It is shown that quantization and superintegrability are not concepts that are inherent to classical Physics alone. Indeed, one may quantize and also detect superintegrability of biological models by means of Noether symmetries. We exemplify the method by using a mathematical model that was proposed by Basener and Ross (2005), and that describes the dynamics of growth and sudden decrease in the population of Easter Island.

We consider a general reducible gauge theory deformed by mass or/and interaction terms violating gauge invariance. It is shown that in the Abelian case, by using the Stueckelberg-type procedure, this theory with broken gauge symmetry can be converted into exactly gauge-invariant theory which under a suitable choice of gauge conditions can be treated within the formalism of minimal wave operators manageable by the covariant Schwinger-DeWitt technique. We carry out quantization of such a theory in general terms when the initial generators of gauge transformations are of the first and second stages of reducibility and derive its partition function in terms of the functional integral with all corresponding ghost fields. This method is applied to quantization of massive fermionic totally antisymmetric tensor field models in $AdS$ space. One-loop quantum effective action for these models is derived in the form of the functional determinants of special Dirac-type differential operators in various dimensions.

The Poincaré symmetry can be contracted in two ways to yield the Galilei symmetry and the Carroll symmetry. The well-known Schrödinger equation exhibits the Galilei symmetry and is a fundamental equation in Galilean quantum mechanics. However, the question remains: what is the quantum equation that corresponds to the Carroll symmetry? In this paper, we derive a novel equation in two dimensions, called the "Carroll-Schrödinger equation", which describes the quantum dynamics in the Carrollian framework. We also construct the so-called "Carroll-Schrödinger algebra" in two dimensions, which is a conformal extension of the centrally extended Carroll algebra with a dynamical exponent of [Formula: see text]. We demonstrate that this algebra is the symmetry algebra of the Carroll-Schrödinger field theory. Moreover, we apply the method of canonical quantization to the theory and utilize it to compute the transition amplitude. Finally, we discuss higher dimensions and identify the so-called "generalized Carroll-Schrödinger equation".

The Faddeev and Jackiw procedure for the quantization of constrained gauge systems is used on the analysis of non-Abelian symmetries. The key point is that the gauge algebra of the non-Abelian constraints under generalized brackets can be reconstructed. This follows from the singular matrix that defines the basic geometric structure of the model and its corresponding zero-modes. The attainment of this algebra, not previously found in the Faddeev-Jackiw formalism for constrained theories, leads to the correct transformation properties for the gauge fields. This construction shows that the zero-modes of the symplectic matrix and the generators of gauge symmetry are closely related. To illustrate the method studied here we consider a simple mechanical model with an underlying non-Abelian symmetry and the field theory of pure Chern-Simons theory in (2+1) dimensions.

We give a complete set of generators for the discrete exceptional U-duality groups of toroidal compactified type II theory and M-theory in d>2. For this, we use the DSZ quantization in d=4 as originally proposed by Hull and Townsend, and determine the discrete group inducing integer shifts on the charge lattice. It is generated by fundamental unipotents, which are constructed by exponentiating the Chevalley generators of the corresponding Lie algebra. We then extend a method suggested by the above authors and used by Sen for the heterotic string to get the discrete U-duality group in d=3, thereby obtaining a quantized symmetry in d=3 from a d=4 quantization condition. This is studied first in a toy model, corresponding to d=5 simple supergravity, and then applied to M-theory. It turns out that, in the toy model, the resulting U-duality group in d=3 is strictly smaller than the one generated by the fundamental unipotents corresponding to all Chevalley generators. However, for M-theory, both groups agree. We illustrate the compactification to d=3 by an embedding of d=4 particle multiplets into the d=3 theory.

the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, the Lagrangian is invariant under these transformations. The term "gauge" refers to any specific mathematical formalism to regulate redundant degrees of free and the f a b c {\displaystyle f^{abc}} are the structure constants of the Lie algebra of the generators of the gauge group. This formulation of the Lagrangian is called a Yang–Mills action. Other gauge invariant actions also exist (e.g., nonlinear electrodynamics, Born–Infeld action, Chern–Simons model, theta term, etc.). In this Lagrangian term there is no field whose transformation counterweighs the one of A {\displaystyle A} . Invariance of this term under gauge transformations is a particular case of a priori classical (geometrical) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated gauge fixing, but even after restriction, gauge transformations may be possible. The complete Lagrangian for the gauge theory is now

This paper investigates the linear reaction–diffusion equation on the unit sphere by means of Lie point symmetry analysis. The determining equations show that the finite–dimensional Lie point symmetry algebra is five-dimensional and is generated by the three rotational Killing fields, time translation, and scaling, together with the infinite–dimensional superposition ideal. An optimal system of one–dimensional subalgebras is constructed and used, together with a two–stage reduction procedure based on commuting generators, to derive inequivalent similarity reductions and explicit invariant solution families. The stationary reductions lead to Legendre-type ordinary differential equations, and the global smoothness requirement on the sphere selects the polynomial branch corresponding to spherical harmonics and the associated eigenvalue quantization. In addition, mixed space–time reductions produce explicit non–stationary invariant patterns, including a locally defined family generated by a combined rotation–time–scaling symmetry. The results provide a symmetry-based framework that complements the classical spectral description of diffusion on the sphere.

It is shown that the BRST charge $Q$ for any gauge model with a Lie algebra symmetry may be decomposed as $$Q=\del+\del~{\dag},\;\;\;\del~2=\del~{\dag 2}=0,\;\;\;[\del, \del~{\dag}]_+=0$$ provided dynamical Lagrange multipliers are used but without introducing other matter variables in $\del$ than the gauge generators in $Q$. Furthermore, $\del$ is shown to have the form $\del=c~{\dag a}\phi_a$ (or $\phi'_ac~{\dag a}$) where $c~a$ are anticommuting expressions in the ghosts and Lagrange multipliers, and where the non-hermitian operators $\phi_a$ satisfy the same Lie algebra as the original gauge generators. By means of a bigrading the BRST condition reduces to $\del|ph\hb=\del~{\dag}|ph\hb=0$ which is naturally solved by $c~a|ph\hb=\phi_a|ph\hb=0$ (or $c~{\dag a}|ph\hb={\phi'_a}~{\dag}|ph\hb=0$). The general solutions are shown to have a very simple form.

The use of chains of groups for the characterization of many particle states, combined with the second quantization formalism has proved very fruitful in atomic and nuclear problems. In these problems the first member of the chain is usually the unitary group of as many dimensions as we have single particle orthonormal states, while the last member is the orthogonal group in three dimensions 0(3) which is a symmetry group of the problem. In molecular problems the set of single particle states is not orthonormal, as the orbitals belonging to different centers overlap. Furthermore 0(3) is not a symmetry group for molecules. In this paper we modify the group theory plus second quantization approach to take into account the novel features of the problem. We note first that with the help of the matrix M of all scalar products of the non-orthonormalized single particle states we can define a dual basis for our problem. We then introduce creation and annihilation operators in both the original and the dual basis, which are related by the linear transformation associated with the matrix M. These operators are used to construct explicitly the generators of a general linear group GL(N), where n-particle states will be characterized by the irreducible representations (IR) of GL(N) and its chain of aubgroups GL(N - 1) ⊃ ... ⊃ GL(1) and thus will be Gelfand states. The one and two particle operators can be expressed in terms of the generators of GL(N) and their matrix elements, with respe

We study nucleon structure in the relativistic quark model based on the Bakamjian-Thomas construction of the Poincare generators for an arbitrary quantization surface. The one body, single particle approximation to the current operators is used to calculate electromagnetic matrix elements. The Lorentz symmetry breaking resulting from such an approximation is fully investigated. The results for the light front and instant quantization limits are detailed. A suggestion for the resolution of the quark model inability to simultaneously describe the positive neutron electric form factor, {ital G}{sup {ital n}}{sub {ital E}}({ital Q}{sup 2}) at small {ital Q}{sup 2} and the negative slope of the neutron to proton structure function ratio at large {ital x} is presented.

We show that both abelian and non-abelian gauge theories admit configurations in which the fields behave as if in the presence of static charge densities, or “shadow charges”. These correspond to nontrivial initial conditions for the fields that generate gauge transformations, the Gauss’ law operators. In non-abelian theories, such configurations seem to demand additional physical fields with exactly static charge densities. In contrast with this expectation, we show that gauge theory alone provides a consistent and gauge-invariant description of shadow charges. Canonical quantization then yields continuous shadow charges for abelian theories and quantized ones for non-abelian theories. In general, our findings indicate that all local conservation laws give rise to gauge symmetries, even in the presence of second-class constraints.