Dirac Brackets Research Articles

Particle on a torus knot: Constrained dynamics and semi-classical quantization in a magnetic field

Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac’s Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied.In the second part of the paper we have considered the quantum theory of a charge moving in a torus knot in the presence of a uniform magnetic field along the axis of the torus in a semiclassical quantization framework. We exploit the Einstein–Brillouin–Keller (EBK) scheme of quantization that is appropriate for multidimensional systems. Embedding of the knot on a specific torus is inherently two dimensional that gives rise to two quantization conditions. This shows that although the system, after imposing the knot condition reduces to a one dimensional system, even then it has manifested non-planar features which shows up again in the study of fractional angular momentum. Finally we compare the results obtained from EBK (multi-dimensional) and Bohr–Sommerfeld (single dimensional) schemes. The energy levels and fractional spin depend on the torus knot parameters that specifies its non-planar features. Interestingly, we show that there can be non-planar corrections to the planar anyon-like fractional spin.

Particle on a Torus Knot: A Hamiltonian Analysis

We have studied the dynamics and symmetries of a particle constrained to move in a torus knot. The Hamiltonian system turns out to be Second Class in Dirac’s formulation and the Dirac brackets yield novel noncommutative structures. The equations of motion are obtained for a path in general where the knot is present in the particle orbit but it is not restricted to a particular torus. We also study the motion when it is restricted to a specific torus. The rotational symmetries are studied as well. We have also considered the behavior of small fluctuations of the particle motion about a fixed torus knot.

Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of $${\mathbb {C}} $$ replaced by $${\mathbb {Z}}_{2}$$ . Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts all required the brackets to take values in $${\mathbb {Z}}_{2}$$ . But the usual QM brackets $$\left\langle \psi |\varphi \right\rangle $$ give the “overlap” between states $$\psi $$ and $$\varphi $$ , so for subsets $$S,T\subseteq U$$ , the natural definition is $$\left\langle S|T\right\rangle =\left| S\cap T\right| $$ (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell’s Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over $${\mathbb {C}} $$ and QM/Sets over $${\mathbb {Z}}_{2}$$ .

Dynamical contents of unconventional supersymmetry

The Dirac Hamiltonian formalism is applied to a system in $(2+1)$-dimensions consisting of a Dirac field $\psi$ minimally coupled to Chern-Simons $U(1)$ and $SO(2,1)$ connections, $A$ and $\omega$, respectively. This theory is connected to a supersymmetric Chern-Simons form in which the gravitino has been projected out (unconventional supersymmetry) and, in the case of a flat background, corresponds to the low energy limit of graphene. The separation between first-class and second-class constraints is performed explicitly, and both the field equations and gauge symmetries of the Lagrangian formalism are fully recovered. The degrees of freedom of the theory in generic sectors shows that the propagating states correspond to fermionic modes in the background determined by the geometry of the graphene sheet and the nondynamical electromagnetic field. This is shown for the following canonical sectors: i) a conformally invariant generic description where the spinor field and the dreibein are locally rescaled; ii) a specific configuration for the Dirac fermion consistent with its spin, where Weyl symmetry is exchanged by time reparametrizations; iii) the vacuum sector $\psi=0$, which is of interest for perturbation theory. For the latter the analysis is adapted to the case of manifolds with boundary, and the corresponding Dirac brackets together with the centrally extended charge algebra are found. Finally, the $SU(2)$ generalization of the gauge group is briefly treated, yielding analogous conclusions for the degrees of freedom.

Is spacetime absolutely or just most probably Lorentzian?

Pre-gauging the cosmological scale factor a(t) does not introduce unphysical degrees of freedom into the exact FLRW classical solution. It seems to lead, however, to a non-dynamical mini superspace. The missing ingredient, a generalized momentum enjoying canonical Dirac (rather than Poisson) brackets with the lapse function n(t), calls for measure scaling which can be realized by means of a scalar field. The latter is essential for establishing a geometrical connection with the five-dimensional Kaluza–Klein Schwarzschild–deSitter black hole. Contrary to the Hartle–Hawking approach, (i) the t-independent wave function is traded for an explicit t-dependent , (ii) the classical FLRW configuration does play a major role in the structure of the ’most classical’ cosmological wave packet, and (iii) the non-singular Euclid/Lorentz crossovers get quantum mechanically smeared.

Modified Nonlinear Model of Arcsin-Electrodynamics

A new modified model of nonlinear arcsin-electrodynamics with two parameters is proposed and analyzed. We obtain the corrections to the Coulomb law. The effect of vacuum birefringence takes place when the external constant magnetic field is present. We calculate indices of refraction for two perpendicular polarizations of electromagnetic waves and estimate bounds on the parameter γ from the BMV and PVLAS experiments. It is shown that the electric field of a point-like charge is finite at the origin. We calculate the finite static electric energy of point-like particles and demonstrate that the electron mass can have the pure electromagnetic nature. The symmetrical Belinfante energy-momentum tensor and dilatation current are found. We show that the dilatation symmetry and dual symmetry are broken in the model suggested. We have investigated the gauge covariant quantization of the nonlinear electrodynamics fields as well as the gauge fixing approach based on Dirac's brackets.

Noncommutative Chern-Simons theory and exotic geometry emerging from the lowest Landau level

We relate the collective dynamic internal geometric degrees of freedom to the gauge fluctuations in $\ensuremath{\nu}=1/m(\mathrm{m}\text{ }\text{ }\mathrm{odd})$ fractional quantum Hall effects. In this way, in the lowest Landau level, a highly nontrivial quantum geometry in two-dimensional guiding center space emerges from these internal geometric modes. Using the Dirac bracket method, we find that this quantum geometric field theory is a topological noncommutative Chern-Simons theory. Topological indices, such as the guiding center angular momentum (also called the shift) and the guiding center spin, which characterize the fractional quantum Hall (FQH) states besides the filling factor, are naturally defined. A noncommutative K-matrix Chern-Simons theory is proposed as a generalization to a large class of Abelian FQH topological orders.

Light-front quantization with the light cone gauge

The Dirac procedure for dealing with constraints is applied to the quantization of gauge theories on the light front. The light cone gauge is used in conjunction with the first-class constraints that arise and the resulting Dirac brackets are found. These gauge conditions are not used to eliminate degrees of freedom from the action prior to applying the Dirac constraint procedure. This approach is illustrated by considering Yang–Mills theory and the superparticle in a 2+1-dimensional target space.

Quantization of 2d Hořava gravity: Nonprojectable case

The quantization of a two-dimensional Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory of gravity without the projectability condition is considered. Our study of the Hamiltonian structure of the theory shows that there are two first-class and two second-class constraints. Then, following Dirac, we quantize the theory by first requiring that the two second-class constraints be strongly equal to zero. This is carried out by replacing the Poisson bracket by the Dirac bracket. The two first-class constraints give rise to the Wheeler-DeWitt equations, which yield uniquely a plane wave solution for the wave function. We also study the classical solutions of the theory and find that the characteristics of classical spacetimes are encoded solely in the phase of the plane wave solution in terms of the extrinsic curvature of the foliations $t=\text{constant}$, where $t$ denotes the globally defined time of the theory.

Hamiltonian analysis of the BFCG theory for the Poincaré 2-group

We perform the full Hamiltonian analysis of the topological BFCG action based on the Poincaré 2-group. The Hamiltonian of the theory is constructed, and the algebra of constraints is computed. The Dirac brackets are evaluated, and the second class constraints are then eliminated from the theory. The results are contrasted with those of the topological Poincaré gauge theory, which is equivalent to the BFCG model at the level of the classical action, but has a very different Hamiltonian structure.

Quaternionic quantization principle in general relativity and supergravity

A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space–time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism, the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to [Formula: see text] supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.

The algebra of observables in Gaußian normal spacetime coordinates

We discuss the canonical structure of a spacetime version of the radial gauge, i.e. Gau{\ss}ian normal spacetime coordinates. While it was found for the spatial version of the radial gauge that a "local" algebra of observables can be constructed, it turns out that this is not possible for the spacetime version. The technical reason for this observation is that the new gauge condition needed to upgrade the spatial to a spacetime radial gauge does not Poisson-commute with the previous gauge conditions. It follows that the involved Dirac bracket is inherently non-local in the sense that no complete set of observables can be found which is constructed locally and at the same time has local Dirac brackets. A locally constructed observable here is defined as a finite polynomial of the canonical variables at a given physical point specified by the Gau{\ss}ian normal spacetime coordinates.

Dirac’s and generalized Faddeev–Jackiw brackets for Einstein’s theory in the [formula omitted] limit

In this paper the Dirac and Faddeev–Jackiw formulation for Einstein’s theory in the G→0 limit is performed; the fundamental Dirac’s and Faddeev–Jackiw brackets for the theory are obtained. First, the Dirac brackets are constructed by eliminating the second class constraints remaining the first class ones, then we fix the gauge and we convert the first class constraints into second class constraints and the new fundamental Dirac’s brackets are computed. Alternatively, we reproduce all relevant Dirac’s results by means of the symplectic method. We identify the Faddeev–Jackiw constraints and we prove that the Dirac and the Faddeev–Jackiw brackets coincide to each other.

Quantized gauged massless Rarita-Schwinger fields

We study the quantization of a minimally gauged massless Rarita--Schwinger field, by both the Dirac bracket and functional integral methods. The Dirac bracket approach in the covariant radiation gauge leads to an anticommutator that has a nonsingular limit as gauge fields approach zero, is manifestly positive semidefinite, and is Lorentz invariant. The constraints also have the form needed to apply the Faddeev--Popov method for deriving a functional integral, using the same constrained Hamiltonian and inverse constraint matrix that appear in the Dirac bracket approach.

Classical gauged massless Rarita-Schwinger fields

We show that, in contrast to known results in the massive case, a minimally gauged massless Rarita-Schwinger field yields consistent classical and quantum theories, with a generalized fermionic gauge invariance. To simplify the algebra, we study a two-component left chiral reduction of the massless theory. We formulate the classical theory in both Lagrangian and Hamiltonian form for a general non-Abelian gauging, and analyze the constraints and the Rarita-Schwinger gauge invariance of the action. An explicit wave front calculation for Abelian gauge fields shows that wave-like modes do not propagate with superluminal velocities. The quantized case is studied in gauge covariant radiation gauge and $\Psi_0=0$ gauge for the Rarita-Schwinger field, by both functional integral and Dirac bracket methods. In $\Psi_0=0$ gauge, the constraints have the form needed to apply the Faddeev-Popov method for deriving a functional integral. The Dirac bracket approach in $\Psi_0=0$ gauge yields consistent Hamilton equations of motion, and in covariant radiation gauge leads to anticommutation relations with the correct positivity properties. We discuss relativistic covariance of the anticommutation relations, and of Rarita-Schwinger scattering from an Abelian potential. We note that fermionic gauge transformations are a canonical transformation, but further details of the transformation between different fermionic gauges are left as an open problem.

Point Particle with Extrinsic Curvature as a Boundary of a Nambu–Goto String: Classical and Quantum Model

It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe–Tucker action for such a “rigid particle” and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable effective mass. The Pauli–Lubanski vector composed with such projected Dirac matrices is equal to the Pauli–Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states.

Dirac Bracket for Spherical Harmonics

We work out the problems of the spherical harmonics and the monopole harmonics using a representation of the Cartesian co-ordinates and the corresponding linear differential operators with the guidance of the Dirac brackets. This formalism reveals the rotational symmetry better than that using the angular variables.

Incomplete Dirac reduction of constrained Hamiltonian systems

First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac’s theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac–Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified.

Dirac Bracket for Pedestrians

We present an exposition explaining the implication of the Dirac bracket as well as its application to derive equations of motion in the full phase space for some simple dynamical systems in classical mechanics.

Non-commutativity and non-associativity of the doubled string in non-geometric backgrounds

We use a T-duality invariant action to investigate the behaviour of a string in non-geometric backgrounds, where there is a non-trivial global $O(D,D)$ patching or monodromy. This action leads to a set of Dirac brackets describing the dynamics of the doubled string, with these brackets determined only by the monodromy. This allows for a simple derivation of non-commutativity and non-associativity in backgrounds which are (even locally) non-geometric. We focus here on the example of the three-torus with H-flux, finding non-commutativity but not non-associativity, and also comment on the relation to the exotic $5^2_2$ brane, which shares the same monodromy.