Papers by Alberto Molgado
Hamiltonian analysis for linearly acceleration-dependent Lagrangians
Journal of Mathematical Physics, 2016
Journal of Non Linear Mathematical Physics, Dec 1, 2007
In this work we investigate a formal mapping between the dynamical properties of the unidimension... more In this work we investigate a formal mapping between the dynamical properties of the unidimensional relativistic oscillator and the asymmetrical rigid top at a classical level. We study the relativistic oscillator within Yamaleev's interpretation of Nambu mechanics. Such interpretation is based on the factorisation of the momenta, and as a consequence of this factorisation we are led to a three dimensional phase space. Solutions of the relativistic oscillator are given in terms of the Jacobian elliptic functions and hence we establish a correspondence of these solutions in terms of well known quantities from the rigid body theory. We also study some mechanical restrictions that appear in the mathematical development of the mapping. In particular, we find a lower bound for the relativistic frequency in order to make the mapping self-consistent and physically legitimate.
Physical Review D Particles and Fields, Jan 14, 2009
We present an alternative geometric inspired derivation of the quantum cosmology arising from a b... more We present an alternative geometric inspired derivation of the quantum cosmology arising from a brane universe in the context of geodetic gravity. We set up the Regge-Teitelboim model to describe our universe, and we recover its original dynamics by thinking of such field theory as a second-order derivative theory. We refer to an Ostrogradski Hamiltonian formalism to prepare the system to its quantization. Our analysis highlights the second-order derivative nature of the RT model and the inherited geometrical aspect of the theory. A canonical transformation brings us to the internal physical geometry of the theory and induces its quantization straightforwardly. By using the Dirac canonical quantization method our approach comprises the management of both first- and second-class constraints where the counting of degrees of freedom follows accordingly. At the quantum level our Wheeler-De Witt equation agrees with previous results recently found. On these lines, we also comment upon the compatibility of our approach with the Hamiltonian approach proposed by Davidson and coworkers.
We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian s... more We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL(2,R). The unreduced phase space is T^*R^{p+q} with p>0 and q>0, and the system has a distinguished classical o(p,q) observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O(p,q). The representation is nontrivial iff (p,q) is not (1,1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantisation that imposes the constraints in the sense H_a Psi = 0 and postulates self-adjointness of the o(p,q) observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantisation gives no quantum theory when (p,q) = (1,2) or (2,1), or when p>1, q>1 and p+q is odd.
Classical and Quantum Gravity, May 19, 2005
We investigate refined algebraic quantization of the constrained Hamiltonian system introduced by... more We investigate refined algebraic quantization of the constrained Hamiltonian system introduced by Boulware as a simplified version of the Ashtekar Horowitz model. The dimension of the physical Hilbert space is finite and asymptotes in the semiclassical limit to (2piplanck)-1 times the volume of the reduced phase space. The representation of the physical observable algebra is irreducible for generic potentials but decomposes into irreducible subrepresentations for certain special potentials. The superselection sectors are related to singularities in the reduced phase space and to the rate of divergence in the formal group averaging integral. There is no tunnelling into the classically forbidden region of the unreduced configuration space, but there can be tunnelling between disconnected components of the classically allowed region.
Causal Poisson bracket via deformation quantization
International Journal of Geometric Methods in Modern Physics, 2016
Starting with the well-defined product of quantum fields at two spacetime points, we explore an a... more Starting with the well-defined product of quantum fields at two spacetime points, we explore an associated Poisson structure for classical field theories within the deformation quantization formalism. We realize that the induced star-product is naturally related to the standard Moyal product through the causal Green functions connecting points in the space of classical solutions to the equations of motion. Our results resemble the Peierls-DeWitt bracket analyzed in the multisymplectic context. Once our star-product is defined we are able to apply the Wigner-Weyl map in order to introduce a generalized version of Wick's theorem. Finally, we include a couple of examples to explicitly test our method: the real scalar field and the bosonic string. For both models we have encountered causal generalizations of the creation/annihilation relations, and also a causal generalization of the Virasoro algebra in the bosonic string case.
We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of... more We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploit the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the -genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schr\"odinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner function, even in the so called equal frequency limit of the model, in agreement with recent results.
We explore the cosmological implications provided by the geodetic brane gravity action corrected ... more We explore the cosmological implications provided by the geodetic brane gravity action corrected by an extrinsic curvature brane term, describing a codimension-1 brane embedded in a 5D fixed Minkowski spacetime. In the geodetic brane gravity action, we accommodate the correction term through a linear term in the extrinsic curvature swept out by the brane. We study the resulting geodetic-type equation of motion. Within a Friedmann-Robertson-Walker metric, we obtain a generalized Friedmann equation describing the associated cosmological evolution. We observe that, when the radiation-like energy contribution from the extra dimension is vanishing, this effective model leads to a self-(non-self)-accelerated expansion of the brane-like universe in dependence on the nature of the concomitant parameter β associated with the correction, which resembles an analogous behaviour in the DGP brane cosmology. Several possibilities in the description for the cosmic evolution of this model are embodied and characterized by the involved density parameters related in turn to the cosmological constant, the geometry characterizing the model, the introduced β parameter as well as the dark-like energy and the matter content on the brane.
We review some techniques from non-linear analysis in order to investigate critical paths for the... more We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Previous attempts to analyse when these are minima ex- ist, but mainly based on physical reasoning and only for a restricted class of models. Our main intention in this regard is to develop precise mathematical conditions for critical paths to be minimum solutions in a variety of situations. Our claim is that, with a few techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models is possible. We present specific models arising in modern physical theories in order to make clear the ideas here exposed.
International Journal of Modern Physics D, 2005
We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian s... more We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL(2, R). The unreduced phase space is T * R p+q with p ≥ 1 and q ≥ 1, and the system has a distinguished classical o(p, q) observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O(p, q). The representation is nontrivial iff (p, q) = (1, 1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantisation that imposes the constraints in the senseĤ a Ψ = 0 and postulates self-adjointness of the o(p, q) observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantisation gives no quantum theory when (p, q) = (1, 2) or (2, 1), or when p ≥ 2, q ≥ 2 and p + q ≡ 1 (mod 2).
Boundary Terms in Cosmological Models and their Quantization
Refined algebraic quantisation: finite dimensional systems
Recibido el 1 de mayo de 2006; aceptado el 1 de noviembre de 2006 We investigate refined algebrai... more Recibido el 1 de mayo de 2006; aceptado el 1 de noviembre de 2006 We investigate refined algebraic quantisation of the constrained Hamiltonian system known as the Ashtekar-Horowitz model. We study two versions of this model which are defined on a two-torus and on a cylinder, respectively. The dimension of the physical Hilbert space depends on the topological structure of the model. In particular, we see that for the compact version of the model the representation of the physical observable algebra is irreducible for generic potentials but decomposes into irreducible subrepresentations for certain special potentials. The superselection sectors are related to singularities in the reduced phase space and to the rate of divergence in the formal group averaging integral. For both versions, there is no tunnelling into the classically forbidden region of the unreduced configuration space.
Ostrogradski Approach for the Regge–Teitelboim Model
The Twelfth Marcel Grossmann Meeting - On Recent Developments in Theoretical and Experimental General Relativity Astrophysics and Relativistic Field Theories - Proceedings of the MG12 Meeting on General Relativity, 2012
General Relativity and Gravitation, 2014
General Relativity and Gravitation, 2014
The quantization of the modified geodetic brane gravity implemented from the Regge-Teitelboim mod... more The quantization of the modified geodetic brane gravity implemented from the Regge-Teitelboim model and the trace of the extrinsic curvature of the brane trajectory, K, is developed. As a secondorder derivative model, on the grounds of the Ostrogradski Hamiltonian method and the Dirac's scheme for constrained systems we find suitable first-and second-class constraints which allow for a proper quantization. The first-class constraints obey a sort of truncated Virasoro algebra. The effective quantum potential emerging in our approach is exhaustively studied where it shows that an embryonic epoch is still present. The quantum nucleation is briefly discussed where we observe that it is driven by an effective cosmological constant.
We present an alternative Hamiltonian description of a branelike universe immersed in a flat back... more We present an alternative Hamiltonian description of a branelike universe immersed in a flat background spacetime. This model is named geodetic brane gravity. We set up the Regge-Teitelboim model to describe our Universe where such field theory is originally thought as a second order derivative theory. We refer to an Ostrogradski Hamiltonian formalism to prepare the system to its quantization. This approach comprize the manage of both first-and second-class constraints and the counting of degrees of freedom follows accordingly.
Physical Review D, 2009
We present an alternative geometric inspired derivation of the quantum cosmology arising from a b... more We present an alternative geometric inspired derivation of the quantum cosmology arising from a brane universe in the context of geodetic gravity. We set up the Regge-Teitelboim model to describe our universe, and we recover its original dynamics by thinking of such field theory as a second-order derivative theory. We refer to an Ostrogradski Hamiltonian formalism to prepare the system to its quantization. Our analysis highlights the second-order derivative nature of the RT model and the inherited geometrical aspect of the theory. A canonical transformation brings us to the internal physical geometry of the theory and induces its quantization straightforwardly. By using the Dirac canonical quantization method our approach comprises the management of both first-and secondclass constraints where the counting of degrees of freedom follows accordingly. At the quantum level our Wheeler-De Witt equation agrees with previous results recently found. On these lines, we also comment upon the compatibility of our approach with the Hamiltonian approach proposed by Davidson and coworkers.
Arxiv preprint arXiv:1109.2332, 2011
We explore the cosmological implications provided by the geodetic brane gravity action corrected ... more We explore the cosmological implications provided by the geodetic brane gravity action corrected by an extrinsic curvature brane term, describing a codimension-1 brane embedded in a 5D fixed Minkowski spacetime. In the geodetic brane gravity action we accommodate the correction term through a linear term in the extrinsic curvature swept out by the brane. We study the resulting geodetictype equation of motion. Within a Friedmann-Robertson-Walker metric, we obtain a generalized Friedmann equation describing the associated cosmological evolution. We observe that, when the radiation-like energy contribution from the extra dimension is vanishing, this effective model leads to a self-(non-self)-accelerated expansion of the brane-like universe in dependence on the nature of the concomitant parameter β associated with the correction, which resembles an analogous behaviour in the DGP brane cosmology. Several possibilities in the description for the cosmic evolution of this model are embodied and characterized by the involved density parameters related in turn to the cosmological constant, the geometry characterizing the model, the introduced β parameter as well as the dark like-energy and the matter content on the brane.
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Papers by Alberto Molgado