- Some possible seminar topics (each with a suggested title and a short description of the goals):
- The mathematics of tensor products and traces
To introduce the concepts of the tensor product of 2 or N Hilbert spaces, the trace of an operator, and the partial trace in a precise manner, formulate the basic theorems about them, and possibly discuss the proofs.
- Density matrices
To introduce the concepts of statistical density matrix and reduced density matrix, explain from the main theorem about POVMs why they are relevant, discuss their properties and relations to Bohmian mechanics and collapse theories.
- The Aharonov-Bohm effect
To introduce the effect, discuss its physical meaning concerning the real existence of vector potentials, and describe it mathematically in terms of (i) potentials, (ii) vector bundles, (iii) covering spaces, and (iv) periodic boundary conditions.
- Anyons
The topological explanation of bosons and fermions (based on the unordered configuration space) suggests that if physical space were 2-dimensional, then further types of wave functions would be possible; the corresponding particles are called anyons.
- Shor's algorithm for factorizing integers on a quantum computer
To explain how Shor's algorithm works; it is a scheme faster than any known classical algorithm and makes use of quantum gates; it has been theoretically studied although current quantum computers only have few qubits. Also possibly an overview of the present status of building quantum computers.
- The Shale-Stinespring theorem
The quantum Dirac field is the many-particle version ("second quantization") of the Dirac equation, where negative energies are excluded via the introduction of anti-particles. The Shale-Stinespring theorem provides a criterion for whether a given external electromagnetic field defines a unitary evolution on the Hilbert space of the quantum Dirac field; that is, for whether only finitely many particle-antiparticle pairs get created in finite time.
- Can we detect whether a wave function has collapsed?
To present theorems about the problem of how well any experimental procedure can in an individual case distinguish between a superposition and a mixture of several mutually orthogonal quantum states; the theorems provide certain limitations to knowledge.
- Empirical tests of collapse theories
Collapse theories make predictions that deviate a little from those of standard quantum mechanics. To give an overview of the literature on how these deviations could be used for testing between collapse theories and standard quantum mechanics.
- The non-relativistic limit of the Dirac equation
To derive the 1-particle Pauli equation from the Dirac equation in the limit c to infinity, along with the Bohmian trajectories; and possibly a discussion of how the Galilean invariance comes out of the Lorentz invariance.
- Relativistic trajectories: the Berndl argument
The argument is a variation of Bell's theorem and shows that no probability distribution over N-tuples of timelike trajectories leads on every spacelike surface S to a joint distribution of the N intersection points with S in agreement with |psi_S|^2; to discuss the physical meaning of this result.
- Interior-boundary conditions and self-adjoint extensions
Interior-boundary conditions are used for defining Hamiltonians with particle creation at a point source. To explain how their mathematical theory connects with the general theory of self-adjoint extensions of symmetric operators, and the physical meaning of different self-adjoint extensions.
- Scattering theory
To give an overview of mathematical approaches toward analyzing the unitary time evolution from t=-infinity to t=+infinity and computing the scattering cross section; possibly the flux-across-surfaces-theorem and the scattering-into-cones-theorem.
- The GAP distribution over wave functions
For any given density matrix rho, GAP(rho) distribution is the most spread-out distribution over the unit sphere of Hilbert space with density matrix rho. To explain its definition, properties, and possibly applications.
- Complex charges
In Hamiltonians with particle creation (as in quantum field theory), the "charge" just means a coefficient in front of the creation term. If this coefficient is complex, the Hamiltonian can still be self-adjoint. To explain what the corresponding time evolution and eigenstates look like.
- Reaching the speed of light
Bohm's equation of motion for a single particle with wave function governed by the Dirac equation has the property that the speed is always less than or equal to c, but not necessarily less. To present, discuss, and possibly prove theorems showing that it is infinitely unlikely for a particle to ever actually reach the speed c.
- Paradoxical reflection in quantum mechanics
This means the phenomenon that a particle can get reflected when encountering a sudden drop (downward step) in the potential. To present different derivations of this phenomenon and discuss its physical meaning.
- Superselection rules
A superselection rule says that one can assume, for a certain observable, that superpositions of different eigenstates do not occur in nature, only mixtures; they occur particularly in quantum field theory. To discuss examples of superselection rules, their physical justification, and their meaning in terms of Bohmian mechanics and collapse theories.
- Quantum mechanics on a network
A network (or graph) consists of several line segments glued together at their end points. Along each segment, we have the usual Schrödinger equation, but the vertices (joint end points of several segments) need special consideration for defining a unitary evolution of wave functions on the network.
- Global existence proofs of Bohmian trajectories
Being solutions of an ordinary differential equation, Bohmian trajectories can fail to exist for all times, for example by running to infinity in finite time. Global existence theorems say that this happens with probability 0.
- Literature: will be recommended individually by topic of the presentation.
