Basic questions I'd like to find answers to about quantum mechanics. Deep philosophical issues:

• The macroscopic universe has some basic properties: It is 4 dimensional. It is Lorentz invariant. It does not support action at a distance. Given a quantum mechanical universe, where do these properties come from?
• Quantum field theories generally rely on successive approximation. What are these computations approximations of? Does the limit of the approximation actually exist, and what properties does it have?
• Is there a model of quantum mechanics whith basic concepts that are fundamentally quantum mechanical? Current practice seems to always involve beginning with a semi-classical problem model, constructing a lagrangian or hamiltonian for the semi-classical system, then using quantization rules to convert this system into a quantum mechanical model. Is there a way to do the reverse -- write down a quantum mechanical system and then compute what it's classical limit behavior would be?

Simple technical issues:

• The claim has been made that a second-quantized electron field gives the same results as an infinite sea of negative energy electrons. Understand this.
• Is "sum over histories" quantum electrodynamics equivalent to canonical variable quantization of the electron and photon fields?
• In what ways is the quantized photon field different from a classical EM field? What experimental tests show that the electromagnetic field is really quantized?
• learn to compute scattering cross-sections.
• What is the geometry of the wave packet of a Dirac particle? What are the differences between the spacial distribution of the field in various spin states? How does this relate to the formal algebraic claims made about the rotational symmetry of spin 1/2?
• How different are the solutions to schrodinger's eqn or dirac's eqn if the EM field of the electron is assumed to act back on the particle? Does QED automatically include this self-force?

Notational issues:

• What happens when quantum mechanics is described in terms of real vector spaces rather than complex?
• Understand the relationship between Hermetian and Unitary operators and conservation of psi^2. Explore what happens if you want to conserve an inner product for some arbitrary operator acting like a metric tensor. Is this the same as quantization by indefinite metric?
• Is there a notation that naturally supports the possiblity of nonlinear operators?

Misc. thoughts:

• what happens if we pick some state variables e.g. the electric charge density, and redefine the notation: there is no "expectation value of charge density". There is only "charge density" which is computed exactly as the "expectation value" is computed now. We claim this is a fundamental quantity which has independent physical existence. Now the whole issue of measurement gets rephrased. There are these fundamental physical properties (such as charge density) that the physics gives values to. If you want to do experimental measurements, you describe your aparatus in terms of these physical properties, and compute what change you expect in your aparatus in response to the system you're measuring.
• the path integral formulation phrases problems in these terms: given an initial state at t0, what are the probability amplitudes for all possible final states at t1? Clearly, for this to be lorentz invariant t0 and t1 need to be arbitrary spacelike hyperplanes (they probably have to be parallel or else they intersect somewhere). Is there a way to rephrase the formulation so it solves on the boundary of an arbitrary compact 4-volume? For example, replace the two infinite hyperplanes t0 and t1 with a boundary surface of a hypercylinder with end caps in t0 and t1. A stranger (but maybe mathermatically easier) example -- given the state of the system at a particular Z position z0 over all x,y,t, compute the amplitudes for all possible states at another position z1.
• Harebrained idea: the Dirac equation can be derived by assuming a first order linear equation, and choosing operators for momentum and energy which are partial derivatives with a non-commuting matrix added in. What happens if instead you assume the same first order linear equation, but instead of matrices and partial derivatives, you use the covariant derivative from differential geometry? Covariant derivatives in general do not commute, and the way in which they do not commute describes the curvature of space. So now you've got an equation that puts some strong constraints on the commutation properties of the derivatives (otherwise the new pseudodirac equation won't be lorentz invariant) which in turn through general relativity pust some constraints on the local mass density. Does the resulting mess have solutions? Do they only exist for particular mass values?