Remembering George Sudarshan

Selected Papers from the 16th International Conference on Squeezed States and Uncertainty Relations (ICSSUR 2019), 17-21 June 2019, Universidad Complutense de Madrid, Spain.In these brief notes we want to render homage to the memory of E.C.G. Sudarshan, adding it to the many contributions devoted to preserve his memory from a personal point of view.This research was funded by the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015/0554). A.I. and F.C. would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD++, S2018/TCS-A4342. G.M. would like to thank the support provided by the Santander/UC3M Excellence Chair Programme 2019/2020

On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics

This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics

Knit product of finite groups and sampling

A finite sampling theory associated with a unitary representation of a finite non-abelian group G on a Hilbert space is established. The non-abelian group G is a knit product N⋈H of two finite subgroups N and H where at least N or H is abelian. Sampling formulas where the samples are indexed by either N or H are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space ℓ2(G) having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

The use of unitary invariant subspaces of a Hilbert space H is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of L-2( R) and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows merging the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows introducing the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both finite/infinite dimensional cases.This work has been supported by the Grant MTM2014-54692-P from the Spanish Ministerio de Economía y Competitividad (MINECO)

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C∗-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C∗-algebraA which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S of a possibly infinitedimensional, unital C∗-algebra A is partitioned into the disjoint union of the orbits of an action of the group G of invertible elements of A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H are smooth, homogeneous Banach manifolds of G = GL(H), and, when A admits a faithful tracial state τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ is a smooth, homogeneous Banach manifold for G .A.I. and G.M. acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015/0554). A.I. would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD++, S2018/TCS-4342. G.M. would like to thank the support provided by the Santander/UC3M Excellence Chair Programme 2019/2020

Evolution of classical and quantum states in the groupoid picture of quantum mechanics

This article belongs to the Special Issue Quantum Mechanics and Its FoundationsThe evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio's theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.The research of A.I. and F.D.C. was funded by Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015/0554). A.I. and F.D.C. would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD++, S2018/TCS-A4342. G.M. would like to thank the support provided by the Santander/UC3M Excellence Chair Programme 2019/2020

Covariant brackets for particles and fields

Acompaña erratum: Modern Physics Letters A, (2017), 32(22), 1792002 (1 page).A geometrical approach to the covariant formulation of the dynamics of relativistic systems is introduced. A realization of Peierls brackets by means of a bivector field over the space of solutions of the Euler-Lagrange equations of a variational principle is presented. The method is illustrated with some relevant examples.We thank to G. Marmo for suggesting the geometric analysis of Peierls brackets. We also thank A. P. Balachandran, for many enlightening discussions. The work of M. A. has been partially supported by the Spanish MINECO/FEDER grant FPA2015-65745-P and DGA-FSE grant 2015-E24/2 and the COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology)

Descriptions of Relativistic Dynamics with World Line Condition

In this paper, a generalized form of relativistic dynamics is presented. A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in R4 . The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincaré algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the "eleventh generator" formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many proposals which evaded the no-interaction theorem.F.D.C. and A.I. would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD++, S2018/TCS-A4342. A.I. and G.M. acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD(SEV-2015/0554). G.M. would like to thank the support provided by the Santander/UC3M Excellence Chair Programme 2019/2020, and he is also a member of the Gruppo Nazionale di Fisica Matematica (INDAM), Italy

Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary

We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth compact boundary. Each of these quadratic forms specifies a semibounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators correspond to semi-bounded self-adjoint extensions of the Laplace-Beltrami operator. This family of extensions is compared with results existing in the literature and various examples and applications are discussed.The first and third name authors are partly supported by the project MTM2010-21186-C02-02 of the Spanish Ministerio de Ciencia e Innovación and QUITEMAD programme P2009 ESP-1594. The second-named author was partially supported by projects DGI MICIIN MTM2012-36372-C03-01 and Severo Ochoa SEV-2011-0087 of the Spanish Ministry of Economy and Competition. The third-named author was also partially supported in 2011 and 2012 by mobility grants of the “Universidad Carlos III de Madrid”

On Self-Adjoint Extensions and Symmetries in Quantum Mechanics

Given a unitary representation of a Lie group G on a Hilbert space H , we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace&-Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L 2-space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace&-Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end.A. Ibort and J. M. Pérez-Pardo are partly supported by the project MTM2010-21186-C02-02 of the spanish Ministerio de Ciencia e Innovación and QUITEMAD programme P2009 ESP-1594. F. Lledó was partially supported by projects DGI MICIIN MTM2012-36372-C03-01 and Severo Ochoa SEV-2011-0087 of the spanish Ministry of Economy and Competition. J. M. Pérez-Pardo was also partially supported in 2011 and 2012 by mobility grants of the “Universidad Carlos III de Madrid”