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.