A complete set of Lorentz-invariant wave packets and modified uncertainty relation

We define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of identity). The position–momentum uncertainty relation for this Lorentz-invariant wave packet deviates from the ordinary Heisenberg uncertainty principle, and reduces to it in the non-relativistic limit.

Relativistic partial waves for celestial amplitudes

The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles. In particular, relativistic partial waves are found as eigenfunctions to the product representation of celestial Poincaré Casimir operators with appropriate eigenvalues. The requirement of hermiticity of Casimir operators is used to fix the corresponding integral inner product, and orthogonality of the obtained relativistic partial waves is verified explicitly. The completeness relation, as well as the relativistic partial wave expansion follow. Example celestial amplitudes of scalars, gluons, gravitons and open superstring gluons are expanded on the basis of relativistic partial waves for demonstration. A connection with the formulation of relativistic partial waves in the bulk of Minkowski space is made in appendices.

An Application of Knowledge Engineering to Mathematics Curricula Organization and Formal Verification

The authors present a theoretical proposal for the organization of mathematical contents, more precisely to curricula development formalization and formal verification, inspired by knowledge engineering techniques. The situation addressed is the following: the starting point is a mathematical “official curriculum” (or part of it), not necessarily completely detailed. In our proposal, a group of experts would have to first build a detailed formulation of this curriculum (including the “prerequisite” relation between contents), which we will denominate “preprocessed official curriculum.” We detail how any “official curriculum development” could then be rigorously formalized and formally verified in a way inspired by rule-based expert system formal verification. We have defined the following terms: “contents soundness,” “contents completeness,” “relation soundness,” “relation completeness,” and “absence of cycles.” We believe that this is a completely new formalization within mathematics teaching theory that, once computer is implemented, would be very helpful. That would be the case, for instance, in countries where government sets the “official curricula” for Primary and Secondary Education and textbook contents have to be manually checked and approved by academic authorities: evaluators would “only” have to extract the textbook contents and set the “prerequisite” relation among them and let the computer do the rest.

Gauge Dependence of the Gauge Boson Projector

The propagator of a gauge boson, like the massless photon or the massive vector bosons W± and Z of the electroweak theory, can be derived in two different ways, namely via Green’s functions (semi-classical approach) or via the vacuum expectation value of the time-ordered product of the field operators (field theoretical approach). Comparing the semi-classical with the field theoretical approach, the central tensorial object can be defined as the gauge boson projector, directly related to the completeness relation for the complete set of polarisation four-vectors. In this paper we explain the relation for this projector to different cases of the Rξ gauge and explain why the unitary gauge is the default gauge for massive gauge bosons.

ON DEFORMED BOSON ALGEBRA AND VACUUM PROJECTION OPERATOR ON NONCOMMUTATIVE PLANE

In the noncommutative phase space, a new mapping is proposed to express the noncommutative coordinate and momentum operators in terms of the ordinary coordinate and momentum operators under the case of large noncommutativity parameters (μν>1). Using this mapping matrix, the deformed boson operators can be expressed in terms of the ordinary boson operators. Thus, the normal ordering expansion form of vacuum projection operator is obtained. As an application, the completeness relation of the two-mode deformed coherent states is verified by using the vacuum projection operator.

THE APPLICATION AND CHARACTERISTICS OF THE COHERENT ENTANGLED STATE |β, x〉

The coherent entangled state |β, x〉 is proposed in Fock space, which exhibits both the properties of the coherent and entangled states. The |β, x〉 makes up a new quantum mechanical representation, and the completeness relation of |β, x〉 is proved by virtue of the technique of integral within an ordered product of operators. The corresponding Schmidt decomposition of |β, x〉 is investigated. Furthermore, a feasible experimental scheme of |β, x〉 is presented, and generalized P-representation is constructed in the coherent entangled state |β, x〉.

DEFORMED BOSON ALGEBRA AND PROJECTION OPERATOR OF VACUUM IN NONCOMMUTATIVE PHASE SPACE

In this paper we introduce a new formalism to analyze Fock space structure of noncommutative phase space (NCPS). Based on this new formalism, we derive deformed boson commutation relations and study corresponding deformed Fock space, especially its vacuum structure, which leads to get a form of the vacuum projection operator. As an example of applications of such an operator, we define two-mode coherent state in the NCPS and show its completeness relation.