**An Introduction to Quantum Logic and Quantum Probabilities**

by**Federico Holik**

**Abstract:** The logic-algebraic approach to quantum mechanics provides a formal framework in which almost all foundational and interpretational problems can be posed and treated in a rigorous way. Its scope is not restricted to non-relativistic quantum mechanics, but can be extended to relativistic quantum field theory providing an axiomatic formulation. It also provides a natural formal framework for the study of a generalized probability theory, which includes Kolmogorovian and quantum probabilities as particular cases of a vast family. In this seminar we will provide an introduction to quantum logic and the algebraic approach to quantum mechanics. Projective valued measures (PVMs) and positive operator valued measures (POVMs) will be discussed, followed by a review of elementary notions of lattice theory. Next, the relationship between lattice theory and von Neumann algebras will be discussed in connection to the problems posed by von Neumann regarding quantum probabilities. In particular, we will discuss non commutative probability theory and the convex operational approach to statistical theories. The seminar will consist of 5 sessions. The above mentioned contents will be covered in first three sessions, while the 4th and 5th are left open to discuss topics inspired by the questions and problems which emerge from the three former ones.

An Introduction to Quantum Logic and Quantum Probabilities |
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Session | Date | Room |
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An Introduction to Quantum Logic and Quantum Probabilities V | 10/07/2013 – 10h30 | Alechinsky, 437A |

An Introduction to Quantum Logic and Quantum Probabilities IV | 03/07/2013 – 10h30 | Alechinsky, 437A |

An Introduction to Quantum Logic and Quantum Probabilities III | 26/06/2013 – 10h30 | Gris, 734A |

An Introduction to Quantum Logic and Quantum Probabilities II | 19/06/2013 – 10h30 | Gris, 734A |

An Introduction to Quantum Logic and Quantum Probabilities I | 12/06/2013 – 10h30 | Gris, 734A |

**Bibliography:**

• M. L. Dalla Chiara, R. Giuntini, and R. Greechie, *Reasoning in Quantum Theory* (Kluwer Acad. Pub., Dordrecht, 2004)

• G. Kalmbach, *Orthomodular Lattices* (Academic Press, San Diego, 1983)

• M. Redei, *The Birkhoff-von Neumann Concept of quantum Logic*

• E. G. Beltrametti and G. Cassinelli, *The Logic of Quantum Mechanics* (Addison-Wesley, Reading, 1981)

• M. Rédei, *Quantum Logic in Algebraic Approach* (Kluwer Academic Publishers, Dordrecht, (1998)

• S. P. Gudder, *Stochastic Methods in Quantum Mechanics* (North Holland, New York – Oxford 1979)

• G. Mackey, *Mathematical foundations of quantum mechanics* (New York: W. A. Benjamin, 1963)

• V. Varadarajan, *Geometry of Quantum Theory I* (van Nostrand, Princeton, 1968)

**An Introduction to Canonical Quantum Gravity**

by**Federico Zalamea**

**Abstract: **Canonical Quantum Gravity relies mainly on two ideas: on the one hand, the fact that fundamental object of General Relativity can be considered to be a connexion rather than a metric, establishing thus a close analogy with Yang-Mills theories; on the other hand, Dirac’s canonical quantization program for constrained Hamiltonian systems (called Refined Algebraic Quantization in its modern version). The goal of this lectures will be to present the first steps in the construction of the theory, paying special attention to the underlying mathematical structures and to the concepts that seem to be most important. More in detail, we will first show how Ashtekar variables (which constitute the starting point of the theory) can be understood in the light of Cartan’s theory of connexions. Then, we will apply Dirac’s procedure in order to build the quantum states space and we will introduce the notion of a spin network.

An Introduction to Canonical Quantum Gravity |
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Session | Date | Room |
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An Introduction to Canonical QG IV | 29/03/2013 – 10:30 | Kupka, 512A |

An Introduction to Canonical QG III | 27/03/2013 – 10:30 | Gris, 734A |

An Introduction to Canonical QG II | 20/03/2013 – 10:30 | Gris, 734A |

An Introduction to Canonical QG I | 13/03/2013 – 10:30 | Gris, 734A |

**Bibliography:**

• S. Gielen and D. Wise, *Lifting General Relativity to Observer Space,* arXiv:1210.0019

• T. Thiemann, *Lectures on Loop Quantum Gravity,* arXiv:gr-qc/0210094

• A. Ashtekar, *Background Independent Quantum Gravity: A Status Report,* arXiv:gr-qc/0404018