Slide by slide talk for Spinstat2008 Oct. 21, 2008. 1. Pauli's exclusion principle in spinor coordinate space. Buon giorno. It is a pleasure to be here. Italy is a favorite place for me and it is especially satisfying to be here to discuss fundamental physics. I began working on the Pauli principle when I first saw the meeting plans in May. The geometry of spin 1/2 particles, that I had been studying, had arrived at a stage of development that would allow me to address the issues of this conference. The machinery of differential geometry is quite powerful and its application to electron theory offered new understanding. To use these tools for electrons, and presumably for other Fermions, spinor coordinates are needed. Since this is a workshop, I will try to avoid analytical calculations and offer explanations that use a minimum of mathematics. Some of what I will say is still being developed, but, I hope to elucidate, for you, the nature of the Pauli exclusion principle and other issues at the foundations of quantum theory. 2. Outline There are three main parts to this talk. First there is a short discussion of the general method that I use to build geometrical theories. It is possible to combine quantum theory and gravity by using a specifically chosen Riemannian geometry. Following that, I will introduce the spinor coordinates and associated structures. Finally, I will discuss the Pauli principle in this context. 3. The problem of derivatives When trying to construct these theories, there is a basic mathematical problem. It comes from the sequence of ideas that have occurred during the historical development of quantum theory. The important commutation relations, when expressed in the language of matrix mechanics, have a validity that is independent of any choice of final operand. Later with the the development of wave mechanics, the matrix equations were converted into differential operators. The mapping is successful, but a scalar wave function must be appended to the formulas to give meaning to the derivative. Unfortunately, the resulting operator calculus, according to the formalism of general relativity, will give ambiguous results when applied to other operands, vector, tensor or spinor. Different approaches may be chosen to resolve the problem. For the theories, that I talk about today, bare operators are rejected as ontological quantities and all physical propositions are to be formulated with complete differential equations. This assures that the problems of covariance will not interfere. As a consequence, the transition from classical physics to quantum physics is not used. It is found that because the classical starting point for quantization cannot be verified the process of quantization remains in doubt. The quantum equations are given a different ontological basis. 4. Conformal waves. In particular, it has been found that the wave structure characteristic of quantum theory can be developed directly from differential geometry. It turns out that a linear wave equation is found in the behavior of the conformal factor. Here, if the wave function, to a particular power, p, is used a a conformal factor in an n-dimensional space, characteristic linear waves result. The required exponent depends on the number of dimensions. this provides a quantum field equation with the property of superposition. 5. Quantum field equation The five dimensional case is very useful, especially since much of the mathematics is understood from earlier studies. The electromagnetic and gravitational fields are introduced through the metric tensor. The conformal wave equation follows as in the previous slide. The quantum particles so described, respond correctly to the external fields. The wave equation has some new terms that allow for covariance. Very small differences may be present and appear to affect the production of particle pairs. As illustrated, the sharp corners in pair production are softened. As far as I know, the experimental situation is still open. 6. Interaction mechanism Other conformal transformations introduce source terms. If a given metric, (which may including gravitational or electromagnetic terms), is assumed to be conformally flat, then the source terms can be derived from the conformal factor. For the five dimensional case, the Einstein and Maxwell field equations are generated. The usual terms appear augmented by new small quantum effects for gravity. These may be related to what is now called dark energy. All interactions are mediated by conformal transformations. It is an unproven conjecture that this particular form for the interaction is required by mass energy equivalence. 7. Second quantization of photons and gravitons We must also deal with operators of second quantization. The essence of second quantization is the description of systems with large or indefinite numbers of photons. A particular formulation is useful for the geometrical theory. For other reasons, in these theories, the electromagnetic potentials are required to be time symmetric. When this happens, explicit free fields are not used. Interactions, equivalent to free fields come from the action of distant particles, sometimes called the 'absorber'. It is a great simplification in principle because without free fields, there is nothing to quantize. All of electromagnetic theory is attached to the source particles. For statistical purposes, it is equivalent to assigning the discreet levels of black body radiation to the quantum states of the cavity walls. The correct statistics are produced and bare operators of second quantization are avoided. Since the only evidence for photons is that they can be emitted and absorbed, the energy constraints can always be assigned to the cavity. This point of view has been worked out at the level of quantum electrodynamics by Davies and others. The construction for gravity is presumed to be entirely similar. 8. Second quantization of electrons This brings us to the last of the essential operators in physics and also to the subject of this workshop. The anti-commutation relations used for Fermions do not originate from any part of classical physics. They cannot be said to come from Poisson brackets. For the geometrical approach, a different attack is needed. The properties of electron spin are essential, and a suitable geometry is required. Fortunately, spin has always had an intrinsic geometrical character. The spinor coordinates, as a starting point for a geometrical theory, satisfy the requirements. In this way, we can begin to look at Fermions in more detail. 9. Local definition of spinor coordinates The spinor coordinates, are characterized by the property that they must transform locally like a spinor under rotations. It is a simple idea, implicit in Cartan's book, but possibly also considered by others, such as van der Waerden or Pauli. The trick is to make it work. Eight coordinates are needed. It is useful to combine them into complex pairs. This allows the Dirac equation to be written in normal form. The relation with space time is specified in differential form. The coefficients must follow from the rules for five dimensional extended Lorentz transformations. The spinor metric is diagonal and combines complex pairs with their conjugates. To satisfy all the conditions, the Dirac matrices are multiplied by the differential elements of spinor space to give the differential elements in space-time. The spinor, zeta, specifies the relative orientation of the spinor space. It is useful to suppose that the Dirac matrices used in the conversion are functions of space time coordinates. 10. Conformal waves in spinor space. As is done in five dimensions, conformal waves in eight dimensions can be used to produce a wave equation. This becomes an equation in space -time if it is assumed that psi is a function of space-time alone. Not every solution of the 8-D conformal wave equation is an electron. Other conditions must be enforced. The Dirac wave function is associated with the spinor-space gradient of the conformal wave scalar. Different spin orientations are implied by the choice of psi and the spin frame orientation. The chain rule applied to the first order equation for the Dirac spinor produces the Dirac equation. In spinor space, the motion is basically a gradient flow derived from a solution of the spin space scalar wave equation, octagon Psi. 11. Local Dirac electron Without interactions, a plane wave particle is associated with a wave having a particular direction in spinor space. The Dirac wave function my be found by spinor space differentiation. The choice of polarization is make with the frame orientation spinor. For more complicated wave functions, this wave form can be used to describe local regions smaller that the packet size. Rotations of the wave function in spinor space produce crossing symmetry. Electrons are converted to positrons and even neutrinos. 12. Transformation theory of interaction The gravitational and electromagnetic interactions are inferred in the spinor space by assigning equivalent effects to the conformal transformations in five and eight dimensions. Both rotations and dilations my be used. Starting with equivalent wave particles, successive transformations produce the given gravito-electromagnetic interaction. The weak interactions may occur in the spinor space but cannot be accessed by transformations in extended space-time. 13. An identified pair Now we begin to talk about two or more electrons and the Pauli principle. Consider this series of vignettes describing successive transformations of an electron as might occur for increasingly stronger electromagnetic fields. Starting with a low velocity "packet" state, successive conformal transformations will increase the velocities and produce a motion that includes high field values and the production of an electron- positron pair and its annihilation. In subsequent frames, the reversals are moved to early and late times, eventually leaving two electrons and one positron. The positron can also be moved away from the experimental area, leaving only the two electrons. These are actually the same particle. I will call them an "identified pair". Both electrons satisfy a Dirac equation. In some sense it must be the same Dirac equation. 14. Parallel electrons. If the spins are parallel, two such electrons must be separate in space time. A single point in spinor space must map to a uniquely to a point in space-time. Consequently, the two electrons must also be separate in spinor space. 15. Anti-parallel electrons For anti-parallel electrons, such as in a helium ground state, a more complicated argument is needed. The atom may be ionized by an electromagnetic field and the electrons brought to separated areas of space time. The spinor space representations must then also be disjoint. Integrating backwards to the original state of the helium, it is easy to see that the electrons cannot be made to overlap in the spinor space. Because the current is the gradient of a scalar, the flow lines cannot cross and the spinor space representation of the two electrons must have always been separated. This must therefor be true even for electrons that have overlapping wave solutions in space-time. Apparently all electrons remain disjoint in spinor space since any one can be removed from the assemblage by electromagnetic forces. 16. Spinor wave propagation. The two electrons in spinor space propagate according to the wave equation "octagon Psi equals zero", which is also the Dirac equation for each one separately. If, in some region, they approach each other, the Pauli principle must come into play. The wave equation dictates the time evolution until contact is made. They cannot overlap in general, all effects occur at the boundary. The process can be visualized by using a Huygens-Fresnel-Feynman type construction. In the usual formulation, anti-symmetrization is applied when the electrons meet, but here, there is no effect as long as they are separated in the spinor coordinate space. It remains possible that they may overlap in space-time. At the instant of contact in spinor space, the effect begins. 17. Boundary development The development of the boundary condition is the essence of the Pauli exclusion principle. Working with the wave front formalism, contributions of one particle to its own new wavefront have a plus sign. Contributions from across the developing boundary have a minus sign. When the time step becomes larger than the particle separation, opposite signs for the resulting wave functions are forced at the boundary. The zero of the boundary is maintained and moves to keep the slopes equal and opposite. Psi-1-prime must equal minus psi-2-prime for equal displacements from the meeting point. The exchange anti-symmetry is fully developed. Now, from the point of view of the wave equation in spinor coordinates, this is a completely natural occurrence. The two electrons actually obey the equation octagon-psi equals zero, and must have a smooth first derivative and well defined second derivative at the boundary. The boundary derivative must be anti-symmetric in the interchange of coordinates and momenta because the Dirac wave function is the spin space derivative of a scalar. A change in sign of the spinor coordinates across the boundary always implies an exchange of both space-time coordinates and spin direction. The Dirac wave function is always odd under this transformation. 18. Multiple electrons in spinor space. This can be extended to more particles. A collection of electrons may appear in spinor space without having any of their interior points in common. They may move or expand by diffraction until such time as they may meet. The anti-symmetrical boundary condition is developed at these junctions. In regions away from the boundaries, the Dirac equation prevails. The anti-symmetry is automatically enforced whenever this happens. The Pauli exclusion principle is thus the result of the internal boundary conditions imposed by the properties of a single differential equation. It provides a possible explanation of the origin and necessity of the Pauli principle. Under these conditions, the Pauli exclusion principle is equivalent to the assumption that all electrons are collectively solutions of the conformal wave equation in spinor space. I have come to think of this as a bowl of spaghetti. (Lunch will be ready soon!) 20. Ongoing considerations These ideas are new and much consideration is to be expected before they might be accepted. But, sometimes, the applications make an important contribution and motivate continuing development. I mention a few possibilities that I have come across. First, there may be some important calculational advantages to spinor space because the large dimensional requirements of configuration space may be simplified. Apparently, more that one electron can be included in a single eight dimensional field. The complexity does not increase as additional electrons are added. the ground and excited states of multi-electron atoms or molecules may be much easier to solve, either analytically or by computer. It is not know what gains might actually be realized by any specific implementation. Applications in nuclear physics might be possible but are more difficult because the form of the quark wave functions is not know in spinor space. The relativistic properties of this construction need to be studied in more detail. What is the relationship with the Feynman constructions of perturbation theory? The characteristic relationship of inter-particle interactions to self-interactions needs to be clarified as it may possibly be related, as Pauli has suggested, to the fine structure constant. What is the continuing role of field operators. Are these analgous to the diffeomorphims of relativity theory? Can electron creation operators be represented as diffeomorphic transformations of mult-ielectron wave functions in spinor space? What is the relationship with the other Fermions? Do muons, tauons and quarks have wave functions in the same spinor space, or is it necessary to return to a product of spinor spaces. What are the experimental tests that might distinguish these cases? A number of problems involving rotation persist in general relativity theory. How is the Dirac-Thirring paradox to be resolved? What happens to Newton's rotating bucket of water if the bucket is held together with the Pauli principle? What can be said about the theories of Mathisson, Corinaldesi and Papapetrou? Is this related to the questions raised by Aharonov an Casher about the rotation of spin vectors by different interactions. I do not yet have answers to any of these questions. 19. Geometry of the Pauli equivalence principle This construction is of crucial importance, to geometrical theories of quantum mechanics. The algebraic structure, usually represented by the bare operators is replaced by differential equations whose solutions contain the physics. The important ideas here are: First, the geometrical description of fundamental physics, second, the relevance of spinor coordinates to Fermion properties and third, the elementary description of the Pauli principle. Closing I would like to thank the organizers for there efforts to make this a useful and effective meeting. I want to thank all of you for coming to listen. I hope you find these ideas and questions to be as interesting as I find them.