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arXiv:1702.04602 · Journal of Physics Conference Series

Working directly with probabilities in quantum field theory

Robert Dickinson, J. R. Forshaw, Peter Millington

Published Aug 1, 2017• 5 citations
arXivHTMLPDF

Abstract

We present a novel approach to computing transition probabilities in quantum field theory, which allows them to be written directly in terms of expectation values of nested commutators and anti-commutators of field operators, rather than squared matrix elements. We show that this leads to a diagrammatic expansion in which the retarded propagator plays a dominant role. As a result, one is able to see clearly how faster-than-light signalling is prevented between sources and detectors. Finally, we comment on potential implications of this approach for dealing with infra-red divergences.

DOI
10.1088/1742-6596/880/1/012041

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We present a novel approach to computing transition probabilities in quantum field theory, which allows them to be written directly in terms of expectation values of nested commutators and anti-commutators of field operators, rather than squared matrix elements. We show that this leads to a diagrammatic expansion in which the retarded propagator plays a dominant role. As a result, one is able to see clearly how faster-than-light signalling is prevented between sources and detectors. Finally, we comment on potential implications of this approach for dealing with infra-red divergences.
We present an approach to computing probabilities in quantum field theory for a wide class of source-detector models. The approach works directly with probabilities and not with squared matrix elements, and the resulting probabilities can be written in terms of expectation values of nested commutators and anticommutators. We present results that help in the evaluation of these, including an expression for the vacuum expectation values of general nestings of commutators and anticommutators in scalar field theory. This approach allows one to see clearly how faster-than-light signalling is prevented, because it leads to a diagrammatic expansion in which the retarded propagator plays a prominent role. We illustrate the formalism using the simple case of the much-studied Fermi two-atom problem.
The diagrammatic method is one of the most powerful tools of theoretical physics. It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory. The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams ( graphs ). Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs . There exist several kinds of diagrams. We will try to present them in a systematic way. In Sect. 20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure. This formalism is used extensively in classical statistical physics. It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables. We use the term “Gaussian integration” in a rather broad sense. Beside commuting “bosonic” variables, we also consider anti-commuting “fermionic” variables, where we use the Berezin integral with respect to a Gaussian weight. Even in the case of commuting variables, the “Gaussian integral” is not necessarily meant in the sense of measure theory. It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight. However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.
DiagrammaticsThe diagrammatic method is one of the most powerful tools of theoretical physics.It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory.The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams (graphs).Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs.There exist several kinds of diagrams.We will try to present them in a systematic way.In Sect.20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure.This formalism is used extensively in classical statistical physics.It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables.We use the term "Gaussian integration" in a rather broad sense.Beside commuting "bosonic" variables, we also consider anti-commuting "fermionic" variables, where we use the Berezin integral with respect to a Gaussian weight.Even in the case of commuting variables, the "Gaussian integral" is not necessarily meant in the sense of measure theory.It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight.However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.We distinguish two kinds of spaces on which we perform the integrals: real and complex.As in many other places in our work, we treat these two cases in parallel.Of course, the difference between the real (i.e.neutral), and the complex (i.e.charged) formalism is mainly that of a different notation.In particular, charged lines need to be equipped with an arrow, whereas neutral lines need not.The terminology that we use is inspired by quantum field theory.Therefore, the variables that enter the integral are associated with "particles"; they are divided into "bosons" and "fermions", each subdivided into "neutral" and "charged" particles.In the main part of the chapter we describe the diagram formalism used in quantum many-body physics and quantum field theory.As a preparation, we include a brief Sect.20.2 devoted to the basic terminology of perturbation theory
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. This consists in dividing the Hamiltonian into one part representing a simpler well-understood problem and a nontrivial part, the effect of which is studied order by order. The expressions resulting from perturbation theory quickly become unwieldy. This chapter deals with propagators. It introduces the retarded and advanced propagator and describes their perturbation theoretic structure in a potential in terms of diagrams. A convenient method of representing perturbative expressions by diagrams was invented by Feynman. Besides the appealing aspect of representing perturbative expressions by drawings, the diagrammatic method can also be used directly for reasoning and problem solving. The easily recognizable topology of diagrams makes the diagrammatic method a powerful tool for constructing approximation schemes as well as exact equations that may hold true beyond perturbation theory.
We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of *Hilb having quantum fields as states and we show that the usual ladder and field operators can be defined as suitable endomorphisms. We deal with relativistic normalisation and we obtain the Lorentz invariant Heisenberg picture operators. By moving to a coherent perspective -- where the classical time and momentum parameters are replaced by wavefunctions over the parameter spaces -- we show that ladder operators and field operators can be obtained by applying the same morphism to plane waves and delta functions respectively. Finally, we formulate the commutation relations diagrammatically and we use them to derive the propagator.
Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.