Abstract We give a mathematically precise review of a diagrammatic language introduced by Friedrichs in order to simplify computations with creation and annihilation operator products. In that language, we establish explicit formulas and algorithms for evaluating bosonic and fermionic commutators. Further, as an application, we demonstrate that the nonlinear Hartree dynamics can be seen as a subset of the diagrams arising in the time evolution of a Bose gas.
Abstract We give a mathematically precise review of a diagrammatic language introduced by Friedrichs in order to simplify computations with creation and annihilation operator products. In that language, we establish explicit formulas and algorithms for evaluating bosonic and fermionic commutators. Further, as an application, we demonstrate that the nonlinear Hartree dynamics can be seen as a subset of the diagrams arising in the time evolution of a Bose gas.
This chapter is an introduction to some computation techniques for fermionic states. After defining Grassmann algebras it is possible to get a classical analogue for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin [23, 26], we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formula for fermionic observables. In particular there exists a Moyal product formula. As an application we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators.
The aim of this paper is to give a self-contained and unified presentation of a fermionic coherent state theory with the necessary mathematical details, discussing their definition, properties and some applications. After defining Grassmann algebras, it is possible to get a classical analog for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin (1966 The Method of Second Quantization (New York: Academic); 1987 Introduction to Superanalysis (Dordrecht: Reidel Publishing Company)), we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formulas for fermionic observables. In particular, there exists a Moyal product formula. As an application, we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators. We prove a Mehler formula for the propagator and Mehlig–Wilkinson-type formulas for the covariant and contravariant symbols of ‘metaplectic’ transformations for fermionic states. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in the language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are quantum systems of finitely many local fermionic modes, with operations that preserve or reverse the parity (number of particles mod 2) of states, and the tensor product, corresponding to the composition of two systems, as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. We conclude by showing, as an example, how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language.
In this work, we establish a direct connection between supplemented Dyck language and the signed expectation value of chains of second quantization operators relatively to the physical vacuum and relatively to a one-determinant state. Inspired by the fact that Dyck language provides an example of the emergence of the Catalan numbers in linguistic framework analysis, we show that these numbers are central when numbering the terms remaining when eliminating vanishing contributions detected by our application of Dyck language to fermionic second quantization. From the translation of creation and annihilation operator - or of pairs of operators - into a bracket alphabet, we derive simple and intuitive sufficient conditions for the nullity of expectation values that does not require an explicit application of Wick's theorem. This is done here with respect to the physical vacuum or relatively to a one-determinant state. We also extend this translation into a diagrammatic framework that allows a visual determination of the signature of fully contracted terms, reproducing the results of Wick's theorem. This approach has been extended to the case of (nested) commutators of pairs of fermionic second quantization including at least one excitation or deexcitation operator. Our results have been implemented in a software, MobiDyck, whose source code is freely available on the web. The algorithmic approach inspired by our work on Dyck language is detailed in this paper. Finally, a comparison of our diagrammatic approach with Goldstone diagrams is provided and closes the article.
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
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.