra structure [ArtinBook, BourbakiAlgebraI] which has became the hallmark of noncommutativity in Quantum Theory [DiracBook]. The operators defined above play the role of elementary processes altering the system by changing its state with respect to the chosen physical characteristic, i.e. they cause a jump between the eigenstates according to the rule and . We shall assume that any change of state can be obtained by the action of some combination of such creation and annihilation acts, making the operators and convenient building blocks describing the transformations of a system.
The creation and annihilation operators can be used to represent elements of the algebra . Indeed, each operator can be seen as an element of the free algebra generated by and , 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 the same operator [CahillGlauber]. To solve this problem, the order of and has to be fixed. Conventionally this is done by choosing the normally ordered form in which all annihilators stand to the right of creators [KlauderBook, AmJPhys]. Consequently, each operator can be uniquely written in the normally ordered form such as111We do not specify limits of summation and constraints on coefficients since it does not affect the algebraic considerations and can be introduced at each step if needed.
In this way elements of the operator algebra are represented in terms of the ladder operators and , 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 .
3 Graphs and their Algebra
Considering combinatorial realizations of operator algebras, we shall specify two classes of graphs and , the latter being the shadow of the former under a suitable forgetful procedure. We shall employ the convenient notion of graph composition to show how these structures are naturally made into algebras, providing the representation of the algebra .
3.1 Graphs Composition
A graph is a collection of vertices connected by lines with internal structure determined by some construction rules. For our purposes, we consider a specific class of graphs defined in the following way.
3.1.1 Vertices Lines.
The basic building blocks of the graphs are vertices attached to two sorts of lines, those coming into, and those going out of, the vertex, having loose ends marked with grey and white arrows respectively. A generic one-vertex graph is characterized by two numbers and counting incoming and outgoing lines respectively, see Fig. 1. We shall denote by 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 attached to vertices are distinguishable.
3.1.2 Construction Rules.
A multi-vertex graph is a set of vertices with additional structure introduced by joining some of the outgoing lines to the incoming ones. The requirement that the original direction of lines is preserved results in a directed structure of graphs indicated by black arrows on the inner lines. We further restrict the class of graphs we consider to those without cycles, i.e. we exclude graphs with closed paths. An example of a multi-vertex graph is shown in Fig. 2.
The rules specified above define the class of graphs denoted by . In a less formal manner, we can describe these graphs as having an inner structure determined by directed connections between vertices and a characteristic set of outer lines marked with grey and white arrows at the loose ends. These graphs can be seen as a kind of process, with the vertices being intermediate steps. This observation can be developed further with the help of the convenient notion of graph composition.
Two graphs can be composed by joining some of the incoming lines (grey arrows) of the first one with some of the outgoing lines (white arrows) of the second one. This operation is inner in 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 possible choices 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 different results.
The notion of graph composition allows for an iterative definition, i.e. any element of can be constructed starting from the void graph by successive composition with one-vertex graphs. Consequently, the class of one-vertex graphs can be seen as representing basic processes or events happening one after another and constituting a composite process – a multi-vertex graph.
3.2 Equivalence of Graphs
In many cases one is not interested in the inner structure of a graph and needs only to focus on the outer lines. This is equivalent to considering the graph’s one-vertex equivalents obtained by replacing all inner vertices and lines by a single vertex and keeping all the outer lines untouched, i.e. where is the graph with outgoing and incoming lines. For example, for the graph in Fig. 2 this gives . The mapping reduces to forgetting about the inner structure of graphs and introduces an equivalence relation in . Accordingly, two graphs are equivalent if and only if both have the same number of incoming and outgoing lines respectively. The simplest choice of representatives of equivalence classes are the one-vertex graphs and so the quotient set is isomorphic to the set of one-vertex graphs .
There are two characteristic mappings between and : the canonical projection map described above and the inclusion map , i.e.