Abstract
This work didactically presents the mathematical procedures required for the construction of the creation and annihilation operators for a free quantum particle considering the coordinates of the light cone. For that, the relationships between the aforementioned coordinates and the coordinates (ct, x, y, z) are listed, in addition to the use of the Klein-Gordon-Fock equation in the formalism of the light cone coordinates. Finally, the temporal evolution operator and the quantum operators of creation and annihilation of the integral type of motion are obtained.
Highlights
Introduced by the English physicist Paul Dirac, the light cone coordinates were presented with the aim of constructing ways to describe the relativistic dynamics of a physical system (Dirac, 1949)
Within the scope of KGFE, as identified by Bagrov et al (1976), by enabling the exchange of the second-order partial differential in time (∂∂t2) for a first-order partial differential in the variable (∂∂u0), the light cone coordinate system facilitates the construction of the temporal evolution operator from which it is possible to establish a new annihilation operator with motion integral status
To obtain the time evolution operator, consider an electrically charged spin-zero free quantum particle whose dynamics is described by KGFE in the light cone coordinate system expressed in Equation 13 which can be rewritten as:
Summary
Introduced by the English physicist Paul Dirac, the light cone coordinates ( called the light front coordinates) were presented with the aim of constructing ways to describe the relativistic dynamics of a physical system (Dirac, 1949). Within the scope of KGFE, as identified by Bagrov et al (1976), by enabling the exchange of the second-order partial differential in time (∂∂t2) for a first-order partial differential in the variable (∂∂u0), the light cone coordinate system facilitates the construction of the temporal evolution operator from which it is possible to establish a new annihilation operator with motion integral status. In the context of semiclassical quantum states, in a relativistic regime, this new operator is suitable for obtaining coherent quantum states of a charged quantum particle under the influence of different classical electromagnetic field configurations This algebraic facilitation has its appeal in supporting theoretical studies of coherent states so useful in modern quantum theory, with immediate applications in quantum field theory, loop quantum gravity, and quantum computation (Pereira & Miranda, 2002; Gazeau, 2009; Bagrov et al, 2015). In the last two sections, the temporal evolution operator and the integral motion type operator are obtained
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