A Generalized Method of Field Quantization
TL;DR: In this article, it was shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate, and that the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated.
Abstract: A method of field quantization is investigated which is more general than the usual methods of quantization in accordance with Bose of Fermi statistics, though these are included in the scheme. The commutation properties and matrix representations of the quantized field amplitudes are determined, and the energy levels of the field are derived in the usual way. It is shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate. With the generalized statistics, the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated. Physical consequences of the assumption that there are particles which obey the generalized statistics are briefly examined.
Citations
More filters
TL;DR: In this article, the authors review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications, including shape invariance and operator transformations, and show that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials.
2,688 citations
TL;DR: A review of studies performed in the field of non-classical states can be found in this article, with a focus on the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields.
Abstract: Seventy five years ago, three remarkable papers by Schr¨ odinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called ‘nonclassical states’ (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the ‘standard citation package’ used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and ‘nonclassical’ states commenced. The 1990s are described in a more ‘squeezed’ manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.
816 citations
TL;DR: In this article, a method is presented to sum over histories for spinor fields which employs the familiar classical c-number expression for the action, predicts anti-commutation rules and Fermi statistics, and retains the invariance of the theory under a change in phase of the complex ψ field.
385 citations
TL;DR: In this paper, it was shown that for fields with integer-valued intrinsic angular momentum, the observed relation between spin and (exchange) statistics follows from continuity alone, parastatistics being excluded.
Abstract: Sufficiently nonlinear classical fields admit modes called kinks, whose number is strictly conserved in virtue of boundary conditions and continuity of the field as a function of space and time. In a quantum theory of such fields, with canonical commutation (not anticommutation) relations, kinks and their conservation still persist, and even if the intrinsic angular momentum is an integer, a rotating kink can have half‐odd angular momentum, if double‐valued state functionals are admitted. We formulate a natural concept of exchange appropriate for kinks. The principal result is that for fields with integer‐valued intrinsic angular momentum, the observed relation between spin and (exchange) statistics follows from continuity alone, parastatistics being excluded. It is likely that in the theories with even (odd) exchange statistics, suitable creation operators will commute (anticommute). We show that, while the rotational spectrum of a kink will in general possess both integer and half‐odd spin states, in fields with integer‐valued intrinsic angular momentum only one of these two possibilities will ever be observed for each kind of kink, and that there is a nonzero ``particle number'' (strictly conserved, additive, scalar quantum number) attached to half‐odd‐spin kinks of each kind. It then follows that a boson and a fermion kink will always differ in at least one particle number, as well as in spin, and that, in particular, every fermion kink will have some nonzero particle number. These results are consistent with the hypothesis that the spinor fields usually employed to describe half‐odd‐spin quanta are not fundamental, but are useful ``point‐limit'' approximations to operators creating or annihilating excitations in a nonlinear field of particular kinds of kinks in particular internal states.
303 citations
TL;DR: In this article, a class of parity and time-reversal-invariant topological states of matter which can arise in correlated electron systems in 2+1-dimensions were described.
204 citations