Cambridge University Press eBooks

The discrete time Dirac equation

George Jaroszkiewicz

Published Apr 17, 2014• 0 citations

Abstract

Introduction In this chapter we apply temporal discretization to mechanical systems described by anticommuting variables rather than the commuting variables normally used in classical mechanics (CM). The anticommuting numbers representing such variables are called Grassmann(ian) numbers by mathematicians, after Hermann G. Grassmann (1809–1877), who did some pioneering work in linear algebra. On that account we shall refer to anticommuting numbers as g-numbers , in contrast to c-numbers ( c for commuting 1 ) when we refer to ordinary real or complex variables. The significant difference between g-numbers and c-numbers is that, where as any two c numbers x and y satisfy the commutative multiplication rule xy = + yx , any two g-numbers θ and ϕ satisfy the anticommutation rule θ ϕ = –ϕ θ . G-numbers and c-numbers can be multiplied together in any order, i.e., g-numbers and c-numbers commute. The product zθ of a c-number z and a g-number θ is a g-number. The values of classical variables in ordinary CM are c-numbers, so we may refer to such variables as c-variables. It is possible to construct forms of CM where the classical variables are g-numbers, in which case we shall refer to such variables as g-variables. G-variables should not be thought of as quantized versions of c-variables. They are just as classical as c-variables but much less familiar in terms of their applications to the real world. G-variables should not be confused with Dirac's q-numbers either.

DOI: 10.1017/cbo9781139525381.024

Related papers

The construction of c-number fields and their nonlinear equations of motion for quantum many-body fermion systems

No author provided

A k-dependent state has been constructed from two pairs of mutually commuting Fermi-Dirac operators by replacing the Grassmann numbers in the conventional definition of a fermion coherent state. The Heisenberg time-dependent form of this state was then combined with a particular spin state to give a resultant state \G], the spin component containing a series of unknown k-dependent weights. A classical c-number field, phi(c),(r, t), was then defined by an expectation value of the quantum field, psi, using \G] and an energy and time-dependent phase. It is demonstrated that the form of the equation of motion for phi(c)(r, t) is identical to that for psi. (C) 1998 Elsevier Science B.V.

Sep 10, 20260 citations

Noncommutative Number Systems for Quantum Information

Otto C. W. Kong

Dirac talked about q-numbers versus c-numbers. Quantum observables are q-number variables that generally do not commute among themselves. He was proposing to have a generalized form of numbers as elements of a noncommutative algebra. That was Dirac's appreciation of the mathematical properties of the physical quantities as presented in Heisenberg's new quantum theory. After all, the familiar real, or complex, number system only came into existence through the history of mathematics. Values of physical quantities having a commutative product is an assumption that is not compatible with quantum physics. The revolutionary idea of Heisenberg and Dirac was pulled back to a much more conservative setting by the work of Schrödinger, followed by Born and Bohr. What Bohr missed is that the real number values we obtained from our measurements are only a consequence of the design of the kind of experiments and our using real numbers to calibrate the output scales of our apparatus. It is only our modeling of the information obtained about the physical quantities rather than what Nature dictates. We have proposed an explicit notion of definite noncommutative values of observables that gives a picture of quantum mechanics as realistic as the classical theory. In this article, we illustrate how matrices can be taken as noncommutative (q-)numbers serving as the values of physical quantities, each to be seen as a piece of quantum information. Our main task is to clarify the subtle issues involved in setting up a conventional scheme assigning matrices as values to the physical quantities.

arXiv:2405.10339May 14, 20240 citations

Trace dynamics: the classical Lagrangian and Hamiltonian dynamics of matrix models

Stephen L. Adler

In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.

Cambridge University Press eBooksAug 26, 20040 citations

Implementation of a Quantized Line Element in Klein-Gordon and Dirac Fields

D.L. Bulathsinghala, K.A.I.L. Wijewardena Gamalath

In this paper an ansatz that the anti-commutation rules hold only as integrated average over time intervals and not at every instant giving rise to a time-discrete form of Klein-Gordon equation is examined. This coarse-grained validation of the anti-commutation rules enables us to show that the relativistic energy-momentum relation holds only over discrete time intervals, fitting well with the time-energy uncertainty relation. When this time-discrete scheme is applied to four vector notations in relativity, the line-element can be quantized and thereby how the physical attributes associated with time, space and matter can be quantized is sketched. This potentially enables us to discuss the Zeno’s arrow paradox within the classical limit. As the solutions of the Dirac equation can be used to construct solutions to the Klein-Gordon equation, this temporal quantization rule is applied to the Dirac equation and the solutions associated with the Dirac equation under such conditions are interpreted. Finally, the general relativistic effects are introduced to a line-element associated with a particle in relativistic motion and a time quantized line-element associated with gravity is obtained.

International Letters of Chemistry Physics and AstronomyMar 1, 20150 citations

Chapter 4

Gaussian Integration, Grassmann Integrals

In this chapter we introduce anticommuting variables and derive the Grassmann integral representations for the partition function and the correlation functions. Grassmann integrals are a suitable algebraic tool for the rearrangement of fermionic perturbation series. We demonstrate this in the ﬁrst section by considering a model with a quadratic perturbation which is explicitely solvable. In that case the perturbation series can be resumed completely. By doing this in two ways, ﬁrst a direct computation without anticommuting variables and then a calculation using Grassmann integrals, we hope to be able to convince the reader of the utility of that formalism.

Chapman & Hall/CRC eBooksAug 30, 20040 citations

Intermission

George Jaroszkiewicz

This chapter forms a natural divide half-way into the book. In the first half, we discussed the principles of discrete time (DT) classical mechanics (CM). In the second half we focus mainly on quantum principles. This chapter is a good place in which to take stock of what we have done, what we plan to do, and how the two halves of the book are related. The common theme is Lee's approach to DT mechanics, discussed in Chapter 10. As we implied previously, theorists whose work is relevant to us can be classified into two types: the applied mathematicians and the mathematical physicists, who may also be called fundamentalists. This division is based not on any value judgements but on the motivations and ambitions driving a theorist's work, which are generally easy to identify. The applied mathematicians fall into two categories. The first consists of those interested in finding better ways of understanding CM and, if necessary, finding ever better approximations to it. They explore DT CM with that in mind and are generally not remotely interested in quantum mechanics (QM). The second category consists of quantum theorists, such as Bender, who develop DT numerical simulation to approximate standard QM. Lattice gauge theorists also fall into the applied group, since their discretization of spacetime is regarded at all times as an approximation to the continuum.

Cambridge University Press eBooksApr 17, 20140 citations