Speaker
Dr
Jerzy Przeszowski
(University of Białystok)
Description
In the canonical quantization procedure the quantum field operators are smeared with a test function of Schwartz class for coordinates on the quantization hypersurface, with a sharp dependence on the temporal parameter. Such quantum operators when acting on the vacuum state produce smeared 1-particle states and higher number particles states. These states, provided their norm is finite, form the Fock space of states.
For the equal-time quantization procedure one needs to diagonalize the Hamiltonian operator,
which can be exactly done mostly only for a free field dynamics. When one imposes the spectral condition that the spectrum of the Hamiltonian operator is non-negative for the Fock space states, then one obtains the definition of the vacuum state and the smeared creation and annihilation operators. This leads to 1-particle states, with a sharp dependence on time parameter, which have a finite, mass dependent norm, so they belong to the Fock space.
For the light-front quantization procedure the conditions that the LF Hamiltonian P^{-} and the longitudinal translation kinematic generator P^{+} have non-negative spectra lead only a restriction on the longitudinal momentum variable k^{+} > 0. Thus the 1-particle states appear with no dependence on mass but with an infinite norm, so they do not belong to the Fock space. However one may evolve these 1-particle states in the light front time by means of the unitary operator and then smear the light front time dependence with a test function. This extra smearing produces 1-particle states with a finite norm, which therefore belong to the Fock space of states. Evidently these final states have mass dependence, from the temporal evolution with the Hamiltonian operator, but the vacuum state remains simple with no reference to the Hamiltonian.
Accordingly the 1-particle states are drastically different within the equal time and the light front formulations, though they lead to the same physical results. For the interacting models, where the Hamiltonian operator changes number of particles, one cannot diagonalize it in the basis of Fock states without introducing some truncation. Thus the equal time approach is bound to be a perturbative formulation. On the contrary the light front procedure gives exact basis for 1-particle states, which then needs to be evolved in light front time - where the perturbation calculation enters. Again the light front vacuum state remains simple.
Primary author
Dr
Jerzy Przeszowski
(University of Białystok)
