Research

Research lines

At present, my main research lines are the following:

The geometrical formulation of Quantum Mechanics

Classical mechanics (CM) can be formulated in several mathematical frameworks each corresponding to a different level of abstraction: Newton equations, the Hamiltonian formalism, the Poisson brackets, etc. Perhaps its more abstract and general formulation is geometrical, in terms of Dirac manifolds. Similarly, Quantum Mechanics (QM) can also be formulated in different ways, some of which resemble its classical counterpart.

The space of physical states are considered as a real differentiable manifold $M_Q$ which is obtained by realifying the complex vector space which is the carrier space of the Hilbert space cotaining the states of the quantum system. The scalar product of the Hilbert space is encoded in three tensors defined on the real vector space $M_Q$. Two of these tensors correspond to a a metric tensor and a symplectic one which allow us to write the expression of the Schrödinger equation as a Hamilton equation, in a form which is completely analogous to the Hamiltonian formulation of Classical Mechanics. The third one is a complex structure tensor on $M_Q$ that, together the two previous tensors, provide $M_Q$ with a Kähler structure.

Physical observables (self--adjoint linear operators) are considered as real quadratic functions defined on the manifold $M_Q$, corresponding to the average value that the given observable takes on each point of $M_Q$. The set of functions can be endowed with a Poisson algebra structure almost equal to the one that characterizes the dynamical variables in CM. Moreover, Schrödinger equation can be recast into Hamiltonian equations form by transforming the complex Hilbert space into a real one of double dimension. The observables are also transformed into dynamical functions in this new phase space, in analogy to the classical one. Finally, a Poisson bracket formulation has also been established for QM, which permits to classify both the classical and the quantum dynamics under the same heading.

Despite its validity for general quantum systems, it has been applied mostly to finite dimensional Hilbert spaces such as those arising in Quantum Information Theory, in particular the characterization of entanglement in geometrical terms.

Nonetheless, the description provides a nice geometrical framework which can be used in different directions, as for instance:

Optimal control theory on Lie groups

Student Supervision

PhD Students

Lígia Raquel dos Santos Abrunheiro, Universidade de Coimbra (Portugal), 14 September 2012
Co-Supervision: Dra. Margarida Camarinha, Universidade de Coimbra (Portugal)
Thesis : Polinomios cubicos riemannianos: uma abordagem hamiltoniana e aplicaçoes
Grade : Aprovado com distinçao e louvor (Highest possible)

Jorge A. Jover-Galtier, Universidad de Zaragoza, 3 July 2017
Co-Supervision: Prof. José F. Cariñena Marzo
Thesis : Open quantum systems: geometric description, dynamics and control
Grade : Sobresaliente cum laude (Highest possible)

Master Students

Teresa Ubieto Puértolas, Universidad de Zaragoza, June 2012
Co-Supervision: Dr. Jacobo Cano Escoriaza
MSc Thesis : Desarrollo de herramientas didácticas para la plataforma Ibercivis
Grade : Sobresaliente (Highest possible)

Jorge Alberto Jover Galtier, Universidad de Zaragoza, 10 July 2013
MSc Thesis : Fenómenos de decoherencia en sistemas moleculares
Grade : Sobresaliente (Highest possible)

Undergraduate Students

Jorge Alberto Jover Galtier, Universidad de Zaragoza, 9 July 2012
TAD Thesis : Geometric Formulation of Quantum Mechanics
Grade : Sobresaliente (Highest possible)

Diego Medrano Jiménez, Universidad de Zaragoza, 26 September 2014
Degree Thesis in Physics: Formulación geométrica de la mecánica cuántica y sus aplicaciones
Grade : Matrícula de honor (highest possible)

Pablo Sala de Torres-Solanot, Universidad de Zaragoza, 9 June 2015
Co-Supervision: Dr. José F. Cariñena Marzo,
Degree Thesis in Mathematics : Formulación simpléctica de la mecánica cuántica
Grade : Matrícula de honor (highest possible)

Adrián Franco Rubio, Universidad de Zaragoza, 10 June 2015
Degree Thesis in Physics : Formulación geométrica de la mecánica cuántica y sus aplicaciones
Grade : Matrícula de honor (highest possible)

Alfonso Lanuza García, Universidad de Zaragoza, 8 July 2016
Co-Supervision: Dr. José F. Cariñena Marzo,
Degree Thesis in Mathematics : Formalismo geométrico de la mecánica cuántica. Integradores Unitarios
Grade : Matrícula de honor (highest possible)

Néstor González Gracia, Universidad de Zaragoza, 7 July 2016
Co-Supervision: Dr. José F. Cariñena Marzo,
Degree Thesis in Physics : Aplicaciones de la formulación geométrica de la Mecánica Cuántica. Evolución Markoviana en sistemas cuánticos.
Grade : Sobresaliente

Carlos Bouthelier Madre, Universidad de Zaragoza, 13 July 2017
Degree Thesis in Physics : Formulación geométrica de la Mecánica Cuántica y Aplicaciones
Grade : Matrícula de Honor (Highest possible)

Cristian-Emanuel Boghiu, Universidad de Zaragoza, 11 July 2018
Degree Thesis in Physics : Formulación geométrica de la dinámica y control de sistemas híbridos clásico-cuánticos
Grade : Matrícula de Honor (Highest possible)

Prizes

The 2012 Award in Physics of the Royal Academy of Sciences of Zaragoza