On the second quantization of a composite model for nonhadrons (original) (raw)

Quantization of second-order Lagrangians: Model problem

Physical Review A, 1991

Many aspects of a model problem, the Lagrangian of which contains a term depending quadratically on the acceleration, are examined in the regime where the classical solution consists of two independent normal modes. It is shown that the techniques of conversion to a problem of Lagrange, generalized mechanics, and Dirac s method for constrained systems all yield the same canonical form for the Hamiltonian. It is also seen that the resultant canonical equations of motion are equivalent to the Euler-Lagrange equations. In canonical form, all of the standard results apply, quantization follows in the usual way, and the interpretation of the results is straightforward. It is also demonstrated that perturbative methods fail, both classically and quantum mechanically, indicating the need for the nonperturbative techniques applied herein. Finally, it is noted that this result may have fundamental implications for certain relativistic theories.

Particle model from quantum foundations

Some ideas are explored concerning the structure of elementary particles (specifically, leptons and hadrons), formulated within the context of a theory of the objective collapse of the wavefunction recently proposed by the authors. In accordance with this hypothesis, to each interaction that induces a discontinuity (quantum jump) in the evolution of the state of an elementary particle, two de Sitter half-spaces are associated, respectively, connected with the outgoing state and its conjugate. It is in these spaces that the structural constituents of the particle lie (quarks in the case of a hadron). The mass of free particles (leptons and hadrons) is given by the energy associated with their time localization in the jump, while the interaction between quarks belonging to a same hadron leads to a chromodynamic coupling constant that ensures both confinement and asymptotic freedom. It is possible to write a toy Hamiltonian in which the organization of quarks in hadrons appears ab initio, and which includes terms both for the exchange of quarks and for the creation/annihilation of pairs, thus avoiding the problem of bottom-up hadronization. In this scenario, the genesis of Regge trajectories is briefly discussed and it is argued that their slope can be quantized; finally, a reinterpretation is suggested of the classical Veneziano and Virasoro amplitudes.

The {\varvec{SL}}(2,\mathbb {R})$$ S L ( 2 , R ) totally constrained model: three quantization approaches

General Relativity and Gravitation, 2014

We provide a detailed comparison of the different approaches available for the quantization of a totally constrained system with a constraint algebra generating the non-compact SL(2, R) group. In particular, we consider three schemes: the Refined Algebraic Quantization, the Master Constraint Programme and the Uniform Discretizations approach. For the latter, we provide a quantum description where we identify semiclassical sectors of the kinematical Hilbert space. We study the quantum dynamics of the system in order to show that it is compatible with the classical continuum evolution. Among these quantization approaches, the Uniform Discretizations provides the simpler description in agreement with the classical theory of this particular model, and it is expected to give new insights about the quantum dynamics of more realistic totally constrained models such as canonical general relativity.

Bound-State Problem in a Model Nonpolynomial-Lagrangian Theory

Physical Review D, 1973

We have shown that the bound-state problem in a nonpolynomial-Lagrangian theory can be solved as in the usual polynomial theory assuming that a Wick rotation is admissible. For definiteness we have assumed the interaction of two scalar fields interacting via the exchange of a superfield to be of the form U (x) = expig Ip(x)], where g denotes the minor coupling constant and Ip(x) is a massless neutral scalar field. The major coupling constant is introduced through the ladder diagrams in the Bethe-Salpeter formalism. We find that the Wick-rotated Bethe-Salpeter equation reduces to a standard Fredholm equation with a modified kernel corresponding to the exchange of the superfield U (x). To study the physical content in the theory we have investigated the equation in the instantaneous approximation. The resulting nonrelativistic equation is projected onto the surface of a four-dimensional sphere by using Fock's transformation variables. The bound-state eigenvalue problem is solved approximately in the weak-binding limit, using Hecke's theorem, leading to a Balmer-type formula. Finally, the fully relativistic equation at E = 0 is considered by transforming it onto the surface of a five-dimensional Euclidean sphere. The approximate-symmetry property of the equation is studied, and the eigenvalue problem is solved in terms of the coupling constants of the theory.

Constraint quantization of a worldline system invariant under reciprocal relativity: II

Journal of Physics a Mathematical and Theoretical, 2008

We consider the worldline quantization of a system invariant under the symmetries of reciprocal relativity. Imposition of the first class constraint, the generator of local time reparametrizations, on physical states enforces identification of the worldline cosmological constant with a fixed value of the quadratic Casimir of the quaplectic symmetry group Q(3, 1) cong U(3, 1)ltimesH(4), the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg group. In our previous paper, J. Phys. A: Math. Theor. 40 (2007) 12095, the 'spin' degrees of freedom were handled as covariant oscillators, leading to a unique choice of cosmological constant, required for projecting out negative-norm states from the physical gauge-invariant states. In the present paper, the spin degrees of freedom are treated as standard oscillators with positive norm states (wherein Lorentz boosts are not number-conserving in the auxiliary space; reciprocal transformations are of course not spin-conserving in general). As in the covariant approach, the spectrum of the square of the energy-momentum vector is continuous over the entire real line, and thus includes tachyonic (spacelike) and null branches. Adopting standard frames, the Wigner method on each branch is implemented, to decompose the auxiliary space into unitary irreducible representations of the respective little algebras and additional degeneracy algebras. The physical state space is vastly enriched as compared with the covariant approach, and contains towers of integer spin massive states, as well as unconventional massless representations of continuous spin type, with continuous Euclidean momentum and arbitrary integer helicity.

Quantisation of O(N) invariant nonlinear sigma model in the Batalin-Tyutin formalism

Nuclear Physics B, 1994

We quantise the O(N ) nonlinear sigma model using the Batalin Tyutin (BT) approach of converting a second class system into first class. It is a nontrivial application of the BT method since the quantisation of this model by the conventional Dirac procedure suffers from operator ordering ambiguities. The first class constraints, the BRST Hamiltonian and the BRST charge are explicitly computed. The partition function is constructed and evaluated in the unitary gauge and a multiplier (ghost) dependent gauge.

Dynamical sectors of a relativistic two particle model

Physical Review D, 2014

We reconsider a model of two relativistic particles interacting via a multiplicative potential, as an example of a simple dynamical system with sectors, or branches, with different dynamics and degrees of freedom. The model can be interpreted as a bigravity model in one dimension. The presence or absence of sectors depends on the values of rest masses. Some aspects of the canonical quantization are described.