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CRC Press eBooks

The Dyson perturbation expansion of the evolution operator

Dževad Belkić

Published Jan 15, 2020• 0 citations
Publisher

Abstract

The first correction in the S-matrix expansion possesses quite an obvious physical interpretation, because it directly involves the asymptotic conditions in the scattering experiment. The physical meaning of the appearance of the times t 1 and t 2 consists of successive inclusion of the interactions. A physical meaning of such a development is customarily established by calculating the concrete matrix elements based upon equation. As Feynman pointed out, the graphical presentations are not only convenient symbols but also could be considered as good candidates for an adequate description of realistic physical observables in scattering events. Thus, for example, in field theory, the places where the partial segments of the zig-zag lines stop, in fact, become the media of the physical events, i.e. the energy and impulse are conserved at the vertices.

DOI
10.1201/9780429146497-10

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Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the funda­mental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that repre­sents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.

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A generalized off-shell unitarity relation for the two-body scattering $T$ matrix in a many-body medium at finite temperature is derived, through a consistent real-time perturbation expansion by means of Feynman diagrams. We comment on perturbation schemes at finite temperature in connection with an erroneous formulation of the Dyson equation in a paper recently published.

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Taylor & Francis eBooks•May 5, 2015•0 citations
The first correction in the S-matrix expansion possesses quite an obvious physical interpretation, because it directly involves the asymptotic conditions in the scattering experiment. The physical meaning of the appearance of the times t 1 and t 2 consists of successive inclusion of the interactions. A physical meaning of such a development is customarily established by calculating the concrete matrix elements based upon equation. As Feynman pointed out, the graphical presentations are not only convenient symbols but also could be considered as good candidates for an adequate description of realistic physical observables in scattering events. Thus, for example, in field theory, the places where the partial segments of the zig-zag lines stop, in fact, become the media of the physical events, i.e. the energy and impulse are conserved at the vertices.
Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the funda­mental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that repre­sents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.
This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.
Almost any of the great textbooks available on scattering theory continues to explain how a collision experiment measures scattering cross sections for different events, and relates the relevant S-matrix elements, or the scattering amplitudes (onshell T-matrix), to those cross sections. We highly recommend Taylor’s book in this respect. In this volume and the volume that will follow, however, our motivation does not lie in (although discussions can be extended to) the theoretical description of collision experiments. We are mostly interested in applications of scattering theory in few-body physics and effective interactions in few- and many-body physics. Therefore, we consider a method that is having huge success in the theoretical and numerical description of collisions in nuclear physics and quantum chromodynamics, as well as ultracold atomic and condensed matter physics recently. The method allows for an estimation of low-energy scattering parameters and binding energies from exact diagonalisation, in the low-energy sector, of the system’s Hamiltonian, and is originally due to Lüscher [2]. Lacking another term for it, we have decided to simply label this and related methods ‘modern scattering theory’.
A generalized off-shell unitarity relation for the two-body scattering TTT matrix in a many-body medium at finite temperature is derived, through a consistent real-time perturbation expansion by means of Feynman diagrams. We comment on perturbation schemes at finite temperature in connection with an erroneous formulation of the Dyson equation in a paper recently published.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are significant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benefits of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantummechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic effects become observable, for which nonrelativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.34) must be used. There are, however, more subtle effects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.