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Worldgraph approach to amplitudes

Peng Dai

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

String theory uses the first-quantized method (quantum mechanics) to derive scattering amplitudes. External states are considered as vertex operators inserted on the worldsheet generated by an internal string and the amplitudes are calculated perturbatively by calculating vacuum expectation values (vevs) of these vertex operators on worldsheets with different topologies. This approach is different from the common approach of particle theories in which the second-quantized method quantum field theory) is adopted to calculate amplitudes. A natural question to ask is whether there exists a first-quantization formalism for particles that gives particle amplitudes to all orders. This thesis presents the recent research in answering this question. In the first-quantized approach for particles, amplitudes are considered as the vevs of vertex operators inserted on different graphs generated by an internal particle. I refer to these graphs as world-graphs and first-quantized approach for particles as worldgraph approach. To evaluate these vevs, vertex operators for several external states and Green functions on different worldgraphs are needed. In this thesis, various vertex operators are considered and a general method to obtain scalar Green functions on different worldgraphs is obtained. Some examples of the worldgraph approach to amplitudes in scalar theory and Yang-Mills theory are presented.

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String theory uses the first-quantized method (quantum mechanics) to derive scattering amplitudes. External states are considered as vertex operators inserted on the worldsheet generated by an internal string and the amplitudes are calculated perturbatively by calculating vacuum expectation values (vevs) of these vertex operators on worldsheets with different topologies. This approach is
different from the common approach of particle theories in which the second-quantized method quantum field theory) is adopted to calculate amplitudes. A natural question to ask is whether there exists a first-quantization formalism for particles that gives particle amplitudes to all orders. This thesis presents the recent research in answering this question. In the first-quantized approach for particles, amplitudes are considered as the vevs of vertex operators inserted on different graphs
generated by an internal particle. I refer to these graphs as world-graphs and first-quantized approach for particles as worldgraph approach. To evaluate these vevs, vertex operators for several external states and Green functions on different worldgraphs are needed.
In this thesis, various vertex operators are considered and a general method to obtain scalar Green functions on different worldgraphs is obtained. Some examples of the worldgraph approach to amplitudes in scalar theory and Yang-Mills theory are presented.
A second quantized formalism for electrons confined to a plane in a strong perpendicular magnetic field is constructed using vertex operators. They are seen to arise naturally from a holomorphic representation of Laughlin’s first quantized wave functions, since they have the unique properties of creating coherent states, satisfying anyonic statistics and factorizing matrix elements. While open string vertex operators are sufficient for representing Laughlin’s “ground state” wave functions, it is shown that the vertex operators appearing in the theory of closed strings are needed in order to represent both types of anyonic excitations (quasi-holes and quasi-electrons) which appear in the theory of the fractional quantum Hall effect.
The scattering amplitude in simple quantum graphs is a well-known process which may be highly complex. In this work, motivated by the Shannon entropy, we propose a methodology that associates a graph with a scattering entropy, which we call the average scattering entropy. It is defined by taking into account the period of the scattering amplitude which we calculate using the Green's function procedure. We first describe the methodology on general grounds and then exemplify our findings considering several distinct groups of graphs. We go on and investigate other possibilities, one that contains groups of graphs with the same number of vertices, with the same degree, and the same number of edges, with the same length, but with distinct topologies and with different entropies. Another possibility we investigate contains graphs of the fishbone type, where the scattering entropy depends on the boundary conditions on the vertices of degree 1, with the corresponding values decreasing and saturating very rapidly as we increase the number of elementary structures in the graphs.
This thesis constructs quantum mechanical theories of finite simple graphs for both labeled and unlabeled graphs.These theories may provide a description for discrete spacetime in a quantum gravity theory. Finite simple graphs and their properties are introduced. The first and second quantization of the standard quantum mechanics of particle systems are reviewed. A quantum mechanical theory of graphs similar to first quantization of standard quantum mechanics is developed via a correspondence of graph edges to particles in particle systems. An algorithm for constructing quantum states of graphs which is independent of graph vertex labeling is developed. It builds antisymmetric graphstates which describe fermiongraphs, the analog of systems of fermions for graphs. A quantum mechanical theory of graphs similar to second quantization of standard quantum mechanics is developed via the same correspondence of graph edges and particles. Edge creation and annihilation operators are defined which insert or delete edges between any pair of grap hvertices. The commutation and anticommutation relations of these operators are defined to produce states which are symmetricand antisymmetric upon any vertex label permutation. Operators for certain graph properties and constructions are defined in this theory. Dynamics are then incorporated into the second quantization approach of the quantum mechanical graph theory by developing two model Hamiltonians, in analogy with the Hamiltonians of the Ising and Heisenberg model sof ferromagnetism. From these Hamiltonians, statistical mechanics can be used to describe the dynamics ofgraphs.
The quantum generalization of the Gelfand-Levitan method is presented for the nonlinear Schr\"odinger model. The basic dispersion relation for operator Jost functions is derived, and the Heisenberg field operator is expressed in terms of scattering-data operators. Construction of Green's functions in the zero-density vacuum is discussed. The four-point function is explicitly calculated from the expression for the field operator and compared with the result of a direct Feynman graph summation. In addition it is proved for any number of particles that the Hamiltonian eigenstates constructed from the quantized scattering data are identical with those previously obtained by means of Bethe's ansatz.
This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. According to Feynman, as described in Chapter 4, instead of formulating quantum theory in terms of operators, the canonical formulation, for calculational purposes quantum dynamics can conveniently be formulated in terms of a few simple stenographic rules, the Feynman rules for propagators and interaction vertices. In Chapters 4 and 5, we showed how to arrive at the Feynman rules of diagrammatic perturbation theory for non-equilibrium states starting from the Hamiltonian defining the theory. The feature of non-equilibrium states, originally carried by the dynamical indices, could be expressed in terms of two simple universal vertex rules for the RAK-components of the matrix Green's functions. We are thus well acquainted with diagrammatics even for the description of non-equilibrium situations. However, for the situations studied using the quantum kinetic equations in Chapters 7 and 8, only the Dyson equation was needed, i.e. the self-energy, the 2-state one-particle irreducible amputated Green's function. No need for higher-order vertex functions was required, and the full flourishing diagrammatics was not put into action. In this chapter we shall proceed the other way around. We shall show that the diagrammatics of a physical theory, including the description of non-equilibrium states, can be obtained by simply stating quantum dynamics, the superposition principle, as the two exclusive options for a particle: to interact or not to interact ! From this simple Shakespearean approach we shall construct the Feynman diagrammatics of non-equilibrium dynamics.